ICSE Mathematics Formulae Booklet 2016-2017

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748

I.C.S.E. (2016 – 2017)

Mathematics(X) By

B.Sc.(Hons);M.Sc.(Mathematics); M.A.(Lings, English & Economics); B.Ed.(Sp. Edu.); GNIIT; S.B.T.C.

    

Ex-Vice Principal (M.V.M. PUBLIC SCHOOL, ALIGARH) Ex-Co-ordinator, Ex-Co-ordinator, I.C.S.E. SCHOOL , SOUTH MUMBAI EXAMINER, ICSE MATHEMATICS H.O.D. MATHEMATICS, I.C.S.E. SCHOOL , SOUTH MUMBAI Ex-Facilitator Ex-Facilitator (IBDP mathematics)

Address: Cluster III, Poonam Estate, Estate, Mira Road Road (East)

Contact:

9224190389/9930350748 1 E-mail: [email protected]


ICSE MATHEMATICS (X)

 

There will be one paper of 2 hours duration carrying 80 marks and Internal Assessment of 20 marks. The paper will be divided into two Sections. Section I (40 marks), Section II (40 marks). Section I: It will consist of compulsory short answer questions. Section II: Candidates will be required to answer four out of seven questions. UNITS & CHAPTERS

1. COMMERCIAL ARITHMETIC 

Compound Interest (Paying back in equal installments in stallments not included)



Sales Tax and Value Added Tax



Banking (Saving Bank Accounts and Recurring Deposit Accounts)

Shares and Dividends (Brokerage and fractional shares not included) 2. ALGEBRA 



Linear Inequations



Quadratic Equations and Solving Problems



Ratio and Proportion



Remainder and Factor Theorems ( f f ( x) x) not to exceed degree 3)

Matrices 3. CO-ORDINATE GEOMETRY 



Reflection



Distance and Section Formulae

Equation of a Straight Line 4. GEOMETRY 



Symmetry



Similarity



Loci (Locus and Its Constructions)



Circles



Tangents and Intersecting Chords

∆

Constructions (tangents to circle, circumscribing & inscribing circle on & reg. hexagon) 5. MENSURATION 



Circumference andAreaof a circle (Area of sectors of circles other than semi-circle and quarter-circle not included)

Surface Area and Volume (of solids) 6. TRIGONOMETRY 



Trigonometrical Identities and Trigonometrical Tables

Heights and Distances (Cases involving more than 2 right angled 7. STATISTICS 

∆

excluded)



Graphical Representation (Histogram and Ogives)



Measures of Central Tendency (Mean, Median, Quartiles and Mode)



Probability 2 E-mail: [email protected]


GENERAL INSTRUCTIONS:  Mathematics needs Continuous Practice right from the start of the year.  Solve each and every ever y question by yourself with understanding. Do similar questions from other

question banks as well. If you understand one sum, you can do hundred similar sums. Tea chers. Ask them for more guidance, academic acad emic help etc.  Be in regular touch with your Teachers.  Remember that without consistent practice you cannot be perfect in Mathematics.  Be prepared for the worst correction. Don‟t give any chance to the examiner to cut your marks for

silly mistakes or careless work.  Don‟t use short cut ways to get answer , make habit of writing all required steps. You never know

which step has to be marked for giving marks. Marking scheme is changed every year. b e substituted  Read the question carefully, write the data first and convert into the same unit if to be for further calculation. v ery common error.  Copy the digits carefully. Check the digits again at the end. Misreading is very  Write 0 and 6 clearly. It has been observed that your 6 is taken as 0 and 0 is taken as 6 by you

only.  If the value is to be substituted in between the sum (e.g., value of r), don‟t keep in decimal form

or round off form. Keep it in fraction form onl y or else this may lead to wrong answer.  If the answer is to be written up to two decimal places, find up to three decimal places and round

off at the end. sometimes it‟s taken as minus sign which provides you  Make proper equal to sign for every step, sometimes wrong answer.  Do Small calculation also (roughly/not mentally) parallel or adjacent to the sum only. (You are

suggested to do it to gain marks, e.g., 12 × 6, 19 + 22, 90 – 90 – 65 65 etc. have been found wrong due to carefree attitude of students.)  Reading time must be used to make the right choice in section „B‟.  Don‟t spend more than 5 or 6 minutes on average for a sum. This will give you spare time for a

quick review at last. If you get stuck, stop and move on. steps. If  Never cross out an answer until you revise it at last. You may get sum marks for correct steps. you attempt a sum twice or thrice, don‟t cancel anything. anything. Let the examiner decide which one is correct. This is very useful tool if you are not confident about your answer.

3 E-mail: [email protected]


COMMERCIAL ARITHMETIC Compound Interest:  Simple interest is computed on the principal.  Compounded interest is computed on the sum of the Principal and Interest previously earned. In

other words, the interest also earns interest.  A=P+I  S.I. =

 × × 100 st

st

 S.I. for 1 year = C.I. for 1 year if compounded annually.  S.I. for 1year

≠

C.I. for 1year if compounded half yearly. (C.I. > S.I.) th

C. I. of n year + Int. on it for 1 year ; R% =  C. I. for (n + 1) year = C.I. th

 − 

( 2

1) ×100

 Amount in (n + 1) year = Amount in n year + Int. on it for 1 year; R% =

    

%, where T = 1yr

 −  ( 2

1) ×100 1

%

      C.I. = P1 +  − 1   1 +   1 +  ; when rates for successive years are different. A = P 1 +   ;when the interest is compounded half-yearly. A = P 1 +   1 +   , If the time is 2 years and the rate is compounded yearly. A = P 1 +   , V = Initial Value, V = Final Value For Growth: V= V 1 +   For Depreciation: V = V 1 −

 A=P 1+ 

1

100

100

1

2

3

100

100

100

×2

2 × 100 2

1

100

1

2 × 100

0

2

0

100

0

100

 Round off Amount (money) up to two decimal places.(61.166 = 61.17, 440.2 = 440.20)  Skip decimal values (after round off) if the amount is calculated to the nearest rupee. nd

 C.I. of 2 year and C.I. of 2 years are two different terms.

Sales Tax and Value Added Tax:  The price at which an Article is marked : List Price/Marked Price/Printed Price/Quoted Price  Sale Price = M.P. – M.P. – Discount, Discount, Discount is calculated on M.P.  Sales Tax is calculated after deducting the discount (on the discounted price).



 Sales Tax =

 Sale-price =  Sale-price =

  

100

 −  −   −  

 Sale-price =

 S.P. =

Rate Rate of Sales Sales Tax Tax ×Sal ×Sales es Pric Price e

100+Pro 100+Profit fit %

× C.P.

100

100 Loss Loss %

× C.P.

100

100 discount 100

100 d%

100+r%

100

100

 VAT paid by a person =

× M.P.

× M.P. , where d = discount, d iscount, r = rate of sales tax

pric price e Adde Added d by the the pers person on ×VAT% ×VAT% 100

= Rate % (S.P. – (S.P. – C.P.) C.P.)

sale – Tax Tax paid on the purchase  VAT = Tax recovered(charged) on the sale –

Banking: 1. SB Account:

a. Withdrawal = Debit b. Deposit = Credit c. Steps for calculation of interest: th

i. Find the minimum balance of each month between 10 day and the last day. ii. Add all the balances. This is the Equivalent Monthly Principal for 1 month. iii. Calculate the SI on the Equivalent E quivalent Monthly Principal with T =

1 12

years. th

iv. No interest is paid for the month in which the account is opened after 10 day or closed on any day (principal for that month is taken as zero). v. If the Amount Received on closing is asked, add the interest to the LAST BALANCE (actual amount available in your account) and not to the Equivalent Monthly Principal. Amount Received on closing the account = Closing Balance + S.I. 2. RD Account:

a. Qualifying sum (P) = b. I =

 

P ×n n+1 ×r 2 ×12 ×100

c. M.V. = P ×



; T =

  , where x where x = = monthly deposit, n = no. of months   years ; P = monthly monthl y deposit, n = no. of months, r = rate% n n+1 2

n n+1

2 ×12



+ I ; Maturity Value = Total deposit (monthly deposit × ) + Interest



Shares and Dividend:  The total money invested by the company is called its capital stock.  The capital stock is divided into a number of equal units. Each unit is a called a share.  Nominal Value is also called Register Value, Printed Value, and Face Value.  The FV of a share always remains the same, while its MV goes on changing.  The part of the profit of a compan y which is distributed amongst the shareholders is known as

dividend.  If the MV = NV, the share is said to be at par.  If the MV >NV, the share is said to be at premium.  If the MV
Investment MV of each each shar share e

=

Tota TotalD lDiv ivid iden end d Divi Divide dend nd per per shar share e

Rate of Dividend Dividend × NV × No. of shares ; total annual income = DNN or FDN Divi Divide dend nd Investment

× 100 %

 Rate of dividend%× NV = Return %

× MV ; DN = PM or DF = PM

 % increase in return on original investment =  % increase in return =

New Dividend Dividend

−

New New Divi Divide dend nd Original Original Investment Investment

Old Old Divi Divide dend nd

Old dividend dividend

× 100 %

× 100 %

ALGEBRA Linear Inequations: Inequations:  The signs

>,< ,

≥ ≤ and

are called signs of inequality.

becomes – ve ve and vice versa.  On transferring + ve term becomes –  If each term is multiplied or divided by + ve number, the sign of inequality inequalit y remains the same.  The sign of inequality reverses:  If each term is multiplied or divided by same negative number.  If the sign of each term on both the sides of an inequation is changed.  On taking reciprocals of both sides, in case both the sides are positive or negative.

≤ ∈

 Always, write the solution set for the inequation, e.g.,{ x : x 3, x

N }, }, solution set = {1, 2, 3}


LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748  To represent the solution on a number line:  Put arrow sign on both the ends of the line and keep extra integers beyond the range.  Use dark dots on the line for each element of N, W and Z. (WIN)  For Q, R: mark range with solid circle

(for

≥ ≤ or

), hollow circle

(for < and >.)

Intersection ( only common elements of the sets). Common range for R  “and” means Intersection ( Union (all elements of the sets without repetition).  “or” means Union (all  Don’t use decimal values e.g., 1.66 or 1.33 on number line, use fraction form 1

2 3

etc only.

1

 Show the number on the number line which you are representing, e.g., 1 . 2

 Solution must be written inside curly brackets { } only.  Show the fraction

 1

1 2

on number line or use an arrow to indicate the written value.

form only.  Remember that solution set for R and Q is always written in set-builder form

Quadratic Equations: 1. Quadratic equation is an equation with one variable, the highest power of the variable is 2. 2. Some useful results: 2

2

2

a) (a + b) = a + b + 2ab 2

2

2

b) (a - b) = a + b - 2ab 2

2

c) a – b b = (a + b) (a – (a – b) b) 2

2

d) (a + b) - (a - b) = 4ab 3

3

2

2

e) a + b =(a + b)(a - ab+ b ) 3

3

2

2

f) a - b =(a - b)(a + ab+ b ) 3

3

3

g) (a + b) = a + b + 3ab(a + b) 3

3

3

h) (a - b) = a - b - 3ab(a - b) 2

2

2

2

i) (a + b + c) = a + b + c + 2ab + 2bc + 2ca 3

3

3

2

2

2

j) a + b + c – 3abc 3abc = (a + b + c) (a + b + c – ab – ab – bc – bc – ca) ca) 7 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748 3. Steps for solving quadratic equation by b y factorization: (PSI system) a. Clear all fractions and brackets if necessary. 2

b. Bring it to the form ax + bx + c = 0 by 0 by transposing terms. c. Factorize the expression by splitting the middle term as a sum of product of a and c. 4. Discriminant (D) =

 −  2

4

a. if D > 0, then the roots are real and unequal b. if D = 0, then the roots are real and equal c. if D < 0, then the roots are not real (imaginary). 2

5. The roots of the quadratic equation ax + bx + c = 0 = 0 ; a x = x =

−    −   2

±

≠

0 can be obtained by using the formula:

4

2

6. Use powers and factorial page to find the square root of the number and round off only in the end as asked in the question. 7. For word problems, keep first the fraction with small denominator. e.g., smaller denominator) and in

− −

2500 5

2500



= 12

 −

2500

2500

= 20 , here ( x is x is

+12

( x – x – 5 5 is smaller denominator)

8. In speed sums, convert minutes to hours because t he speeds of train , plane or o r cars are in km/hr.  There is mark for correct formula with correct substitution. Wrong formula leads to zero.  Use log table to find square root. Don‟t find square root by division method, it‟s risky and leads to

wrong answer.  There is difference between answers 2 decimal pla ces and 2 significant figures.

(2.21 is up to 2 d.p. and 2.2 is up to 2 s.f.)

Ratio and Proportion: 

A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Since ratio is a number, so it has no units. 8 E-mail: [email protected]

LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748 

To find the ratio between two quantities, qu antities, change them to the same units.



To compare two ratios, convert them into like fractions.



In the ratio, a : b, a is a is called antecedent and b is called consequent.



For equal ratios , = = =

 

      ; each ratio =             + +

+ +

 =  = k (say) , then a = bk and c = dk used for proving sums    =  =  =  (), then d = d, c = dk, b = dk , a = dk used for proving sums    2

3



Compound ratio of a : b : b and and c : d is is (a (a × c) c) : (b (b × d )



Duplicate ratio of a : b is a is a : b



Triplicate ratio of a : b is a : b



Sub-duplicate ratio of a : b : b is



Sub-triplicate ratio of a : b is



Reciprocal ratio of a : b : b is b : a or



Proportion- An equality of two ratios is called a proportion. Written as: a : b :: b :: c : d or or



Product of extreme terms = product of middle terms, if a, b, c, d are in proportion then ad = bc



Continued Proportion- a : b :: b : c or a : b = b : c ; mean proportion (b (b) =



Invertendo:

If a : b = c : d , then b : a = d : : c



Alternendo :

If a : b = c : d , then a : c = c = b : d



Componendo :

If a : b = c : d , then a + b : b = b = c + d : d



Dividendo :

If a : b = c : d , then a - b : b = b = c - d : d



Componendo and Dividendo :

2

2

3

3

        ∶ :

3

:

3

1

1

=  

     2

=

If a : b = c : d , then a + b : b : a – b= b=c + d : : c – d

     ∶ ∶

 Write final answer in ratio form 25 : 16 .  Ratio must be written in simplified form.

25 16

15 25 = 3 5

led in denominator ( + or -)  In C & D whatever is cancelled in numerator , doesn’t get cancel led 9 E-mail: [email protected]


Remainder and Factor Theorem:

∈

1. If f If f ( x) x) is a polynomial, which is divisible by b y ( x – x – a), a), a R, R, then the remainder is f is f (a (a). 2. If the remainder on dividing a polynomial f polynomial f ( x) x) by ( x – x – a), a), f (a (a) = 0, then ( x x - a) a) is a factor of f of f ( x). x).

−  ≠  ≠ − 

3. When f When f ( x) x) is divided by (ax ( ax + b), b), then remainder is f is f 4. When f When f ( x)is x)is divided by (ax (ax - b), b), then remainder is f is f

,a

,a

 If (2x + 3) is a factor of f (x), then never skip to write f

0

0

3

2

=0

 If remainder is given, don’t dare to replace it by zero.

(x – 3). 3). [(x + 1), (2x+3) ,(x  Write all factors in product form, e.g., (x + 1 ) (2x+3) (x – ,(x – 3) is wrong]

Matrices: 

A rectangular arrangement of numbers, in the form o f horizontal (rows) and vertical lines (columns) is called a matrix. Each number of a matrix is called its element. The elements of a matrix are enclosed in brackets [ ].



The order of a matrix = No. of rows × No. of columns



Row matrix: Only 1 row. E.g.,



Column matrix: Only 1 column. E.g.,

   



Square matrix: No. of rows = No. of columns. E.g.,



Rectangular matrix: No. of rows



Zero matrix: All elements are zero. E.g.,

≠

       

No. of columns. E.g.,



0 0

0 0

0 0


LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748 

Diagonal matrix: A square matrix with all the elements zero except the elements on the leading

    0

diagonal. E.g., 0 0

0 0

0



Unit matrix (I): A diagonal matrix with all the elements on the leading diagonal = 1; I =



1 Transpose of a matrix: If A = 2



Addition or subtraction of matrices is possible iff they are of the same order.



Addition of two matrices:



Multiplication of matrix by a real number:i number:i×



Multiplication of 2 matrices:



Multiplication process:

2 3

1 5 t then A = 2 6 5



  1 0

0 1

2 3 6

 

                             +

x × y ×

×

+ +

=

=

b× b× a ,

=

+ +

× ×

× ×

y = b ,order of the product matrix = ( x × a) ,

× ×

+ +

×r ×r

× + ×s , run& run& fall ×q+d×s

 Don‟t use curly brackets for writing matrices. 2

 Matrix M is not the square of each element in the matrix M.  Two matrices can‟t de divided by one another.

    ÷

is wrong.

 Working for multiplication is must.  Be very careful in doing multiplication, even one wrong element will lead to zero.

COORDINATE GEOMETRY

Reflection:

  − ( x, x, -y) -y)  M   − (- x, x, y)y)



M x



y



   (- x, x, -y)-y) 



Mo



X- axis means y means y = 0



Y- axis means x means x = = 0



Any point that remains unaltered under a given transformation is called an invariant point.



M x=a( x, x, y) y)



M x=b

    (2a (2a - x - x,, y ) y )  ( x, x, y) y)      ( x, x, 2b - y)y) =

=

 Write co-ordinates of the reflected point on the graph paper and name it specifically.

co-ordinates in square brackets.  Don‟t write co-ordinates  Learn to recognize the geometrical shapes: kite, arrow, arrow head, hexagon, octagon, trapezium,

rhombus, etc. Also learn to find the areas of triangle and quadrilaterals.  There is difference between arrow and arrow head.

More Coordinate Geometry:

  −    −   2

+



Distance formula: Distance between 2given points ( x x1 , y1) and ( x x2 , y2) =



Distance between the origin (0, 0) and any point ( x, x, y) y) =



Three points A, B and C are collinear if AB + BC = AC or AC + CB = AB or CA + AB = CB



To show an equilateral triangle or right angled triangle, find all three sides.



To show the quadrilateral as a parallelogram or rhombu s, find all four sides.



To show the quadrilateral as a rectangle or squ are, find all four sides and both the diagonals.



Section formula: Coordinates of a point P( x, x, y) = y) =





 

   2

2

1

2

1

2

2

+

  ,      ; ratio = m : m      ,  Midpoint formula: Coordinates of the midpoint M( x, x, y) y) of a line segment =     ,   The co-ordinates of the centroid of a triangle G( x, x, y) y) =  1 2+ 1+

1 2+

2 1

1+

2

2 1

1

2

1+

2

1+

2+

3

3

1+ 2+

2

1+

2

2

2

3

3

 Substitute the correct values with sign in correct formula.


LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748 wh ile writing the final answer. Mistake of sign is very  Check the sign of coordinates carefully while common error.

Equation of a Line: 

Every straight line can be represented by a linear equation.



Any point, which satisfies the equation of a line, lies on that line.



Inclination of a line is the angle which the part of the line makes with x with x-axis. -axis.



Inclination is positive in anti-clockwise direction and negative in clockwise direction.



Slope or gradient of any inclined plane p lane is ratio of vertical rise and horizontal d istance.



Slope of a line (m) =



Inclination of x of x-axis -axis and every line parallel to it is 0°.



Inclination of y of y-axis -axis and every line parallel to it is 90°.



The slope of a vertical line segment is not n ot defined.



Slope of a line which passes through any two points P( x x1 , y1) and Q( x x2, y2) =



  

  = tan  



Slopes of two parallel lines are equal or m1 = m2.



Product of the slopes of two perpendicular lines= lines= - 1 or m1 × m2 = -1.



The equation of x-axis is y = 0 and 0 and the equation of y-axis is y-axis is x x = 0,



Equation of a line:

 −−   2

1

2

1

.

y = mx + c

:

(Slope-intercept form : m = slope, c = y-intercept) y-intercept)

o

(y – (y – y y1 ) = m( m( x – x – x x1)

:

(Slope-point form : ( x x1, y1) = co-ordinates of the point)

o

−  =  −  −   − 

:

[Two points form ( x x1, y1) ,( x x2, y2)]

o

1

2

1

1

2

1



GEOMETRY

Symmetry: 

A figure is said to have line symmetry if on folding the figure about this line, the two parts of the figure exactly coincide. Geometrical Name

Line(s) of Symmetry

Line segment

2 lines of symmetry – symmetry – line line itself and perpendicular bisector of it.

Angle with equal arms

1 line of symmetry – symmetry – the the angle bisector

A pair of equal parallel

2 lines of symmetry – symmetry – line line midway and perpendicular bisector of them.

line segments A scalene triangle

Nil

An isosceles triangle

1 – the the bisector of the vertical angle which is bisector of the base.

An equilateral triangle

3 – the the angle bisectors which are also side

An isosceles trapezium

1 – the the line joining midpoints of the two parallel sides.

A parallelogram

Nil

A Rhombus

2 – the the diagonals

A rectangle

2 - the lines joining midpoints of the opposite sides.

A square

4 – the the diagonals , lines joining midpoints of o f the opposite sides.

A kite

1 – the the diagonal that bisects the pair of angles contained by equal sides.

A circle

Infinite – Infinite – all all the diameters

A semicircle

1 – the the bisector of the diameter

A regular pentagon

5 - the angle bisectors or the

A regular hexagon

6 - the angle bisectors, the bisectors of the sides.

⊥

⊥

⊥

⊥

bisectors.

⊥

bisectors of the sides.



Similarity: 

Criteria for similarity – similarity – 1. 1. AA or AAA



A drawn from vertex of a rt- d divides the into 2 similar , also to original triangle.



BPT – BPT – A A line drawn || to an y side of a divides other two sides proportionally.



The areas of 2 similar



Median divides a triangle into 2



If



Scale factor = k, k =

⊥

∠∆

∆

∆

∆

2. SAS

3. SSS

∆

∆∆

are proportional to the square of their corresponding sides.

∆

of equal area.

have common vertex & are between same ||, ratio of their areas = ratio of bases.

   ; k =    ; k =    .          2

3

Loci: 

The locus is the set of all points which satisfy the given geometrical condition.



Locus of a point equidistant from 2 fixed points A and Bis bisector of line segment joining them.



Locus of a point equidistant from 2 intersecting lines AB and BCis angle bisector between the lines.



Locus of a pointAat a constant distance from a fixed point p oint is circle.



Locus of a point equidistant from a given lineABis a pair of lines parallel to the given line and at the

⊥

given distance from it. 

For equilateral triangle, centroid = incentre = circumcentre = o rthocentre

Circle: 



⊥

A line drawn from centre of a circle to bisect the chord is to the chord. A perpendicular line drawn to a chord from the centre of the circle bisects the chord.

⊥



The bisector of a chord passes through the centre of the circle.



One and only one circle can be drawn passing through 3 non-collinear points.



Equal chords are equidistant from the centre. 15 E-mail: [email protected]


Chords which are equidistant from the centre are equ al in length.



If the parallel chords are drawn in a circle, then the line through the midpoints of the chords passes through the centre.



Greater the size of chord, lesser is its distance from the centre.



Angle at the centre = 2 × angle on the circumference.



Angles in the same segment are equal.



Angle in a semicircle is a right angle.



The opposite angles of a cyclic quadrilateral are supplementary.



If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.



Angle in the major segment is acute and in the minor segment is obtuse. ob tuse.



Exterior angle of a cyclic quadrilateral = Interior opposite angle.



In equal or same circle. If two arcs subtend equal angle at the centre, then they are equal.



In equal circle, if two arcs are equal, then they subtend equal angle at the centre.



In equal circle, if two chords are equal, th ey cut off equal arcs.



In equal circle, if two arcs are equal, the chords of the arcs are also equal.



The tangent at any point of a circle & the radius through this point are to each other.



If two tangents are drawn to a circle from an exterior point,

⊥

o

The tangents are equal,

o

They subtend equal angle at the centre of the circle,

o

They are equally inclined to the line joining the point and the centre of the circle.



If two chords of a circle intersect internally/externally, the product of their segments is equal.



Angles in the alternate segment are equal.



Tangent = product of the lengths of the segments of the chord.



Incentre : Point of intersection of the angle bisectors.



Cicumcentre : Point of intersection of the bisectors of the sides.

2

⊥


LIST OF FORMULAE, ICSE MATH (X), By Rakesh Kushwaha Sir (M.Sc.; B.Ed.): 09224190389/0993035 0748  Make habit of copying diagram to find the values of different angles.  Mark 90° for angle in semi-circle; equal angles in same segment; central angle is twice the angle

on circumference, opposite angles of cyclic; exterior angle of cyclic, etc. in the diagram.  Reasons must be written wherever possible. (No reason leads to no mark.)

Constructions: 

Standard Angles should be made by using scale and compass only not by using protractor.



Two angle bisectors and a Perpendicular from centre to any side of regular polygon pol ygon are required to draw incircle.



Any two side perpendicular bisectors are required to draw circumcircle.

MENSURATION

Circumference and Area of a Circle: Geometric Name

Figure

Area

   

Circle

r

1

Semi-circle

2 1

Quarter-circle

4

Circular ring 





Perimeter

  

2 r 2

r

2

r

(R – (R – r r )

r + + 2r 2r

1 2

r + + 2r 2r

---

Distance travelled by a wheel in one revolution = Its circumference No. of Revolutions =

        

1

Area of a triangle = × b × h 2



  −  −  − 



Area of scalene triangle =



Area of equilateral triangle =

, s =

 + + 2

 a 3

2

4

Surface Area and Volume: Solid Cuboid

Shape

Curved Surface Area

Total Surface Area

Volume

Area of 4 walls

2(lb + bh + hl)

l× b×h

6 side

side

2(l + b)× h Cube

Area of 4 walls 2

4 side



2 r(h + r)

2 h(R+r)



External + Internal

Cylinder

2 rh

Cylinder

Cone

Sphere

Hemi-



rl



rh

2 rh + 2 Rh +

   



(R - r )h (sol (soliid

2 2 2 (R - r )

enclosed between)

1

r(l + r)

--





4

2

2 r





3

2

4

r

3

2

2

3 r

3



r h



r



r



(R - r ) (solid

2

3

3

sphere Spherical Shell

--



4 (

2

2

+ r )

External + Internal

4 3

3

3

enclosed between)





  

2 2 2 (R + r )

Hemi-

2 2 2 (R + r ) +

Spherical

2 3

2

2



3

3

(R - r )

(R – r r )

Shell

2

2

= (3R + r )

     2

+

2

+

2



Diagonalof a cuboid =



Diagonal of a cube =a =a 3



Slant height of a right circular cone, l = =

   2

+

2

leave the answer in decimal form if asked  Marks are given for correct answer only, so don’t leave the to round off to integer.  Make habit of writing all possible ways, e.g.,

3 2

1

= 1 = 1.5 , if money then 1.50, let the 2

examiner choose the correct one. There Th ere is no harm in writing decimal as well as fraction form. There are always exceptions, somewhere fraction is demanded, elsewhere decimal.

Trigonometry: 



 OR  ;         sin

| cos

|

| sec

|

SOH CAH TOA or OSH ACH OTA



Trigonometric ratios of standard angles

Adj side



0°

°



sin

0 4

cos

30°

  1

=0

4

1

tan

=

3

0

45° 1 2

 2 4

=

1

 2

1

2



1

1



60°

2

3



Hypotenuse

Opp side



sin =

1 cosec



,



cosec =

1 sin



E-mail: [email protected]

 3 4

=

1

90°

 3

2

 4 4

=1

0

2

 3

n.d.

19




















             − − − ≠ 1

cos =

sec

1

tan =

cot sin

tan =

cos

  

,

sec =

,

cot =

,

cot =

1 cos 1 tan cos sin

   

Sin2 + cos2 = 1 ( mutual understanding)





cosec2 - cot 2 = 1 or 1 + cot 2 = cosec2 ( cosec is big brother) 2

2

- tan

sin(90°

2

= 1 or 1 + tan



) = cos ,

cosec(90° tan(90°

2

= sec

cos(90°



) = sec ,

sec(90°



cot(90°

) = cot ,

sin 3A

 

3 sinA

≠

3

∠

( sec is big brother)

−  −  −  ) = sin

) = cosec

) = tan

1

3

sin A , if A = 30°, then sin 3A = sin 90° = 1, 3 sinA= 3 × , sin A = 2



1 3 2

1

= . 8

 In proving sums, don‟t assume LHS = RHS in the beginning, Solve Separately. Transfer T ransfer of

digits from one side to other is strictly prohibited.  Don‟t do direct conversion of trigonometric ratios of complementary angles. Show the steps.  Write reasons for the conversion without fail. Don‟t forget to write the iden tity whichever is used.

Height and Distance:

angle





> angle



Angle of depression

 Angle of elevation



Bigger angle is always inside and smaller is outside.



Height is vertical length and distance is horizontal length.





=

 =  =    



tan



Remember the values of tan 30°, 45° and 60°

  1

45° ,

       1

3

30°,

3

. It‟s very easy to remember. Just remember

1

 . 3

60°

 Diagram is must in Heights and Distances. Mark angles co rrectly. Outside angle is always  

 

smaller. Angle of elevation decreases when distan ce increases. Make habit of writing the standard values of tan as per given in the question. E.g., you get 1 mark just for writing tan 45° = 1. You may easily get 3 marks in heights and distances sums: 1 mark for correct diagram, 1mark for the value of tan 30°, 45° or 60°and 1 mark for writing correct formula with correct substitution. There is only one mark for big b ig calculation. So don‟t omit this important topic. Use log table to write the values of angles (22°, 32°, 48 etc) other than standard angles in finding heights and distances. Keep unknown side as numerator while using tan , to avoid division by big decimal number. Multiplication is always easier. Use complementary angle if required.





STATISTICS Statistics:



∑ 



Arithmetic mean of non-tabulated data: = =



Arithmetic mean of tabulated data(Direct Method): = =



Arithmetic mean by Short-cut Method: = =













∑ ; x = mid value v alue (C.I.) ∑

∑ + A ; A = assumed mean , d = x = x – – A A ∑ ∑ ×   + A ; i= class width , t =  Arithmetic mean by Step-deviation Method:  = =  ∑    term For raw data, if n is odd, Median = 



+1 2

    For raw data, if n is even, Median =   From ogive, Median =   term. 2

term term +

   2

+1

term

2

2

From ogive Lower quartile, Q1 =

 4

term 21

E-mail: [email protected]


  3



From ogive Upper quartile, Q3 =



Inter Quartile Range, IQR = Q3 – Q Q1



Semi Inter Quartile Range =



Mode is the variate which has the maximum frequency.



The class with maximum frequency is called the moda l class. (e.g., 20 – 20 – 30) 30)



To estimate mode from histogram: draw two straight lines from the corners of the rectangles on either

4

term

 – 3

1

2

sides of the highest rectangle to the opposite co rners of the highest rectangle. Through the point of intersection of the two straight lines, draw a vertical line to meet the x-axis at the point M (say). The variate at the point M is the required mode. 

To find median of grouped data, draw ogive (cumulative frequency curve).  For finding mean of raw data, be very careful in counting number of data (n).  Use kink (if required), show scale and label the axes to get marks. Don‟t dare to dare to change the scale

if already given in the question.  For finding median, Q1, Q3 or number of desired range (below or above) a bove) using ogive,

perpendiculars are must. No mark is allotted without perpendiculars.  Write the answer in decimal form not in fraction.  In Ogive, it should be the smooth curve without halt.

Probability: 

Probability is a measure of uncertainty.



An Experiment is an action which results in some (well-defined) outcomes.



Sample space is the collection of all possible outcomes of an experiment. n(S)



An Event is a subset of the sample space associated with a random experiment. n(E)



An Event occurs when the outcome of an experiment satisfies the condition mentioned in the event. 22 E-mail: [email protected]


The outcomes which ensure the occurrence of an event are called favorable outcomes to that event.



The probability of an event E, written as P(E), is defined as P (E) =



P(E) =



The value of probability is always between 0 and 1.



The probability of sure (certain) event is 1.



The probability of an impossible event is 0.



An elementary event is an event which has one (favorable) outcome from the sample space.



A Compound event is an event which has more than one outcome from the sample space.



If E is an event, then the event „not E‟ is complementary event of E and denoted by E.



0



P(E) + P(E) = 1



In a pack (deck) of playing pla ying cards, there are 52 cards which are divided into 4 suits of 13 cards each – each –

 

( ) ( )



≤  ≥  ( )

spades (

         

1

), hearts (

), diamonds (

) and clubs (

). Spades and clubs are black in colour,

while hearts and diamonds are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. Kings, queens and jacks are called face (picture/court) cards. The cards bearing number 10, 9, 8, 7, 6, 5, 4, 3, 2 are called numbered cards. Thus a pack of playing cards has 4 aces, 12 face cards and 36 numbered cards. The aces together with face cards (= 16).are called cards of honour. 

When a coin is tossed, it may show head h ead (H) up or tail (T) up. Thus Th us the outcomes are: {H, T}.



When two coins are tossed simultaneously, then the o utcomes are: {HH, HT, TH, TT}. [n(S) = 2 ]



When a die is thrown once the outcomes are: {1, 2, 3, 4, 5, 6}.



When two dice are thrown simultaneously, then the outcomes are: {(1, 1), (1, 2)…….(6, 6)}. 6)}.

n

 Write the answer in simplified form.



9

30

=

3 10

n

[n(S) = 6 ]



 Don‟t forget to write sample space. At times, there is special 1 mark for it, which is ignored by

most of the students.


ICSE Mathematics Formulae Booklet 2016-2017

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