MATHS - IIB QUESTION BANK - Chapter Wise Important Questions for IPE

SENIOR–MATHS-IIB Chapter wise Important Questions for IPE by MN Rao

Circles VERY SHORT ANSWER TYPE QUESTIONS: 1.

Find the equations of circles with center C and radius r where i) C = ( 2, −3 ) , r = 4

5 2 Find the equation of the circle passing through (3,4) and having the center at (-3,4) Find the center and radius of each of the circle whose equations are given below ii) 2 x 2 + 2 y 2 − 3x + 2 y − 1 = 0 i) 3x 2 + 3 y 2 − 5 x − 6 y + 4 = 0 Find the equations of the circle for which the points given below are the end points of a diameter i) (1,2),(8,6) ii) (7,-3),(3,5) Obtain the parametric equation of each of the following circles 2 2 i) x 2 + y 2 = 4 ii) ( x − 3) + ( y − 4) = 82 iii) x 2 + y 2 − 6 x + 4 y − 12 = 0

iii) C = ( cos α ,sin α ) , r = 1 2. 3.

4. 5.

6. 7. 8. 9. 10. 11.

12. 13.

ii) C = ( −a , −b ) , r = a 2 − b 2 ( a > b ) C = (1, 7 ) , r =

Show that A(3,-1) lies on the circle x 2 + y 2 − 2 x + 4 y = 0 . Also find the other end of the diameter through A. Locate the position of the point P with respect to the circle S=0 when i) P ( 3, 4 ) and S ≡ x 2 + y 2 − 4 x − 6 y − 12 = 0 ii) P ( 4, 2 ) and S ≡ 2 x 2 + 2 y 2 − 5 x − 4 y − 3 = 0 Find the power of the point P with respect to the circle S=0 when i) P = ( −1,1) and S ≡ x 2 + y 2 − 6 x + 4 y − 12 ii) P = ( 2, 4 ) and S ≡ x 2 + y 2 − 4 x − 6 y − 12 Find the length of tangent from P to the circle S= 0 when i) P = ( 2,5 ) and S ≡ x 2 + y 2 − 5 x + 4 y − 5 If the length of the tangent from (5,4) to the circle x 2 + y 2 + 2ky = 0 then find k If a point P is moving such that the length of tangents drawn from P to the circles x 2 + y 2 − 4 x − 6 y − 12 = 0 and x 2 + y 2 + 6 y + 18 y + 26 = 0 are in the ratio 2:3 then find the equation of the locus of P. Find the equation of the tangent at P of the circles S=0 where P and S are given by i) P = ( 3, 4 ) S ≡ x 2 + y 2 − 4 x − 6 y + 11 Find the equation of the normal at P of the circle S=0 where P and S are given by i) P = ( 3, −4 ) S ≡ x 2 + y 2 + x + y − 24 ii) P = (1,3) , S ≡ 3 ( x 2 + y 2 ) − 19 x − 29 y + 76

14. 15. 16. 17 18 19.

Find the length of the chord intercepted by the circle x 2 + y 2 − x + 3 y − 22 = 0 on the line y=x-3 Find the equation of the circle with center (2,3) and touching the line 3x-4y+1=0 Find the equation of tangent and normal at (3,2) is the circle x 2 + y 2 − x − 3 y − 4 = 0 Find the chord of contact of (1,1) to the circle x 2 + y 2 = 9 Find the polar of (1,-2) with respect to x 2 + y 2 − 10 x − 10 y + 25 = 0 Find the pole of ax + by + c = 0 ( c ≠ 0 ) with respect to x 2 + y 2 = r 2

20. 21. 22. 23.

Find the pole of x − 2 y + 22 = 0 with respect to x 2 + y 2 − 5 x + 8 y + 6 = 0 Show that the points (4,2) and (3,-5) are conjugate points with respect to the circle 2 2 x + y − 3x − 5 y + 1 = 0 Find the angle between the tangents drawn from (3,2) to the circle x 2 + y 2 − 6 x + 4 y − 2 = 0 Fin the angle between the pair of tangent drawn from (0,0) to the circle x 2 + y 2 − 14 x + 2 y + 25 = 0

24.

Find the locus of P where tangents drawn from P to x 2 + y 2 = a 2 are perpendicular to each other

Vanithatv – GUIDE education and career live show daily at 8.30pm – MATHS 2B

1

25.

26.

27. 28. 29.

Discuss the relative position of the following pair of circle i) x 2 + y 2 − 4 x − 6 y − 12 = 0 , x 2 + y 2 + 6 x + 18 y + 26 = 0 ii) x 2 + y 2 − 2 x + 4 y − 4 = 0 , x 2 + y 2 + 4 x − 6 y − 3 = 0 Find the number of possible common tangents that exists for the following pairs of circles i) x 2 + y 2 + 6 x + 6 y + 14 = 0 , x 2 + y 2 − 2 x − 4 y − 4 = 0 ii) x 2 + y 2 = 4 , x 2 + y 2 − 6 x − 8 y + 16 = 0 Find the internal center of similitude for the circles i) x 2 + y 2 + 6 x − 2 y + 1 = 0 and x 2 + y 2 − 2 x − 6 y + 9 = 0 Find the external center of similitude for the circles i) x 2 + y 2 − 2 x − 6 y + 9 = 0 and x 2 + y 2 = 4 Find the direct common tangents of the circles x 2 + y 2 + 22 x − 4 y − 100 = 0 and x 2 + y 2 − 22 x + 4 y + 100 = 0 .

Find the transverse common tangents of the circle x 2 + y 2 − 4 x − 10 y + 28 = 0 and x2 + y2 + 4x − 6 y + 4 = 0 31. Find the center and radius of the sphere x 2 + y 2 + z 2 − 2 x + 4 y − 6 z − 2 = 0 32. If (2,3,5) is one end of a diameter of the sphere x 2 + y 2 + z 2 − 6 x − 12 y − 2 z + 20 = 0 , then find the coordinates of the other end of the diameter 33. Find the center and radius of the sphere passing through the points (0,0,0) (0,4,0), (-2,0,0) and (0,0,-4) SHORT ANSWER TYPE QUESTIONS: 30.

1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12.

Find the values of a,b if ax 2 + bxy + 3 y 2 − 5x + 2 y − 3 = 0 represents a circle . Also find the radius and center of the circle. If x 2 + y 2 + 2 gx + 2 fy = 0 represents a circle with center (-4,-3) then find g,f and the radius of the circle If the abscissae of points A,B are the roots of the equation x 2 + 2ax − b2 = 0 ordinates of A,B are roots of y 2 + 2 py − q 2 = 0 then find the equation of a circle for which AB is a diameter Find the equation of a circle which passes through (4,1) (6,5) and having the center on 4x+3y-24=0 Find the equation of a circle which is concentric with x 2 + y 2 − 6 x − 4 y − 12 = 0 and passing through (-2,14) Find the equation of circle passing through each of the following three points i) (3,4) ,(3,2), (1,4) ii) (2,1),(5,5),(-6,7), Find the equation of the circle passing through (0,0) and making intercept 6 units on x- axis and intercept 4 units on Y- axis. Show that the following four points in each of the following are concyclic and find the equation of the circle on which they lie. i) (9,1),(7,9)(-2,12),(6,10 If (2,0),(0,1),(4,5) and (0,c) are concyclic then find c. Show that the locus of the point of intersection of the lines x cos α + y sin α = a, x sin α − y cos α = b ( α is a parameter) is a circle . x y If x 2 + y 2 = c 2 and + = 1 intersect at A and B then find AB . Hence deduce the condition that the a b line touches the circle If y = mx + c and x 2 + y 2 = a 2 i) intersect at A and B ii) AB = 2λ then show that c 2 = (1 + m2 )( a 2 − λ 2 )

13.

Find the equations of the tangents to the circle x 2 + y 2 − 4 x + 6 y − 12 = 0 which are parallel to x + y −8 = 0.

14.

Show that x + y + 1 = 0 touches the circle x 2 + y 2 − 3x + 7 y + 14 = 0 and find its point of contact


2

15. 16.

Find the chord of contact of (1,1) to the circle x 2 + y 2 = 9 Find the value of k if x + y − 5 = 0 and 2 x + ky − 8 = 0 are conjugate with respect to the circle x2 + y2 − 2 x − 2 y −1 = 0

17.

If the polar of the points on the circle x 2 + y 2 = a 2 with respect to the circle x 2 + b 2 = b 2 touches the circle x 2 + y 2 = c 2 then prove that a,b,c are in Geometrical progression

18.

Tangents are drawn to the circle x 2 + y 2 = 16 from the point p (3,5).Find the area of the triangle formed by these tangents and the chord of contact of P. Show that x 2 + y 2 − 6 x − 9 y + 13 = 0, x 2 + y 2 − 2 x − 16 y = 0 touch each other .Find the point of contact and the equation of common tangent at their point of contact Find the pair of tangent drawn from (1,3) to the circle x 2 + y 2 − 2 x + 4 y − 11 = 0 and also find the angle between them

19. 20.

LONG ANSWER TYPE QUESTIONS:

1.

2. 3. 4.

Finds the equation of the circum – circle of the triangle formed by the straight lines given in each of the following i) 2 x + y = 4, x + y = 6, x + 2 y = 5 ii) 5x − 3 y + 4 = 0, 2 x + 3 y − 5 = 0, x + y = 0 Find all common tangents of the following pairs of circles i) x 2 + y 2 + 4 x + 2 y − 4 = 0 and x 2 + y 2 − 4 x − 2 y + 4 = 0 Find the equation of the sphere containing the points (4,-1,2) (0,-2,3),(1,-5,-1) and (2,0,1) A sphere of radius r ( r > 0 ) passes through the origin and meets the coordinate axes at A,B,C. Prove that the centroid of ∆ ABC lies on the sphere of

(x

2

+ y 2 + z 2 ) = 4r 2

SYSTEM OF CIRCLES VERY SHORT ANSWER TYPE QUESTIONS 1. Find k if the following pair of circles are orthogonal i) x 2 + y 2 + 2by − k = 0, x 2 + y 2 + 2ax + 8 = 0 2. Find the angle between the circles given by the equations i) x 2 + y 2 − 12 x − 6 y + 41 = 0, x 2 + y 2 + 4 x + 6 y − 59 = 0 Show that the circles given by the following interrect each other orthogonally 3. i) x 2 + y 2 − 2 x − 2 y − 7 = 0, 3x 2 + 3 y 2 − 8 x + 29 y = 0 Find the equation of the radical axis of the following circles 4. i) x 2 + y 2 − 3x − 4 y + 5 = 0,3 ( x 2 + y 2 ) − 7 x + 8 y + 11 = 0 ii) x 2 + y 2 − 2 x − 4 y − 1 = 0, x 2 + y 2 − 4 x − 6 y + 5 = 0 5. Show that the circles x 2 + y 2 − 8x − 2 y + 8 = 0 and x 2 + y 2 − 2 x + 6 y + 6 = 0 touch each other and find the point of contact . 6. Find the equation of the circle coaxial with the circles x 2 + y 2 − 2 x + 2 y + 1 = 0 and x 2 + y 2 + 8 x − 6 y = 0 which passes through the point (-1,-2) 7. If (3,1) is one of the limiting points of a coaxial system of which the radical axis is x − y + 2 = 0 . Find the other limiting point SHORT ANSWER TYPE QUESTIONS 1. Find the equation of the circle which passes through the origin and intersects the circles below, orthogonally i) x 2 + y 2 − 4 x + 6 y + 10 = 0, x 2 + y 2 + 12 y + 6 = 0 2. Find the equation of the circle which passes through the points (2,0), (0,2) and orthogonal to the circle 2 x2 + 2 y 2 + 5x − 6 y + 4 = 0


3

4. 5. 6.

Find the radical center of the following circles. i) x 2 + y 2 − 4 x − 6 y + 5 = 0 , x 2 + y 2 − 2 x − 4 y − 1 = 0, x 2 + y 2 − 6 x − 2 y = 0 Find the equation and length of the common chord of the following circles i) x 2 + y 2 − 5 x − 6 y + 4 = 0, x 2 + y 2 − 2 x − 2 = 0 Find the radical axis of the coaxial system having the limiting points (2,1) and (-5,-6)

LONG ANSWER TYPE QUESTIONS: 1 1 1 + 2 = 2 a b c

1.

Show that the circles x 2 + y 2 + 2ax + c = 0 and x 2 + y 2 + 2by + c = 0 touch each other if

2.

Find the equation of the circle which cuts the following circles orthogonally i) s x 2 + y 2 + 4 x + 2 y + 1 = 0, 2( x 2 + y 2 ) + 8x + 6 y − 3 = 0, x 2 + y 2 + 6 x − 2 y − 3 = 0 Find the coordinates of the limiting points of the coaxial system to which the circles x 2 + y 2 + 10 x − 4 y − 1 = 0 and x 2 + y 2 + 5 x + y + 4 = 0 are two members If the straight line 2 x + 3 y = 1 intersects the circle x 2 + y 2 = 4 at the points A and B then find the equation of the circle having AB as diameter If x + y = 3 is the equation of the chord AB of circle x 2 + y 2 − 2 x + 4 y − 8 = 0 ,find the equation of the

3.

4 5 6.

7. 8. 9.

circle having AB as diameter If the origin is a limiting point of a system of coaxial circles of which x 2 + y 2 + 4 x + 4 y + 4 = 0 is a member then find the other limiting point If (1,4) and (2,-1) are the limiting points of a coaxial system of circles then find the radical axis . The equation of orthogonal system to the coaxial system of which x 2 + y 2 + 2 x + 4 y + 7 = 0 is a member and (-2,-1) is limiting point . Find the coaxial system which is orthogonal to the coaxial system x 2 + y 2 + 2 x − 3 y − 1 + λ ( x − 4 y + 1) =0, λ is a parameter

The origin is a limiting point of a system of coaxial circles of which x 2 + y 2 + 2 x + 2 y − 4 = 0 is a member . find the equation of the coaxial system orthogonal to the given coaxial system. 11. Find the equation of a coaxial system in the simple form 12. Th: The equation of coaxial system of circles orthogonal to the coaxial system If (3,5) is limiting point of a coaxial system for which x 2 + y 2 + 2 x + 2 y − 24 = 0 is a member then find 13. the other limiting point of the coaxial system. 14. Show that the circle x 2 + y 2 + 4 x + 4 y − 1 = 0 bisects the circumference of the circle x 2 + y 2 + 2 x − 3 = 0 PARABOLA VERY SHORT ANSWER TYPE QUESTIONS: 10.

6.

Find the vertex and focus of x 2 − 6 x − 6 y + 6 = 0 Find the equation of axis and directrix of the parabola y 2 + 6 y − 2 x + 5 = 0 Find the equation of the parabola whose focus is S (3,5) and vertex is A (1,3) Find the equations of the tangent and normal to the parabola y 2 = 6 x at the positive end of the latus rectum.  3 Find the equation of the tangent and normal to the parabola x 2 − 4 x − 8 y + 12 = 0 at  4,   2 2 Show that the line 2 x − y + 2 = 0 is a tangent to the parabola y = 16 x . Find the point of contact also

7. 8. 9.

Find the pole of the line 2 x + 3 y + 4 = 0 with respect to y 2 = 8 x Find the value of k if the lines 2 x + 3 y + 4 = 0 and x + y + k = 0 are conjugate w.r.t y 2 = 8 x Find the value of k, if the line 2 x + 3 y + 12 = 0 and x − y + 4k = 0 are conjugate with respect to the

1. 2. 3. 4. 5.

parabola y 2 = 8 x 10. Find the equation of the tangent and normal to the parabola y 2 − x + y + 1 = 0 at (3,1) Vanithatv – GUIDE education and career live show daily at 8.30pm – MATHS 2B

4

SHORT ANSWER TYPE QUESTIONS:

1. 2. 3. 4.

If L and L' are the ends of the latus rectum of the parabola x 2 = 6 y, find the equations of OL and OL' where ‘O’ is the origin . Also find the angle between them. Find the equation of the parabola whose axis is parallel to x – axis and which passes through the points (-2,1),(1,2) and (-1,3). If lx + my + n = 0 is a normal to the parabola y 2 = 4ax , then show that al 3 + 2alm2 + nm2 = 0 Show that the equation of common tangents to the circle x 2 + y 2 = 2a 2 and the parabola y 2 = 8ax are y = ± ( x + 2a )

5.

Prove that the polars of all points on the directrix of a parabola y 2 = 4ax ( a > 0 ) are concurrent at

focus 6. If the polar of P with respect to the parabola y 2 = 4ax touches the circle x 2 + y 2 = 4a 2 then , show that P lies on the curve x 2 − y 2 = 4a 2 LONG ANSWER TYPE QUESTIONS: 1.

2.

3. 4. 5

Prove that the area of the triangle inscribed in the parabola y 2 = 4ax is 1 ( y1 − y2 )( y2 − y3 )( y3 − y1 ) sq. units where y1 , y2 , y3 are the ordinates of its vertices . 8a Find the coordinates of the vertex and focus, the equation of the directrix and axis of the following parabolas. i) y 2 + 4 x + 4 y − 3 = 0 ii) x 2 − 2 x + 4 y − 3 = 0 Show that the common tangents to the circle 2 x 2 + 2 y 2 = a 2 and the parabola y 2 = 4ax intersect at the focus of the parabola y 2 = −4ax Show that the poles of the chords of a parabola y 2 = 4ax which subtend a right angle at vertex, lies on a line parallel to its directrix. From an external point P. tangent are drawn to the parabola y 2 = 4ax and these tangents make angle θ1 , θ 2 with its axis , such that tan θ1 + tan θ 2 is a constant b. Then show that P lies on the line y = bx

( x1 , y1 ) , ( x2 , y2 )

and

( x3 , y3 )

6.

Prove that the area of the triangle formed by the tangent at

to the

7.

1 ( y1 − y2 )( y2 − y3 )( y3 − y1 ) sq. units. 16a If a normal chord a point ‘t’ on the parabola y 2 = 4ax subtends a right angle at vertex then prove that

8.

t=± 2 Show that the area of an equilateral triangle inscribed in the parabola y 2 = 4ax with one vertex is at the

parabola y 2 = 4ax ( a > 0 ) is

9. 10. 11.

12. 13.

origin is 48 3 a 2 sq. units. Find the equation of a parabola in the standard form If lx + my + n = 0 is a normal to the parabola y 2 = 4ax ,then show that al 3 + 2alm 2 + nm 2 = 0 Show that the equation of common tangents to the circle x 2 + y 2 = 2a 2 and the parabola y 2 = 8ax are y = ± ( x + 2a ) Find the equation of the parabola whose vertex and focus are on the positive x –axis at a distance ‘a’ and a ' from the origin respectively If L and L' are the ends of the latus rectum of the parabola x 2 = 6 y , find the equations of OL and OL' where ‘O’ is the origin . Also find the angle between them


5

ELLIPSE VERY SHORT ANSWER TYPE QUESTIONS:

1.

Find the equation of the ellipse in the standard form whose distance between foci is 2 and the length of latus rectum is 15/2 2. Find the equation of the ellipse in the standard form such that distance between foci is 8 and distance between directrices is 32. 3. Find the eccentricity of the ellipse (in standard form) if its length of the latus rectum is equal to half of its major axis . 2 2 x − h) ( y − k ) ( 4. Find the equation of the ellipse in the form + = 1 , given the following data. a2 b2 i) Centre (2,-1), e = 1/ 2 , length of latus rectum 4 5. Find the equation of tangent and normal to the ellipse x 2 + 8 y 2 = 33 at (-1,2) 7. Find the pole of the line 5x + 7 y + 8 = 0 w.r.t 5 x 2 + 7 y 2 = 8 8. Find the value of k if the lines 2 x + 3 y + 1 = 0 , x + y + k = 0 are conjugate w.r.t ellipse 3x 2 + 4 y 2 = 12 9. Find the values of k if (1, 2 ) ( k , −1) are conjugate points w.r.t the 2 x 2 + 3 y 2 = 6 10. Ex- If the length of the latus rectum is equal to half of tis minor axis of an ellipse in the standard form, then find the eccentricity of the ellipse SHORT ANSWER TYPE QUESTIONS: 1. 2.

3. 4.

Find the equation of the ellipse with focus at (1,-1) , e = 2 / 3 and directrix as x+y+2=0 Find the length of major axis, minor axis, latus rectum, eccentricity , coordinates of center , foci and the equations of directrices of the following ellipse i) 9 x 2 + 16 y 2 = 144 ii) 4 x 2 + y 2 − 8 x + 2 y + 1 = 0 Find the equation of the tangents to 9 x 2 + 16 y 2 = 144 , which makes equal interecepts on the coordinates axis Find the equation of the tangents to the ellipse 2 x 2 + y 2 = 8 which are i) parallel to x − 2 y − 4 = 0 ii) perpendicular to x + y + 2 = 0 iii) which makes an angle

5.

π

with x- axis. 4 Show that the foot of the perpendicular drawn from the center on any tangent to the ellipse on the curve

(x

2

2

+ y 2 ) = a 2 x2 + b2 y 2 x2 y2 + = 1 lies on a 2 b2

6.

Show that the poles of the tangents to the circle x 2 + y 2 = a 2 + b 2 w.r.t the ellipse

7. 8.

x2 y2 1 + 4 = 2 2 4 a b a +b Ex- Equation of ellipse in standard form Ex- If θ1 , θ2 are the eccentric angles of the extremeties of a focal chord of the ellipse

9. 10.

x2 y 2 + = 1( a > b ) nd e its eccentricity . Then show that. a 2 b2 Ex- Find the equation of tangent and normal to the ellipse 9 x 2 + 16 y 2 = 144 and the end of the latus rectum in the first quadrant Find the condition for the line x2 y2 i) lx + my + n = 0 to be a tangent to the ellipse 2 + 2 = 1 a b


6

x2 y2 + =1 a 2 b2 x2 y 2 If the normal at one end of a latus rectum of the ellipse 2 + 2 = 1 a b 4 Passes through one end of the minor axis, then show that e + e2 = 1 (e is the eccentricity of the ellipse) Show that the pole of the line lx + my + n = 0, ( n ≠ 0 ) with respect to the ellipse S = 0 is

ii) lx + my + n = 0 to be normal to the ellipse 11. 12.

13.

14.

15. 16.

 a 2l b 2m  ,− −  n   n Show that the poles of the tangents to the auxillary circle w.r.t the ellipse S=0 lie on the curve x2 y 2 1 + = a4 b4 a2 A chord PQ of an ellipse S=0 subtends a right angle at the center of the ellipse show that the point of x2 y 2 1 1 intersection of tangents at P and Q lies on another ellipse 4 + 4 = 2 + 2 a b a b x2 y2 Show that the poles of the tangents of y 2 = 4 kx( k > 0) w.r.t the ellipse 2 + 2 = 1 lies on a parabola a b Find the equation of the Ellipse in the standard form

HYPERBOLA VERY SHORT ANSWER TYPE QUESTIONS:

1. 2.

5 , then find the eccentricity of its conjugate hyperbola 4 Find the equation of th hyperbola whose asymptotes are 3x = ±5 y and the vertices are ( ±5, 0 )

If the eccentricity of hyperbola is


1.

2. 3. 4. 5. 6. 7.

8.

Find the center , foci, eccentricity , equation of the directrices , length of the latus rectum of the following hyperbolas 9 x 2 − 16 y 2 + 72 x − 32 y − 16 = 0 Show that the difference of the focal distance of any point on the hyperbola is constant 1 1 If e1 , e2 are the eccentricities of a hyperbola and its conjugate hyperbola prove that 2 + 2 = 1 e e1 Find the equations of the tangents to the hyperbola 3 x 2 − 4 y 2 = 12 which are (i) parallel and (ii) perpendicular to the line y=x-7 Find the equation of normal to the hyperbola x 2 − 3 y 2 = 144 at the end of the latus rectum (first quadrant ). Prove that the product of the perpendicular distance from any point on a hyperbola to its asymptotes is constant. x2 y2 (i) If the line lx + my + n = 0 is a tangent to the hyperbola 2 − 2 = 1 then show that a 2 l 2 − b 2 m 2 = n2 a b 2 2 2 x y a2 b2 (ii) If the lx + my = 1 is a normal to the hyperbola 2 + 2 = 1 then show that 2 − 2 = ( a 2 + b2 ) a b l m Find the condition that the lines l1 x + m1 y + n1 = 0 and l2 x + m2 y + n2 = 0 are conjugate with respect to x2 y2 the hyperbola 2 − 2 = 1 a b


7

POLAR COORDINATES VERY SHORT ANSWER TYPE QUESTIONS:

1.

2.

3. 4. 5. 6.

7.

Taking origin as the pole and the positive X-axis initial ray, convert the following polar equations into Cartesian forms θ iii) r = a sin 2 i) r 2 cos 2θ = a 2 ii) θ = Tan −1 ( m ) 2 Find the distance between the following pairs of points with polar coordinates  π   2π  i)  2,  ,  4,   6  3  Show that the points with polar coordinates (0,0), ( 3, π / 2 ) and ( 3, π / 6 ) form an equilateral triangle .

π  π  Find the polar equations of the line joining the points  5,  and  −5,  6  2  Taking origin as the pole and the positive X- axis s the initial ray find the Cartesian coordinates of the point P whose polar coordinates are (1, −π / 4 ) Taking origin as the pole and positive X –axis as initial ray, find the polar coordinates of the point P 3   3 whose Cartesian coordinates are  ,−  2  2 Find the polar equation of a straight line passing through ( 4, 200 ) and making an angle 1400 with the initial ray

 π Find the polar equation of the circle with center at  5,  and passing through pole.  2 SHORT ANSWER TYPE QUESTIONS:

8.

1.

Find the area of the triangle formed by the following points with polar coordinates i) ( a ,θ ) , ( 2a ,θ + π / 3 ) , ( 3a,θ + 2π / 3 ) ii) ( −5, −2π / 3) , ( 4, 5π / 6 ) ,(0,0)

3.

 π Find the polar equations of the lines passing though  2,   4 i) parallel to ii) perpendicular to the straight line 7 / r = 4 cos θ + 3sin θ Find the center and radius of the circle r 2 − 2 r ( 3cos θ + 4sin θ ) = 39

4.

If

2.

PP' and

QQ ' are two perpendicular focal chords of conic, prove that

1

+

1

( SP ) ( SP ) ( SQ ) ( SQ ' ) '

is

constant

7.

 π  π  π Find the area of the triangle formed by the points with polar coordinates 1,  ,  2,  ,  3,   6  3  2 l Show that the polar equation of a conic in the standard form is = 1 + e cos θ (‘ l ’ is semilatus rectum, r e is eccentricity ) Find the polar equation of circle for which ( r1 , θ1 ) and ( r2 ,θ 2 ) are extremities of a diameter.

8.

Find the center and radius of the circle r 2 − 8r

9.

If PSQ is chord passing through the focus S of a conic and ‘ l ’ is semi latus rectum, show that 1 1 2 + = SP SQ l 1 1 If PP ' , QQ ' are perpendicular focal chords of a conic then show that + = a (constant) PP ' QQ '

5. 6.

10.

(

)

3 cos θ + sin θ + 15 = 0


8

SUCCESSIVE DIFFERENTIATION VERY SHORT ANSWER TYPE QUESTIONS : 1. Find the nth derivative of sin 5x. sin 3x 2. Find the nth derivative of e x cos x.cos 2 x 2x + 1 3. If y = 2 , find yn . x −4

4.

Find the nth derivative of x i). 2 ( x − 1) ( x + 1)

ii) log ( 4 x 2 − 9 )

π

5.

Find D3 ( sin x log x ) when x =

6.

Find the nth derivative of f ( x ) = log ( 8 x3 + 36 x 2 + 54 x + 27 ) for all x > −

2 3 2

SHOR ANSWER TYPE QUESTIONS

1.

If y = ax n +1 + bx − n , then show that x 2 y2 = n ( n + 1) y .

2.

If y = aenx + be− nx , then show that y2 = n 2 y

3.

If y 1 + x 2 = log x + 1 + x 2 , then show that (1 + x 2 ) y1 + xy = 1

4.

If y = a cos x + ( b + 2 x ) sin x, then show that y2 + y = 4cos x .

5.

If y = a + be−4 x , then show that y2 + 4 y1 = 0

6. 7.

If ay 4 = ( x + b ) , then show that 5yy2 = y12 Find the nth derivative of the following, using Leibnitz Theorem. i) e 2 x cos 2 x ii) e 2 x sin x cos 5 x If y = x n log x, then show that xyn +1 = n !

8.

(

)

5

n

9.

10.

( −1) n! 1 1 1 log x ( x > 0 ) , then show that yn = n+1  log x − 1 − − .. −  . x x 2 3 n  2 2 If y = cos ( m log x ) , x > 0, then show that x y2 + xy1 + m y = 0 and hence deduce that If y =

x 2 yn + 2 + ( 2n + 1) xyn +1 + ( m 2 + n 2 ) yn = 0

11.

If y =

sinh −1 x 1+ x

(1 + x ) y

13.

, then show that (1 + x 2 ) y2 + 3 xy1 + y = 0 and hence deduce that 2

2

12.

2

n+ 2

+ ( 2n + 3) xyn +1 + ( n + 1) yn = 0

 y x If Cos −1   = n log   , x > 0 , then show that x 2 y2 + xy1 + n 2 y = 0 b n and hence deduce that x 2 yn + 2 + ( 2n + 1) xyn +1 + 2n 2 yn = 0 If y = esin x , then show that (1 − x 2 ) y2 − xy1 − y = 0 and hence deduce that −1

(1 − x ) y 2

n+ 2

− ( 2n + 1) xyn +1 − ( n 2 + 1) yn = 0

2

14.

If y = e− x , then prove that yn + 2 + 2 xyn +1 + 2 ( n + 1) yn = 0

15.

If y = em sin x , then prove that

−1

(1 − x ) y 2

n+2

− ( 2 n + 1) xyn +1 − ( n 2 + m 2 ) yn = 0


9

16.

Prove that the nth derivative of f ( x ) = Tan −1 x, ( x ∈ R ) is n −1

( −1) ( n − 1)!

(x 17.

+ 1)

2

n/ 2

sin nθ , where θ ∈ [ −π , π ] is the angle satisfying cos θ =

x 2

and sin θ =

x +1

1 x2 + 1

If y = a cos ( log x ) + b sin ( log x ) .x > 0 then prove that x 2 yn+ 2 + ( 2n + 1) xyn+1 + ( n 2 + 1) yn = 0

INTEGRATION VERY SHORT ANSWER QUESTIONS Evaluate the following integrals

1. 2. 3.

4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

1− x dx on ( 0, ∞ ) x  x 1 2  ∫  e − x + x2 − 1  dx on I ⊂ R \ [ −1,1]

∫



∫ 

1 1 − x2

+

  dx on ( −1,1) 1 + x2  2

 1 2 3  + −  ∫  x x2 − 1 2 x 2  dx on (1, ∞ ) 3   nπ   2   ∫  sec x tan x + x − 4 dx, x ∈ I ⊂ R \   2 : n is an odd integer  ∪ {0} 

∫

(a

x

− bx )

2

dx, ( a > 0, a ≠ 1 and b > 0, b ≠ 1) on R a xbx 1 + cos 2 x ∫ 1 − cos 2 x dx on I ⊂ R \ {nπ : n ∈ Z } 1 ∫ cosh x + sinh x dx on R sin (Tan −1 x ) ∫ 1 + x 2 dx, x ∈ R 2 ∫ 25 + 9 x 2 dx on R 3 4 ∫ x sin x dx on R cos x

∫ (1 + sin x )

2

3π   dx on I ⊂ R \ 2nπ + : n∈Z . 2  

x8 ∫ 1 + x18 dx on R 1 ∫ x log x log ( log x ) dx on (1, ∞ ).   1  nπ ∫ 1 + sin 2 x dx on I ⊂ R \  2 + ( −1)

 :n∈Z 2  x x ∫ e cot e dx, x ∈ I ⊂ R \ {log nπ : n ∈ Z } n

π

∫ cos x dx on R ∫ cos x cos 3x dx on R 3


10

Evaluate the following integrals.



1 

19.

∫ ( tan x + log sec x )e dx on   2n − 2  π  , n ∈ Z

20.

∫e

x

21.

 1 + x log x    dx on ( 0, ∞ ) x   x −1 ∫ x Tan xdx, x ∈ R

22.

∫ cosh

x

−1

x dx on [1, ∞ ] 

23.

24. 25.

1

1   x+   Evaluate ∫  1 − 2 e x  dx on I where I = 0 ( o, ∞ )  x  1 Evaluate ∫ dx on R 1 + 4 x2  ( 2n + 1) π  log (1+ tan 2 x ) : n∈ Z e dx on I ⊂ R \  ∫ 2  

 ( 2n ± 1) π  : n∈ Z 2 

1 − cos 2 x

26.

∫ 1 + cos 2 x dx I ⊂ R \ 

27.

∫ x (4 + x )

28.

∫ 

29. 30. 31.

2 2

3



dx on R

 + 2x + e x  dx on (-1,1) 1− x  x   ∫  1 + sin 2  dx on R 1 ∫ 1 + e− x dx on R 2 3 ∫ x sin x dx on R

1

2



cot x

π

33.

∫ log ( sin x ) dx on I ⊂  2nπ , 2nπ + 2  , n ∈ z ∫ x sin ( − x ) dx on R

34.

∫

32.

35. 36. 37. 38.

2

1 cos x dx on ( 0, ∞ ) x 1 ∫ 9 x 2 + 16 dx on R x2 ∫ x6 + 2 x3 + 2 dx on R 1 + sin 2 x π   ∫ 1 + cos 2 x dx on I ⊂ R \ ( 2n ± 1) 2 : n ∈ Z  cos x + sin x π 3π   ∫ 1 + sin 2 x dx on I ⊂  2nπ − 4 , 2nπ + 4  , n ∈ Z


11

SHORT ANSWER TYPE QUESTIONS

1.

e x (1 + x )

∫ cos ( xe ) dx on I ⊂ R \ {x ∈ R : cos ( xe ) = 0} 2

x

x

3.

Evaluate the following integrals 2 ( log x ) dx on 0, ∞ ( ) ∫ x Evaluate the following integrals. n ∫ x log x dx on ( 0, ∞ ) , n is a real number and n ≠ −1 .

4.

∫ x cos

5.

∫ x sec

2.

2

xdx on R

6.

π   2 x dx on I ⊂ R \ ( 2 nπ + 1) : n ∈ Z  . 4   −1 ∫ x Tan x dx, x ∈ R

7.

∫e

8.

∫

9. 10. 11. 12.

ax

2

sin ( bx + c ) dx, ( a, b, c ∈ R, b ≠ 0 ) on R sin θ

dθ 2 − cos 2 θ cos x ∫ sin 2 x + 4 sin x + 5 dx dx ∫ 2 sin 2 x + 3 cos2 x 1 ∫ 2 − 3cos 2 x dx

∫x

1 + x − x 2 dx x2

13.

∫ ( x + 1)( x + 2 )

14.

∫ ( x − 1) ( x

15.

∫ (1 − x ) ( 4 + x ) dx

16.

∫ x ( x + 1)( x + 2 )

17.

∫ ( x − 1) ( x + 2 )dx

18.

∫ ( x − a )( x − b )( x − c )

19.

∫ ( x + 3) ( x

20. 21.

∫ cos ∫x e

22.

Find

x+3 + 1)

2

dx

2

dx

1

2

dx

7x − 4 2

1

2x + 3

2

2

+ 4)

dx

sin x cos x dx x + 3cos x + 2 −3 x dx 2

∫

x2 dx on ( −5, ∞ ) x +5


12

23. 24. 25. 26. 27. 28. 29. 30. 31.

Evaluate

dx

∫

1 + x − x2 dx Evaluate ∫ 5 + 4 cos x dx Evaluate ∫ ( x + 5) x + 4

∫ 3 + 8 x − 3x dx Evaluate ∫ x + x +1

Evaluate

2

dx

2

 1 − sin x  Evaluate ∫ e x   dx on I ⊂ R \ {2 nπ : n ∈ Z }  1 − cos x  1− x dx, on ( −1,1) Evaluate ∫ Tan −1 1+ x 1 ∫ e x + e− x dx on R sec x π   ∫ ( sec x + tan x )2 dx on I ⊂ R \ ( 2n + 1) 2 : n ∈ z 

cos x

32.

∫

1.

∫ 4sin x + 5cos x dx

2.

∫ 4 cos x + 5sin x dx

3.

∫ 1 + sin x + cos x dx

4. 5.

∫ 3sin x + 4 cos x + 5 Show that ∫ x e dx = − x e

6.

If I n = ∫ cos n x dx, then show that I n =

dx on R 3sin x − 4sin x + 5 LONG ANSWER TYPE QUESTIONS

7.

2

9cos x − sin x

2cos x + 3sin x 1

2sin x + 3cos x + 4 n −x

n −x

+ n ∫ x n −1e − x dx

1 n −1 cos n −1 x sin x + I n− 2 , ∫ n n Obtain the reduction formula for I n = ∫ cosec n xxdx, n being a positive integer, n ≥ 2 and deduce the 5

value of ∫ cosec x dx 8.

If I m ,n = ∫ sin m x cos n xdx, then show that sin m −1 x cos n +1 x m − 1 + I m −2 n , for a positive integer n and an integer m ≥ 2 . m+n m+n dx Find ∫ 3cos x + 4 sin x + 6 cos x + 3sin x + 7 Evaluate ∫ dx cos x + sin x + 1 cos x + 3sin x + 7 Evaluate ∫ cos x + sin x + 1 dx I m,n = −

9. 10. 11. 12.

Reduction formula for ∫ sin m x cos n x dx for a positive integer m and an integer n ≥ 2


13

13.

Reduction formula for ∫ secn x dx for an integer n ≥ 2

14.

Evaluate ∫ sec5 x dx

15.

Find

π

x sin x

∫ 1 + sin x dx 0

DIFINITE INTEGRATION EVALUATE THE FOLLOWING DEFINITE INTEGRALS VERY SHORT ANSWER TYPE QUESTIONS 2

1.

∫ 1 − x dx 0 π 2

2.

cos x

∫π 1 + e

−

x

∫

2

dx

x + 16

0 1

4.

dx

2

3

3.

x

∫ xe

− x2

dx

0 4

5.

x2 ∫0 1 + x dx 4

6.

∫ 2 − x dx 0

π 2

7.

Show that

x

∫ sin x + cos x dx = 2

π

0

8.

2

log

(

)

2 +1

Evaluate the following integrals π

π

2

i)

1 ∫0 1 + tan x dx π

iii) ∫ cos 3 x sin 4 x dx 0

esin x ∫0 esin x + ecos x dx 2

ii)

log 2

1

iv)

∫ xe 0

x2

dx

v)

∫

cosh 2x dx

0


14

SHORT ANSWER TYPE QUESTIONS

Evaluate the following integrals. π 2

1.

dx

∫ 4 + 5cos x 0 a

2.

∫ x(a − x)

n

dx

0 2

3.

∫x

3/ 2

2 − x dx

0

π

4.

∫ x sin dx 3

0

log (1 + x ) dx 1 + x2 0

1

5.

∫ 1

6.

∫ 0

1− x dx 1+ x

e

7.

∫ x log x dx 1 1

8.

xe x

∫ (1 + x )

dx

2

0

1

9.

 2− x 

∫ log  2 + x  dx

−1 1

10.

∫ 1+ 0

11.

x x

dx

Evaluate the following integrals π

i) ∫ x sin

2π

5

6

x cos xdx

ii) ∫ x sin

a 6

0

0

5

x cos xdx

iii) ∫ x+ 0

dx a2 − x2

LONG ANSWER TYPE QUESTIONS

Evaluate the following integrals π 4

1.

sin x + cos x

∫ 9 + 16sin 2 x dx

1.

0

π

2.

x sin 3 x ∫0 1 + cos2 x dx

π

3.

x sin x

∫ 1 + cos 0

2

x

dx

π

sin 2 x ∫0 cos+ sin x dx Find the values of the following integrals. 2

4.


15

π 2

5.

∫ sin

10

x dx

0

π 2

6.

∫ cos

11

x dx

0

π 2

7.

∫ cos

7

x sin 2 x dx

4

x cos 4 x dx

0

π 2

8.

∫ sin 0

π 2

9.

∫π sin

3

θ cos3 θ dθ s

− 2 a

10.

7

2 2 ∫ x ( a − x ) 2 dx 0 2

11.

∫x

2 − xdx

0 4

12.

0 1

13.

5 2 2

∫ (16 − x ) ∫ 0

dx

dx x +1 + x

1

14.

∫ x dx −1

15.

Th- Let m and n be positive integers. Then  1 n =1  m + 1 , if  n −3 2 1  n −1 . ... . if 1 ≠ n is odd  m+n m+n−2 m+3 m + 1, π  2  n −1 n −3 1 m −1 1 π m n sin x cos x dx = . ... . ... . , if nis even and m is even  ∫0 m+n−2 m+2 m 2 2  m+n  n −1 n −3 1 m −1 1 . ... . ... , if n is even,1 ≠ m is odd  m+n−2 m+2 m 3  m+n  1  n + 1 , if m = 1 NUMERICAL INTEGRATION

VERY SHORT ANSWER TYPE QUESTIONS: 1. Find the area of the region enclosed by the given curves y = e X , y = x, x = 0, x = 1

2. 3. 4.

π . 2 Find the area bounded by y = sin x and y = cos x between any two consecutive points of intersection. Find the area enclosed by the curves y = 3x and y = 6x – x2. Find the area of the region enclosed by the given curves y = sin x, y = cos x, x = 0, x =

SHORT ANSWER TYPE QUESTIONS: 1. Find the area if the region enclosed by the given curves x 2 = 4 y , x = 2, y = 0 . Vanithatv – GUIDE education and career live show daily at 8.30pm – MATHS 2B

16

2. 3.

Find the area of the region enclosed by the given curves y = x 2 , y = 2 x Trapezoidal rules and Simpson’s rule – Simple applications

LONG ANSWER TYPE QUESTIONS:

1.

Find the area of the region enclosed by the given curves y 2 = 4 x, y 2 = 4 ( 4 − x ) .

2.

Show that the area enclosed between the curves y 2 = 12 ( x + 3 ) and y 2 = 20 ( 5 − x ) is 64

3.

Show that the area of the region bounded by

5 . 3

x2 y 2 + = 1 (ellipse) is π ab. Also deduce the area of the a 2 b2

circle x 2 + y 2 = a 2 4

4.

Given that e = 2.72, e2 = 7.39, e3 = 20.09., e4 = 54.60 evaluate ∫ e x dx approximately using Simpson’s 0

rule. 1

5.

Find the approximate value of π from

1

∫ 1+ x

2

dx by using Simpson’s rule by dividing [0, 1] into 4 equal

0

parts. 7

6.

Use Simpson’s rule to evaluate

1

∫ x dx approximately by taking n = 6 and hence find the approximate 7 1

value of log e 7 to three places of decimals. 8.

Find the area bounded between the curves y = x 2 , y = x Find the area bounded between the curves y 2 = 4ax, x 2 = 4by ( a > 0, b > 0 ) .

9.

Find the area enclosed within the curve x + y = 1 .

10. 11.

A river is 40 meters wide. The depth d in meters at a distance x meters from a bank is given in the following table. Using Simpson’s rule, estimate the area of the cross section of the river. Find the area bounded by the X-axis and the curve y = 4 x − x 2 − 3

12.

Find the area of the region enclosed by the given curves. y 2 = 2 x, y = 4 x − 1

7.


17

DIFFERENTIAL EQUATIONS VERY SHORT ANSWER TYPE QUESTIONS: 1. Find the order of the differential equation of the family of all circles with their centres at the origin.  y  dy 2. Express xdy − ydx = x 2 + y 2 dx in the form F   = .  x  dx Solve the following differential equations. 3. ( x 2 + y 2 ) dy = 2 xydx

4.

(x

2

)

(

)

y − 2 xy dx + 2 xy − x 2 dy = 0 2

5. 6.

7. 8.

dy y y + = dx x x 2

y  Solve: x dy =  y + x cos 2  dx x  Solve the following differential equations. dy 4 x + 6 y + 5 = dx 3 y + 2 x + 4 Express the following differential equations in the form f ( x ) dx + g ( y ) dy = 0

10.

dy = e x− y + x 2 e − y dx dy ( cos x ) + y sin x = tan x dx 2 Find the order of the differential equation corresponding to y = c ( x − c ) , where c is an arbitrary

11.

constant Form the differential equation corresponding to y = A cos 3 x + B sin 3x , where A and B are parameters

9.


1.

xy = ae x + be− x ; ( a, b ) From the differential equations of the following family of curves where

2.

parameters are given in brackets. y = a cos(nx + b); ( a, b ) From the differential equations of the following family of curves where

3.

4.

5. 6. 7. 8.

parameters are given in brackets. Form the differential equations of the following family of curves where parameters are given in brackets: y = ae3 x + be 4 x ; Find the general solution of 1 − x 2 dy + 1 − y 2 dx = 0 Solve the following differential equations. dy y − x e dx 1 + x 2 dx + 1 + y 2 dy = 0

 dy  sin −1   = x + y .  dx  dy = tan 2 ( x + y ) . dx Solve the following differential equations .


18

9. 10.

(x (x

)

2

+ y 2 dy = 2 xydx

2

y − 2 xy 2 dx = x 3 − 32 y dy

)

(

)

2

11.

12. 13. 14. 15. 16. 17. 18.

dy y y + = dx x x 2 Solve the following differential equations. y  xdy =  y + x cos 2  dx x  dy 4 x + 6 y + 5 Solve = dx 3 y + 2 x + 4 −1 + y = eTan x (1 + x 2 ) dy dx dy 2 y + = 2x2 dx x dy + y = Tan −1 x (1 + x 2 ) dx dy ( x + y + 1) = 1 dx Form the differential equation corresponding to the family of circles of radius r given by 2 2 ( x − a ) + ( y − b) = r 2 ,

LONG ANSWER TYPE QUESTIONS :

1. 2. 3. 4.

Solve the following differential equations. dy 6 x + 6 x + 5 y − 7 = dx 2 x + 18 y + 14 2 dy Solve ( x − y ) = a2 dx dy y 2 − 2 xy = 2 Solve dx x − xy dy Solve sin 2 x. + y = cot x . dx


19

MATHS - IIB QUESTION BANK - Chapter Wise Important Questions for IPE

Recommend Documents