Math Formula Sheet AIEEE

MATHS FORMULA - POCKET BOOK

QUADRATIC EQUATION & EXPRESSION 1.

MATHS FORMULA - POCKET BOOK 4.

Conjugate roots : Irrational roots and complex roots occur in conjugate pairs i.e.

Quadratic expression : A polynomial of degree two of the form ax2 + bx + c, a ≠ 0 is called a quadratic expression in x.

if one root α + iβ, then other root α iβ if one root α +

2.

β , then other root α

β

Quadratic equation : An equation ax2 + bx + c = 0, a ≠ 0, a, b, c ∈ R has two and only two roots, given by α=

3.

−b + b2 − 4ac 2a

and

β=

5.

Sum of roots :

−b − b2 − 4ac 2a

S=α+β=

−Coefficient of x −b = Coefficient of x2 a

Product of roots :

Nature of roots : Nature of the roots of the given equation depends upon the nature of its discriminant D i.e. b2 4ac.

P = αβ =

cons tant term c = Coefficient of x2 a

Suppose a, b, c ∈ R, a ≠ 0 then (i)

If D > 0

⇒

roots are real and distinct (unequal)

(ii)

If D = 0

⇒

roots are real and equal (Coincident)

(iii)

If D < 0

Formation of an equation with given roots : x2 Sx + P = 0

roots are imaginary and unequal i.e. ⇒ non real complex numbers. Suppose a, b, c ∈ Q a ≠ 0 then

If D > 0 and D is a perfect square ⇒ roots are rational

(i)

6.

⇒ 7.

Roots under particular cases : For the equation ax2 + bx + c = 0, a ≠ 0

& unequal (ii)

(i)

If D > 0 and D is not a perfect square ⇒ roots are

For a quadratic equation their will exist exactly 2 roots real or imaginary. If the equation ax2 + bx + c = 0 is satisfied for more than 2 distinct values of x, then it will be an identity & will be satisfied by all x. Also in this case a = b = c = 0.

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If b = 0 ⇒ roots are of equal magnitude but of opposite sign.

irrational and unequal.

E

x2 (Sum of roots) x + Product of roots = 0

(ii)

If c = 0 ⇒ one root is zero and other is b/a

(iii)

If b = c = 0 ⇒ both roots are zero

(iv) If a = c ⇒ roots are reciprocal to each other.

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MATHS FORMULA - POCKET BOOK (v)

MATHS FORMULA - POCKET BOOK (vi) α4 + β4 = (α2 + β2)2 2α2β2

If a > 0, c < 0 or a < 0, c > 0 ⇒ roots are of opposite signs

2

(vi) If a > 0, b > 0, c > 0 or a < 0, b < 0, c < 0 ⇒ both

={(α + β)2 2αβ}2 2α2β2

roots are ve (vii) If a > 0, b < 0 , c > 0 or a < 0, b > 0, c < 0

If roots of quadratic equation ax2 + bx + c, a ≠ 0 are α and β, then (α β) =

(ii)

α2 + β2 = (α + β)2 2αβ =

(iii)

α2 β2 = (α + β)

2

(α + β) − 4αβ = ±

b2 − 4ac a

b2 − 2ac a2

2

(α + β) − 4αβ =

−b b2 − 4ac a2

9.

=

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2c2 a2

−b(b2 − 2ac) b2 − 4ac a4

(viii) α2 + αβ + β2 = (α + β)2 αβ =

b2 + ac a2

(ix)

α α 2 + β2 (α + β)2 − 2αβ β + = = β αβ αβ α

(x)

FG α IJ H βK

2

+

FG β IJ H αK

2

=

α4 + β4 [(b2 − 2ac)2 − 2a2c2 ] = α 2 β2 a2c 2

b1c2 − b2c1 c1a2 − c2a1 c1a2 − c2a1 = a1b2 − a2b1

(i)

One common root if

(ii)

a1 b1 c1 Both roots common if a = b = c 2 2 2

(α + β)2 − 4αβ [α2+ β2 αβ] (b2 − ac) b2 − 4ac a3


The equations a1 x2 + b1 x + c1 = 0 and a2x2 + b2x + c2 = 0 have

α 3 β3 = (α β) [α2+ β2 αβ] =

2

Condition for common roots :

−b(b2 − 3ac) (iv) α3 + β3 = (α + β)3 3(α + β) αβ = a3 (v)

I JK

(vii) α4 β4 =(α2 + β2) (α2 β2) =

Symmetric function of the roots :

(i)

− 2ac a2

⇒ both

roots are +ve. 8.

Fb =G H

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10. Maximum and Minimum value of quadratic expression :

In a quadratic expression ax

2

LF b I + bx + c = a MGH x + 2a JK MN

2

−

(v)

OP PQ

D 4a2 ,

D ≥ 0, a.f(k1) > 0, a.f(k2) > 0, k1 <

If a > 0, quadratic

a.f(k1) < 0, a.f(k2) < 0

expression has minimum value

−b 4ac − b2 at x = and there is no maximum value. 2a 4a (ii)

(vii) λ will be the repeated root of f(x) = 0 if f(λ) = 0 and f'(λ) = 0

If a < 0, quadratic expression has maximum value −b 4ac − b2 at x = and there is no minimum value. 2a 4a

12. For cubic equation ax3 + bx2 + cx + d = 0 : We have α + β + γ =

11. Location of roots : (i)

If k lies between the roots then a.f(k) < 0 (necessary & sufficient)

(ii)

If between k1 & k2 their is exactly one root of k1, k2 themselves are not roots f(k1) . f(k2) < 0

13. For biquadratic equation ax4 + bx3 + cx2 + dx + e = 0 : We have α + β + γ + δ =

−d b , αβγ + βγδ + γδα + γδβ = a a

(necessary & sufficient)

If both the roots are less than a number k D ≥ 0, a.f(k) > 0,

−b c −d , αβ + βγ + γα = and αβγ = a a a

where α, β, γ are its roots.

Let f(x) = ax2 + bx + c, a ≠ 0 then w.r.to f(x) = 0

(iii)

−b < k2 2a

(vi) If k1, k2 lies between the roots

Where D = b2 4ac (i)

If both the roots lies in the interval (k1, k2)

−b
αβ + αγ + αδ + βγ + βδ + γδ =

c e and αβγδ = a a


(iv) If both the roots are greater than k D ≥ 0, a.f(k) > 0,

−b >k 2a



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COMPLEX NUMBER 1.

Complex Number : A number of the form z = x + iy (x, y ∈ R, i =

−1 ) is called a complex number, where x is called a real part i.e. x = Re(z) and y is called an imaginary part i.e. y = Im(z).

Modulus |z| =

y . x

Polar representation : x = r cosθ, y = r sinθ, r = |z| =

(ii)

z z = 2i Im(z) = purely imaginary

*

z z = |z|2

*

z1 + z2 +....+zn = z 1 + z 2 + .......... + z n

*

z1 − z2 = z 1 z 2

*

z1z2 = z 1 z 2

*

FG z IJ Hz K

*

ez j

= ( z )n

*

c zh

= z

*

If α = f(z), then α = f( z )

x2 + y2 ,

amplitude or amp(z) = arg(z) = θ = tan1 (i)

*

x2 + y2

Exponential form : z = reiθ , where r = |z|, θ = amp.(z)

1 2

n

→

P(x, y) then its vector representation is z = OP

4.

2 3 4 −1 , i = 1, i = i , i = 1

(provided z2 ≠ 0)

*

z + z = 0 or z = z ⇒ z = 0 or z is purely imaginary

*

z= z

⇒

z is purely real

Modulus of a complex number : Magnitude of a complex number z is denoted as |z| and is defined as

Hence i4n+1 = i, i4n+2 = 1, i4n+3 = i, i4n or i4(n+1) = 1 3.

|z| =

Complex conjugate of z :

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(Re(z))2 + (Im(z))2 , |z| ≥ 0

If z = x + iy, then z = x iy is called complex conjugate of z

(i)

z z = |z|2 = | z |2

*

z is the mirror image of z in the real axis.

(ii)

z1 =

*

|z| = | z |

*

z + z = 2Re(z) = purely real

(iii)

|z1 ± z2|2 = |z1|2 + |z2|2 ± 2 Re (z1 z2 )


2

Integral Power of lota : i=

E

1

Where α = f(z) is a function in a complex variable with real coefficients.

(iii) Vector representation :

2.

FG z IJ Hz K

=

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z |z|2


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6.

(iv) |z1 + z2|2 + |z1 z2|2 = 2 [|z1|2 + |z2|2] (v)

MATHS FORMULA - POCKET BOOK Square root of a complex no.

|z1 ± z2| ≤ |z1| + |z2| a + ib = ±

(vi) |z1 ± z2| ≥ |z1| |z2| 5.

Argument of a complex number : = ±

Argument of a complex number z is the ∠ made by its radius vector with +ve direction of real axis. arg z

= θ,

z ∈ 1st quad.

7.

= π θ , z ∈ 2nd quad. = θ,

arg (any real ve no.) = π

(iii)

arg (z z ) = ± π/2

and (cosθ + isinθ)n = cos nθ i sin nθ 8.

Euler's formulae as z = reiθ, where eiθ = cosθ + isinθ and eiθ = cosθ i sinθ

(iv) arg (z1.z2) = arg z1 + arg z2 + 2 k π (v)

arg

FG z IJ Hz K 1 2

FG 1 IJ H zK

9.

nth roots of complex number z1/n

LM FG 2mπ + θ IJ + i sinFG 2mπ + θ IJ OP N H n K H n KQ ,

= r1/n cos

, if z is non real

where m = 0, 1, 2, ......(n 1)

= arg z, if z is real (vii) arg ( z) = arg z + π, arg z ∈ ( π , 0] = arg z π, arg z ∈ (0, π ] (viii) arg (zn) = n arg z + 2 k π (ix)

eiθ + eiθ = 2cosθ and eiθ eiθ = 2 isinθ

∴

= arg z1 arg z2 + 2 k π

(vi) arg ( z ) = arg z = arg

(i)

Sum of all roots of z1/n is always equal to zero

(ii)

Product of all roots of z1/n = (1)n1 z

10. Cube root of unity :

arg z + arg z = 0

cube roots of unity are 1, ω, ω2 where

argument function behaves like log function. ω=


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OP PQ

|z|+a |z|−a −i , for b < 0 2 2

(cosθ + isinθ)n = cosθ + isin nθ

= θ π , z ∈ 4 quad. (ii)

LM MN

It states that if n is rational number, then

th

arg (any real + ve no.) = 0

OP PQ

|z|+a |z|−a +i , for b > 0 2 2

De-Moiver's Theorem :

z ∈ 3rd quad.

(i)

LM MN

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−1 + i 3 and 1 + ω + ω2 = 0, ω3 = 1 2


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11. Some important result : If z = cosθ + isinθ (i)

z+

1 = 2cosθ z

(ii)

z

1 = 2 isinθ z

(iii)

zn +

or

z−

z1 + z 2 2

=

|z1 − z 2 | 2

|z z1|2 + |z z2|2 = |z1 z2|2

Where z1, z2 are end points of diameter and z is any point on circle.

1 = 2cosnθ zn

(a)

1 1 1 + + =0 y x z

(b) yz + zx + xy = 0

(c)

x +y +z =0

(d) x + y + z = 3xyz

2

z − z1 z − z1 z − z2 + z − z2 = 0

or

(iv) If x = cosα + isinα , y = cos β + i sin β & z = cosγ + isinγ and given x + y + z = 0, then

2

or

2

3

3

13. Some important points : (i)

Distance formula PQ = |z2 z1|

(ii)

Section formula For internal division =

3

m1z2 + m2 z1 m1 + m2

MATHS FORMULA - POCKET BOOK MATHS FORMULA - P 12. Equation of Circle : *

|z z1| = r represents a circle with centre z1 and radius r.

*

|z| = r represents circle with centre at origin.

*

|z z1| < r and |z z1| > r represents interior and exterior of circle |z z1| = r.

*

z z + a z + a z + b = 0 represents a general circle where a ∈ c and b ∈ R.

*

Let |z| = r be the given circle, then equation of tangent at the point z1 is z z 1 + z z1 = 2r2

*

diametric form of circle : arg

FG z − z IJ Hz − z K 1 2

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(iii)

m1z2 − m2 z1 m1 − m2

Equation of straight line. * Parametric form z = tz1 + (1 t)z2 where t ∈ R

* Non parametric form

z z1 z2

z 1 z1 1 z2 1

= 0.

* Three points z1, z2, z3 are collinear if

π = ± , 2


For external division =

z1 z2 z3

z1 1 z2 1 = 0 z3 1

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z − z1 (iv) The complex equation z − z 2

MATHS FORMULA - POCKET BOOK (xii) If z1, z2, z3 be the vertices of a triangle, then the triangle is equilateral

= k represents a circle

iff (z1 z2)2 + (z2 z3)2 + (z3 z1)2 = 0.

if k ≠ 1 and a straight line if k = 1. (v)

(xiii) If z1, z2, z3 are the vertices of an isosceles triangle, right angled at z2,

The triangle whose vertices are the points represented by complex numbers z1, z2, z3 is equilateral if

then z12 + z22 + z32 = 2z2 (z1 + z3). (xiv) z1, z2, z3. z4 are vertices of a parallelogram then

1 1 1 z2 − z3 + z3 − z1 + z1 − z2 = 0

z1+ z3 = z2 + z4

i.e. if z12 + z22 + z32 = z1z2 + z2z3 + z1z3. (vi) |z z1| = |z z2| = λ , represents an ellipse if |z1 z2| < λ , having the points z1 and z2 as its foci and if |z1 z2| = λ , then z lies on a line segment connecting z1 & z2 (vii) |z z1| ~ |z z2| = λ represents a hyperbola if |z1 z2| > λ , having the points z1 and z2 as its foci, and if |z1 z2| = λ , then z lies on the line passing through z1 and z2 excluding the points between z1 & z2. (viii) If four points z1, z2, z3, z4 are concyclic, then

FG z Hz

1

1

− z2 − z4

IJ FG z K Hz

3 3

− z4 − z2

IJ K

is purely real.

(ix)

If three complex numbers are in A.P., then they lie on a straight line in the complex plane.

(x)

If z1, z2, z3 be the vertices of an equilateral triangle and z0 be the circumcentre, then z12 + z22 + z32 = 3z02.

(xi)

If z1, z3, z3 ....... zn be the vertices of a regular polygon of n sides & z0 be its centroid, then z12 + z22 + ......... + zn2 = nz02.


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PERMUTATION & COMBINATION 1.

(iv) If out of n objects, 'a' are alike of one kind, 'b' are alike of second kind and 'c' are alike of third kind and the rest distinct, then the number of ways of permuting

Factorial notation The continuous product of first n natural numbers is called factorial i.e. n or n! = 1. 2. 3........(n 1).n n! = n(n 1)! = n(n 1)(n 2)! & so on

the n objects is 4.

Restricted Permutations -

n! n (n 1)......... (n r + 1) = (n − r)!

or

Here 0! = 1 and (n)! = meaningless. 2.

Fundamental principle of counting (i) Addition rule : If there are two operations such that they can be done independently in m and n ways respectively, then either (any one) of these two operations can be done by (m + n) ways. Addition ⇒ OR (or) Option (ii) Multiplication rule : Let there are two tasks of an operation and if these two tasks can be performed in m and n different number of ways respectively, then the two tasks together can be done in m × n ways. Multiplication ⇒ And (or) Condition (iii)

3.

(iii)

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The number of permutations of n dissimilar things taken r at a time, when m particular things always occupy definite places = nmprm

(ii)

The number of permutations of n different things taken r at a time, when m particular things are always to be excluded (included) nm

Pr (nmCrm × r!)

Circular Permutations When clockwise & anticlockwise orders are treated as different. (i)

The number of circular permutations of n different things taken r at a time

(ii)

n

Pr r

The number of circular permutations of n different things taken altogether

n

Pn = (n 1)! n

When clockwise & anticlockwise orders are treated as same. (i)

The number of circular permutations of n different things

n! (n − r)!

n

taken r at a time

The number of permutations of n dissimilar things taken all at a time is npn = n! The number of permutations of n distinct objects taken r at a time, when repetition of objects is allowed is nr.


5.

Permutations (Arrangement of objects) (i) The number of permutations of n different things taken

(ii)

(i)

=

Bijection Rule : Number of favourable cases = Total number of cases Unfavourable number of cases.

r at a time is npr =

n! a! b! c!

(ii)

The number of circular permutations of n different things taken all together

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Pr 2r n

1 Pn = (n 1)! 2 2n


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Combination (selection of objects) -

(iv) Total number of selections of zero or more objects from n identical objects is n + 1.

The number of combinations of n different things taken r at a time is denoted by nCr or C (n, r) Cr =

n

7.

(i)

n

(ii)

n

(iii)

n

(iv)

n

(v)

n

(vi)

n

(vii)

n

n! r !(n − r)!

=

(v)

n

Pr r!

=

Cr + nCr1 = n+1Cr

⇒

r = s or r + s = n

C0 = Cn = 1 n

n r

Cr =

1 (n r + 1) nCr1 r

n1

(vii) The number of selections taking atleast one out of a1 + a2 + a3 + ....... + an + k objects when a1 are alike (of one kind), a2 are alike (of second kind), ........ an are alike (of kth kind) and k are distinct is

Cr1

[(a1 + 1) (a2 + 1) (a3 + 1) .......... (an + 1)] 2k 1 9.

Restricted combinations -

Division and distribution (i)

The number of combinations of n distinct objects taken r at a time, when k particular objects are always to be (i)

included is nkCrk

(ii)

excluded is nkCr

(iii)

included and s particular things are to be excluded is nks

8.

n, & p different objects respectively is

(i)

The number of selections of n identical objects, taken at least one = n

(ii)

The number of selections from n different objects, taken at least one

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(m + n + p)! m! n! p!

The total number of ways in which n different objects are to be divided into r groups of group sizes n1, n2, n3, ............. nr respectively such that size of no two groups

n! is same is n ! n !............n ! . 1 2 r (iii)

Cn = 2n 1

n

The total number of ways in which n different objects are to be divided into groups such that k1 groups have group size n1, k2 groups have group size n2 and so on, kr groups have group size nr, is given as

n!

The number of selections of r objects out of n identical objects is 1.


(ii)

Total number of combinations in different cases -

(iii)

E

The number of ways in which (m + n + p) different objects can be divided into there groups containing m,

Crk

= nC1 + nC2 + nC3 + ....... +

Cn = 2n

n

[(a1 + 1) (a2 + 1) (a3 + 1) + ...... + (an + 1)] 1

C1 = nCn1 = n Cr =

C0 + nC1 + nC2 + nC3 + ....... +

n

(vi) The total number of selections of at least one out of a1 + a2 + ...... + an objects where a1 are alike (of one kind), a2 are alike (of second kind), ......... an are alike (of nth kind) is

Cr = nCnr Cr = nCs

Total number of selections of zero or more objects out of n different objects

k1

(n1 !) (n2 !) .............(nr !)k r k1 ! k 2 !............ k r !

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k2


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(iv) The total number of ways in which n different objects are divided into k groups of fixed group size and are distributed among k persons (one group to each) is given as

(b)

C3

n

(c)

(number of ways of group formation) × k!

The coefficient of xn in the expansion of (1 xr) is equal to n + r 1Cr 1

(ii)

The number of solution of the equation x1 + x2 + .......... + xr = n, n ∈ N under the condition n1 ≤ x1 ≤ n'1,

If m parallel lines in a plane are intersected by a family of other n parallel lines. Then total number of parallelogram so formed is mC2 × nC2.

(e)

Given n points on the circumference of a circle, then number of straight lines nC2 number of triangles nC3 number of quadrilaterals nC4

where all x'is are integers is given as Coefficient of xn is n1 +1

+...+x

n'1

j ex

n2

+x

n2 +1

+...+x

n'2

j...ex

nr

+x

nr +1

+...+ x

n'r

jOQP

11. Derangement Theorem (i)

Number of diagonals in a polygon of n sides is

(d)

n2 ≤ x2 ≤ n'2 , ................ nr ≤ xr ≤ n'r

+x

C3.

C2 n.

(i)

n1

m

n

10. Selection of light objects and multinomial theorem -

LMex N

Number of total triangles formed by joining the n points on a plane of which m(< n) are collinear is

(f)

If n straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. Then the number of part into which these lines divide the plane is = 1 + Σn

(g)

Number of rectangles of any size in a square of n × n

If n things are arranged in a row, then the number of ways in which they can be rearranged so that no one of them occupies the place assigned to it is

n

n

is

L 1 + 1 − 1 + 1 −....+(−1) 1 OP = n! M1 − n!Q N 1! 2! 3! 4!

(h)

∑ r3

r =1

and number of squares of any size is

∑ r2 .

r =1

Number of rectangles of any size in a rectangle of

n

(ii)

n × p is

If n things are arranged at n places then the number of ways to rearrange exactly r things at right places is n! = r

LM1 − 1 + 1 − 1 + 1 +....+(−1) N 1! 2! 3! 4!

n− r

1 (n − r)!

np (n + 1) (p + 1) and number of squares 4 n

of any size is

OP Q

∑

r =1

(n + 1 r) (p + 1 r).

12. Some Important results (a)

Number of total different straight lines formed by joining the n points on a plane of which m(

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PROBABILITY 1.

(vi) P(AB) ≤ P(A) P(B) ≤ P(A + B) ≤ P(A) + P(B) (vii) P(Exactly one event) = P(A B ) + P( A B)

Mathematical definition of probability : Probability of an event =

= P(A + B) P(AB)

No. of favourable cases to event A Total no. of cases

Note :

2.

(viii) P( A + B ) = 1 P(AB) = P(A) + P(B) 2P(AB)

(i) (ii)

0 ≤ P (A) ≤ 1 Probability of an impossible event is zero

(iii)

Probability of a sure event is one.

(iv)

P(A) + P(Not A) = 1 i.e. P(A) + P( A ) = 1

(ix)

P(neither A nor B) = P ( A B ) = 1 P(A + B)

(x)

When a coin is tossed n times or n coins are tossed once, the probability of each simple event is

(xi)

Odds for an event : If P(A) =

When a dice is rolled n times or n dice are rolled once, the probability of each simple event is

m n−m and P( A ) = n n

Then odds in favour of A =

P(A) m = P(A) n−m

and odds in against of A =

P(A) n−m = P(A) m

1 . 6n

(xii) When n cards are drawn (1 ≤ n ≤ 52) from well shuffled deck of 52 cards, the probability of each simple event is

1 . Cn

52

(xiii) If n cards are drawn one after the other with replacement, the probability of each simple event is

3.

Set theoretical notation of probability and some important results :

(xiv) P(none) = 1 P (atleast one)

(i)

P(A + B) = 1 P( A B )

(xv) Playing cards :

(ii)

P(AB) P(A/B) = P(B)

(iii)

P(A + B) = P(AB) + P( A B) + P(A B )

(iv) A ⊂ B ⇒ P(A) ≤ P(B) (v)

P( AB ) = P(B) P(AB)


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1 . 2n

PAGE # 21

(a)

Total cards : 52 (26 red, 26 black)

(b)

Four suits : Heart, diamond, spade, club (13 cards each)

(c)

Court (face) cards : 12 (4 kings, 4 queens, 4 jacks)

(d)

Honour cards : 16 (4 Aces, 4 kings, 4 queens, 4 Jacks)


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1 . (52)n

PAGE # 22



(xvi) Probability regarding n letters and their envelopes :

Conditional probability : P(A/B) = Probability of occurrence of A, given that B has

If n letters corresponding to n envelopes are placed in the envelopes at random, then (a)

Probability that all the letters are in right envelopes =

(b)

already happened =

Probability that all letters are not in right enve-

Note :

1 n!

Probability that no letters are in right envelope

4.

LM 1 − 1 + 1 +.....+(−1) N2! 3! 4!

n− r

1 (n − r)!

OP Q

i.e. n (A ∩ B) = 0

⇒

= P(E1) P(E2/E1) P(E3/E1 ∩ E2) P(E4/E1 ∩ E2 ∩ E3) ......... If events are independent, then

P(A ∩ B) = 0

P(E1 ∩ E2 ∩ E3 ∩ ....... ∩ En) = P(E1) P(E2) ....... P(En) 6.

Probability of at least one of the n Independent events : If P1, P2, ....... Pn are the probabilities of n independent events A, A2, .... An then the probability of happening of at least one of these event is.

P(A ∩ B) ≠ 0 ∴

P(A ∪ B) = P(A) + P(B) P(A ∩ B)

or

P(A + B) = P(A) + P(B) P(AB)

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1 [(1 P1) (1 P2)......(1 Pn)]

When events are independent i.e. P(A ∩ B) = P(A) P (B)

or

P(A + B) = P(A) + P(B) P(A) P(B)


P(A ∩ B) = P(B/A) P(A), P(A) ≠ 0

P(E1 ∩ E2 ∩ E3 ∩ ............... ∩ En)

When events are not mutually exclusive i.e.

∴

Multiplication Theorem :

Generalized :

∴ P(A ∪ B) = P(A) + P(B)

(iii)

(ii)

or

When events are mutually exclusive

(ii)

If A and B are independent event, then P(A/B) = P(A) and P(B/A) = P(B) P(A ∩ B) = P(A/B). P(B), P(B) ≠ 0

Addition Theorem of Probability : (i)

No. of sample pts. in A ∩ B . No. of pts. in B

(i)

Probability that exactly r letters are in right

1 envelopes = r!

P(A ∩ B) P(A)

If the outcomes of the experiment are equally

likely, then P(A/B) =

1 1 1 1 = + .... + (1)n 2! 3! 4! n!

(d)

P(A ∩ B) P(B)

P(B/A) = Probability of occurrence of B, given that A has

1 n!

lopes = 1 (c)

already happened =

P(A1 + A2 + ... + An) = 1 P ( A 1 ) P ( A 2 ) .... P( A n )

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Total Probability :

(a)

Let A1, A2, ............. An are n mutually exclusive & set of exhaustive events and event A can occur through any one of these events, then probability of occurence of A

(c)

Variance E(x2) (E(x))2 = npq

(d)

Standard deviation = npq

P(A) = P(A ∩ A1) + P(A ∩ A2) + ............. + P(A ∩ An) n

=

8.


∑

r =1

(b)

(i)

P(Ar) P(A/Ar)

If two persons A and B speaks truth with the probability p1 & p2 respectively and if they agree on a statement, then the probability that they are speaking truth will be given by

p1p2 p1p2 + (1 − p1 ) (1 − p2 ) .

Let A1, A2, A3 be any three mutually exclusive & exhaustive events (i.e. A1 ∪ A2 ∪ A3 = sample space & A1 ∩ A2 ∩ A3 = φ) an sample space S and B is any other event on sample space then,

(ii)

P(B / A i)P(A i ) P(Ai/B) = P(B / A )P(A ) + P(B / A )P(A ) + P(B / A )P(A ) , 1 1 2 2 3 3

αp1p2 αp1p2 + (1 − α) (1 − p1) (1 − p2 )

Probability distribution :

(iii)

If a random variable x assumes values x1, x2, ......xn with probabilities P1, P2, ..... Pn respectively then (a) (b) (c)

(ii)

If A and B both assert that an event has occurred, probability of occurrence of which is α then the probability that event has occurred. Given that the probability of A & B speaking truth is p1, p2.

i = 1, 2, 3

(i)

E (x2) = npq + n2 p2

10. Truth of the statement :

Baye's Rule :

9.

mean E(x) = np

(jhuth) coincides is β then from above case required probability will be

P1 + P2 + P3 + ..... + Pn = 1

αp1p2 + − αp1p2 (1 α) (1 − p1) (1 − p2 ) β .

mean E(x) = Σ Pixi

Variance = Σx Pi (mean) = Σ (x ) (E(x)) 2

2

2

If in the second part the probability that their lies

2

Binomial distribution : If an experiment is repeated n times, the successive trials being independent of one another, then the probability of r success is nCr Pr qnr n

atleast r success is

∑

k =r

Ck Pk qnk

n

where p is probability of success in a single trial, q = 1 p , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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PROGRESSION AND SERIES 1.

(iv) If A1, A2,...... An are n A.M's between a and b, then A1 = a + d, A2 = a + 2d,...... An = a + nd,

Arithmetic Progression (A.P.) : (a)

where d =

General A.P. a, a + d, a + 2d, ...... , a + (n 1) d where a is the first term and d is the common difference

(b)

General (nth) term of an A.P.

(v)

Tn = a + (n 1)d [nth term from the beginning] = a + (m n)d

term i.e. an =

Sum of n terms of an A.P. Sn

=

n n [2a + (n 1)d] = [a + Tn] 2 2

2.

where Sn1 is sum of (n 1) terms.

(e)

Supposition of terms in A.P. (i)

Three terms as a - d, a, a + d

(ii)

Four terms as a 3d, a d, a + d, a + 3d

(iii)

Five terms as a 2d, a d, a, a + d, a + 2d A.M. of n numbers A1, A2, ................ An is defined as

A.M. = (ii)

(iii)

For an A.P., A.M. of the terms taken symmetrically from the beginning and from the end will always be constant and will be equal to middle term or A.M. of middle term.

a+b i.e. 2

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1 (a + an+r), r < n 2 nr

a − Tnr a(1 − r n ) = , r<1 1−r 1−r Tnr − a a(r n − 1) = ,r>1 r −1 r −1

(d)

Sum of an infinite G.P. S ∞ =

(e)

Supposition of terms in G.P.

If A is the A.M. between two given nos. a and b, then 2A = a + b


= =

A1 + A 2 +.........+ A n ΣA i Sum of numbers = = n n n

A=

E

Sn

Arithmetic mean (A.M.) : (i)

n (a + b) 2

Geometric Progression (G.P.) (a) General G.P. a, ar, ar2 , ...... where a is the first term and r is the common ratio (b) General (nth) term of a G.P. Tn = arn1 If a G.P. having m terms then nth term from end = armn (c) Sum of n terms of a G.P.

Note : If sum of n terms i.e. Sn is given then Tn = Sn Sn1 (d)

Sum of n A.M's inserted between a and b is

(vi) Any term of an A.P. (except first term) is equal to the half of the sum of term equidistant from the

If an A.P. having m terms, then nth term from end (c)

b−a n+1

PAGE # 27

a , 1−r

a , a, ar r

(i)

Three terms as

(ii)

Four terms as

a a , , ar, ar3 r3 r

(iii)

Five terms as

a a , , a, ar, ar2 r2 r


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|r|<1

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MATHS FORMULA - POCKET BOOK (f)


Geometric Mean (G.M.) (i)

Geometrical mean of n numbers x1, x2, .......... xn is defined as G.M. = (x1 x2 ............... xn)1/n.

(i)

If G is the G.M. between two given numbers a and b, then G2 = ab ⇒ G =

(ii)

ab

If G1, G2, .............. Gn are n G.M's between a and b, then G1 = ar, G2 = ar ,..... Gn 2

(iii)

3.

F bI = ar , where r = G J H aK

(d)

Σa = a + a + .... + (n times) = na

(e)

Σ(2n 1) = 1 + 3 + 5 + .... (2n 1) = n2

(f)

Σ2n = 2 + 4 + 6 + .... + 2n = n (n + 1)

(b)

General (nth) term Tn = [a + (n 1) d] rn1

(c)

Sum of n terms of an A.G.P Sn =

(d)

Sum of infinite terms of an A.G.P.

= (c)

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2ab a+b

(i)

If H is the H.M. between a and b, then H =

(ii)

If H1, H2,......,Hn are n H.M's between a and b,

ab(n + 1) ab(n + 1) , ....., Hn = bn + a na + b 1 1 & , then their a b

reciprocal will be required H.M's. Relation Between A.M., G.M. and H.M. (i)

AH = G2

(ii)

A ≥ G ≥ H If A and G are A.M. and G.M. respectively between two +ve numbers, then these numbers are

(iii)

n(n + 1)(2n + 1) 6

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 C

Harmonic Mean (H.M.)

or first find n A.M.'s between

n(n + 1) 2

Σn2 = 12 + 22 + 32 + ..... + n2 =

1 1 = th n term coresponding to A.P. a + (n − 1)d

then H1 =

6.

(b)

1 1 1 , , +...... a a + d a + 2d

General (nth term) of a H.P. Tn

d(1 − r n−1) a + r. (1 − r)2 1−r

Sum standard results : Σn = 1 + 2 + 3 + ..... + n =

3

(b)

dr a + (1 − r)2 1−r

(a)

3

General H.P.

(a + 2d) r2, .............

General form a, (a + d)r,

3

(a)

Arithmetico - Geometric Progression (A.G.P.) :

4.

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1/n+1

n

(a)

2

Σn = 1 + 2 + 3 + .... + n 3

Harmonic Progression (H.P)

Product of the n G.M.'s inserted between a & b is (ab)n/2

S∞ =

E

5.

Ln(n + 1) OP = M N 2 Q

(c)

3

A±

PAGE # 29

A 2 − G2


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BINOMIAL THEOREM 1.


Properties of Binomial coefficients :

n

(x + a) n = nC0 xn + nC1 xn1 a + nC2 xn2 a2 + ....... + nCr xnr ar + .... + nCn an

∑

Cr xnr ar

n

r=0

(i)

General term - Tr+1 = nCr xnr ar is the (r + 1)th term from beginning.

(ii)

(m + 1) term from the end = (n m + 1) from beginning = Tnm+1

(iii)

middle term

th

(b) If n is odd then middle term =

FG n + 1IJ H 2K

term

and

term

To determine a particular term in the given expasion : α

±

1 xβ

IJ K

C0 + C1 + C2 + ..... + Cn = 2n

*

C0 C 1 + C2 C3 + ..... + C n = 0

*

C0 + C 2 + C4 + ..... = C1 + C 3 + C5 + .... = 2n1

*

n

*

2n

*

n

*

C1 + 2C2 + 3C3 + ... + nCn = n.2n1

*

C1 2C2 + 3C3 ......... = 0

*

C0 + 2C 1 + 3C2 + ......+ (n + 1)Cn = (n + 2)2n1

*

C02 + C 12 + C22 + ..... + Cn2 =

n

, if xn occurs in

*

C02 C 12 + C22 C32 + .....

Tr+1 (r + 1)th term then r is given by n α r (α + β) = m


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Cr =

n n −1 nn−1 Cr −1 = r r r −1

Cn + r =

n−2

Cr −2 and so on ...

Cr +

n

2n! n−r ! n+r !

c

hc

Cr − 1 =

n+1

h

Cr

=

and for x0, n α r (α + β) = 0

E

Cr ..... n Cn are usually denoted by C0, C1 ,.....

th

th

F Let the given expansion be G x H

C1 ,

th

Binomial coefficient of middle term is the greatest binomial coefficient. 2.

n

n

*

th

Fn I (a) If n is even then middle term = G + 1J H2 K FG n + 3IJ H 2 K

C2 .....

C0 ,

C r .......... Cn respectively.

n

=

n

For the sake of convenience the coefficients

Binomial Theorem for any +ve integral index :

PAGE # 31

|RS T|c−1h

c2nh! cn!h 2

0,

n/2 n

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2n

Cn

if n is odd

Cn/2 , if n is even


=

PAGE # 32


Note : 2n + 1

4.

2n + 1

2n + 1

C0 +

2n + 1

Cn + 2 + .....

C1 + .... +

C2n + 1 = 2

2n + 1

Cn =

2n + 1


Cn + 1 +

(ii)

2n

(x + y + z)n =

∑

r + s + t =n

n! xr ys z t s! r !t !

Generalized (x1 + x2 +..... xk)n

*

C0 +

C1 C2 Cn 2n+1 − 1 + + ..... + = 2 3 n+1 n+1

*

C0

C1 C2 C3 (−1)n C n + .... + 2 3 4 n+1

=

= 1 n+1

6.

∑

r1 +r2 +...rk =n

n! r1 r2 rk r1 ! r2 !....rk ! x1 x2 ..... xk

Total no. of terms in the expansion (x1 + x2 +... xn)m is m+n1 C n1

Greatest term : (i)

If

(n + 1)a ∈ Z (integer) then the expansion has two x+a

greatest terms. These are kth and (k + 1)th where x & a are +ve real nos. (ii)

If

(n + 1)a ∉ Z then the expansion has only one greatx+a

est term. This is (k + 1)th term k =

LM(n + 1)aOP , N x+a Q

{[.] denotes greatest integer less than or equal to x} 5.

Multinomial Theorem : n

(x + a)n = ∑

(i)

r =0

n

= ∑

r =0

n

Cr xnr ar, n ∈N

n! n! xnr ar = r +∑ x s ar , s =n s ! r ! (n − r)! r !

where s = n r , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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TRIGONOMETRIC RATIO AND IDENTITIES 1.

Trigonometric identities : (i) sin2θ + cos2θ = 1 (ii) cosec2θ cot2θ = 1 (iii) sec2θ tan2θ = 1

4.

Sign convention :

Arc length AB = r θ Area of circular sector =

(ii)

1 2 r θ 2

y

π n

(a)

Internal angle of polygon = (n 2)

(b)

Sum of all internal angles = (n 2) π

(c)

Radius of incircle of this polygon r =

(d)

Radius of circumcircle of this polygon R

1 na2 cot 4

(e)

Area of the polygon =

(f)

Area of triangle =

(g)

F a πI Area of incircle = π G cot J H 2 nK

(h)

2.

II quadrant sin & cosec are +ve

For a regular polygon of side a and number of sides n

x' III quadrant tan & cot are +ve

5.

FG π IJ H nK

2

6.

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sinθ

cosθ

cosθ

cosθ

tanθ

tanθ

m sinθ m cotθ

cotθ

cotθ

m sinθ cosθ ±tanθ

cosθ

±sinθ

±sinθ

cosθ

m cotθ m tanθ

±tanθ

±cotθ ±cotθ m tanθ secθ ±cosecθ secθ m cosecθ secθ cosecθ secθ ±cosecθ m cosecθ sec θ

Sum & differences of angles of t-ratios : (i) sin(A ± B) = sinA cosB ± cosA sinB (ii) cos(A ± B) = cosA cosB ± sinA sinB

& π radian = 1800


sinθ

secθ cosecθ

Relation between system of measurement of angles :

D G 2C = = π 90 100

IV quadrant cos & sec are +ve

T-ratios of allied angles : The signs of trigonometrical ratio in different quadrant. Allied∠ of (θ) 900 ± θ 1800 ± θ 2700 ± θ 3600 ± θ T-ratios

2

FG a cos ec π IJ H2 nK

x

y'

π 1 2 a cos 4 n

Area of circumcircle = π

I quadrant All +ve O

π a cot 2 n

a π = cosec 2 n

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Some important results : (i)

E


(iii)

PAGE # 35

tan (A ± B) =

tan A ± tanB 1 m tan A tan B


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MATHS FORMULA - POCKET BOOK (iv) cot (A ± B) =


cot A cot B m 1 cot B ± cot A

Formulaes for product into sum or difference and viceversa :

(v) sin(A + B) sin(A B) = sin2A sin2B = cos2 B cos2 A (vi) cos(A + B) cos (A B) = cos2A sin2B = cos2B sin2A (vii) tan(A + B + C) =

S1 − S3 + S5 − S7 +...... = 1 − S + S − S + S −...... 2 4 6 8 S1 = Σ tan A S2 = Σ tan A tan B, S3 = Σ tan A tan B tan C & so on (viii) sin (A + B + C) = Σ sin A cos B cos C Π sin A = Π cos A (Numerator of tan (A + B + C)) cos (A + B + C) = Π cos A Σ sin A sin B cos C = Π cos A (Denominator of tan (A + B + C)) for a triangle A + B + C = π 8.

Σ tan A = Π tan A Σ sin A = Σ sin A cos B cos C 1 + Π cos A = Σ sin A sin B cos C (viii) sin75 = 0

= cos15

2 2

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3 −1

(x)

tan75 = 2 +

3 = cot15

(xi)

cot750 = 2

0 3 = tan15

N

S

sinC + sinD = 2sin

(ix)

tanA + tanB =

sin(A + B) cos A cos B

T-ratios of multiple and submultiple angles : (i)

sin2A = 2sinA cosA =

(ii)

cos750 =

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(v)

2 tan A 1 + tan2 A

⇒ sinA = 2sinA/2 cosA/2 = = sin150

PAGE # 37

2 tan A / 2 1 + tan2 A / 2

cos2A = cos2A sin2A = 2cos2A 1 = 1 2sin2A =

0


2cosA cosB = cos(A + B) + cos(A B)

0

(ix)

0

(iii)

= (sin A + cos A)2 1 = 1 (sin A cos A)2

3 +1

2 2

2cosA sinB = sin(A + B) sin(A B)

FG C + D IJ cos FG C − D IJ H 2 K H 2 K F C + D I F C − DI (vi) sinC sinD = 2cos GH 2 JK sin GH 2 JK F C + D IJ cos FG C − D IJ (vii) cosC + cosD = 2cos GH H 2 K 2 K F C + D IJ sin FG D − C IJ (viii) cosC cosD = 2sin GH H 2 K 2 K

Generalized tan (A + B + C + ......... )

(ix)

2sinA cosB = sin(A + B) + sin(A B)

(ii)

(iv) 2sinA sinB = cos(A B) cos(A + B)

S1 − S3 tan A + tanB + tan C − tan A tan B tan C = 1−S 1 − tan A tan B − tanB tan C − tan C tan A 2

Where

(i)

1 − tan2 A 1 + tan2 A


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(iii)

2 tan A tan2A = 1 − tan2 A


2 tan A / 2 ⇒ tanA = 1 − tan2 A / 2

(iii)

sinA + sinB + sinC = 4cosA/2 cosB/2 cosC/2

(v)

= sin θ (2 cos θ 1) (2 cos θ + 1)

(vii) tanA + tanB + tanC = tanA tanB tanC

= 4cos(60 θ ) cos(60 + θ ) cos θ

(viii) cotB cotC + cotC cotA + cotA cotB = 1

= cos θ (1 2 sin θ ) (1 + 2 sin θ )

(ix)

0

3 tan A − tan3 A (vi) tan3A = 1 − 3 tan2 A = tan(600 A) tan(600 + A)tanA (vii) sinA/2 =

(xi)

Σ cot A/2 = Π cot A/2

sinα + sin(α + β) + sin(α + 2β) + .... + to nterms

(i)

1 − cos A 2

LM FG n − 1IJβOP sinLnβ O N H 2 K Q MN 2 PQ , β ≠ 2nπ

sin α +

1 + cos A 2

(ix)

1 − cos A 1 − cos A = , A ≠ (2n + 1)π sin A 1 + cos A

tanA/2 =

(x)

Σ tan A/2 tan B/2 = 1 Σ cot A cot B = 1

11. Some useful series :

(viii) cosA/2 =

=

(ii)

sin

β 2

cosα + cos(α + β) + cos(α + 2β) + .... + to nterms

LM FG n − 1IJ βOP sin nβ N H 2 K Q 2 β ≠ 2nπ

cos α +

Maximum and minimum value of the expression : acosθ + bsinθ

=

Maximum (greatest) Value = a2 + b2 Minimum (Least) value = a2 + b2

(iii)

sin

β 2

cosα .cos2α . cos22α ....cos(2n1 α) =

sin 2n α , α ≠ nπ 2n sin α

= 1 , α = 2kπ

10. Conditional trigonometric identities :

= 1 , α = (2k+1)π

If A, B, C are angles of triangle i.e. A + B + C = π, then (i)

sin2A + sin2B + sin2C = 1 2sinA sinB cosC

(vi) cos2A + cos2B + cos2C = 1 2cosA cosB cosC

cos3θ = 4cos3θ 3cosθ 0

9.

cos2A + cos2B + cos2C = 1 4cosA cosB cosC

(iv) cosA + cosB + cosC = 1 + 4 sinA/2 sinB/2 sinC/2

(iv) sin3θ = 3sinθ 4sin3θ = 4sin(600 θ ) sin(600 + θ ) sin θ (v)

(ii)

sin2A + sin2B + sin2C = 4sinA sinB sinC i.e. Σ sin 2A = 4 Π (sin A)


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TRIGONOMETRIC EQUATIONS 1.

Thus the equation reduces to form cos(θ α) =

General solution of the equations of the form

2.

(i)

sinθ = 0

⇒

θ = nπ,

n∈I

(ii)

cosθ = 0

⇒

θ = (2n + 1)

(iii)

tanθ = 0

⇒

θ = nπ,

(iv) sinθ = 1

⇒

θ = 2nπ +

(v)

cosθ = 1

⇒

θ = 2πn

(vi) sinθ = 1

⇒

θ = 2nπ

(vii) cosθ = 1

⇒

θ = (2n + 1)π

(viii) sinθ = sinα

⇒

θ = nπ + (1)nα

c 2

a + b2

= cosβ(say)

now solve using above formula

π , n∈I 2

3.

Some important points :

n∈I

π 2

(i)

If while solving an equation, we have to square it, then the roots found after squaring must be checked wheather they satisfy the original equation or not.

(ii)

If two equations are given then find the common values of θ between 0 & 2π and then add 2nπ to this common solution (value).

π 3π or 2nπ + 2 2

θ = 2nπ ± α

(ix)

cosθ = cosα

⇒

(x)

tanθ = tanα ⇒

θ = nπ + α

(xi)

sin2θ = sin2α

⇒

θ = nπ ± α

(xii) cos2θ = cos2α

⇒

θ = nπ ± α

(xiii) tan2θ = tan2α

⇒

θ = nπ ± α

For general solution of the equation of the form a cosθ + bsinθ = c, where c ≤

a2 + b2 , divide both side by

a2 + b2 and put

a 2

a +b

2

= cosα,

b 2

a + b2

= sinα.


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INVERSE TRIGONOMETRIC FUNCTIONS 1.

If y = sin x, then x = sin1 y, similarly for other inverse Tfunctions.

2.

(ii)

Range (R)

1 ≤ x ≤ 1

π 2

cos1 x

1 ≤ x ≤ 1

0 ≤ θ ≤ π

tan1 x

∞ < x < ∞

cot1 x

∞ < x < ∞

0 < θ < π

sec1 x

x ≤ 1, x ≥ 1

0 ≤ θ ≤ π, θ ≠

x ≤ 1, x ≥ 1

π π , θ ≠ 0 ≤ θ ≤ 2 2

sin

x

cosec1 x

≤ θ ≤

π 2

sin (sin1 x) = x provided 1 ≤ x ≤ 1 cos (cos1 x) = x provided 1 ≤ x ≤ 1

Domain (D)

1

≤ θ < 0

or 0 < θ ≤

Domain and Range of Inverse T-functions : Function

π 2

cosec1 (cosec θ ) = θ provided

tan (tan1 x) = x provided ∞ < x < ∞

π 2

cot (cot1 x) = x provided ∞ < x < ∞ sec (sec1 x) = x provided ∞ < x ≤ 1 or 1 ≤ x < ∞ cosec (cosec1 x) = x provided ∞ < x ≤ 1

π π < θ < 2 2

or 1 ≤ x < ∞ (iii)

π 2

sin1 ( x) = sin1 x, cos1 ( x) = π cos1 x tan1 ( x) = tan1 x cot1 ( x) = π cot1 x cosec1 ( x) = cosec1 x

3.

Properties of Inverse T-functions : sin1 (sin θ ) = θ provided

(i)

sec1 ( x) = π sec1 x

π 2

π ≤ θ ≤ 2

cos1 (cos θ ) = θ provided θ ≤ θ ≤ π tan1 (tan θ ) = θ provided cot

1

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∀ x ∈ [ 1, 1]

tan1 x + cot1 x =

π , 2

∀ x ∈ R

(cot θ ) = θ provided 0 < θ < π

π π or < θ ≤ π 2 2


π , 2

π π < θ < 2 2

sec1 (sec θ ) = θ provided 0 ≤ θ <

E

(iv) sin1 x + cos1 x =

sec1 x + cosec1 x =

PAGE # 43

π , x ∈ ( ∞ , 1] ∪ [1, ∞ ) 2 ∀


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Value of one inverse function in terms of another inverse function : (i)

sin1 x = cos1

1

= sec1

(ii)

1−x

1−x

2

2

= tan1

(iii)

tan1 x = sin1

= sec1

(iv) sin1

(v)

cos1

(vi) tan1

1

1 = cosec1 x

FG 1 IJ H xK FG 1 IJ H xK FG 1 IJ H xK

1 − x2

x 1 + x2

1 − x2

1 − x2 = cot1 x

1 1 + x2

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1 − x2 x

(i)

x

1

(iii)

1 − x2 (iv)

= cot1

(v)

1 x

FG x + y IJ ; if x > 0, y > 0, xy < 1 H 1 − xy K F x + y IJ ; if x > 0, y > 0, xy > 1 tan x + tan y = π + tan G H 1 − xy K F x−yI tan x tan y = tan GH 1 + xy JK ; if xy > 1 F x − y IJ ; if x > 0, y < 0, xy < 1 tan x tan y = π + tan G H 1 + xy K F x + y + z − xyz IJ tan x + tan y + tan z = tan G H 1 − xy − yz − zx K L O sin x ± sin y = sin Mx 1 − y ± y 1 − x P ; N Q tan1x + tan1y = tan1

(ii)

(vi)

1 + x2 , x ≥ 0 x

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

if x,y ≥ 0 & x2 + y2 ≤ 1

2

LM N

OP Q

2 2 (vii) sin1x ± sin1y = π sin1 x 1 − y ± y 1 − x ;

= cosec1 x , ∀ x ∈ ( ∞ , 1] ∪ [1, ∞ )

if x,y ≥ 0 & x2 + y2 > 1

RS cot x |T− π + cot x −1

=

−1

LM N

OP Q

2 2 (viii) cos1x ± cos1y = cos1 xy m 1 − x 1 − y ;

= sec1 x, ∀ x ∈ ( ∞ , 1] ∪ [1, ∞ )

if x,y > 0 & x2 + y2 ≤ 1

LM N

OP Q

2 2 (ix) cos1x ± cos1y = π cos1 xy m 1 − x 1 − y ;

for x > 0 for x < 0


Formulae for sum and difference of inverse trigonometric function :

, 0 ≤ x ≤ 1

= cos1

1 1 + x2 = cosec

= cot1

5.

1 , 0 ≤ x ≤ 1 x

= cosec1

cos1 x = sin1 1 − x2 = tan1

= sec1

x


if x,y > 0 & x2 + y2 > 1

PAGE # 45


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PROPERTIES & SOLUTION OF TRIANGLE

Inverse trigonometric ratios of multiple angles (i)

2sin1x = sin1(2x

(ii)

2cos1x = cos1(2x2 1), if 1 ≤ x ≤ 1

1 − x2 ), if 1 ≤ x ≤ 1

Properties of triangle :

FG 2x IJ = sin FG 2x IJ = cos FG 1 − x IJ H1 − x K H1 + x K H1 + x K

1.

A triangle has three sides and three angles. In any ∆ABC, we write BC = a, AB = c, AC = b

2

(iii)

2tan1x = tan1

1

2

2

1

A

2

A

(iv) 3sin1x = sin1(3x 4x3) (v)

c

3cos x = cos (4x 3x) 1

1

B

F 3x − x I GH 1 − 3x JK

B

3

(vi) 3tan1x = tan1

b

3

C a

C

and ∠BAC = ∠A, ∠ABC = ∠B, ∠ACB = ∠C

2

In ∆ABC : (i) A + B + C = π (ii) a + b > c, b +c > a, c + a > b

2.

(iii) 3.

a > 0, b > 0, c > 0

Sine formula : a b c = = = k(say) sin A sinB sin C sin A sinB sinC = = = k (say) a b c

or

4.


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Cosine formula : cos A =

b2 + c2 − a2 2bc

cos B =

c2 + a2 − b2 2ac

cos C =

a2 + b2 − c2 2ab

PAGE # 47


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Projection formula : a = b cos C + c cos B b = c cos A + a cos C c = a cos B + b cos A

6.

(c)

Napier's Analogies : A −B a−b = cot 2 a+b

C 2

tan

B−C b−c = cot 2 b+c

A 2

tan

C−A c−a = cot 2 c+a

B 2

tan

(a)

sin

sin

sin

(b)

A = 2

B = 2 C = 2

(s − a) (s − b) ab

A = 2

s (s − a) bc

B cos = 2

s (s − b) ca

C = 2

s (s − c) ab

cos

cos

9.

C

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(s − b) (s − c) s (s − a)

tan

B = 2

(s − c) (s − a) s (s − b)

tan

C = 2

(s − b) (s − a) s (s − c)

1 1 1 ∆ = 2 ab sin C = 2 bc sin A = 2 ca sin B

(ii)

∆ =

s(s − a) (s − b) (s − c)

tan

A B s −c tan = 2 2 s

tan

B C s −a tan = 2 2 s

tan

C tan 2

where 2s = a + b + c


A = 2

(i)

(s − b) (s − c) bc (s − c) (s − a) ca

tan

∆, Area of triangle :

8.

Half angled formula - In any ∆ABC :

7.

E


A s −b = 2 s

10. Circumcircle of triangle and its radius : (i)

R=

a b c = = 2sin A 2sinB 2sin C

(ii)

R=

abc 4∆

PAGE # 49

Where R is circumradius


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11. Incircle of a triangle and its radius : ∆ s

(iii)

r=

(iv)

r = (s a) tan

(v)

(iv) r1 + r2 + r3 r = 4R (v)

r = 4R sin

A B C = (s b) tan = (s c) tan 2 2 2

(vi)

A B C sin sin 2 2 2

(vii)

(vi) cos A + cos B + cos C = 1 +

r R

1 1 1 1 r1 + r2 + r3 = r 1 r12

+

(ix)

A B , r2 = s tan , r3 = s tan 2 2

r1 = s tan

(iii)

r1 = 4R sin

A B C cos cos , 2 2 2

r2 = 4R cos

A B C sin cos , 2 2 2

r3 = 4R cos

A B cos sin 2 2

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+

1 r2

=

a2 + b2 + c2 ∆2

∆ = 2R2 sin A sin B sin C = 4Rr cos

A B C cos cos 2 2 2

B C C A b cos cos cos 2 2 2 2 , r2 = , A B cos cos 2 2

(x)

r1 =

A B cos 2 2 C cos 2

c cos r3 = C 2

C 2


r32

a cos

∆ ∆ ∆ , r2 = , r3 = s−a s −b s−c

(ii)

1

1 1 1 1 + + = 2Rr bc ca ab

12. The radii of the escribed circles are given by : r1 =

+

r22

(viii) r1r2 + r2r3 + r3r1 = s2

B C A C B A a sin sin b sin sin c sin sin 2 2 2 2 2 2 (vii) r = = = A B C cos cos cos 2 2 2

(i)

1

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HEIGHT AND DISTANCE 1.

(ii)

d = h (cotα cotβ)

Angle of elevation and depression :

h

If an observer is at O and object is at P then ∠ XOP is called angle of elevation of P as seen from O.

α

β

d

If an observer is at P and object is at O, then ∠ QPO is called angle of depression of O as seen from P. 2.

Some useful result : (i)

In any triangle ABC if AD : DB = m : n

∠ ACD = α , ∠ BCD = β & ∠ BDC = θ then (m + n) cotθ = m cotα ncot β

C α β

θ

A A

m

D

B n

B

= ncotA mcotB [m n Theorem] , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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MATHS FORMULA - POCKET BOOK (C)

POINT 1.

Distance formula : Distance between two points P(x1, y1) and Q(x2, y2) is given by d(P, Q) = PQ = =

(x 2 − x1)2 + (y2 − y1 )2

(Difference of x coordinate)2 + (Difference of y coordinate)2

Note :

(i) d(P, Q) ≥ 0 (ii) d(P, Q) = 0 ⇔ P = Q (iii) d(P, Q) = d(Q, P) (iv) Distance of a point (x, y) from origin (0, 0) =

2.

x2 + y2

AP m = = λ , Here λ > 0 BP n

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n

m A(x 1 , y1 )

P (ii)

FG mx + nx H m+n 2

1

,

B(x 2 , y2 )

P

my 2 + ny1 m+n

IJ K

Externally : m

AP m = = λ BP n

P

In Parallelogram : Calculate AB, BC, CD and AD. (i) If AB = CD, AD = BC, then ABCD is a parallelogram. (ii) If AB = CD, AD = BC and AC = BD, then ABCD is a rectangle. (iii) If AB = BC = CD = AD, then ABCD is a rhombus. (iv) If AB = BC = CD = AD and AC = BD, then ABCD is a square.


Section formula : (i) Internally :

Use of Distance Formula : (a) In Triangle : Calculate AB, BC, CA (i) If AB = BC = CA, then ∆ is equilateral. (ii) If any two sides are equal then ∆ is isosceles. (iii) If sum of square of any two sides is equal to the third, then ∆ is right triangle. (iv) Sum of any two equal to left third they do not form a triangle i.e. AB = BC + CA or BC = AC + AB or AC = AB + BC. Here points are collinear. (b)

E

3.

For circumcentre of a triangle : Circumcentre of a triangle is equidistant from vertices i.e. PA = PB = PC. Here P is circumcentre and PA is radius. (i) Circumcentre of an acute angled triangle is inside the triangle. (ii) Circumcentre of a right triangle is mid point of the hypotenuse. (iii) Circumcentre of an obtuse angled triangle is outside the triangle.

(iii)

FG mx − nx H m−n 2

1

,

n P B(x 2, y 2 )

A(x 1, y 1 )

my 2 − ny1 m−n

IJ K

Coordinates of mid point of PQ are

FG x H

1

+ x2 y1 + y2 , 2 2

IJ K

(iv) The line ax + by + c = 0 divides the line joining the points

(ax1 + by + c) (x1, y1) & (x2, y2) in the ratio = (ax + by + c) 2 2

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For parallelogram midpoint of diagonal AC = mid point of diagonal BD

(vi) Coordinates of centroid G

FG x H

1

+ x 2 + x 3 y1 + y 2 + y 3 , 3 3

(vii) Coordinates of incentre I

FG ax + bx + cx H a+b+c 1

2

3

ay + by 2 + cy 3 , 1 a+b+c


Area of Polygon : Area of polygon having vertices (x 1, y1), (x2, y2), (x3, y3) ........ (xn, yn) is given by area

IJ K

IJ K =

(viii) Coordinates of orthocentre are obtained by solving the equation of any two altitudes. 4.

Area of Triangle : The area of triangle ABC with vertices A(x1, y1), B(x2, y2) and C(x3, y3).

∆=

=

1 2

x1

y1 1

x2

y2 1

x3

y3 1

x1

y1

x2 1 2 x3

y2

x1

y1

y3

6.

1 [x y + x2y3 + x3y1 x2y1 x3y2 x1y3] 2 1 2


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y2

x3

y3

M

M

xn

yn

x1

y1

. Points must be taken in order.

B

y

B

x' → cosθ

sinθ

y' → sinθ

cosθ

Some important points : (i) Three pts. A, B, C are collinear, if area of triangle is zero (ii) Centroid G of ∆ABC divides the median AD or BE or CF in the ratio 2 : 1 (iii) In an equilateral triangle, orthocentre, centroid, circumcentre, incentre coincide. (iv) Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining orthocentre and circumcentre in the ratio 2 : 1 (v) Area of triangle formed by coordinate axes & the line

Note : (i) Three points A, B, C are collinear if area of triangle is zero. (ii) If in a triangle point arrange in anticlockwise then value of ∆ be +ve and if in clockwise then ∆ will be ve.

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x

(Determinant method)

[Stair method]

E

y1

Rotational Transformation : If coordinates of any point P(x, y) with reference to new axis will be (x', y') then

7. =

1 2

x1

ax + by + c = 0 is

PAGE # 57

c2 . 2ab


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MATHS FORMULA - POCKET BOOK (ix)

STRAIGHT LINE 1.

(x)

making an angle θ , 0 ≤ θ ≤ π , θ ≠

a b

(xi)

(vi) Slope of two parallel lines are equal. (vii) If m1 & m2 are slopes of two ⊥ lines then m1m2 = 1. 2.

Standard form of the equation of a line : (i) Equation of x-axis is y = 0 (ii) Equation of y-axis is x = 0 (iii) Equation of a straight line || to x-axis at a distance b from it is y = b (iv) Equation of a straight line || to y-axis at a distance a from it is x = a (v) Slope form : Equation of a line through the origin and having slope m is y = mx. (vi) Slope Intercept form : Equation of a line with slope m and making an intercept c on the y-axis is y = mx + c. (vii) Point slope form : Equation of a line with slope m and passing through the point (x1, y1) is y y1 = m(x x1) (viii) Two point form : Equation of a line passing through the points (x1, y1) & (x2, y2) is

3.


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x − x1 y − y1 = = r cos θ sinθ x = x1 + r cos θ , y = y1 + r sin θ ⇒ Where r is the distance of any point P(x, y) on the line from the point (x1, y1) Normal or perpendicular form : Equation of a line such that the length of the perpendicular from the origin on it is p and the angle which the perpendicular makes with the +ve direction of x-axis is α , is x cos α + y sin α = p.

Angle between two lines : (i) Two lines a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0 are

y − y1 x − x1 y2 − y1 = x2 − x1

E

π with the 2

+ve direction of x-axis is

y2 − y1 B(x2, y2) is x − x . 2 1 Slope of the line ax + by + c = 0, b ≠ 0 is

x y + = 1. a b Parametric or distance or symmetrical form of the line : Equation of a line passing through (x1, y1) and

and b respectively on x-axis and y-axis is

Slope of a Line : The tangent of the angle that a line makes with +ve direction of the x-axis in the anticlockwise sense is called slope or gradient of the line and is generally denoted by m. Thus m = tan θ . (i) Slope of line || to x-axis is m = 0 (ii) Slope of line || to y-axis is m = ∞ (not defined) (iii) Slope of the line equally inclined with the axes is 1 or 1 (iv) Slope of the line through the points A(x1, y1) and

(v)

Intercept form : Equation of a line making intercepts a

(a)

a1 b1 c1 Parallel if a = b ≠ c 2 2 2

(b)

Perpendicular if a1a2 + b1b2 = 0

(c)

a1 b1 c1 Identical or coincident if a = b = c 2 2 2

(d)

a2b1 − a1b2 If not above three, then θ = tan1 a a − b b 1 2 1 2

(ii) (a) (b)

Two lines y = m1 x + c and y = m2 x + c are Parallel if m1 = m2 Perpendicular if m1m2 = 1

(c)

m1 − m2 If not above two, then θ = tan1 1 + m m 1 2

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Position of a point with respect to a straight line : The line L(xi, yi) i = 1, 2 will be of same sign or of opposite sign according to the point A(x1, y1) & B (x2, y2) lie on same side or on opposite side of L (x, y) respectively.

5.

Equation of a line parallel (or perpendicular) to the line ax + by + c = 0 is ax + by + c' = 0 (or bx ay + λ = 0)

6.

Equation of st. lines through (x1,y1) making an angle α with y = mx + c is

MATHS FORMULA - POCKET BOOK 12. Homogeneous equation : If y = m1x and y = m2x be the two equations represented by ax2 + 2hxy + by2 = 0 , then m1 + m2 = 2h/b and m1m2 = a/b 13. General equation of second degree : ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represent a pair of

a h g h b f =0 straight line if ∆ ≡ g f c

m ± tan α (x x1) 1 m m tan α

y y1 =

If y = m1x + c & y = m2x + c represents two straight lines

length of perpendicular from (x1, y1) on ax + by + c = 0

7.

|ax1 + by1 + c|

is 8.

then m1 + m2 =

a2 + b2

Distance between two parallel lines ax + by + ci = 0, i = 1, 2 is

9.

|c1 − c 2| a2 + b2

14. Angle between pair of straight lines : The angle between the lines represented by ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 or ax2 + 2hxy + by2 = 0

2 h2 − ab is tanθ = (a + b)

Condition of concurrency for three straight lines Li ≡ ai x + bi y + ci = 0, i = 1, 2, 3 is

a1 b1 a2 b2 a3 b3

(i)

c1 c2 = 0 c3

(ii)

10. Equation of bisectors of angles between two lines : a1x + b1y + c1 2 1

2 1

a +b

=±

a2 x + b2 y + c 2 a22 + b22

11. Family of straight lines : The general equation of family of straight line will be written in one parameter The equation of straight line which passes through point of intersection of two given lines L1 and L2 can be taken as L1 + λ L2 = 0 , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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−2h a , m1m2 = . b b

The two lines given by ax2 + 2hxy + by2 = 0 are (a) Parallel and coincident iff h2 ab = 0 (b) Perpendicular iff a + b = 0 The two line given by ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 are (a) Parallel if h2 ab = 0 & af2 = bg2 (b) Perpendicular iff a + b = 0 (c) Coincident iff g2 ac = 0

13. Combined equation of angle bisector of the angle between the lines ax2 + 2hxy + by2 = 0 is 2

x −y a−b

PAGE # 61

2

=

xy h


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CIRCLE 1.

General equation of a circle : x2 + y2 + 2gx + 2fy + c = 0 where g, f and c are constants (i) Centre of the cirle is (g, f) i.e. (ii)

2.

4.

5.

Concentric circles : Two circles having same centre C(h, k) but different radii r1 & r2 respectively are called concentric circles.

6.

Position of a point w.r.t. a circle : A point (x1, y1 ) lies outside, on or inside a circle

FG − 1 coeff. of x, −1 coeff. of yIJ H 2 K 2

Radius is

S ≡ x2 + y2 + 2gx + 2fy + c = 0 according as S1 ≡ x12 + y12 + 2gx1 + 2fy1 + c is +ve, zero or ve

g2 + f 2 − c

Central (Centre radius) form of a circle : (i) (x h)2 + (y k)2 = r2 , where (h, k) is circle centre and r is the radius. (ii) x2 + y2 = r2 , where (0, 0) origin is circle centre and r is the radius.

3.


7.

Chord length (length of intercept) = 2 r2 − p2

8.

Intercepts made on coordinate axes by the circle : (i)

x axis = 2 g2 − c

(ii)

y axis = 2 f 2 − c

Diameter form : If (x1, y1) and (x2, y2) are end pts. of a diameter of a circle, then its equation is (x x1) (x x2) + (y y1) (y y2) = 0

9.

Parametric equations : (i) The parametric equations of the circle x2 + y2 = r2 are x = rcosθ, y = r sinθ ,

10. Length of the intercept made by line : y = mx + c with the circle x2 + y2 = a2 is

Length of tangent = S1

where point θ ≡ (r cos θ , r sin θ ) (ii) (iii)

The parametric equations of the circle (x h)2 + (y k)2 = r2 are x = h + rcosθ, y = k + rsinθ The parametric equations of the circle x2 + y2 + 2gx + 2fy + c = 0 are x = g +

(iv)

g2 + f 2 − c cosθ, y = f +

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(1 + m2) |x1 x2|

where |x1 x2| = difference of roots i.e.

D a

11. Condition of Tangency : Circle x2 + y2 = a2 will touch the line y = mx + c if c = ±a

θ1 + θ2 θ + θ2 θ − θ2 + y sin 1 = r cos 1 . 2 2 2


or

g2 + f 2 − c sinθ

For circle x2 + y2 = a2, equation of chord joining θ 1 & θ 2 is

x cos

a2 (1 + m2 ) − c2 1 + m2

2

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1 + m2


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MATHS FORMULA - POCKET BOOK 12. Equation of tangent, T = 0 : (i)

MATHS FORMULA - POCKET BOOK 15. The point of intersection of tangents drawn to the circle x2 + y2 = r2 at point θ 1 & θ 2 is given as

Equation of tangent to the circle

F r cos θ + θ GG 2 θ −θ GH cos 2

x2 + y2 + 2gx + 2fy + c = 0 at any point (x1, y1) is

1

xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0 (ii)

Equation of tangent to the circle x2 + y2 = a2 at any point (x1, y1) is xx1 + yy1 = a2

(iii)

In slope form : From the condition of tangency for every value of m. The line y = mx ± a 1 + m2 is a tangent to the circle

±am 1 + m2

,

±a 1 + m2

I JK

(iv) Equation of tangent at (a cos

16. Equation of the chord of contact of the tangents drawn from point P outside the circle is T = 0

radius is

θ , a sin

2 times the radius of the given circle.

θ ) to the 19.

Equation of polar of point (x1, y1) w.r.t. the circle S = 0 is T = 0

20. Coordinates of pole : Coordinates of pole of the line

13. Equation of normal :

F −a l −a mI are G n , n J H K 2

Equation of normal to the circle

lx + my + n = 0 w.r.t the circle x + y = a 2

x2 + y2 + 2gx + 2fy + c = 0 at any point P(x1, y1) is

y1 + f y y1 = x + g (x x1) 1 (ii)

I JJ JK

18. Director circle : Equation of director circle for x2 + y2 = a2 is x2 + y2 = 2a2. Director circle is a concentric circle whose

circle x2 + y2 = a2 is x cos θ + y sin θ = a.

(i)

2

θ1 + θ 2 2 , θ1 − θ2 cos 2 r sin

17. Equation of a chord whose middle pt. is given by T = S1

x2 + y2 = a2 and its point of contact is

F GH

1

2

2

2

2

21. Family of Circles : S + λS' = 0 represents a family of circles passing through the pts. of intersection of

(i)

Equation of normal to the circle x + y = a point (x1, y1) is xy1 x1y = 0 2

2

2

at any

S = 0 & S' = 0 if λ ≠ 1

14. Equation of pair of tangents SS1 = T2

(ii)

S + λ L = 0 represent a family of circles passing through the point of intersection of S = 0 & L = 0

(iii)

Equation of circle which touches the given straight line L = 0 at the given point (x1, y1) is given as (x x1)2 + (y y1)2 + λL = 0.


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MATHS FORMULA - POCKET BOOK (iv) Equation of circle passing through two points A(x1, y1) & B(x2, y2) is given as

(x x1) (x x2) + (y y1) (y y2) + λ

x

y

x1

y1 1

x2

1

MATHS FORMULA - POCKET BOOK 25. Equation of tangent at point of contact of circle is S 1 S2 = 0 26. Radical axis and radical centre :

= 0.

y2 1

22. Equation of Common Chord is S S1 = 0.

(i)

Equation of radical axis is S S1 = 0

(ii)

The point of concurrency of the three radical axis of three circles taken in pairs is called radical centre of three circles.

27. Orthogonality condition : If two circles S ≡ x2 + y2 + 2gx + 2fy + c = 0

23. The angle θ of intersection of two circles with centres C1 & C2 and radii r1 & r2 is given by

and S' = x2 + y2 + 2g'x + 2f'y + c' = 0 intersect each other orthogonally, then 2gg' + 2ff' = c + c'.

r12 + r12 − d2 cosθ = , where d = C1C2 2r1r2

24. Position of two circles : Let two circles with centres C1, C2 and radii r1, r2 . Then following cases arise as (i)

C1 C2 > r1 + r2 ⇒ do not intersect or one outside the other, 4 common tangents.

(ii)

C1 C2 = r1 + r2 ⇒ Circles touch externally, 3 common tangents.

(iii)

|r1 r2| < C1 C2 < r1 + r2 ⇒ Intersection at two real points, 2 common tangents.

(iv) C1 C2 = |r1 r2| ⇒ internal touch, 1 common tangent. (v)

C1 C2 < |r1 + r2| ⇒ one inside the other, no tangent.

Note : Point of contact divides C1 C2 in the ratio r1 : r2 internally or externally as the case may be , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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PARABOLA 1.

Standard Parabola : Imp. Terms

y2 = 4ax y2 = 4ax x2 = 4ay

x2 = 4ay

Vertex (v)

(0, 0)

(0, 0)

(0, 0)

(0, 0)

Focus (f)

(a, 0)

(a, 0)

(0, a)

(0, a)

Directrix (D)

x = a

x = a

y = a

y = a

Axis

y = 0

y = 0

x = 0

x = 0

L.R.

4a

4a

4a

4a

Focal

x + a

a x

y + a

a y

(at2, 2at) ( at2, 2at)

(2at, at2)

(2at, at2)

Parametric

x = at2

x = at2

x = 2at

x = 2at

Equations

y = 2at

y = 2at

y = 2at2

y = at2

y2 = 4ax

distance Parametric Coordinates

x2 = 4ay

y2 = 4ax

x2 = 4ay , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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Special Form of Parabola * Parabola which has vertex at (h, k), latus rectum l and axis parallel to x-axis is (y k)2 = l (x h)

⇒ *

axis is y = k and focus at


Equations of tangent in different forms : (i)

Equations of tangent of all other standard parabolas at (x1, y1) / at t (parameter)

FGh + l , kIJ H 4 K

Equation

Parabola which has vertex at (h, k), latus rectum l and axis parallel to y-axis is (x h)2 = l (y k)

⇒ *

axis is x = h and focus at

FGh, k + l IJ H 4K

i.e.

FG H

F − b , 4ac − b I GH 2a 4a JK

IJ K

y 2=4ax

yy 1 =2a(x+x1 )

(at2, 2at)

y 2 =4ax

yy 1=2a(x+x 1 ) (at2, 2at)

ty=x+at2

x =4ay

xx 1 =2a(y+y1 )

tx=y + at2

x 2 =4ay

xx 1=2a(y+y 1 ) (2at, at2)

Equation

2

of parabolas

,with vertex

and axes parallel to y-axis

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 A T

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(2at, at ) 2

tx =y+at2

Point of

Equations

Condition of

contact in terms of

of tangent in terms of

Tangency

slope(m) y2 = 4ax

Position of a point (x1, y1) and a line w.r.t. parabola y2 = 4ax. * The point (x1, y1) lies outside, on or inside the parabola y2 = 4ax according as y12 4ax1 >, = or < 0 * The line y = mx + c does not intersect, touches, intersect a parabola y2 = 4ax according as c > = < a/m

C

ty=x+at2

Equations of tangent of all other parabolas in slope form

y2 = 4ax x2 = 4ay x

5.

2

= 4ay

FG a Hm

2

FG − a H m

,

2

,

2a m

slope (m)

IJ K

2a m

IJ K

y = mx +

a m

y = mx

a m

c =

a m

c =

a m

(2am, am2)

y = mx am2

c = am2

(2am, am )

y = mx + am

c = am2

2

2

Point of intersection of tangents at any two points P(at12, 2at1) and Q(at22, 2at2) on the parabola y2 = 4ax is (at1t2, a(t1 + t2)) i.e. (a(G.M.)2, a(2A.M.))

6.

Combined equation of the pair of tangents drawn from a point to a parabola is SS' = T2, where S = y2 4ax,

Note : Condition of tangency for parabola y2 = 4ax, we have c = a/m and for other parabolas check disc. D = 0.

D U

Tangent of 't'

(ii) Slope form

Note : Parametric equation of parabola (y k)2 = 4a(x h) are x = h + at2, y = k + 2at

E

Parametric coordinates't'

2

3.

Tangent at

of parabola (x 1, y1)

2

Equation of the form ax2 + bx + c = y represents parabola. b 4ac − b2 y = a x+ 2a 4a

Point Form / Parametric form

S' = y12 4ax1 and T = yy1 2a(x + x1)

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MATHS FORMULA - POCKET BOOK Note : (i) In circle normal is radius itself. (ii) Sum of ordinates (y coordinate) of foot of normals through a point is zero. (iii) The centroid of the triangle formed by taking the foot of normals as a vertices of concurrent normals of y2 = 4ax lies on x-axis.

Equations of normal in different forms (i)

Point Form / Parametric form Equations of normals of all other standard parabolas at (x1, y1) / at t (parameter) Eqn. of

Normal

Point

Normals

parabola

at (x1, y1)

't'

at 't'

y2 = 4ax

yy1 =

−y1 (xx1) (at2, 2at) 2a

y+tx = 2at+at3

y2 = 4ax

yy1 =

y1 (xx1) 2a

ytx = 2at+at3

x2 = 4ay

x2 = 4ay

(ii)

yy1 =

yy1 =

2a x1

2a x1

(xx1)

(at2, 2at)

(2at, at2)

x+ty = 2at+at3

(xx1)

(2at, at2)

Eqn. of

Point of

Equations

Condition of

parabola

contact

of normal

Normality

y2 = 4ax

(am2, 2am) y = mx2amam3 c = 2amam3

x2 = 4ay

FG − 2a , a IJ H m mK 2

FG 2a , − a IJ Hm m K 2

y = mx+2a+

a 2

m

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S

(iii)

We get one normal if h ≤ 2a. If point lies on x-axis, then one normal will be x-axis itself.

(i)

If normal of y2 = 4ax at t1 meet the parabola again

c = 2a+

y = mx2a

a 2

The normals to y2 = 4ax at t1 and t2 intersect each other at the same parabola at t3, then t1t2 = 2 and t3 = t1 t2

10. (i)

Equation of focal chord of parabola y2 = 4ax at t1 is y =

(ii)

a

m

2t1

t12

−1

a m2

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(x a)

If focal chord of y2 = 4ax cut (intersect) at t1 and t2 then t1t2 = 1 (t1 must not be zero) Angle formed by focal chord at vertex of parabola is tan θ =

2

m

c = 2a

2 t1

(ii)

(iii)


9.

= 4ax (am2, 2am) y = mx+2am+am3 c = am+am3

x2 = 4ay

E

(ii)

xty = 2at+at3

Equations of normal, point of contact, and condition of normality in terms of slope (m)

y

Condition for three normals from a point (h, 0) on x-axis to parabola y2 = 4ax (i) We get 3 normals if h > 2a

at t2 then t2 = t1

Slope form

2

8.

2 |t2 t1| 3

Intersecting point of normals at t1 and t2 on the parabola y2 = 4ax is (2a + a(t12 + t22 + t1t2), at1t2 (t1 + t2))


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11. Equation of chord of parabola y2 = 4ax which is bisected at (x1, y1) is given by T = S1

(v)

12. The locus of the mid point of a system of parallel chords

Angle included between focal radius of a point and perpendicular from a point to directrix will be bisected of tangent at that point also the external angle will be bisected by normal.

2a . m

(vi) Intercepted portion of a tangent between the point of tangency and directrix will make right angle at focus.

13. Equation of polar at the point (x1, y1) with respect to parabola y2 = 4ax is same as chord of contact and is given by

(vii) Circle drawn on any focal radius as diameter will touch tangent at vertex.

of a parabola is called its diameter. Its equation is y =

(viii) Circle drawn on any focal chord as diameter will touch directrix.

T = 0 i.e. yy1 = 2a(x + x1) Coordinates of pole of the line l x + my + n = 0 w.r.t. the parabola y2 = 4ax is

FG n , −2amIJ Hl l K

14. Diameter : It is locus of mid point of set of parallel chords and equation is given by T = S1 15. Important results for Tangent : (i)

Angle made by focal radius of a point will be twice the angle made by tangent of the point with axis of parabola

(ii)

The locus of foot of perpendicular drop from focus to any tangent will be tangent at vertex.

(iii)

If tangents drawn at ends point of a focal chord are mutually perpendicular then their point of intersection will lie on directrix.

(iv) Any light ray travelling parallel to axis of the parabola will pass through focus after reflection through parabola.


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MATHS FORMULA - POCKET BOOK Note : If P is any point on ellipse and length of perpendiculars from to minor axis and major axis are p1 & p2, then |xp| = p1 , |yp| = p2

ELLIPSE 1.

Standard Ellipse (e < 1)

R|S x T| a

2

Ellipse

2

Imp. terms

+

y

2

b2

U| = 1V W|

⇒

For a > b

For b > a

Centre

(0, 0)

(0, 0)

Vertices

(±a, 0)

(0, ±b)

Length of major axis

2a

2b

Length of minor axis

2b

2a

Foci

(±ae, 0)

(0, ±be)

Equation of directrices

x = ±a/e

y = ±b/e

Relation in a, b and e

b2 = a2(1 e2)

a2 = b2(1 e2)

Length of latus rectum

2b2/a

2a2/b

F ±ae, ± b I GH a JK 2

Ends of latus rectum

Parametric coordinates

p12 a2

+

p22 b2

= 1

a > b

F ± a , ± beI GH b JK 2

(a cos φ , b sin φ ) (a cos φ , b sin φ ) 0 ≤ φ < 2π

Focal radii

SP = a ex1

SP = b ey1

S'P = a + ex1

S'P = b + ey1

Sum of focal radii

SP + S'P =

2a 2b

Distance bt

2ae

2be

Distance btn directrices

2a/e

2b/e

Tangents at the vertices

x = a, x = a

y = b, y = b

n

foci


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Special form of ellipse :

(ii)

If the centre of an ellipse is at point (h, k) and the directions of the axes are parallel to the coordinate axes,

cx − hh

2

then its equation is

3.

a2

+

(y − k)2 b2

ellipse

= 1.

If

a2

+

y2

x2 a2

(iii)

The point lies outside, on or inside the ellipse if S1 =

*

+

a

y12 2

5.

xx1 a2

+

x

a2

+

x2 a2

7.

yy1 b2

y

(i)

A T

I O

N

S

±b2 a2m2 + b2

I JK .

Parametric form : The equation of tangent at any

+

y2 b2

= 1 is given by SS1 = T2

Point form : The equation of the normal at (x1, y1) to the ellipse

= 1 at the point (x1, y1) is

x2 2

a

+

y2 b2

= 1 is

a2 x b2x = a2 b2. y1 x1

= 1.


a2m2 + b2

,

Equation of normal in different forms :

2

b2

= 1 at

±a2m

Equation of pair of tangents from (x1, y1) to an ellipse

Point form : The equation of the tangent to the ellipse

D U

6.

Equation of tangent in different forms : 2

b2

F GH

x y cos φ + sin φ = 1. a b

The line y = mx + c does not intersect, touches, intersect, the ellipse if

(i)

y2

a2m2 + b2 touches the ellipse

point (a cos φ , b sin φ ) is

1 > , = or < 0

b

a2m2 + b2 < = > c2

E

+

Position of a point and a line w.r.t. an ellipse : x12 2

= 1, then c2 = a2m2 + b2. Hence, the

b2

Line y = mx ±

Note : Ellipse is locus of a point which moves in such a way that it divides the normal of a point on diameter of a point of circle in fixed ratio.

*

+

Point of contact :

x2 + y2 = a2.

4.

a2

y2

a2m2 + b2 always represents the tangents to the ellipse.

= 1 is an ellipse then its auxillary circle is

b2

x2

straight line y = mx ±

Auxillary Circle : The circle described by taking centre of an ellipse as centre and major axis as a diameter is called an auxillary circle of the ellipse. x2

Slope form : If the line y = mx + c touches the

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MATHS FORMULA - POCKET BOOK (ii)


Parametric form : The equation of the normal to the ellipse

x2 a2

+

y2 b2

= 1 at (a cos φ , b sin φ ) is

(vi) If a light ray originates from one of focii, then it will pass through the other focus after reflection from ellipse.

ax sec φ by cosec φ = a b . 2

(iii)

2

Slope form : If m is the slope of the normal to the ellipse

x2 a2

+

is y = mx ±

y2 b2

= 1, then the equation of normal

m (a2 − b2 ) a2 + b2m2

Sum of square of intercept made by auxillary circle on any two perpendicular tangents of an ellipse will be constant.

9.

Equation of chord of contact of the tangents drawn from the external point (x1, y1) to an ellipse is given by

xx1

.

2

a

yy1

+

b2

= 0 i.e. T = 0.

The co-ordinates of the point of contact are

F GH

±a2 a2 + b2m2

,

±mb2 a2 + b2m2

I JK .

(x1, y1) to an ellipse

a2

+

y2 b2

Properties of tangents & normals : (i)

Product of length of perpendicular from either focii to any tangent to the ellipse will be equal to square of semi minor axis.

x cos a

(ii)

The locus of foot of perpendicular drawn from either focii to any tangent lies on auxillary circle.

(i)

(iii)

The circle drawn on any focal radius as diameter will touch auxillary circle.


C

A T

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N

S

θ+φ y + sin b 2

b2

= 1 whose

x2 a2

θ+φ = cos 2

+

(ii)

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A T

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N

= 1 is

θ−φ 2

tan

θ1 , tan 2

θ2 ±e − 1 = 2 1±e

θ 1 + θ 2 + θ 3 + θ 4 = (2n + 1) π .


b2

Sum of feet of eccentric angles is odd π. i.e.

E

y2

Relation between eccentric angles of focal chord

⇒

(iv) The protion of the tangent intercepted between the point and directrix makes right angle at corresponding focus. E

y2

11. Equation of chord joining the points (a cos θ , b sin θ ) and

= 1.

(a cos φ , b sin φ ) on the ellipse 8.

a2

+

mid point is (x1, y1) is T = S1.

Note : In general three normals can be drawn from a point

x2

x2

10. The equation of a chord of an ellipse

S

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12. Equation of polar of the point (x1, y1) w.r.t. the ellipse x2 a2

y2

+

= 1 is given by

b2

xx1

yy1

+

a2

b2

(c)

= 0 i.e. T = 0.

the ellipse

+

a2

F −a l , −b nI GH n n JK . 2

y2

= 1 is

b2

(d)

2

13. Eccentric angles of the extremities of latus rectum of the ellipse 14. (i)

x2

+

2

a

y2 2

b

FG ± b IJ . H aeK

= 1 are tan1

Equation of the diameter bisecting the chords of slope in the ellipse

y = (ii)

b2 a2m

x2 a2

+

y2 b2

x2 a2

+

y2 b2

= 1 and S, S' be two foci

of the ellipse, then SP.S'P = CQ2

The pole of the line l x + my + n = 0 w.r.t. the ellipse x2

If CP, CQ be two conjugate semi-diameters of

The tangents at the ends of a pair of conjugate diameters of an ellipse form a parallelogram.

15. The area of the parallelogram formed by the tangents at the ends of conjugate diameters of an ellipse is constant and is equal to the product of the axis i.e. 4ab. 16. Length of subtangent and subnormal at p(x1, y1) to the ellipse x2

= 1 is

a2

+

y2 b2

a2 = 1 is x1 & (1 e2) x1 x1

x

Conjugate Diameters : The straight lines y = m1x, y = m2x are conjugate diameters of the ellipse x2 a2

(iii)

+

y2 b2

= 1 if m1m2 =

b2 a2

.

Properties of conjugate diameters : (a) If CP and CQ be two conjugate semi-diameters x2

of the ellipse

a2

+

y2 b2

= 1, then

CP2 + CQ2 = a2 + b2 (b)

If θ and φ are the eccentric angles of the extremities of two conjugate diameters, then

θ φ = ±

π 2


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HYPERBOLA 1.

Standard Hyperbola : Hyperbola

x2 a2

y2 b2

= 1

Imp. terms

x2 a2

or

(0, 0)

2a

2b

2b (±ae, 0)

2a (0, ±be)

x = ±a/e

y = ± b/e

= 1

e =

Length of L.R. Parametric

2b2/a

2a2/b

co-ordinates

(a sec φ , b tan φ )

(b sec φ , a tan φ )

0 ≤ φ < 2π

0 ≤ φ < 2π

SP = ex1 a S'P = ex1 + a 2a

SP = ey1 b S'P = ey1 + b 2b

Fa + b I GH a JK 2

2

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N

S

Hyperbola

Fa + b I GH b JK 2

e =

2

2

y = b, y = b x = 0 Conjugate Hyperbola

y = 0


b2

(0, 0)

S'P SP Tangents at the vertices x = a, x = a Equation of the y = 0 transverse axis Equation of the x = 0 conjugate axis

C

y2

Eccentricity

Focal radii

D U

a2

= 1

b2

Centre Length of transverse axis Length of conjugate axis Foci Equation of directrices

2

E

x2

y2

+

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Special form of hyperbola :

(b)

If the centre of hyperbola is (h, k) and axes are parallel to the co-ordinate axes, then its equation is (x − h)2 a2

3.

(y − k)2 b2

x2 a2

= 1.

(c)

The equations x = a sec φ and y = b tan φ are known as the parametric equations of hyperbola

x2 2

a

y2 2

b

x2 a2

a2

b2

F± GH

= 1

according as

x12

a2

y12

b2

6.

according as c2 <, =, > a2m2 b2.

x2 a2

at (x1, y1) is

y2

b2

7.

xx1 2

a

yy1 2

b

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y2 b2

= 1 are y = mx ±

a2m2 − b2 and the

a2m a2m2 − b2

,±

b2 a2m2 − b2

I JK .

y2

= 1 is given by SS1 = T2

b2

Equations of normals in different forms : (a)

Point form : The equation of normal to the hyperbola x2 a2

(b)

= 1

y2 b2

= 1 at (x1, y1) is

a2 x b2y + = a2 + b2. y1 x1

Parametric form : The equation of normal at (a sec θ , b tan θ ) to the hyperbola x2

= 1.


a2

Point form : The equation of the tangent to the hyperbola

E

x2

Equations of tangents in different forms : (a)

= 1 at (a sec φ , b tan φ ) is

Equation of pair of tangents from (x1, y1) to the hyperbola

1 is +ve, zero or ve.

The line y = mx + c does not intersect, touches, intersect the hyperbola

5.

b2

co-ordinates of points of contacts are

The point (x1, y1) lies inside, on or outside the hyperbola y2

y2

Slope form : The equations of tangents of slope m to the hyperbola

= 1

Position of a point and a line w.r.t. a hyperbola : x2

x y sec φ tan φ = 1. b a

Parametric equations of hyperbola :

4.

Parametric form : The equation of tangent to the hyperbola

a2

PAGE # 87

y2 b2

= 1 is ax cos θ + by cot θ = a2 + b2


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MATHS FORMULA - POCKET BOOK (c)

Slope form : The equation of the normal to the hyperbola

x2 2

a

y2 b

a2

y2 b2

F± GH

a2 − b2m2

a2 a2 − b2m2

or c2 =

m(a2 + b2 )2

a2

9.

y2 b2

(a2 − m2b2)

, which

,m

mb2 a2 − b2m2

I JK .

a2

yy1 b2

2

2

a2

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N

S

IJ K

y sin b

FG φ H

1

+ φ2 2

IJ K

= cos

FG φ H

1

IJ K

+ φ2 . 2

y2

= 1 is y =

b2

*

2

b2 a2m

x.

y2 b2

b2 a2

The equations of asymptotes of the hyperbola

a2

a2

= 1 is

b2

x2

x2

F − a l , b mI GH n n JK 2

y2

15. Asymptotes of a hyperbola :

= 1.


m1m2 =

y2 b2

= 1 are y = ±

b x. a

Asymptote to a curve touches the curve at infinity. = 1

*

The asymptote of a hyperbola passes through the centre of the hyperbola.

whose mid point is (x1, y1) is T = S1.

D U

− φ2 2

14. The diameters y = m1x and y = m2x are conjugate if

2

10. The equation of chord of the hyperbola

E

1

13. The equation of a diameter of the hyperbola x2

Equation of chord of contact of the tangents drawn from the external point (x1, y1) to the hyperbola is given by

xx1

a2

= 1 is x + y = a b . 2

FG φ H

12. Equation of polar of the point (x1, y1) w.r.t. the hyperbola is given by T = 0.

x2

The equation of director circle of hyperbola x2

φ 2, b tan φ 2) is

The pole of the line l x + my + n = 0 w.r.t.

is condition of normality. Points of contact : Co-ordinates of points of contact are

8.

a2 − b2m2

= 1, m (a2 + b2 )

then c = m (e)

x cos a

2

Condition for normality : If y = mx + c is the normal x2

and Q(a sec

m (a + b )

the normal is y = mx m

of

11. Equation of chord joining the points P(a sec φ 1, b tan φ 1)

= 1 in terms of the slope m of

2

2

(d)


PAGE # 89


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MATHS FORMULA - POCKET BOOK *


The combined equation of the asymptotes of the hyperbola

*

x2 a2

y2 b2

x2

= 1 is

a2

y2 b2

Equation of tangent at (x1, y1) to xy

*

= 0.

x y = c2 is x + y = 2. 1 1

The angle between the asymptotes of

x

2

2

a *

2

y

2

b

= 1 is 2 tan1

y

Equation of tangent at t is x + yt2 = 2ct

2

b2

or 2 sec1 e.

*

xx1 yy1 = x12 y12

A hyperbola and its conjugate hyperbola have the same asymptotes.

*

The bisector of the angles between the asymptotes are the coordinate axes.

*

Equation of hyperbola Equation of asymptotes = Equation of asymptotes Equation of conjugate hyperbola = constant.

Equation of normal at (x1, y1) to xy = c2 is

*

Equation of normal at t on xy = c2 is xt3 yt ct4 + c = 0. (This results shows that four normal can be drawn from a point to the hyperbola xy = c2)

*

If a triangle is inscribed in a rectangular hyperbola then its orthocentre lies on the hyperbola.

16. Rectangular or Equilateral Hyperbola : *

A hyperbola for which a = b is said to be rectangular hyperbola, its equation is x2 y2 = a2

*

Equation of chord of the hyperbola xy = c2 whose middle point is given is T = S1

*

xy = c2 represents a rectangular hyperbola with asymptotes x = 0, y = 0.

*

Point of intersection of tangents at t1 & t2 to the

*

Eccentricity of rectangular hyperbola is

*

Parametric equation of the hyperbola xy = c2 are

hyperbola xy = c2 is

2 and angle between asymptotes of rectangular hyperbola is 90º.

x = ct, y = *

F 2c t t GH t + t

1 2

1

2

,

2c t1 + t 2

I JK

c , where t is a parameter. t

Equation of chord joining t1, t2 on xy = c2 is x + y t1t2 = c(t1 + t2)


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MEASURES OF CENTRAL TENDENCY AND DISPERSION 1.


Geometric Mean : (i) For ungrouped data G.M. = (x1 x2 x3 .....xn)1/n

Arithmetic mean : (i)

n

For ungrouped data (individual series)

or

x =

n

(ii)

x1 + x2 +......+ xn Σ xi = i=1 n(no. of terms) n (ii)

Direct method x =

i =1 n

Σ fi

For grouped data

e

j

1 N

n

, where N =

∑f

i

i= 1

F f log x I GG ∑ JJ = antilog G GH ∑ f JJK

n

Σ fixi

i

i =1

G.M. = x1f1 x2f2 .... xnfn

For grouped data (continuous series)

(a)

F1 I G.M. = antilog G n ∑ log x J H K

n

, where xi , i = 1 .... n

i

i

i =1

n

i=1

i

be n observations and fi be their corresponding frequencies (b)

Σfidi short cut method : x = A + Σf , i

i= 1

4.

Harmonic Mean - Harmonic Mean is reciprocal of arithmetic mean of reciprocals.

where A = assumed mean, di = xi A = deviation for each term 2.

(i)

If x is the mean of x1, x2, ...... xn. The mean of ax1, ax2 .....axn is a x where a is any number different from zero.


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S

∑f

i

(ii)

For grouped data H.M. =

n

Ff I

i= 1

∑ GH x JK i

i =1

5.

i

Relation between A.M., G.M and H.M.

(iv) Arithmetic mean is independent of origin i.e. it is x effected by any change in origin.

C

i

n

If each of the n given observation be doubled, then their mean is doubled

(iii)

1

∑x i=1

In a statistical data, the sum of the deviation of items from A.M. is always zero.

(ii)

D U

n

Properties of A.M. (i)

E

n

For ungrouped data H.M. =

A.M. ≥ G.M. ≥ H.M. Equality holds only when all the observations in the series are same.

PAGE # 93


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Median : (a) Individual series (ungrouped data) : If data is raw, arrange in ascending or descending order and n be the no. of observations. If n is odd, Median = Value of

FG n + 1IJ H 2 K

Median = u -

th

th

7.

Discrete series : First find cumulative frequencies of the variables arranged in ascending or descending order and Median =

(c)

FG n + 1IJ H 2 K

(i)

For individual series : In the case of individual series, the value which is repeated maximum number of times is the mode of the series.

(ii)

For discrete frequency distribution series : In the case of discrete frequency distribution, mode is the value of the variate corresponding to the maximum frequency.

(iii)

For continuous frequency distribution : first find the model class i.e. the class which has maximum frequency.

observation, where n is cumulative

Where

f

C

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N

S

LM f − f OP N2f − f − f Q 1

1

0

0

2

× i

Where l 1 = Lower limit of the model class. f1 = Frequency of the model class. class.

× i

l = Lower limit of the median class. f = Frequency of the median class. N = Sum of all frequencies. i = The width of the median class C = Cumulative frequency of the class preceding to median class.


Mode = l 1 +

frequency. Continuous distribution (grouped data) (i) For series in ascending order

Median = l +

E

For continuous series

th

FG N − CIJ H2 K

× i

f

Mode :

th

(b)

FG N − CIJ H2 K

where u = upper limit of median class.

observation

FG nIJ + value of H 2K FG n + 1IJ ] observation. H2 K

1 If n is even, Median = [Value of 2

For series in descending order

class.

f0 = Frequency of the class preceding model f2 = Frequency of the class succeeding model i = Size of the model class.

8.

Relation between Mean, Mode & Median : (i)

In symmetrical distribution : Mean = Mode = Median

(ii)

In Moderately symmetrical distribution : Mode = 3 Median 2 Mean

PAGE # 95


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Measure of Dispersion : The degree to which numerical data tend to spread about an average value is called variation or dispersion. Popular methods of measure of dispersion. 1.

fi = Frequency of the corresponding xi (ii)

σ =

Mean deviation : The arithmetic average of deviations from the mean, median or mode is known as mean deviation. (a) Individual series (ungrouped data) Mean deviation =

(b)

Σ|x − S| n

2

or

σ=

FG IJ H K

Σd2 Σd − N N

2

Variance Square of standard direction i.e. variance = (S.D.)2 = (σ)2 Coefficient of variance = Coefficient of S.D. × 100

Σ f | x − s| Σf |x − s| = Σf N

=

Note : Mean deviation is the least when measured from the median. 2.

FG IJ H K

Σfd2 Σfd − N N

Where d = x A = Derivation from assumed mean A f = Frequency of item (term) N = Σf = Total frequency.

Where n = number of terms, S = deviation of variate from mean mode, median. Continuous series (grouped data). Mean deviation =

N = Σ f = Total frequency Short cut method

σ × 100 x

Standard Deviation : S.D. (σ) is the square root of the arithmetic mean of the squares of the deviations of the terms from their A.M. (a) For individual series (ungrouped data) σ =

Σ(x − x)2 N

where x = Arithmetic mean of

the series (b)

N = Total frequency For continuous series (grouped data) (i)

Direct method σ = Where

Σfi (xi − x)2 N

x = Arithmetic mean of series xi = Mid value of the class


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MATRICES AND DETERMINANTS

1.

2.


Trace of a matrix : Sum of the elements in the principal diagonal is called the trace of a matrix.

MATRICES :

trace (A ± B) = trace A ± trace B

Matrix - A system or set of elements arranged in a rectangular form of array is called a matrix.

trace kA = k trace A trace A = trace AT trace In = n when In is identity matrix.

Order of matrix : If a matrix A has m rows & n columns then A is of order m × n.

trace On = O

The number of rows is written first and then number of columns. Horizontal line is row & vertical line is column 3.

Types of matrices : A matrix A = (aij)m×n A matrix A = (aij)mxn over the field of complex numbers is said to be

trace AB ≠ trace A trace B. 5.

Addition & subtraction of matrices : If A and B are two matrices each of order same, then A + B (or A B) is defined and is obtained by adding (or subtracting) each element of B from corresponding element of A

6.

Multiplication of a matrix by a scalar :

Name

Properties

A row matrix

if m = 1

A column matrix

if n = 1

A rectangular matrix

if m ≠ n

A square matrix

if m = n

Properties :

A null or zero matrix

if aij = 0 ∀ i j. It is denoted by O.

A diagonal matrix

if m = n and aij = 0 for i ≠ j.

(i)

K(A + B) = KA + KB

A scalar matrix

if m = n and aij = 0 for i ≠ j

(ii)

(K1 K2)A = K1(K2 A) = K2(K1A)

(iii)

(K1 + K2)A = K1A + K2A

KA = K (aij)m×n

= k for i = j i.e. a11 = a22 ....... = ann = k (cons.) Identity or unit matrix

if m = n and aij = 0 for i ≠ j

7.

Upper Triangular matrix

if m = n and aij = 0 for i > j

Lower Triangular matrix

if m = n and aij = 0 for i < j

Properties :

Symmetric matrix

if m = n and aij = aji for all i, j or AT = A

(i)

Skew symmetric matrix

if m = n and aij = aji ∀ i, j


A T

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N

S

In general matrix multiplication is not commutative i.e. AB ≠ BA.

(ii)

or AT = A D U

= (Ka)m×n where K is constant.

Multiplication of Matrices : Two matrices A & B can be multiplied only if the number of columns in A is same as the number of rows in B.

= 1 for i = j

E

On is null matrix.

PAGE # 99

A(BC) = (AB)C

[Associative law]


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MATHS FORMULA - POCKET BOOK (iii)

A.(B + C) = AB + AC

[Distributive law]

DETERMINANT : 1.

/ B=C (iv) If AB = AC ⇒ (v)


Minor & cofactor : If A = (aij)3×3, then minor of a11 is

If AB = 0, then it is not necessary A = 0 or B = 0

a22 M11 = a 32

(vi) AI = A = IA (vii) Matrix multiplication is commutative for +ve integral i.e. Am+1 = Am A = AAm 8.

cofactor of an element aij is denoted by Cij or Fij and is equal to (1)i+j Mij or

Transpose of a matrix : A' or A T is obtained by interchanging rows into columns or columns into rows Properties : (AT)T = A

(ii)

(A ± B)T = AT ± BT

(iii)

(AB) = B A T

T

9.

expansion of determinant of order 3 × 3

a1 b1 a2 b2 ⇒ a3 b3

IT = I

or

AAT = In = ATA

(i)

Orthogonal matrix : if

(ii)

Idempotent matrix : if A2 = A

(iii)

Involutory matrix : if A = I

A T

c1 c2 c3

b1 = a2 b 3

b2 = a1 b 3

c2 a2 b 1 c3 a3

c1 a1 + b 2 c3 a3

c2 a2 b2 + c 1 c3 a3 b3

c1 a1 b1 c3 c2 a3 b3

Properties : (i)

|AT| = |A|

(iv) Nilpotent matrix : if ∃ p ∈ N such that Ap = 0

(ii)

Hermitian matrix : if Aθ = A i.e. aij = a ji

By interchanging two rows (or columns), value of determinant differ by ve sign.

(iii)

If two rows (or columns) are identical then |A| = 0

2

or A

1

=A


if i ≠ j

T

(vi) Skew - Hermitian matrix : if A = Aθ

D U

= Mij,

Determinant : if A is a square matrix then determinant of matrix is denoted by det A or |A|.

Some special cases of square matrices : A square matrix is called

(v)

E

if i = j

and a11 F21 + a12 F22 + a13 F23 = 0

(iv) (KA)T = KAT (v)

Cij = Mij,

Note : |A| = a11F11 + a12 F12 + a13 F13 2.

(i)

a23 a33 and so.

I O

N

S

(iv) |KA| = Kn det A, A is matrix of order n × n

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MATHS FORMULA - POCKET BOOK Properties :

If same multiple of elements of any row (or column) of a determinant are added to the corresponding elements of any other row (or column), then the value of the new determinant remain unchanged.

(vi) Determinant of :

3.

(a)

A nilpotent matrix is 0.

(b)

An orthogonal matrix is 1 or 1

(c)

A unitary matrix is of modulus unity.

(d)

A Hermitian matrix is purely real.

(e)

An identity matrix is one i.e. |In| = 1, where In is a unit matrix of order n.

(f)

A zero matrix is zero i.e. |0n| = 0, where 0n is a zero matrix of order n

(g)

A diagonal matrix = product of its diagonal elements.

(h)

Skew symmetric matrix of odd order is zero.

2.

N

S

(adj AT) = (adjA)T (adj KA) = Kn1(adj A)

(i)

A1 exists if A is non singular i.e. |A| ≠ 0

(ii)

A1 =

(iii)

A1A = In = A A1

adjA , |A| ≠ 0 | A|

(A1)1 = A

1 | A|

(vii) If A & B are invertible square matrices then

a1m1 + b1m2 a2m1 + b2m2

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 I O

(iii)

(vi) |A1| = |A|1 =

(AB)1 = B1 A1 3.

Rank of a matrix : A non zero matrix A is said to have rank r, if

adj A = (Cij)T , where Cij is cofactor of aij

A T

(adjAB) = (adjB) (adjA)

(iv) (AT)1 = (A1)T

Adjoint of a matrix :

C

(iii)

Inverse of a matrix :

MATRICES AND DETERMINANTS :

D U

|adj A| = |A|n1

(v)

If order is different then for their multiplication, express them firstly in the same order.

E

(ii)

(v)

Multiplication of two second order determinants is defined as follows.

1.

A(adj A) = (adjA) A = |A|In

(iv) adj(adjA) = |A|n2

Multiplication of two determinants :

a1 b1 l1 m1 a1l1 + b1l2 × a2 b2 l2 m2 = a2 l1 + b2 l2

(i)

(i)

Every square sub matrix of order (r + 1) or more is singular

(ii)

There exists at least one square submatrix of order r which is non singular.

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Homogeneous & non homogeneous system of linear equations :

(b) Solution of homogeneous system of linear equations :

A system of equations Ax = B is called a homogeneous system if B = 0. If B ≠ 0, then it is called non homogeneous system equations.

The homogeneous system Ax = B, B = 0 of n equations in n variables is (i)

5.

(a) Solution of non homogeneous system of linear equations : (i)

Consistent (with unique solution) if |A| ≠ 0 and for each i = 1, 2, ......... n xi = 0 is called trivial solution.

Cramer's rule : Determinant method

(ii)

The non homogeneous system Ax = B, B ≠ 0 of n equations in n variables is -

Consistent (with infinitely many solution), if |A| = 0

(a) |A| = |Ai| = 0

Consistent (with unique solution) if |A| ≠ 0 and for each i = 1, 2, ........ n,

(for determinant method)

(b) |A| = 0, (adj A) B = 0

(for matrix method)

NOTE : A homogeneous system of equations is never inconsistent.

det A i xi = , where Ai is the matrix obtained det A from A by replacing ith column with B. Inconsistent (with no solution) if |A| = 0 and at least one of the det (Ai) is non zero. Consistent (With infinite many solution), if |A| = 0 and all det (Ai) are zero. (ii)

Matrix method : The non homogeneous system Ax = B, B ≠ 0 of n equations in n variables is Consistent (with unique solution) if |A| ≠ 0 i.e. if A is non singular, x = A1 B. Inconsistent (with no solution), if |A| = 0 and (adj A) B is a non null matrix. Consistent (with infinitely many solutions), if |A| = 0 and (adj A) B is a null matrix.


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MATHS FORMULA - POCKET BOOK Properties :

FUNCTION 1.

Modulus function :

R|x, x > 0 |x| = S−x, x < 0 |T0, x = 0

(i)

loga 1 = 0

(ii)

loga a = 1

(iii)

aloga b = b if k > 0,

Properties : |x| ≠ ± x |xy| = |x||y|

(i) (ii) (iii)

(iv) loga b1 + loga b2 + ...... + loga bn = loga (b1 b2 ........bn) (v)

x |x| = y |y|

|x| = a

⇒ x=±a

|x| = a

⇒ no solution

|x| > a

⇒ x < a or x > a

|x| ≤ a |x| < a

⇒ a ≤ x ≤ a ⇒ No solution.

|x| > a

⇒ x∈R

logb a to be defined a > 0, b > 0, b ≠ 1

(ii)

loga b = c

(iii)

loga b > c

⇒

b=a

b > ac, ⇒ or b < ac, (iv) loga b > loga c b > c, ⇒

a>1

or

b < c,

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S

= loga b loga c

FG IJ H K

1 (viii) loga b

3.

or

1 loga b = log a b

n loga b m

(vii) logam bn =

c

= loga b = log1/a b

FG bIJ H cK

(ix)

log1/a

= loga

(x)

alogb c = clogb a

FG c IJ H bK

Greatest Integer function :

0
f(x) = [x], where [.]denotes greatest integer function equal or less than x.

if a > 1

i.e., defined as [4.2] = 4, [4.2] = 5

if 0 < a < 1

Period of [x] = 1


FG bIJ H cK

logc b loga b = log a c

Logarithmic Function : (i)

loga

(vi) Base change formulae

(iv) |x + y| ≤ |x| + |y| (v) |x y| ≥ |x| |y| or ≤ |x| + |y| (vi) ||a| |b|| ≤ |a b| for equality a.b ≥ 0. (vii) If a > 0

2.

k = blogb k

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Properties : (i) x 1 < [x] ≤ x (ii) [x + I] = [x] + I [x + y] ≠ [x] + [y] (iii) [x] + [x] = 0, x ∈ I = 1, x ∉ I (iv) [x] = I, where I is an integer x ∈ [I, I + 1) (v) [x] ≥ I, x ∈ [I, ∞ )

Definition : Let A and B be two given sets and if each element a ∈ A is associated with a unique element b ∈ B under a rule f, then this relation (mapping) is called a function. Graphically - no vertical line should intersect the graph of the function more than once. Here set A is called domain and set of all f images of the elements of A is called range.

(vi) [x] ≤ I, x ∈ ( ∞ , I + 1] (vii) [x] > I, [x] ≥ I + 1, x ∈ [I + 1, ∞ ) (viii) [x] < I, [x] ≤ I 1, x ∈ ( ∞ , I) 4.

Table : Domain and Range of some standard functions -

or

f(x)

=

|x| , x

= 0,

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Domain

Range

Polynomial function

R

R

Identity function x

R

R

Constant function K

R

(K)

R0

R0

x2, |x| (modulus function)

R

R+ ∪{x}

x3, x|x|

R

R

R

{-1, 0, 1}

Signum function

1 x

|x| x

, x ∈R−

x +|x|

R

R+ ∪{x}

x=0

x -|x|

R

R- ∪{x}

[x] (greatest integer function)

R

1

x - {x}

R

[0, 1]

[0, ∞ )

[0, ∞ ]

ax (exponential function)

R

R+

log x (logarithmic function)

R+

R

,

, x ∈R+

x

x ≠ 0 x=0


Functions

Reciprocal function

Signum function :

R|−1 |S 0 f(x) = sgn (x) = | |T 1

D U

Range = For all values of x, all possible values of f(x).

Fractional part function : f(x) = {x} = difference between number & its integral part = x [x]. Properties : (i) {x}, x ∈ [0, 1) (ii) {x + I} = {x} {x + y} ≠ {x} + {y} (iii) {x} + {x} = 0, x ∈ I = 1, x ∉ I (iv) [{x}] = 0, {{x}} = {x}, {[x]} = 0

5.

E

i.e., Domain = All possible values of x for which f(x) exists.

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MATHS FORMULA - POCKET BOOK Trigonometric


Domain

Range

sin x

R

[-1, 1]

cos x

R

[-1, 1]

tan x

R-

RS± π , ± 3π ,...UV T2 2 W

R

cot x

R- {0, ± π , ± 2 π ,...}

R

sec x

R -

RS± π , ± 3π ,...UV T2 2 W

R - (-1,1)

cosec x

R- {0, ± π , ± 2 π }

R - (-1,1)

Inverse

Domain

Range

Kinds of functions :

or

-1

x

(-1, 1]

Graphically-no horizontal line intersects with the graph of the function more than once. (ii) (iii)

Graphically - atleast one horizontal line intersects with the graph of the function more than once.

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FG −π , π IJ H 2 2K

cot-1 x

R

(0, π )

sec-1 x

R -(-1,1)

π [0, π ]2


LM −π , π OP N 2 2Q

R

R - (-1,1)

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Many one function : f : A → B is a many one function if there exist x, y ∈ A s.t. x ≠ y but f(x) = f(y)

(v)

tan-1 x

cosec-1 x

i.e. if to each y ∈ B ∃ x ∈ A s.t. f(x) = y

(iv) Into function : f is said to be into function if R(f) < B

[0, π ]

x

Onto function (surjection) - f : A → B is onto if R (f) = B

[-1,1]

cos

-1

a≠b

⇒ f(a) ≠ f(b), a, b ∈ A

Trigo Functions sin

One-one (injection) function - f : A → B is one-one if f(a) = f(b) ⇒ a = b

(i)

Functions

One-one-onto function (Bijective) - A function which is both one-one and onto is called bijective function.

Inverse function : f1 exists iff f is one-one & onto both f1 : B → A, f1(b) = a

9.

⇒ f(a) = b

Transformation of curves : (i)

Replacing x by (x a) entire graph will be shifted parallel to x-axis with |a| units.

RS UV TW

If

a is +ve it moves towards right. a is ve it moves toward left.

Similarly if y is replace by (y a), the graph will be shifted parallel to y-axis,

LM− π , π OP- {0} N 2 2Q

upward if a is +ve downward if a is ve.

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(iii)


Replacing x by x, take reflection of entire curve is yaxis.

(g)

Zero function i.e. f(x) = 0 is the only function which is even and odd both.

Similarly if y is replaced by y then take reflection of entire curve in x-axis.

(h)

If f(x) is odd (even) function then f'(x) is even (odd) function provided f(x) is differentiable on R.

Replacing x by |x|, remove the portion of the curve corresponding to ve x (on left hand side of y-axis) and take reflection of right hand side on LHS.

(i)

A given function can be expressed as sum of even & odd function. i.e.

(iv) Replace f(x) by |f(x)|, if on L.H.S. y is present and mode is taken on R.H.S. then portion of the curve below x-axis will be reflected above x-axis. (v)

Replace x by ax (a > 0), then divide all the value on xaxis by a. Similarly if y is replaced by ay (a > 0) then divide all the values of y-axis by a.

Even function if f(x) = f(x) and

(ii)

Odd function if f(x) = f(x).

= even function + odd function. 12. Increasing function : A function f(x) is an increasing function in the domain D if the value of the function does not decrease by increasing the value of x.

14. Periodic function: Function f(x) will be periodic if a +ve real number T exist such that

∀ x ∈ Domain. There may be infinitely many such T which satisfy the above equality. Such a least +ve no. T is called period of f(x).

11. Properties of even & odd function : The graph of an even function is always symmetric about y-axis.

(b)

The graph of an odd function is always symmetric about origin.

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(d)

Sum & difference of two even (odd) function is an even (odd) function.

(e)

Product of an even or odd function is an odd function.

(f)

Sum of even and odd function is neither even nor odd function.

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(i)

If a function f(x) has period T, then Period of f(xn + a) = T/n and Period of (x/n + a) = nT

Product of two even or odd function is an even function.


f(x + T) = f(x),

(a)

(c)

1 1 [f(x) + f(x)] + [f(x) f(x)] 2 2

13. Decreasing function : A function f(x) is a decreasing function in the domain D if the value of function does not increase by increasing the value of x.

10. Even and odd function : A function is said to be (i)

f(x) =

(ii)

If the period of f(x) is T1 & g(x) has T2 then the period of f(x) ± g(x) will be L.C.M. of T1 & T2 provided it satisfies definition of periodic function.

(iii)

If period of f(x) & g(x) are same T, then the period of af(x) + bg(x) will also be T.

PAGE # 113


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MATHS FORMULA - POCKET BOOK Function

Period

sin x, cos x

2π

MATHS FORMULA - POCKET BOOK 15. Composite function : If f : X → Y and g : Y → Z are two function, then the composite function of f and g, gof : X → Z will be defined as

sec x, cosec x

gof(x) = g(f(x)), ∀ x ∈ X

tan x, cot x

π

In general gof ≠ fog

sin (x/3)

6π π/4

If both f and g are bijective function, then so is gof.

tan 4x cos 2πx

1 π

|cos x| sin x + cos x 4

2 cos

4

FG x − π IJ H 3 K

π/2 6π

sin3 x + cos3 x

2π/3

sin x + cos x

2π

sin x sin5x

2π

tan2 x cot2 x

π

x [x]

1

[x]

1

3

4

NON PERIODIC FUNCTIONS : 2 3 x, x , x , 5

cos x2 x + sin x x cos x cos x , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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LIMIT 1.

Limit of a function : xlim → a f(x) = l (finite quantity)

2.

lim lim Existence of limit : xlim → a f(x) exists iff x → a− f(x) = x → a+ f(x) = l

3.

Indeterminate forms :

4.

0 ∞ , , ∞ ∞ , ∞ × 0, ∞ 0 , 0∞ , 1∞ 0 ∞


Limit of the greatest integer function : Let c be any real number Case I : If c is not an integer, then xlim → c [x] = [c] Case II: If c is an integer, then xlim [x] = c 1, xlim [x] = c → c− → c+ [x] = does not exist and xlim →c

6.

Methods of evaluation of limits :

f(x) 0 is of form g(x) 0 then factorize num. & devo. separately and cancel the Factorisation method : If xlim →a

(i)

Theorems on limits : lim (k f(x)) = k lim f(x), x→a

(i)

x→a

(ii)

x→a

(iii)

x→a

(iv)

lim

(v)

k is a constant.

lim (f(x) ± g(x)) = lim f(x) ± Lim g(x) x→a x→a

(ii)

lim f(x).g(x) = lim f(x). Lim g(x) x→a x→a

(iii)

x→a

lim f(x) x→a f(x) = lim g(x) , provided xlim → a g(x) ≠ 0 g(x) x→a

FH

the indeterminate form, replace

IK

(iv)

lim f(g(x)) = f lim g(x) , provided value of x→a

x→a

0 form. 0 Rationalization method : If we have fractional powers on the expression in num, deno or in both, we rationalize the factor and simplify. When x → ∞ : Divide num. & deno. by the highest power of x present in the expression and then after removing common factor which is participating in making

(v)

n n lim x − a = nan1 x→a x −a By using standard results (limits) :

g(x) function f(x) is continuous. (vi)

lim [f(x) + k] = lim f(x) + k x→a

FH

IK

g(x) (viii) xlim = → a (f(x))

LM lim f(x)OP N Q

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x→0

(b)

x→0

(c)

x→0

(d)

x→0

x lim tan x = 1 = lim x→0 x tan x lim sinx = 0

lim g(x)

x→a

x→a


lim x lim sinx = 1 = x → 0 x sin x

(a)

x→a

lim f(x) (vii) xlim → a log(f(x)) = log x → a

1 1 , 2 ,.. by 0. x x

PAGE # 117

lim cosx = lim x→0

1 =1 cos x


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(e)

0 lim sin x = π x→0 180 x

(f)

−1 x lim sin x = 1 = lim −1 x→0 x→0 sin x x

(g)

−1 x lim tan x = 1 = lim x→0 x→0 tan−1 x x

MATHS FORMULA - POCKET BOOK (vi) By substitution : (a) If x → a, then we can substitute x=a+t ⇒ t=xa If x → a, t → 0. (b) When x → ∞ substitute x = t ⇒ t → ∞ (c)

When x → ∞ substitute t =

(vii) By using some expansion :

x

(h)

lim a − 1 = log a x→0 e x

ex = 1 + x +

x2 x3 + + ..... 2! 3!

(i)

x lim e − 1 = 1 x→0 x

ex = 1 x +

x2 x3 + ..... 2! 3!

(j)

lim log(1 + x) = 1 x→0 x

log(1 + x) = x

x2 x3 + ...... 2 3

(k)

1 lim loga (1 + x) = x→0 loga x

log(1 x) = x

x2 x3 ..... 2 3

ex ln a = ax = 1 + xlogea +

n

(l) (m)

(n)

(o)

(p)

lim (1 + x) − 1 = n x

x→0

lim sinx = lim cos x = 0 x→∞ x x

lim

x→∞

1 x

1 x

=1

FG1 + 1 IJ H xK F aI = lim G1 + J H xK

lim (1 + x)1/x = e = lim x→∞

x→0

1/x lim = ea x → 0 (1 + ax)

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(x loge a)2 (x loge a)3 + + ...... 2! 3!

x3 x5 + ....... 3! 5!

cosx = 1

x2 x4 + ...... 2! 4!

tanx = x +

2 x3 + x5 + ..... 15 3

n(n − 1) 2 x + ..... 2! Sandwich Theorem : In the neighbour hood of x = a f(x) < g(x) < h(x) (1 + x)n = 1 + nx +

x

7.

lim f(x) = lim h(x) = l, then lim g(x) = l. x →a x →a x →a

x

x→∞


sinx = x

x→∞

sin

1 ⇒ t → 0+ x

⇒ PAGE # 119

l < lim g(x) < l. x →a


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DIFFERENTIATION 1.


FUNDAMENTAL RULES FOR DIFFERENTIATION :

SOME STANDARD DIFFERENTIATION : Function

(i)

d f(x) = 0 if and only if f(x) = constant dx

Derivative

Function

Derivative

0

xn

nxn 1

(ii)

d dx

ccf(x)h

1 x loge a

loge x

1 x

(iii)

d dx

cf(x) ± g(x)h

ax loge a

ex

ex

sin x

cos x

cos x

sin x

(iv)

d dv du (uv) = u + v , where u & v are functions dx dx dx

tan x

sec2 x

cot x

cosec2 x

cosec x cot x

sec x

sec x tan x

A cons. (k)

loga x ax

cosec x

sin1 x

1 1−x

2

,1
cos1x

1 − x2

tan1 x

1 |x| 1 − x

1 1+x

2

2

,|x|>1

,x ∈ R

0, x ∉ I

[x]

cosec1 x

1 |x| 1 − x2

cot1 x

|x|

1 1 + x2

,1|x|>|

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d g(x) dx

If

d d f(x) = φ(x), then f (ax + b) dx dx

= a φ(ax + b)

, x ∈R

(vi)

x , x≠ 0 | x|

d dx

FG u IJ H vK

=

v

du dv −u dx dx 2 v

(quotient rule)

(vii) If y = f(u), u = g(x) [chain rule or differential coefficient of a function of a function]

d [x] does not exist at any integral Point. dx


d f(x) ± dx

d du dv dw (uvw) = vw + uw + uv . dx dx dx dx

then NOTE :

=

,1
(v) sec1 x

d f(x), where c is a constant. dx

of x. (Product rule) or

1

= c

dy dy du = × dx dx du

llly If y = f(u), u = g(v), v = h(x), then

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MATHS FORMULA - POCKET BOOK (xii) Differentiation of implicit function : If f (x, y) = 0, differentiate w.r.t. x and collect the terms containing

dy dy du dv = × × dx du dv dx i.e if y = un ⇒

dy at one side and find dx

dy du = nun1 dx dx

OR

[The relation f(x, y) = 0 in which y is not expressible explicitly in terms of x are called implicit functions]

(viii) Differentiation of composite functions Suppose a function is given in form of fog(x) or f[g(x)], then differentiate applying chain rule

(xiii) Differentiation of parametric functions : If x = f(t) and y = g(t), where t is a parameter, then

d f[g(x)] = f'g(x) . g'(x) i.e., dx

FG 1IJ H uK

−1

dy g'(t) dy dt = = f'(t) dx dx dt

(ix)

d dx

(x)

u du d |u| = , | u| dx dx

(xi)

Logarithmic Differentiation : If a function is in the

=

u2

du , u ≠ 0 dx

u ≠ 0 (xiv) Differentiation of a function w.r.t. another function : Let y = f(x) and z = g(x), then differentiation of y w.r.t. z is

f1 ( x ) f2 ( x ).... form (f(x))g(x) or g ( x ) g ( x ).... We first take log on 1 2

dy / dx f'(x) dy = = g'(x) dz / dx dz

both sides and then differentiate. (a)

loge (mn) = logem + logen

(b)

loge

(c)

loge (m)n = nlogem

(xv) Differentiation of inverse Trigonometric functions

m = logm logen n

m (e) logan xm = loga x n

(g) loge e = 1

(h)

(d)

logn m

(f)

aloga

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x

using Trigonometrical Transformation : To solve the problems involving inverse trigonometric functions

logm n = 1

first try for a suitable substitution to simplify it and then differentiate. If no such substitution is found

= x

then differentiate directly by using trigonometrical formula frequently.

loge m logn m = log n e


dy . dx

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Important Trigonometrical Formula : (i)

sin2x = 2sinx. cosx =

(ii) cos2x =

(viii) tan2x = (iii)

1 − tan2 x 1 + tan2 x

(xvi) sin1 sin (x) = x, for

2 tan x 1 + tan2 x

= 2 cos2 x 1 = 1 2 sin2 x

tan1 (tan x) = x, for

1 − tan2 x

cos1 (x) = π cos1 x

sin3x = 3sinx 4sin3 x (xviii) sin1

3 tan x − tan3 x

(ix)

tan3x =

(x)

sin1 x + cos1 x = π /2

(xi)

sec1 x + cosec1 x = π /2

tan1

1 − 3 tan2 x

(xiii) tan1 x ± tan1 y = tan1

sec1

FG x ± y IJ H 1 m xy K

F H

(xv) cos1 x ± cos1 y = cos1

FH xy m

FG 1 IJ H xK FG 1 IJ H xK FG 1IJ H xK

= cosec1 x, cos1

= cot1 x, cot1

1 − x2

I K

1 − y2

cos1 (sin θ ) = cos1

IK

tan1 (cot θ ) = tan1


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FG 1 IJ H xK

FG 1 IJ H xK

= cos1 x, cosec1

(xix) sin1 (cos θ ) = sin1

(xiv) sin1 x ± sin1 y = sin1 x 1 − y2 ± y 1 − x2

C

π π < x < 2 2

(xvii) sin1 (x) = sin1 x, tan1 (x) = tan1 x,

2 tan x

(xii) tan1 x + cot1 x = π /2

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π 2

≤ x ≤

cos1 (cos x) = x, for 0 ≤ x ≤ π

(vi) cos3x = 4cos3 x 3cosx

E

π 2

= sec1 x,

= tan1 x,

FG 1IJ H xK

Fsin FG π − θIJ I GH H 2 K JK

= sin1 x

=

Fcos FG π − θIJ I GH H 2 K JK FG tan FG π − θIJ IJ H H 2 KK


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π θ 2

=

π θ 2

=

π θ 2

PAGE # 126


Some Useful Substitutions :

Part B

Part A

Expression Substitution

Formula

Result

a2 + x2

3x 4x3

x = sinθ

3sinθ 4sin3 θ

sin3θ

4x3 3x

x = cosθ

4cos3 θ 3cosθ

cos3θ

a− x a+ x or a+ x a− x

x = a tan θ

a2 x2

x = a sin θ or x = a cos θ

1 − 3x 2

2x 1+x

2

2x 1−x

2

x = tanθ

x = tanθ

x = tanθ

3 tan θ − tan3 θ 1 − 3 tan2 θ

2 tan θ 1 + tan2 θ 2 tan θ 2

1 − tan θ

2x2 1

x = cosθ

2cos2 θ 1

cos2θ

1 x2

x = sinθ

1 sin2 θ

cos2 θ

x = cosθ

1 cos2 θ

sin2 θ

x = secθ

sec θ 1

tan θ

x = cosecθ

cosec2 θ 1

cot 2 θ

x = tanθ

1 + tan2 θ

sec2 θ

x = cotθ

1 + cot2 θ

cosec2 θ

1

, 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 I O

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a−x a+x

x = a cos θ

x 2 a2

a2 + x2

tan2θ cos2θ

1 + x2

A T

sin2θ

1 2sin2 θ

x

C

a+x or a−x

x = sinθ

2

x = a tan θ or x = a cot θ

tan3θ

1 2x2

2

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Substitution

Expression

3x − x3

E


a2 − x2

5.

x = a sec θ or x=acosec θ

a2 − x2

or

x2 = a2 cos θ

a2 + x2

Successive differentiations or higher order derivatives : (a)

If y = f(x) then

dy = f'(x) is called the first derivadx

tive of y w.r.t. x

2

⇒

d2 y dx

2

=

d dx

FG dy IJ H dx K

=

c

h

d f'(x) dx

is called the second derivative of y w.r.t. x

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llly

d3 y dx3

=

d2 dx2

cf'(x)h


etc......

Thus, This process can be continued and we can

If y = cos (ax + b), then yn = an cos

obtain derivatives of higher order Note : To obtain higher order derivative of parametric functions we use chain rule

6.

dn

(i)

⇒ (b)

dxn

dy = t dx d2 y dx 2

=

d dx

FG dy IJ H dx K

(ii) =

nπ 2

IJ K

FGax + b + nπ IJ H 2K

nth Derivatives of Some Functions :

i.e. if x = 2t, y = t2

⇒

FG H

If y = sin (ax + b), then yn = an sin ax + b +

(f)

d dt 1 (t) = 1. = dx dx t

(iii)

If y = (ax + b)m m ∉ I, then

dn dx

n

dn dx

n

ex j n

= n!

csin xh

FG H

nπ = sin x + 2

FG H

IJ K

(cos x) = cos x +

πn 2

IJ K

yn = m(m1) (m2) ..... (mn+1) (ax + b)mn .an (c)

If m ∈I, then

(iv)

ym = m! am and ym+1 = 0

(d)

(−1)n n! 1 If y = , then yn = an (ax + b)n+1 ax + b

(v)

dn dx n dn dx n

NOTE : (e)

If y = log (ax + b), then yn =


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n−1

(−1)

(n − 1)!

(ax + b)n

(emx ) = mn emx

(log x) = ( 1)n1 (n1)! xn If u = g(x) is such that g'(x) = K (constant)

an then

PAGE # 129

dn dx

n

c h

f g(x)

LM d MNdu

n

= Kn

n


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OP PQ

f(u)

u= g(x)

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Differentiation of Infinite Series : method is illustrated


Differentiation of Determinant :

with the help of example x if y = x

x− −∞

then function becomes y = xy now taking log

∆ =

on both sides

R1 R2 R3

= |C1 C2 C3|

i.e logy = y log x, differentiating both sides w.r.t. x we get

dy = dx

⇒

8.

1 y

R'1 ∆' = R 2 R3

dy 1 dy = y + logx dx dx x

y x

FG 1 − log xIJ Hy K

R1 '2 R + R3

R1 R 2 + R'3

= |C'1 C2 C3| + |C1 C'2 C3| + |C1 C2 C'3|

y2 = x(1 − y log x)

L-hospital rule : if as x → a f(x) & g(x) either both → 0 or both → ∞, then lim

f(x) f'(x) = lim x →a g'(x) g(x)

(a)

it can be applied only on 0/0 or ∞/∞ form

x →a

(b) Numerator & denominator are differentiated separately not (c)

u formulae. v

If R.H.S. exist or d'not exist because value → ∞, then L.H rule can be applied. But if value fluctuate on R.H.S. then L.H. rule can't be applied. If it is applied continuously then at each step 0/0 or ∞/∞ should be checked.


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APPLICATION OF DERIVATIVES


Length of intercepts made on axes by the tangent :

TANGENT AND NORMAL : 1.

Geometrically f'(a) represents the slope of the tangent to the curve y = f(x) at the point (a, f(a))

2.

If the tangent makes an angle ψ (say) with +ve x direction

x intercept = x1

FG dy IJ H dx K

(x1 , y1)

FG dy IJ H dx K

⇒

8.

(x , y ) 1 1

(x1 , y1)

Length of perpendicular from origin to the tangent :

= 0.

⇒ 5.

FG dy IJ H dx K

6.

(x1 , y1)

→ ∞

(x1 , y1)

y y1 =

FG dy IJ H dx K

(x1 , y1)

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Slope of the normal

=

=

1 Slope of the tan gent

FG dx IJ H dy K

(x 1 , y 1 )

10. If normal makes an angle of φ with +ve direction of x-axis,

(x x1)


9.

= ± 1.

Equation of the tangent to the curve y = f(x) at a point (x1, y1) is

C

=

(x1 ,y1 )

2

(x1 , y1 )

If the tangent line makes equal angle with the axes, then

FG dy IJ H dx K

FG dy IJ H dx K F dy I 1+G J H dx K

y1 − x1

π If the tangent is perpendicular to x-axis, ψ = 2

4.

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FG dy IJ H dx K

y intercept = y1 x1

= tan ψ = slope of the tangent.

U| |V || W

If the tangent is parallel to x-axis, ψ = 0

3.

E

1

(x , y ) 1 1

then f'(x) =

R| | y S F dy I || GH dx JK T

then

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dy = cot φ . dx


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11. If the normal is parallel to x-axis ⇒

FG dy IJ H dx K

12. If the normal is perpendicular to x-axis ⇒

MATHS FORMULA - POCKET BOOK 17. Angle of intersection of the two curves :

= 0.

(x1 , y1)

FG dy IJ H dx K

(x1 , y1)

tanθ = ±

= 0.

FG dy IJ H dx K

1

2

1

13. If normal is equally inclined from both the axes or cuts equal intercept then

FG dy IJ − FG dy IJ H dx K H dx K F dy I F dy I 1−G J G J H dx K H dx K

14. The equation of the normal to the curve y = f(x) at a point (x1, y1) is

FG dy IJ H dx K

y intercept = y1 + x1

FG dy IJ H dx K

(x1 ,y1 )

FG dx IJ H dy K

(x1 ,y1 )

F dy I +y G J H dx K F dy I 1+G J H dx K

Length of normal = y

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FG dy IJ H dx K

2

dy dx

Length of tangent =

1

1+

FG dy IJ H dx K

2

Length of sub-tangent =

y dy / dx

Length of sub-normal = y

dy dx

(x1 ,y1 ) 2


2

y 1+

16. Length of perpendicular from origin to normal :

E

2

18. Length of tangent, normal, subtangent & subnormal :

(x1 , y1 )

x intercept = x1 + y1

=

is the slope of first

(x x1)

15. Length of intercept made on axes by the normal :

x1

1

2

1

y y1 =

FG dy IJ H dx K

FG dy IJ of second. If both curves intersect orthogoH dx K F dy I F dy I nally then GH dx JK GH dx JK = 1 curve &

= ± 1.

1

where

PAGE # 135


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MONOTONICITY, MAXIMA & MINIMA : 1.

A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain

2.

At a point function f(x) is monotonic increasing if f'(a) > 0 At a point function f(x) is monotonic decreasing if f'(a) < 0

3.

(ii)

6.

Working rule for finding local maxima & Local Minima : (i) Find the differential coefficient of f(x) w.r.to x, i.e. f'(x) and equate it to zero. (ii) Solve the equation f'(x) = 0 and let its real roots (critical points) be a, b, c ...... (iii) Now differentiate f'(x) w.r.to x and substitute the critical points in it and get the sign of f"(x) for each critical point. (iv) If f"(a) < 0, then the value of f(x) is maximum at x = 0 and if f"(a) > 0, then the value of f(x) is minimum at x = a. Similarly by getting the sign of f"(x) for other critical points (b, c, ......) we can find the points of maxima and minima.

7.

Absolute (Greatest and Least) values of a function in a given interval : (i) A minimum value of a function f(x) in an interval [a, b] is not necessarily its greatest value in that interval. Similarly a minimum value may not be the least value of the function. (ii) If a function f(x) is defined in an interval [a, b], then greatest or least values of this function occurs either at x = a or x = b or at those values of x for which f'(x) = 0. Thus greatest value of f(x) in interval [a, b] = max [f(a), f(b), f(c), f(d)] Least value of f(x) in interval [a, b] = min. [f(a), f(b), f(c), f(d)] Where x = c, x = d are those points for which f'(x) = 0.

In an interval [a, b], a function f(x) is Monotonic increasing if f'(x) ≥ 0 Monotonic decreasing if f'(x) ≤ 0 constant if f'(x) = 0 ∀ x ∈ (a, b) Strictly increasing if f'(x) > 0 Strictly decreasing if f'(x) < 0

4.

Maximum & Minimum Points : Maxima : A function f(x) is said to be maximum at x = a, if there exists a very small +ve number h, such that f(x) < f(a), ∀ x ∈ (a h, a + h), x ≠ a. Minima : A function f(x) is said to be minimum at x = b, if there exists a very small +ve number h, such that f(x) > f(b), ∀ x ∈ (b h, b + h), x ≠ b. Remark :

5.

(a)

The maximum & minimum points are also known as extreme points.

(b)

A function may have more than one maximum & minimum points.

Conditions for Maxima & Minima of a function : (i)

Necessary condition : A point x = a is an extreme point of a function f(x) if f'(a) = 0, provided f'(a) exists. , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510

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Sufficient condition : (a) The value of the function f(x) at x = a is maximum if f'(a) = 0 and f"(a) < 0. (b) The value of the function f(x) at x = a is minimum if f'(a) = 0 and f"(a) > 0.

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Some Geometrical Results : In Usual Notations Area of equilateral and its perimeter

Results

3 (side)2. 4 3 (side)

Area of square

(side)2

Perimeter

4(side)

Area of rectangle

l × b

Perimeter

2(l × b)

Area of trapezium

1 (sum of parallel sides) 2

MATHS FORMULA - POCKET BOOK ROLLE'S THEOREM & LAGRANGES THEOREM: 1. Rolle's Theorem : If f(x) is such that (a) It is continuous on [a, b] (b) It is differentiable on (a, b) and (c) f(a) = f(b), then there exists at least one point c ∈ (a, b) such that f'(c) = 0. 2. Mean value theorem [Lagrange's theorem] : (i) If f(x) is such that (a) It is continuous on [a, b] (b) It is differentiable on (a, b), then there exists at least one c ∈ (a, b) such that f(b) − f(a) = f'(c) b−a

× (distance between them) Area of circle

πr2

Perimeter

2πr

Volume of sphere

4 3 πr 3

Surface area of sphere

4πr2

Volume of cone

1 2 πr h 3

Surface area of cone

πrl

Volume of cylinder

πr2h

Curved surface area

2πrh

Total surface area

2πr(h + r)

Volume of cuboid

l × b × h

Surface area of cuboid

2(lb + bh + hl)

Area of four walls

2(l × b) h

Volume of cube

l3

Surface area of cube

6l2

Area of four walls of cube

4l2


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(ii)

If for c in lagrange's theorem (a < c < b) we can say that c = a + θ h where 0 < θ < 1 and h = b a the theorem can be written as f(a + h) = f(a) + h f'(a + θ h), 0 < θ < 1, h = b a

PAGE # 139


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MATHS FORMULA - POCKET BOOK Function

INDEFINITE INTEGRATION 1.

d If F(x) = f(x), then dx

(i)

Here

zm r

z cos x

zch

z

f x dx = F(x) + c

z z z z z z z z

dx is the notation of integration, f(x) is the

integrand, c is any real no. (integrating constant)

d (ii) dx (iii) (iv)

2.

f x dx = f(x)

z ch z ch

f' x dx = f(x) + c, c ∈ R

z

k f x dx = k f(x) dx

z

(v)

zch

ch ch

(f x ± g x ) dx =

zch zch

f x dx ± g x dx

FUNDAMENTAL FORMULAE : Function Integration

z zc

n+1

x + c, n ≠ 1 n+1

n

x dx

h

n

ax + b dx

z

z z z z

c

h

1 ax + b . a n+1

+ c, n ≠ 1

1 dx x

log|x| + c

1 dx ax + b

1 (log|ax + b|) + c a

ex dx

ex + c

ax dx

ax + c loge a

sinx dx

cos x + c


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z z z

n+1

dx

Integration sin x + c

sec2 x dx

tan x + c

cos ec2x dx

cot x + c

sec x tan x dx

sec x + c

cos ec x cot xdx

cosec x + c

tanx dx

log|cos x| + c = log|sec x| + c

cot x dx

log|sin x| + c = log|cosec x| + c

sec x dx

log|sec x + tan x|+c = log tan

cos ec x dx

x log|cosec x cot x|+c = log tan +c 2

z z

dx

sin1 x + c = cos1x + c

1 − x2 dx 2

a −x

2

dx 1 + x2

dx a2 + x2 dx |x| x2 − 1 dx |x| x2 − a2

PAGE # 141

sin1

x x + c = cos1 + c a a

tan1x + c = cot1x + c x 1 x −1 tan1 + c = cot1 a + c a a a

sec1x + c = cosec1x + c 1 x sec1 + c = a a

x −1 cosec1 a + c a


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FG π + x IJ +c H 4 2K

PAGE # 142


INTEGRATION BY SUBSTITUTION :

SOME RECOMMENDED SUBSTITUTION :

z

By suitable substitution, the variable x in


Function

ch

f x dx is changed

zc h zch ch z d c hi c h z cc hh

f ax + b dx

f x f' x dx

fφx

f' x f x

φ x dx

dx

z d c hi c h z cc hh f x

n

f' x dx

Substitution ax + b = t

Integration

φ(x) = t

cf(x)h

c c

dx

f(x) = t

+ c, n ≠ 1

c

a− x , a+ x

2[f(x)]1/2 + c


C

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h

x = a tanθ or x = a sinhθ

, x 2 a2

x = a sec θ

c

PAGE # 143

h

x = a tan2 θ

1

c

x a− x

x = a sin2 θ

h

x−a , x

h

x x−a ,

x−α , β−x

E

, x2 + a2

a− x , x

x , x−a

f' x

fx

h

x a−x ,

n+1

n+1

x2 − a2

x = a sin θ or a cos θ

a+ x , x

x , a− x

f(x) = t

1

1 x a+ x . x a+ x

f t dt

log|f(x)| + c

x + a2

2

x , a+ x

z ch

f(x) = t

1 2

, a2 x 2

or x = a cosh θ

+c

2

a2 − x2

x −a , 2

1 F(ax + b) + c a

dfcxhi

2

x2 + a2 ,

2

f(x) = t

1

a −x , 2

into another variable t so that the integrand f(x) is changed into F(t) which is some standard integral. Some following suggestions will prove useful. Function

Substitution

1

c

x a− x

h

a+x a−x

cx − αh cβ − xh ,(β > α)

x = a sec2 θ

x = a cos 2θ x = α cos2 θ + β sin2 θ


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IMPORTANT RESULTS USING STANDARD SUBSTITUTIONS : Function

Integration

z

x−a 1 log x+a 2a

1 2

2

x −a

=

z

1 2

a −x

2

dx

INTEGRATION OF FUNCTIONS USING ABOVE STANDARD RESULTS : Function

z z

+ c

x −1 coth1 a + c when x > a a

a+ x 1 log a− x 2a

+ c

z

2

2

x −a

log{|x +

= cosh1

z

dx 2

2

x +a

z

FG IJ H K x a

FG x IJ + c H aK

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ax + bx + c

dx or

px + q ax2 + bx + c

dx or

(px + q) (ax2 + bx + c) dx

z ax

x − a dx

x − a |} + c

x2 + a2 dx

1 1 x x2 + a2 + a2 log {|x + 2 2

x2 + a2 |} + c

2

px + q 2

LMF MNGH

a x+

b 2a

IJ K

2

+

4ac − b2 2

4a

OP PQ

then use appropriate formula

Express : px + q

= λ

d (ax2 + bx + c) + µ dx

evaluate λ & µ by equat

constant, the integral reduces to known form

1 1 x x2 − a2 a2 log {|x + 2 2

2

ax2 + bx + c

dx or

+ c

1 1 2 x a2 − x2 + a sin1 2 2

2

1

Express : ax2 + bx + c =

ing coefficient of x and


z

x2 + a2 |} + c

a − x dx 2

z z

FG x IJ + c H aK

log{|x +

= sinh1

2

dx or

(ax2 + bx + c) dx

z z

x − a |} + c 2

ax + bx + c

z

x 1 = tanh1 a + c, when x < a a dx

1 2

Method

2

2

P(x) 2

+ bx + c

dx ,

Apply division rule and express it

in form Q(x) +

polynomial of degree

The integral reduces to known

2 or more

form

PAGE # 145


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ch

R x

where P(x) is a

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2

ax + bx + c

PAGE # 146


z

1 2

a sin x + b cos2 x + c

zc

or

z

z

dx Divide numerator & denominator by cos x, 2

1 a sin x + b cos x

h

2

dx

dx dx a sin x + b cos x + c

then put tanx = t & solve.

Replace sin x =

cos x =

z


2 tan x / 2 1 + tan2 x / 2

,

dx x + kx2 + a2

1 − tan2 x / 2 1 + tan2 x / 2

4.

INTEGRATION BY PARTS : when integrand involves more than one type of functions the formula of integration by parts is used to integrate the product of the functions i.e.

z

Express : num. = λ(deno.) + (i)

Express : Num. = λ(deno.) + = (1st fun)

d (deno.) + ν Evaluate λ, µ, ν. dx Thus integral reduces to known form. x 2 ± a2 x 4 + kx 2 + a 4

dx

(ii)

Divide numerator & denominator

Fx ± a I GH x JK


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h c

du dx

IK OP dx Q

υ dx

h

1st fun. . 2nd fun. dx

z

2nd fun. dx

z LMNFGH

IJ ez 2nd fun.dxjOP dx K Q

d 1st fun. dx

Rule to choose the first function : first fun. should be choosen in the following order of preference (ILATE). [The fun. on the left is normally chosen as first function] L Logarithmic function

= t, the

A Algebraic function

integral becomes one of standard forms. E

zc

z LMN FHz

I Inverse trigonometric function

2

by x2 and put

z

u. υ dx = u. υ dx

or

µ

z

Divide num & deno. by 2a2 and

4

then add & sub x2. Thus the form reduces to the known form.

d (deno.) Evaluate λ & µ. Thus dx integral reduces to known form.

a sin x + b cos x + c dx p sin x + q cos x + r

Divide numerator & denominator by 2 and then add & sub. a2. Thus the form reduces as above.

µ

z

dx

x + kx 2 + a 4

z

then put tan x/2 = t and replace 1 + tan2 x/2 = sec2 x/2 a sin x + b cos x dx c sin x + d cos x

x2 4

T Trigonometric function E Exponential function

PAGE # 147


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PAGE # 148

(iii)

z z z

(a) (b)

(c)

(iv)

z


ch ch

5.

ex f x + f' x dx = ex f(x) + c

ch ch

mx

mf x + f' x dx = e

e

LM f' cxh OP dx MN c h m PQ

emx f x +

mx

INTEGRATION OF RATIONAL ALGEBRAIC FUNCTIONS USING PARTIAL FRACTION : Every Rational fun. may be represented in the form

f(x) + c

ch

emx f x

=


where P(x), Q(x) are polynomials. If degree of numerator is less than that of denominator, the rational fun. is said to be proper other wise it is improper. If deg (num.) ≥ deg(deno.) apply division rule

+ c.

m

i.e.

ch ch

xf' x + f x dx = x f(x) + c.

z

eax sinbx dx and e ax

=

a2 + b2 e

ax

e

=

h

eax sin bx + c dx

e ax a +b

and

2

z

ax

e

cacos bx + b sinbxh

eax a2 + b2

c

h

c

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c

h

hc

hc

h

A B C + + x−a x −b x−c

≠ b

A + x−a

c

px2 + qx + r

j

Bx + C A + 2 x−a x + bx + c

he

x − a x2 + bx + c , where

ex h

B

+

cx − ah

2

C x−b

x2 + bx + c can not be factorised

+ k

a cos bx + c + b sin bx + c

A B + x−a x −b

h

cx − ah cx − bh , a

cos bx + c dx


c

px2 + qx + r , x−a x−b x−c

hc

px2 + qx + r

cos bx dx and

2

c

Types of partial fractions

a, b, c are distinct

(a sin bx b cos bx) + k and

[a sin (bx + c) b cos(bx + c)] + k1

2

c h , for integrating rcxh , resolve the gcxh gcxh r x

px + q x−a x−b , a ≠ b

2

a +b

z

c

= q(x) +

Types of proper rational functions

ax

2

(vi)

z

ch gcxh f x

fraction into partial factors. The following table illustrate the method.

NOTE : Breaking (iii) & (iv) integral into two integrals. Integrate one integral by parts and keeping other integral as it is by doing so we get the result (integral).

(v)

c h, Qcxh Px

px3 + qx2 + rx + s 2

je

j

+ ax + b x2 + cx + d ,

Ax + B 2

x + ax + b

+

Cx + D 2

x + cx + d

where x + ax + b, x2 + cx + d can not be factorised 2

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INTEGRATION OF IRRATIONAL ALGEBRAIC FUNCTIONS : (i)

If integrand is a function of x & (ax + b)1/n then put (ax + b) = tn

(ii)

If integrand is a function of x, (ax + b)1/n and

(viii) To evaluate

1 or quadratic

where p = (L.C.M. of m & n).

(iv) To evaluate

(v) To evaluate

or

or

z z z

put

linear linear dx quad. linear

linear. quadratic

zc

h

linear = t

form :

put linear = t2

dx

h

7.

put linear = 1/t

dx

linear . quadratic dx pure quad. pure quad

put

pure quad = t

z

dx pure quad. pure quad

then is the resulting integral, put


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FG linear IJ H quadratic K

z

linear quad. quad

dx

into partial fractions and

dx linear quad.

1 − cos 2 mx 2

(i)

sin2 mx =

(ii)

cos2 mx =

(iii)

sin mx = 2sin

(iv)

sin3 mx =

(v)

cos3 mx =

(vi) (vii) (viii) (ix) (x)

tan2 mx cot2 mx 2 cos A 2 sin A 2 sin A

1 + cos 2mx 2

x dx

2

z

z

INTEGRATION USING TRIGONOMETRICAL IDENTITIES : (A) To evaluate trigonometric functions transform the function into standard integrals using trigonometric identities as

linear . quadratic

(vii) To evaluate

quad. quad

2

(vi) To evaluate

or

split the integral into two, each of which is of the

dx

zc

dx

and if the quadratic not under the square root can be resolved into real linear factors, then resolve

(ax + b)1/m then put (ax + b) = tp

(iii) To evaluate

z

put x =

1 and t

pure quad = u

PAGE # 151

mx mx cos 2 2

3 sin mx − sin 3mx 4 3 cos mx + cos 3mx 4

= sec2 mx 1 = cosec2 mx 1 cos B = cos (A + B) + cos (A B) cos B = sin (A + B) + sin (A B) sin B = cos (A B) cos (A + B)


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(B)

z


z z z

sinm x cosm xdx .

(i) (ii) (iii) (iv)

if m is odd put cos x = t if m is even put sin x = t if m & n both odd put sin x or cos x as t if m & n both even use the formula of sin2x & cos2x

(v)

if m & n rational no. &

m+n−2 is ve integer 2

put tan x = t 8.

z z z z z z

cos ecnx dx

sinm x cosn x dx

xneaxdx , n ∈ N

z

C

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I O

xn cos x + nxn1sin x n(n 1) In2

sinn x dx

cosn x dx

n−1 cosn−1 x sin x + In2 n n

n−1 sinn−1 cos x + In2 n n

tann x dx

ctanxh

cotn x dx

S

h

xn−1eax dx

xn sin x dx

N

c cm + nh

cosn−1 x sinm+1 x + n − 1 Im, n−2

NOTE : These formulae are specifically useful when m & n are both even nos.

1 n ax n x e I a a n1 where In1 =

n−2 cos ecn−2 x cot x + I n − 1 n2 n−1

sinm1x cosn+1x + (m 1) Im2,n

Integration

n−1

In2

n−1

ccot xh

n−1

n−1

In2


n−2 secn−2 x tan x + I n − 1 n2 n−1

INTEGRATION BY SUCCESSIVE REDUCTION (REDUCTION FORMULA) : Function

E

secn x dx

PAGE # 153


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DEFINITE INTEGRATION 1.

z

z b

ch

III.

ch

ch

b a

zch zc b

z

a

V.

z

a

*

{uz v.dx}

b a

z FGH z

b

a

−a

IJ K

zch

f x dx =

0

VII. PROPERTIES OF DEFINITE INTEGRAL :

zch b

f x dx =

a

z b

a

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f x dx =

0

zc a

h

f a − x dx

0

c h

f x + f −x dx

ch , if f cxh is an odd function

, if f x is an even function

R| S|2 fcxh dx |T 0

z

a

0

c h ch , if f c2a − xh = − f cxh , if f 2a − x = f x

If f(x) is a periodic function with period T, Then

zch

nT

zch T

f x dx = n f x dx

ch

f t dt


0

2a

VI.

z ch a

R|2 f x dx S| z c h T 0

=

du . v. dx dx dx

h

0

a

When we use method of substitution. We note that while changing the independent variable in a definite integral, the limits of integration must also we changed accordingly.

I.

ch

f x dx =

zch a

f a + b − x dx or

b

For integration by parts in definite integral we use following rule.

uv dx =

f x dx where a < c < b

c

a

f x dx =

IV.

To evaluate definite integral of f(x). First obtain the indefinite integral of f(x) and then apply the upper and lower limit.

b

a

a

Remarks :

*

zch b

f x dx +

This property is mainly used for modulus function, greatest integer function & breakable function

= F(b) F(a) is called definite integral

of f(x) w.r.t. x from x = a to x = b Here a is called lower limit and b is called upper limit. *

zch c

ch

a

f x dx = F x + c

a

z

f x dx =

b

b

f x dx = F(x) + c, then

zch a

ch

a

Definite Integration : If

z

f x dx = f x dx

b

II.

0

0

and further if a ∈ R+, then PAGE # 155


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zch

a+nT

zch zch a

f x dx =

f x dx ,

nT

(ii)

zch

nT

0


T

f x dx = (n m) f x dx ,

mT

0

If the function φ(x) and ψ(x) are defined on [a, b] and differentiable at a point x ∈ (a, b), and f(x, t) is continuous, then,

zch zch

b +nT

d dx

b

f x dx =

a+ nT

f x dx

a

ψ(x))

VIII. If m and M are the smallest and greatest values of a function f(x) on an interval [a, b], then 3.

zch b

m(b a) <

z b

a

ch

f x dx

|f x dx|

a

=

z a

2.

ch

f x dx ≤

zch b

A T

z

ψ (x)

f (x, t) dt +

φ(x)

RSdψ (x) UV f(x, T dx W

φ(x)).

z

π /2

cosn x dx =

sinn x dx

0

R| n − 1 . n − 3 ..... 2 .1, S|n −n1 nn−−32 13 π |T n . n − 2 ...... 2 . 2 ,

z

if n is odd if n is even

(ii)

For integration

sinm x cosn x dx follow the following

0

Leibnitz's Rule :

steps

(i)

(a)

If m is odd put cos x = t

(b)

If n is odd put sin x = t

(c)

If m and n are even use sin2x = 1 cos2x

If f(x) is continuous and u(x), v(x) are differentiable


OP PQ =

π /2

a

z

v(x)

f(t) dt =

or cos2x = 1 sin2x and then use

u(x)

z

I O

N

S

0

PAGE # 157

z

π /2

π /2

d d f{v(x)} {v(x)} f{u(x)} {u(x)}. dx dx

D U

RSdφ(x) UV f(x, T dx W

g x dx

Differentiation Under Integral Sign :

d functions in the interval [a, b], then, dx

E

φ(x)

a

z ch

If f(x) < g(x) on [a, b], then

f (x, t) dt

π /2

(i)

b

X.

z

f x dx < M(b a)

b

<

z

ψ (x)

Reduction Formulae :

a

IX.

LM MN

n

sin x dx or

cosn x dx

0


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PAGE # 158


z

∞

(iii)

e

− ax

cos bx dx =

0

z z

Summation of series by Definite integral or limit as a sum :

a2 + b2

zch b

(i)

b

e−ax sinbx dx =

lim h[f(a) + f(a + h) + f(a + 2h) +..... f x dx = h→0

a

2

a + b2

0

+f(a + (n 1)h] where nh = b a.

∞

(v)

4.

a

∞

(iv)


e −ax xndx =

0

z

n! n

a +1

(ii)

lim

n→∞

FI ∑ GH JK

zch 1

n

r 1 f n r =1 n

f x dx

=

0

π /2

(vi)

sinn x cosm x dx

[i.e. exp. the given series in the form

0

LM m − 1 . m − 3 .... 2 . 1 MM m + n m + n − 2 3 + n 1 + n MM mm +− 1n. mm+ n− 3− 2.... 2 1+ n. n n− 1. nn −− 32 .... 23 MMm − 1. m − 3 .... 1 . n − 1. n − 3.... 1 . π Nm + n m + n − 2 2 + n n n − 2 2 2

=

OP PP PP PP Q

; if m is odd and n may be even or odd ; if m is enen and n is odd ; if m is even and n is even

replace

z

f x dx ]

z

π /2

*

[(m − 1) (m − 3)....] [(n − 1) (n − 3) .....] (m − n) (m + n − 2) ....

to be multiplied

zch

Key Results :

0

=

r 1 by x and by dx and the limit of the n n

0

5.

sinm x cos n x dx

by

1

1

sum is

These formulae can be expressed as a single formula : π /2

FrI

∑ n f GH nJK

0

z

π /2

π when m and n are both even 2

*

0

z

π /2

logsin x dx =

logcos x dx =

0

c h f csin xh + fccos xh dx = f sin x

zc

π /2

0

−π log2 2

c h h c h

f cos x

f sin x + f cos x dx

integers. , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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z

π /2

=

=

c

h c

h

c

zc

f cos ec x

h c

f cos ec x + f sec x

sinmx sin nx dx =

R|0 S| π T2

z z 0

z a

*

0

z

x2 a2 − x2 dx =

z

x2

x a − x2

FG H

π 2 a2 − x2 − dx = a3 4 3 a+ x

a

cos mx . cos nx dx

*

0

z

2a

*

If n ∈ N, then

*

x

a2 − x2

C

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j

n

z

x−a

z

x−a π b−a dx = a+x 2

b

(i)

dx

a

(ii) 2

πa 3a + 6 8


if a > 0

zc a

(iii)

PAGE # 161

c

b−x

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=π

c

hc

h

h

c

h

x − a b − x dx = π b − a 2 2


c h

2. 4. 6...... 2n 2n+1 dx = 3.5 . 7..... 2n + 1 a

If a < b then

dx = a

dx =

a2 − x2

0

*

2

ze a

π 2 a 4

IJ K

πa2 2

2ax − x2 dx =

0

2

πa4 16

0

0

E

j

.

2a2

a

*

a

2

1

dx =

3/2

0

π 1 dx = 2 a2 − x2

a

*

0

c h dx = π/4. fctan xh + fccot xh f cot x

if m = n

z 0

z

x dx

a2 + x2

0

if m, n are different + ve int egers

0

*

z

h

dx=

ze a

*

0

a2 − x2 dx =

a

0

c h dx f csec xh + f ccos ecxh f sec x

π /2

a

*

dx =

π /2

0

=

h

h

π /2

*

z

π /2

f tan x + f cot x

0

z

c

f tan x


N

S

PAGE # 162


z b

(iv) *

a

dx

cx − ahcb − xh

x

π

=

*

2

ab

(i)

z z

a− x a π −2 a + x dx = 2

c

c

0

z z a

(iii)

0

a

(iv)

0

a+ x a− x

dx =

a+ x a − x dx =

z

∞

(i)

x e −ax dx =

0

z

∞

(ii)

e −r

0

z

∞

(iii)

0

h

I O

N

S

f x dx = f(c) [b a]. The no.

zch

fun. f(x) on the interval [a, b]. The above result is called the first mean value theorem for integrals.

2 2

x

dx =

zd

2k

*

Q x [x] is a periodic function with period 1. If f(x) is a periodic fun. with period T, then

*

zch z c

a+ T

f x dx is independent of a.

FG π + 1IJ a H2 K π

i

x − x dx = k, where k ∈ I,

0

10 a a 3

2a a

1 f x dx is called the mean value of the b−a a

h

a

π /4

h

log 1 + tan x dx =

*

0

π log2 8

a > 0

π (r > 0) 2r

e−ax − e −bx dx = loge(b/a) (a, b > 0) x


f(c) =

If a > 0, n ∈ N, then

*

C

point c ∈ (a, b) s.t

a

a a+x π +2 dx = 2 a−x

a

(ii)

zch b

ab > 0

b

0

D U

If f(x) is continuous on [a, b] then there exists a

If a > 0 then a

E


PAGE # 163


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DIFFERENTIAL EQUATIONS 1.

(B)

dy = f(x) g(y) dx This can be integrated as

the form

Order of a differential equation : The order of a differential equation is the order of the highest derivative occurring in it.

2.

z

Degree of a differential equation : The degree of a differential equation is the degree of the highest order derivative occurring in it when the derivatives are made free from the radical sign. d2 y

Eg. (i)

dx

2

+

(ii)

(iii)

F d yI GH dx JK 3

2

+

3

(C)

F dy I 1+G J H dx K

FG1 + dy IJ H dx K

2

+ 5y = 0

or

z

dy = f(y)

z

z

z

D U

Equations Reducible to Homogeneous form and variable separable form

*

Form

C

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N

S

ax + by + c dy = Ax + By + C dx

........... (1)

a b ≠ A B This is non Homogeneous Put x = X + h and y = Y + k in (1) where

dy dY = Put ah + bk + c = 0, Ah + Bk + C = 0, dX dx find h, k

f(x) dx

dY aX + bY = . This is homogeneous. dX AX + BY Solve it and then put X = x h, Y = y k we shall get the solution.

Then

dx to get its solution.


(D)

dy = f(x) or dx

dy =

FG IJ H K

x or yn f y .

Such an equation can be solved by putting y = vx or x = vy. After substituting y = vx or x = vy. The given equation will have variables separable in v and x.

∴

Integrate both sides i.e.

Homogeneous Equations : It is a differential equation

FG IJ H K

SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND FIRST DEGREE :

dy = f(y) dx

f(x) dx + c

y of degree n if it can be written as xn f x

2

(A) Differential equation of the form

z

f(x, y) dy = , where f(x, y) and g(x, g(x, y) dx y) are homogeneous functions of x and y of the same degree. A function f(x, y) is said to be homogeneous

order of (i) 2 (ii) 1 & (iii) 3, degree of (i) 1 (ii) 2 & (iii) 2 3.

dy = g(y)

of the form

dy + 5y = 0 dx

dy y = x + dx

Variable Separable Form : Differential equation of

PAGE # 165


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Form where

ax + by + c dy = + By + C Ax dx


..... (1),

*

In x :

dx + Rx = S, where R, S are functions of y dy

alone or constant.

a b = = k say A B

its solution x e

k (Ax + By) + c dy = ∴ Ax + By + C dx

where

dy dz A + B = dx dx

Put

Ax + By = z

⇒

dz kz + c = A + B dx z+c

⇒

e

z

R.dy

z

R dy

=

z

S. e

z

R.dy

is called the integrating factor (I.F.) of

the equation. (F) Equation reducible to linear form :

This is variable separable form and can be solved. *

dy Form = f(ax + by + c) dx

*

⇒

Put

ax + by = z

∴

dz = a + b f(z) dx

a + b

Differential equation of the form

Linear equation :

*

In y :

dy dz = dx dx

yn Put

where

e

z

P dx

P dx

=

z

Qe

z

P dx

dy + pyn dx yn

+ 1

+ 1

= Q

= z

Note : In general solution of differential equation we can take integrating constant c as tan1 c, ec, log c etc. according to our convenience.

alone or constant.

z

+ Py = Qyn

⇒ The given equation will be linear in z and can be solved in the usual manner.

dy + Py = Q, where P, Q are function of x dx

its solution ye

dy dx

where P and Q are functions of x or constant is called Bernoulli's equation. On dividing through out by yn, we get

This is variable separable form and can be solved. (E)

dy + c

dx + c

is called the integrating factor (I.F.) of

the equation. , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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VECTORS 1.


Vectors in terms of position vectors of end points -

AB = OB OA = Position vector of B position vector of A i.e. any vector = p.v. of terminal pt p.v. of initial pt.

Types of vectors : (a) Zero or null vector : A vector whose magnitude is zero is called zero or null vector. r a Vector a (b) Unit vector : a$ = | | = a Magnitude of a (c)

2.

5.

Multiplication of a vector by a scalar : r r If a is a vector and m is a scalar, then m a is a vector and r magnitude of m a = m|a| r $ and if a = a1 $i + a2 j + a3 k$

Equal vector : Two vectors a and b are said to be equal if |a| = |b| and they have the same direction.

Triangle law of addition : AB + BC = AC

c = a + b

c =

b a +

r $ then m a = (ma1) $i + (ma2) j + (ma3) k$

6.

Distance between two points : Distance between points A(x1, y1, z1) and B(x2, y2, z2)

C

→

= Magnitude of AB =

b

7. A

3.

a

Position vector of a dividing point : r (i) If A( a ) & B( b ) be two distinct pts, the p.v. c of the point C dividing [AB] in ratio m1 : m2 is given by r r m1b + m2a r c = m +m 1 2

B

Parallelogram law of addition : OA + OB = OC

a + b = c B

C

b

D

a

p.v. of the mid point of [AB] is

(iii)

If point C divides AB in the ratio m1 : m2 externally, then p.v. of C is c =

A


C

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N

S

1 [p.v. of A + p.v. of B] 2

(ii)

where OC is a diagonal of the parallelogram OABC

E

(x2 − x1 )2 + (y2 − y1)2 + (z2 − z1 )2

PAGE # 169

m1 b − m2 a m1 − m2


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MATHS FORMULA - POCKET BOOK (iv) p.v. of centriod of triangle formed by the points A( a ),

MATHS FORMULA - POCKET BOOK 10. Coplanar and non coplanar vector : (i)

r r a+b+c B( b ) and C ( c ) is 3 (v)

If a , b , c be three non coplanar non zero vector then x a + y b + z c = 0

⇒ x = 0, y = 0, z = 0

p.v. of the incentre of the triangle formed by the points r r r A( α ), B( β ) and C( γ ) is

(ii)

If a , b , c be three coplanar vectors, then a vector

c can be expressed uniquely as linear combination of aα + bβ + cγ where a = |BC|, b = |CA|, c = |AB| a+b+c 8.

remaining two vectors i.e. c = λ a + µ b (iii)

Some results : (i)

b and c i.e. r = x a + y b + z c

If D, E, F are the mid points of sides BC, CA & AB respectively, then AD + BE + CF = 0

Any vector r can be expressed uniquely as inner combination of three non coplanar & non zero vectors a ,

11. Products of vectors :

(ii)

If G is the centriod of ∆ABC, then GA + GB + GC = 0

(I)

Scalar or dot product of two vectors :

(iii)

If O is the circumcentre of a ∆ABC, then

(i)

a . b = |a| |b| cosθ

OA + OB + OC = 3 OG = OH where G is centriod and H is orthocentre of ∆ABC.

(ii)

Projection of a in the direction of b =

a. b | b|

(iv) If H is orthocentre of ∆ABC, then a. b & Projection of b in the direction of a = | a|

HA + HB + HC = 3 HG = OH 9.

Collinearity of three points : (i)

Three points A, B and C are collinear if AB = λ AC for some non zero scalar λ.

(ii)

The necessary and sufficient condition for three points

(iii)

Component of r ⊥

with p.v. a , b , c to be collinear is that there exist three scalars l, m, n all non zero such that l a + m b + n c = 0, l + m + n = 0


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FG IJ H Ka F r. a IJ to a = r G H|a| K a

r. a Component of r on a = |a|2

(iv)

$i . $i = $j . $j = k$ . k$ = 1

(v)

$i . $j = $j . k$ = k$ . $i = 0

PAGE # 171

2


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MATHS FORMULA - POCKET BOOK (vii) a × ( b × c ) = ( a × b ) × c

(vi) If a and b are like vectors, then a . b = | a || b | and

(viii) a × ( b + c ) = ( a × b ) + ( a × c )

If a and b are unlike vectors, then a . b = | a || b | (vii) a , b are ⊥ ⇔ a . b = 0

(ix)

$i × $i = $j × $j = k$ × k$ = 0 , $i × $j = k$ ,

(x)

$j $ × k$ = $i , k$ × $i = j Area of triangle :

(viii) ( a . b ). b is not defined (ix)

( a ± b )2 = a2 ± 2 a . b + b2

(x)

| a + b | = | a| + | b |

⇒ a || b

(xi)

| a + b |2 = |a|2 + |b|2

⇒ a ⊥ b

(xii) | a + b | = | a b |

⇒

a ⊥ b

(a)

1 2

(b)

If a , b , c are p.v. of vertices of ∆ABC,

(xiii) work done by the force : work done = F . d , is displacement vector.

then =

where F is force vector and d

(xi)

(II) Vector or cross product of two vectors :

a × b = |a| |b| sinθ n$

(ii)

if a , b are parallel

(iii)

a × b = ( b × a )

(iv)

n$ = ±

(v)

$ $ let a = a1 $i + a2 j + a3 k$ & b = b1 $i + b2 j + b3 k$ , then

(b) = 0

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1 |a × b | 2

(xii) Moment of Force :

a×b

Moment of the force F acting at a point A about O is

| a × b|

Moment of force = OA × F = r × F

a. a a. b (xiii) Lagrange's identity : | a × b |2 = a. b b. b (III) Scalar triple product : r r (i) If a = a1 $i + a2 $j + a3 k$ , b = b1 $i + b2 $j + b3 k$ and r c = c1 $i + c2 $j + c3 k$ then

a × a =0


If a and b are two diagonals of a parallelogram, then area =

$i $j k$ a a2 a3 a × b = 1 b1 b2 b3 (vi)

If a & b are two adjacent sides of a parallelogram, then area = | a × b |

(i)

a × b

1 |( a × b ) + ( b × c ) + ( c × a )| 2

Area of parallelogram : (a)

⇔

AB × AC

PAGE # 173


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r r r r r r ( a × b ). c = [ a b c ] =

MATHS FORMULA - POCKET BOOK (d)

a1 a2 a3 b1 b2 b3 c1 c 2 c 3

r r r and [ a b c ] = volume of the parallelopiped whose r r r coterminus edges are formed by a , b , c r r r r r r r r r (ii) [ a b c ] = [ b c a ] = [ c a b ], r r r r r r r r r but [ a b c ] = [ b a c ] = [ a c b ] etc. r r r r r (iii) [ a b c ] = 0 if any two of the three vectors a , b , r c are collinear or equal. r r r r r r (iv) ( a × b ). c = a .( b × c ) etc.

(v)

→

→

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r r b c,

a b c r Direction cosines of r = ai$ + bj$ + ck$ are r , r , r . |r | |r | |r |

(i)

→

r r r r r c + a ] = 2[ a b c ] r r c a] = 0 r r r r r c × a ] = [ a b c ]2


→

(ii)

Incentre formula : The position vector of the incentre r r r aa + bb + cc of ∆ ABC is . a+b+c

(iii)

Orthocentre formula : The position vector of the r r r a tan A + b tan B + c tan C orthocentre of ∆ ABC is tan A + tan B + tan C

→

[ AB AC AD ] = 0 r r r r (xii) (a) [ a + b b + c r r r r (b) [ a b b c r r r r (c) [ a × b b × c

r r a × b,

12. Application of Vector in Geometry :

1 | AB × AC . AD | 6 r r r r (x) Four points with p.v. a , b , c , d will be coplanar if r r r r r r r r r r r r [d b c ] + [d c a] + [d a b] = [a b c ] (xi) Four points A, B, C, D are coplanar if

(ix) Volume of tetrahedron ABCD is

→

r r r a , b , c are coplanar, then so are r r r × c , c × a and r r r r r r r + b , b + c , c + a and a b , r a are also coplanar.

(IV) Vector triple Product : r r r r r r If a , b , c be any three vectors, then ( a × b ) × c r r r and a × ( b × c ) are known as vector triple product and is defined as r r r r r r r r r ( a × b ) × c = ( a . c ) b (b . c ) a r r r r r r r r r and a × ( b × c ) = ( a . c ) b ( a . b ) c r r r r r r Clearly in general a × ( b × c ) ≠ ( a × b ) × c but r r r r r r r r ( a × b ) × c = a × ( b × c ) if and only if a , b r & c are collinear

[ $i $j k$ ] = 1

r r r r r r (vi) If λ is a scalar, then [λ a b c ] = λ[ a b c ] r r r r r r r r r r (vii) [ a + d b c ] = [ a b c ] + [ d b c ] r r r r r r (viii) a , b , c are coplanar ⇔ [ a b c ] = 0

If r b r a r c

(iv) Vector equation of a straight line passing through a r fixed point with position vector a and parallel to a r r r r given vector b is r = a + λb . PAGE # 175


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MATHS FORMULA - POCKET BOOK (viii) The equation of the plane passing through a point r r r having position vector a and parallel to b and c is r r r r rrr rrr r = a + λb + µc or [ r bc ] = [ abc ], where λ and µ are scalars. (ix) Vector equation of a plane passing through a point r r r r rrr abc is r = 1 − s − t a + sbt + c r r r r r r r rrr or r. b × c + c × a + a × b = [ abc ].

The vector equation of a line passing through two r r points with position vectors a and b is r r r r r =a+ λb−a .

e

j

(vi) Shortest distance between two parallel lines : Let l1 and l2 be two lines whose equations are l 1 : r r r r r r r = a1 + λb1 and l2 : r = a2 + µb2 respectively.

e

Then, shortest distance

PQ =

cb

1

(x)

h c

× b2 . a2 − a1 |b1 × b2 |

c

h=

b1 b2 a2 − a1 |b1 × b2 |

h

r r r r = a1 + λb1 and

h

r r r r = a2 + µb2 intersect,

n1 . n2 r r r2 .n2 = d2 is given by cos θ = ± |n ||n | . 1 2 (xiii) The equation of the planes bisecting the angles r r between the planes r1 .n1 = d1 r r r r |r.n1 − d1| |r.n2 − d2| r r = r r and r2 .n2 = d2 are |n1| |n2| r r r r (xiv) The plane r .n = d touches the sphere | r − a | = R, r r |a.n − d| r if = R. |n|

h

Therefore, [ b1 b2 a2 − a1 ] = 0

⇒

r r r r [ a2 − a1 b1b2 ] = 0

c

h

⇒

car

2

r r r − a1 . b1 × b2

h e

j = 0.

r (vii) Vector equation of a plane normal to unit vector n and at a distance d from the origin is r r . n$ = d. r If n is not a unit vector, then to reduce the equation r r r r . n = d to normal form we divide both sides by | n | r r n r d d r = r r . to obtain or r . n$ = r . |n| |n| |n| , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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h

constant. The perpendicular distance of a point having position r r r vector a from the plane r.n = d is given by r r |a.n − d| r p= . |n| r r (xii) An angle θ between the planes r1.n1 = d1 and

then the shortest distance between them is zero.

c

j

(xi)

r r r | a2 − a1 × b| r r r r and r = a2 + µb is given by d = . |b| If the lines

h

The equation of any plane through the intersection r r r r of planes r . n1 = d1 and r . n2 = d2 is r r r . n1 + λn2 = d1 + λd2, where λ is an arbitrary

c

shortest distance between two parallel lines : The r r r shortest distance between the parallel lines r = a1 + λb

c

c

(xv)

If the position vectors of the extremities of a diamr r eter of a sphere are a and b , then its equation is r r r r r r r r r ( r − a ).( rr − b ) = 0 or | r |2 r. a − b + a.b = 0.

e

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THREE DIMENSIONAL GEOMETRY 1.

*

Coordinates of the centroid of a triangle are

FG x H

1

Points in Space : (i)

Origin is (0, 0, 0)

(ii)

Equation of x-axis is y = 0, z = 0

(iii)

Equation of y-axis is z = 0, x = 0

*

Equation of YOZ plane is x = 0

FG x H

y1 1 y 2 Where ∆x = 2 y 3

Distance formula : Distance between two points A(x1, y1, z1) and B(x2, y2, z2) is given by AB = (ii)

(x 2 − x1 )2 + (y 2 − y1 )2 + (z 2 − z1 )2

(iii)

3.

Distance of a point p(x, y, z) from coordinate axes OX, OY, OZ is given by

z2 + x2 and

*

4.

Section formula :

*

Internally are

*

Externally are

FG mx + nx H m+n FG mx − nx H m−n

1

,

my 2 + ny1 mz 2 + nz1 , m+n m+n

2

1

,

my2 − ny1 m−n

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2

∆2x + ∆2y + ∆2z

and so.

1 6

x1 x2 x3 x4

y1 y2 y3 y4

z1 z2 z3 z4

1 1 1 1

Direction cosines and direction ratios of a line : *

If

α , β , γ are the angles which a directed line

segment makes with the +ve direction of the coordinate axes, then l = cos α , m = cos β , n = cos γ are

IJ K mz − nz I , J m−n K

2


Volume of tetrahedron =

x2 + y2

The coordinates of a point which divides the join of (x1, y1, z1) and (x2, y2, z2) in the ratio m : n

E

z1 1 z2 1 z3 1

x12 + y12 + z12

y 2 + z2 ,

IJ K

x1 − x2 y1 − y2 z1 − z2 Condition of collinearity x − x = y − y = z − z 2 3 2 3 2 3

*

Distance between origin (0, 0, 0) & point (x, y, z) =

+ x2 + x3 + x 4 y1 + y2 + y3 + y 4 z1 + z 2 + z 3 + z 4 , , 4 4 4

Area of triangle is given by ∆ =

*

(vii) Equation of XOY plane is z = 0

(i)

1

Note :

(vi) Equation of ZOX plane is y = 0

2.

IJ K

Coordinates of centroid of a tetrahedron

(iv) Equation of z-axis is x = 0, y = 0 (v)

+ x 2 + x3 y1 + y2 + y 3 z1 + z 2 + z3 , , 3 3 3

called direction cosines of the line and cos2 α + cos2 β + cos2 γ = 1 i.e. l 2 + m2 + n2 = 1, where 0 ≤ α, β, γ

1

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≤ π


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x − x1 y − y1 z − z1 x2 − x1 = y2 − y1 = z2 − z1

If l , m, n are direction cosines of a line and a, b, c are proportional to l , m, n respectively, then a, b, c are called direction ratios of the line and

m n l = = =± a b c

l 2 + m2 + n2 a2 + b2 + c 2

=±

1 2

a + b2 + c2

The angle θ between the lines whose d.c.'s are l 1, m1, n1 and l 2, m2, n2 is given by

.

*

Direction cosines of x-axis are 1, 0, 0, similarly direction cosines of y-axis and z-axis are respectively 0, 1, 0 and 0, 0, 1.

*

If l , m, n are d.c.s of a line OP and (x, y, z) are coordinates of P then x = l r, y = mr and z = nr where r = OP. Direction cosines of PQ = r, where P is (x1, y1, z1) and Q(x2, y2, z2) are

*

*

cos θ = l 1 l 2 + m1m2 + n1n2. The lines are || if

The lines are ⊥ if l 1 l 2 + m1m2 + n1n2 = 0 *

The angle θ between the lines whose d.r.s are a1, b1, c1 and a2, b2, c2 is given by cos θ = ±

x2 − x1 y2 − y1 z 2 − z1 , , r r r

*

The lines are || if

If a, b, c are direction no. of a line, then a2 + b2 + c2 need not to be equal to 1.

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a1 b1 c1 = = a2 b2 c2 and

= (x2 x1) l + (y2 y1)m + (z2 z1)n, where p(x1, y1, z1) and Q(x2, y2, z2). *

Straight line in space : * Equation of a straight line passing through a fixed point and having d.r.'s a, b, c is

Two straight lines in space (not in same plane) which are neither parallel nor intersecting are called skew lines.

*

Shortest distance between two skew lines,

x − x1 y − y1 z − z1 = = and m1 n1 l1

I O

x − x2 y − y2 z − z2 = = is given m2 n2 l2

form) Equation of a line passing through two points is


a22 + b22 + c22

L = l (x2 x1) + m(y2 y1) + n(z2 z1)

*

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a12 + b12 + c12

l , m, n

x − x1 y − y1 z − z1 = = (is the symmetrical a c b

E

a1a2 + b1b2 + c1c2

The lines are ⊥ if a1a2 + b1b2 + c1c2 = 0 Length of the projection of PQ upon AB with d.c.,

*

Note : Direction cosines of a line are unique but the direction ratios of line are not unique. If P(x1, y1, z1) & Q(x2, y2, z2) be two points and L be a line with d.c.'s l , m, n, then projection of [PQ] on

5.

m1 n1 l1 l2 = m2 = n2 and

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x2 − x1 s.d. = ±

*

l1 l2

*

y2 − y1 z2 − z1 m1 n1 m2 n2

Normal form of the equation of plane is l x + my + nz = p, where l , m, n are the d.c.'s of the normal to the plane and p is the length of perpendicular from the origin.

(m1n2 − m2n1)2 + (n1 l2 − l1n2 )2 + (l1m2 − m1 l2 )2

*

ax + by + cz + k = 0 represents a plane || to the plane ax + by + cz + d = 0 and ⊥ to the line

Two straight lines are coplanar if they are intersecting or parallel

x2 − x1 condition

6.


l1 l2

y2 − y1 m1 m2

x y z = = . a c b

z2 − z1 n1 = 0 n2

*

Equation of plane through three non collinear points is

x x1 x2 x3

Plane : A plane is a surface such that if two points are taken in it, straight line joining them lies wholly in the surface. *

Ax + By + Cz + D = 0 represents a plane whose normal has d.c.s proportional to A, B, C.

*

Equation of plane through origin is given by Ax + By + Cz = 0.

*

Equation of plane passing through a point (x1, y1, z1) is A(x x1) + B(y y1) + C(z z1) = 0, where A, B, C are d.r.'s of a normal to the plane.

*

Equation of plane through the intersection of two planes

or *


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= 0

a1a2 + b1b2 + c1c2 a12 + b12 + c12 a22 + b22 + c22

where θ is the angle between the normals.

Equation of plane which cuts off intercepts a, b, c respectively on the axes x, y and z is

plane are ⊥ if a1a2 + b1b2 + c1c2 = 0

a1 b1 c1 plane are || if a = b = c = 0. 2 2 2

x y z + + = 1. a c b

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= 0

y − y1 z − z1 y2 − y1 z2 − z1 y3 − y1 z3 − z1

cos θ = ±

Q ≡ a2x + b2y + c2z + d2 = 0 is P + λ Q = 0.

E

y3

z 1 z1 1 z2 1 z3 1

The angle between the two planes is given by

P ≡ a1x + b1y + c1z + d1 = 0 and *

x − x1 x2 − x1 x3 − x1

y y1 y2

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If AP be the ⊥ from A to the given plane, then it is || to the normal, so that its equation is


Line and Plane : If ax + by + cz + d = 0 represents a plane and

x − x1 y − y1 z − z1 = = represents a straight line, then m n l

x−α y−β z−γ = = = r (say) a c b

Any point P on it is (ar + α , br + β , cr + γ ) Length of the ⊥ from P(x1, y1, z1) to a plane

*

ax + by + cz + d = 0 is given by

p= *

The line is ⊥ to the plane if

*

The line is || to the plane if a l + bm + cn = 0.

*

The line lies in the plane if a l + bm + cn = 0 and ax1 + by1 + cz1 + d = 0

ax1 + by1 + cz1 + d

*

a2 + b2 + c 2

The angle θ between the line and the plane is given by

Distance between two parallel planes (ax + by + cz + d1 = 0, ax + by + cz + d2 = 0) is given by

*

sin θ =

d2 − d1

*

2

a + b2 + c2

al + bm + cn a2 + b2 + c2

l2 + m2 + n2

General equation of the plane containing the line x − x1 y − y1 z − z1 = = is n m l

Two points A(x1, y1, z1) and B(x2, y2, z2) lie on the same or different sides of the plane

A(x x1) + B(y y1) + C(z z1) = 0. where

ax + by + cz + d = 0, according as the expression

A l + Bm + Cn = 0.

ax1 + by1 + cz1 + d and ax2 + by2 + cz2 + d are of same or different sign. *

a b c = = m n l

*

*

Length of the perpendicular from a point (x1, y1, z1) to the line

Bisector of the angles between the planes a1x + b1y + c1z + d1 = 0

p2 = (x1 α )2 + (y1 β )2 + (z1 γ )2 [ l (x1 α )

and a2x + b2y + c2z + d2 = 0 are

a1x + b1y + c1z + d1 a12 + b12 + c12

= ±

x−α y−β z−γ = = is given by m n l + m(y1 β ) + n(z1 γ )]2

a2 x + b2 y + c2 z + d2 a22 + b22 + c22

if a1a2 + b1b2 + c1c2 is ve then origin lies in the acute angle between the planes provided d1 and d2 are of same sign. , 608-A, TALWANDI KOTA (RAJ.) Ph. 0744 - 6450883, 2405510 E

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Math Formula Sheet AIEEE

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