Maths-2B Previous question papers [Intermediate Education, Andhra Pradesh]

Intermediate Previous papers Mathematics-IIB

1

MAY–2012

SECTION-A (10 × 2 = 20)

1) Find the other end of the diameter of the circle 2 2 8 8 27 0x y x y+ − − + = if one end of it is (2, 3)

2) Find the equation of the sphere whose center is (2, –3, and 4) and radius is 5. 3) Find the equation of the parabola whose focus is S (1, –7) and vertex is A (1, –2).

4) Show that the angle between the two asymptotes of a hyperbola 2 2

2 21

x y

a b− = is 12 tan

b

a−

or 12s ( )ec c−

5) Find the nth derivative of ( ) ( )3 2log 8 36 54 27f x x x x= + + + for all3

2x > − .

6) Evaluate

1

2

11 e d

xx x

x

+ −

∫ 7) Evaluate ( )

1d

3 2x

x x+ +∫ on ( )2,I ⊂ − ∞

8) Evaluate 2

3

2 2

30

sin cosd

sin cos

x xx

x x

π−+∫ 9) Find order and degree of

11 3 41 2 2

2

d d0

d d

y y

x x

+ =

∕∕

10) Find the area bounded between the curves 2 1 2y x− = and 0x = .

SECTION-B (5 × 4 = 20)

11) Find the condition that the tangents drawn from the exterior point ( ),g f to 2 2 2 2 0s x y gx fy c= + + + + =

are perpendicular to each other. 12) Find the equation of the parabola whose axis is parallel to Y-axis and which passes through the points

(4, 5), (–2, 11) and (–4, 21).

13) Find the eccentricity, foci and equation of the directrices of the hyperbola 2 25 4 20 8 4x y x y− + + =

14) If PP′ and QQ′ are two perpendicular focal chords of a conic, prove that ( )( ) ( )( )

1 1

SP SP SQ SQ+

′′ is

constant.

15) Evaluate 21 dx x x x+ −∫ 16) Solve ( )dtan 1

d

yx y x

x− − = 17) Solve ( )3 d

2d

yx y y

x+ =

SECTION-C (5 × 7 = 35) 18) Find the equation and center of the circle passing through the points (–2, 3), (2, –1) and (4, 0).

19) Find the equation of the circle which cuts the circles 2 2 2 4 1 0x y x y+ + + + = ,

2 22 2 6 8 3 0x y x y+ + + − = and 2 2 2 6 3 0x y x y+ − + − = orthogonally.

20) The tangent and normal to the ellipse 2 24 4x y+ = at a point ( )P θ on it meets the major axis in Q and

R respectively. If 0 2πθ< < and 2QR= , then show that ( )1 2cos 3θ −= .

21) If 1

2

sin

1

h xy

x

−=

+then show that ( )2

2 11 3 0x y xy y+ + + = and hence by using Leibnitz theorem, deduce

that ( )2 22 11 (2 3) ( 1) 0n n nx y n xy n y+ ++ + + + + =

22) Evaluate 1

dsin 3 cos

xx x+∫ 23) Evaluate

( )1

20

log 1d

1

xx

x

+

+∫

24) The velocity of a train which starts from rest is given by the following table, the time being recorded in minutes from the start and the speed in kilometers. Estimate approximately the total distance run in 20 minutes by Simpson’s rule and Trapezoidal rule.

Minutes 2 4 6 8 10 12 14 16 18 20 Kmph 10 18 25 29 32 20 11 5 2 0


2

MARCH–2012

SECTION-A (10 × 2 = 20) 1) Find the equation of the circle passing through (3, 4) and having the centre at (–3, 4).

2) Find the centre and radius of the sphere 2 2 2 2 4 6 2 0x y z x y z+ + − + − − =

3) Find the value of ‘K’ if points (1, 2), (K, –1) are conjugate with respect to the parabola 2 8y x= .

4) If the eccentricity of a hyperbola is 5/4, then find the eccentricity of its conjugate hyperbola.

5) Find the nth derivative of ( ) sin7 cosf x x x= x R∀ ∈ .

6) Evaluate3

1dx x

x

+ ∫ , 0x > 7) Evaluate ( )( )

d

1 2

x

x x+ +∫ 8) Evaluate 3

22

2d

1

xx

x+∫

9) Find the area of the region enclosed between 3 3, 0, 1, 2y x y x x= + = = − =

10) Form the differential equation corresponding to 22y cx c= − , where ' 'c is a parameter.

SECTION-B (5 × 4 = 20)

11) Find the angle between the tangents drawn from (3, 2) to the circle 2 2 6 4 2 0x y x y+ − + − =

12) Find the condition for the line y mx c= + to be a tangent to the parabola 2 4x ay= .

13) Find the pole of the line 21 16 12 0x y− − = with respect to the ellipse 2 23 4 12x y+ = .

14) If PSQ is a chord passing through the focus S of a conic and ‘l’ is semi latus rectum, show that 1 1 2

SP SQ l+ =

15) Evaluate d

5 4cos

x

x+∫

16) Solve the differential equation (2 ) (2 )x y dy y x dx− = −

17) Solve the differential equation tan sindy

y x xdx

+ =

SECTION-C (5 × 7 = 35) 18) Find the equation of a circle which passes through (4, 1), (6, 5) and having the centre on

4 3 24 0x y+ − = .

19) Find the coordinates of the limiting points of the coaxial system to which the circles 2 2 10 4 1 0x y x y+ + − − = and

2 2 5 4 0x y x y+ + + + = are two members.

20) Show that the poles of the tangents to the circle 2 2 2 2x y a b+ = + with respect to the ellipse

2 2

2 21

x y

a b+ = lies on

2 2

4 4 2 2

1x y

a b a b+ =

+.

21) If cos( log ), 0y m x x= > , then show that 2 22 1 0x y xy m y+ + = and hence deduce that

2 2 22 1(2 1) ( ) 0n n nx y n xy m n y+ ++ + + + =

22) Obtain reduction formula for (tan ) ,nnI x dx= ∫ n being a positive integer, 2n ≥ and deduce the value

of 6(tan )x dx∫ .

23) Show that2

0

d log( 2 1)sin cos 2 2

xx

x x

ππ= +

+∫ .

24) Calculate the approximate value of

5

11

dx

x+∫ , by taking n = 4 in the Simpson’s rule.


3

MAY–2011

SECTION-A (10 × 2 = 20)

1) If 2 2 2 2 12 0x y gx fy+ + + − = represents a circle with center (2, 3), then find ,g f and its radius.

2) Find the equation of the sphere that passes through the point (4, 3, –1) and having center at (3, 8, 1).

3) Find pole of the line 2 3 4 0x y+ + = with respect to 2 8y x= .

4) Find the equation of hyperbola whose foci are at ( 5,0)± and with transverse axis length 8.

5) Find nth derivative of sin 5 .sin 3x x

6) Evaluate 2 2

1 2

1 1dx

x x

+

− + ∫ 7) Evaluate

1

( 1)( 2)dx

x x

+ + ∫ 8) Evaluate

2

0

1 xdx−∫

9) Calculate the approximate value of

92

1

x dx∫ using Trapezoidal rule with 4 parts.

10) Find order and degree of differential equation

62 5

26

d y dyy

dxdx

+ =

SECTION-B (5 × 4 = 20)

11) Find length of chord intercepted by the circle 2 2 3 22 0x y x y+ − + − = on the line 3y x= −

12) Find the equation of the tangent and normal to the parabola 2 8y x= at (2, 4).

13) Find eccentricity, foci, length of latusrectum and equation directrices of the hyperbola 2 24 4x y− = .

14) Find the condition that straight line cos sink

A Br

θ θ= + may touch the circle 2 cosr a θ= .

15) Evaluate 21 3x x dx+ −∫ 16) Solve 2 2( )dy

x y xydx

− = 17) Solve

3tan cosdy

y x xdx

+ =

SECTION-C (5 × 7 = 35) 18) If (1, 2) (3, –4), (5, -–6) and (c, 8) are concyclic, find “c”.

19) If (3, 5) is a limiting point of coaxial system of circle of which 2 2 2 2 24 0x y x y+ + + − = find other

limiting point. 20) Show that the points of intersection of perpendicular tangents to an ellipse lies on a circle.

21) If1sin xy e

−= , then show that 2

2 1(1 ) 0x y xy y− − − = and hence deduce that

2 22 1(1 ) (2 1) ( 1) 0n n nx y n xy n y+ +− − + − + = .

22) Find reduction formula of (cot )n x dx∫ , hence find 4(cot )x dx∫

23) Find

01 sin

xdx

x

π

+∫

24) Find area enclosed between curves 2 24 , 4(4 )y x y x= = −


4

MARCH–2011

SECTION-A (10 × 2 = 20)

1) Obtain the parametric equation of the circle 2 2 2( 3) ( 4) 8x y− + − = .

2) If (2, 3, 5) is one end of a diameter of the sphere 2 2 2 6 12 2 20 0x y z x y z+ + − − − + = , then find the

coordinates of the other end of the diameter.

3) Find the points on the parabola 2 8y x= whose focal distance is 10.

4) If 1,e e are the eccentricities of a hyperbola and its conjugate hyperbola, then prove that2 2

1

1 11

e e+ = .

5) Find the 3rd derivative of cosxe x.

6) Evaluate 4

6

sin

cos

xdx

x∫ 7) Evaluate 1

log [log(log )]dx

x x x∫ 8) Evaluate

1 3

20

1

xdx

x +∫

9) Find the area of the region enclosed by the curves 24 , 0x y x= − = .

10) Form the differential equation of the family of all circles with their centers at the origin and also find its order. SECTION-B (5 × 4 = 20) 11) Find the equation of the circle with center (–2, 3) cutting a chord length 2 units on 3 4 4 0x y+ + = .

12) If the polar of P with respect to the parabola 2 4y ax= touches the circle 2 2 24x y a+ = , then show that P

lies on the curve 2 2 24x y a− = .

13) Find the equations of the tangents to the hyperbola 2 24 4x y− = which are:

i) parallel to and ii) perpendicular to the line 2 0x y+ = .

14) Find the area of triangle formed by points with the polar coordinates ( ),a θ , 2 ,3

aπθ +

,

23 ,

3a

πθ +

.

15) Evaluate 5 4cos2

dx

x+∫

16) Solve 2cosy

xdy y x dxx

= +

17) Solve 2 3( ) 1dy

x y xydx

+ =

SECTION-C (5 × 7 = 35)

18) Show that the circles 2 2 6 2 1 0x y x y+ − − + = , 2 2 2 8 13 0x y x y+ + − + = touch each other. Find the

point of contact and the equation of the common tangent at their point of contact. 19) Find the equation of the circle which passes through the origin and belongs to the coaxial system of

which the limiting points are (1, 2) and (4, 3). 20) Find the length of major axis, minor axis, latusrectum, eccentricity, coordinates of center, foci and the

equation of directrices of the ellipse 2 24 8 2 1 0x y x y+ − + + = .

21) If1sinm xy e

−= , then prove that 2 2 2

2 1(1 ) (2 1) ( ) 0n n nx y n xy n m y+ +− − + − + = .

22) Evaluate 2sin 3cos 4

3sin 4cos 5

x xdx

x x

+ ++ +∫

23) Evaluate 3

20

sin

1 cos

x xdx

x

π

+∫

24) A curve is drawn to pass through the points given by the following table. Using Simpson's rule, find the approximate area bounded by the curve, the X-axis and the lines 1x = and 4x = .

x 1 1.5 2 2.5 3 3.5 4 y 2 2.4 2.7 2.8 3 2.6 2.1


5

MAY–2010

SECTION-A (10 × 2 = 20)

1) Find the centre and radius of the circle 2 2 21 ( ) 2 2 0m x y cx mcy+ + − − = .

2) Find the equation of the sphere that passes through the point (4, 3, –1) and having its centre (3, 8, 1).

3) If 1

,22

is one extremity of a focal chord of the parabola 2 8y x= , find the coordinates of the other extremity.

4) Find the eccentricity of the Ellipse (in Standard Form) whose length of the latusrectum is half of its minor axis.

5) Find the nth derivative of 2cosy x=

6) Evaluate 2

(1 )

cos ( )

x

x

e xdx

xe

+∫ 7) Evaluate ( )logx dx∫ 8) Evaluate

1 2

20

1

xdx

x

+ ∫

9) Find the area bounded by 3 3, , 1, 2y x X axis x x= + − = − =

10) Form the differential equation corresponding to cos3 sin3y A x B x= + where A B< are parameters.

SECTION-B (5 × 4 = 20)

11) Show that the tangent at (–1, 2) of the circle 2 2 4 8 7 0x y x y+ − − + = touches the circle

2 2 4 6 0x y x y+ + + = and also find its point of tangency.

12) Find the value of k if the lines 2 0x y+ + = and 2 0x y k− + = are conjugate w.r.t 2 4 2 3 0y x y+ − − = .

13) Find eccentricity, coordinates of foci and equations of directrices of the ellipse 2 216 9 144y x− = .

14) If PSQ is a chord passing through the focus S of a conic and ‘l’ is semi latus rectum, show that

1 1 2

SP SQ l+ =

15) Evaluate 1

2 3cosdx

x−∫

16) Solve ( 1) ( 1) 0xe ydy y dx+ + + = 17) Solve sec tandy

y x xdx

+ =

SECTION-C (5 × 7 = 35) 18) Find the equation of a circle which passes through the points (5, 7), (8, 1) and (1, 3). 19) Find the coordinates of the limiting points of the coaxial system to which the circles

2 2 10 4 1 0x y x y+ + − − = and 2 2 5 4 0x y x y+ + + + = are two members.

20) Show that the equation of the parabola in standard form is 2 4y ax= .

21) If1sinm xy e

−= , then show that 2 2 2

2 1(1 ) (2 1) ( ) 0n n nx y n xy n m y+ +− − + − + = .

22) Find the reduction formula for (sin )n x dx∫ ( 2)n ≥ and hence find 4(sin )x dx∫

23) Show that 2

0

log( 2 1)sin cos 2 2

xdx

x x

ππ= +

+∫

24) Show that the area of the region bounded by 2 2

2 21

x y

a b+ = (ellipse) is abπ . Also deduce the area of the

circle 2 2 2x y a+ =


6

MARCH–2010 SECTION-A (10 × 2 = 20)

1) Obtain the parametric equation of each of the following circle 2 2 6 4 12 0x y x y+ − + − = .

2) Find the center and radius of the sphere 2 2 2 2 4 6 11x y z x y z+ + − − − = .

3) Find the value of k if the lines 2 3 4 0x y+ + = and 0x y k+ + = are conjugate w.r.t. 2 8y x= .

4) Find the eccentricity of the hyperbola 1xy = .

5) Find the nth derivative of 2log(4 )x−

6) Evaluate 1 logx x

dxx

+ ∫ on ( )0,∞ 7) Evaluate

25 4

0

cos sinx xdx

π

∫

8) Evaluate 2

(1 )

cos ( )

x

x

e xdx

xe

+∫ on { }: cos( ) 0xI R x R xe⊂ − ∈ =

9) Find the area of the region enclosed by the given curves 24 , 0x y x= − =

10) Find the order and degree of

5 322

21 0

d y dy

dxdx

+ + =

SECTION-B (5 × 4 = 20) 11) Find the equation of the circle whose center lies on X-axis and passing through the points (–2, 3), (4, 5).

12) Show that the equations of the common tangents to the circle 2 2 22x y a+ = and the parabola 2 8y ax=

are ( )2y x a= ± +

13) Find the eccentricity, foci and the equations of directrices of the following ellipse: 2 24 8 2 1 0x y x y+ − + + =

14) Show that the polar equation of a conic in the standard form is 1 cosl

er

θ= + .

(‘ l ’ is semi-latusrectum, ‘ e ’ is eccentricity)

15) Evaluate 5 4cos

dx

x+∫ 16) Solve ( )2 2 2x y dy xydx+ = 17) Solve ( ) 12 tan1 xdyx y e

dx

−+ + =

SECTION-C (5 × 7 = 35)

18) Show that the circles 2 2 6 2 1 0x y x y+ − − + = ; 2 2 2 8 13 0x y x y+ + − + = touch each other. Find the

point of contact and the equation of common tangent at the point of contact. 19) Find the limiting points of the coaxial system determined by the circles

2 2 2 210 4 1 0, 5 4 0x y x y x y x y+ + − − = + + + + =

20) If the polar of P with respect to the parabola 2 4y ax= touches the circle 2 2 24x y a+ = , then show that

P lies on the curve 2 2 24x y a− = .

21) If cos( log ), 0y m x x= > , then show that 2 22 1 0x y xy m y+ + = and hence deduce that

2 2 22 1(2 1) ( ) 0n n nx y n xy m n y+ ++ + + + =

22) Evaluate ( )2

1

3 12

xdx

x x

+

+ +∫ 23) Evaluate

1

20

log(1 )

1

xdx

x

++∫

24) Find the approximate value of π from

1

20

1

1dx

x+∫ by using Simpson’s rule by dividing [0, 1] into 4 equal

parts.


7

MAY–2009 SECTION-A (10 × 2 = 20)

1) If the center of the circle 2 2 12 0x y ax by+ + + − = is (2, 3), find the values of ,a b and the radius of the circle.

2) Find the equation of the sphere that passes through the point (4, 3, –1) and having its centre (3, 8, 1).

3) Find the coordinates of the points on the parabola 2 2y x= whose focal distance is 5/2.

4) Find the equations of the tangents to the hyperbola 2 23 4 12x y− = which is parallel to the line 7y x= − .

5) Find the nth derivative of 3 2( ) log(8 36 54 27)f x x x x= + + +

6) Evaluate ( )2 2sec .cosx ec x dx∫ 7) Evaluate 2

(1 )

(2 )

xe xdx

x

++∫ 8) Evaluate

22 4

2

(sin .cos )x x dx

π

π−∫

9) Find the area of the enclosed by the curve ( ) sinf x x= in the interval [ ]0, 2π .

10) Form the differential equation corresponding to 22y cx c= − , where ‘ c ’ is a parameter.

SECTION-B (5 × 4 = 20)

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

12) Prove that the poles of tangents to the parabola 2 4y ax= w.r.t the parabola 2 4y bx= lie on parabola.

13) One focus of hyperbola located at (1, –3) and corresponding directrix in the line 2y = . Find the equation

of hyperbola if its eccentricity is 3/2.

14) If PSQ is chord passing through the focus S of a conic and ‘ l ’ is semi latusrectum, show that

1 1 2

SP SQ l+ =

15) Evaluate 2

1

(1 )(4 )dx

x x− +∫

16) Solve 2 2( ) 0x y dx xydy− − = 17) Solve 2 1(1 ) (tan )y dx y x dy−+ = −

SECTION-C (5 × 7 = 35)

18) Find the equation of the circle whose centre lies on X-axis and passing through the points ( 2,3), (4,5)− .

19) In the limiting points of the coaxial system determined by the circles 2 2 2 6 0x y x y+ + − = and

2 22 2 10 5 0x y y+ − + = .

20) Find eccentricity, coordinates of foci and equations of directories of the ellipse2 29 16 36 32 92 0x y x y+ − + − = .

21) If 1

2

sin

1

h xy

x

−=



that ( )2 22 11 (2 3) ( 1) 0n n nx y n xy n y+ ++ + + + + = .

22) Obtain the reduction formula for sinnI xdx= ∫ , n being a positive integer, 2n ≥ and deduce the value

of 4sin xdx∫ .

23) Show that 2

0


xdx

x x

ππ= +

+∫

24) Dividing [0,6] into 6 equal parts, evaluate

63

0

x dx∫ approximately by using Trapezoidal rule and

Simpson's rule.


8

MARCH–2009 SECTION-A (10 × 2 = 20)

1) If the equation 2 2 4 6 0x y x y c+ − + + = represents a circle with radius 6, find c .

2) Find the centre and radius of the sphere 2 2 2 2 4 6 11x y z x y z+ + − − − = .

3) Find the coordinates of the points on the parabola 2 2y x= whose focal distance is 5/2.

4) Find the equation of the Hyperbola whose foci are (4, 2), (8, 2) and eccentricity is 2.

5) If nx nxy ae be−= + , then show that 22y n y= .

6) Evaluate 1 cos 2xdx−∫ 7) Evaluate 8

181

xdx

x+∫ on ℝ 8) Find the value of

2

0

1 xdx−∫

9) Find the area bounded by the parabola 2y x= , the x-axis and the lines 1, 2x x= − = .

10) Find the order and degree of

632 5

26

d y dyy

dxdx

+ =

.

SECTION-B (5 × 4 = 20)

11) If a point P is moving such that the lengths of tangents drawn from P to 2 2 6 18 26 0x y x y+ + + + = are

in the ratio 2:3, then find the equation of the locus of P.

12) Show that the equations of the common tangents to the circle 2 2 22x y a+ = and the parabola 2 8y ax=

are ( 2 )y x a= ± + .

13) Find eccentricity, coordinates of foci and equations of directrices of the ellipse2 29 16 36 32 92 0x y x y+ − + − = .

14) Show that the points with polar coordinates ( )0,0 , 3,2

π

and 3,6

π

form an equilateral triangle

15) Evaluate 1( cos )x x dx−∫

16) Solve 2 21 1 0x dx y dy+ + + = 17) Solve tan secxdyy x e x

dx− =

SECTION-C (5 × 7 = 35)

18) Show that the circles 2 2 6 2 1 0x y x y+ − − + = , 2 2 2 8 13 0x y x y+ + − + = touch each other. Find the

point of contact and the equation of common tangent at the point of contact.

19) Find the limiting points of the coaxial system determined by the circles 2 2 10 4 1 0x y x y+ + − − = ,

2 2 5 4 0x y x y+ + + + = .

20) If the polar of P with respect to the Parabola 2 4y ax= touches the circle 2 2 24x y a+ = , then show that

P lies on the curve 2 2 24x y a− = .

21) If 1

2

sin

1

h xy

x

−=



that ( )2 22 11 (2 3) ( 1) 0n n nx y n xy n y+ ++ + + + + = .

22) Evaluate 2(6 5) 6 2x x xdx+ − +∫ 23) Show that 2

0


xdx

x x

ππ= +

+∫

24) Find the approximate value ofπ from

1

20

1

1dx

x+∫ by using Simpson’s rule by dividing [0, 1] into 4 equal

parts.


9

MAY–2008

SECTION-A (10 × 2 = 20) 1) Find the equation of the circle passing through (2, –1) and having the center at (2, 3).

2) Find the centre and radius of the sphere 2 2 2 2 4 6 11x y z x y z+ + − − − = .

3) If (1,2)and ( , 1)k − are conjugate points w.r.t to the parabola 2 8y x= , then find k .

4) If the eccentricity of a hyperbola be 5/4, then find the eccentricity of its conjugate hyperbola.

5) Find the nth derivative of 2log(4 9)x − .

6) Evaluate 1 logx x x

e dxx

+ ∫ on (0, )∞ 7) Evaluate

2

8

2

1

xdx

x+∫ 8) Evaluate

4

0

2 x dx−∫

9) Find the area of the region enclosed by the given curves 3 3, 0, 1, 2y x y x y x= + = = − = .

10) Form the differential equations of the following family of curves where parameters are given in brackets3 4x xy ae be= + ; ( , )a b

SECTION-B (5 × 4 = 20)

11) Find the equation of tangent and normal at (3, 2) of the circle 2 2 4 0x y x y+ − − − = .

12) If the focal chord of the parabola 2 4y ax= meets it at P , Q and if S the focus then prove that

1 1 1

SP SQ a+ = .

13) Find the equation of tangent to the ellipse 2 22 8x y+ = which is parallel and perpendicular to 2 4x y− =

14) Show that the following points form an equilateral triangle ( )0,0 , 5,18

π

,7

5,18

π

.

15) Evaluate 2

1

2 3 1dx

x x− +∫ 16) Solve 2

2

10

1

dy y y

dx x x

+ ++ =+ +

17) Solve sec tandy

y x xdx

+ =

SECTION-C (5 × 7 = 35)

18) Find the equations of transverse common tangents of 2 2 4 10 28 0x y x y+ − − + = and

2 2 4 6 4 0x y x y+ + − + = .

19) Find the equation of the circle which is orthogonal to each of the following circles2 2 2 17 4 0x y x y+ + + + = , 2 2 7 6 11 0x y x y+ + + + = , 2 2 22 3 0x y x y+ − + + = .

20) Derive the equation of the hyperbola in standard form.

21) If 1siny x−= , then show that 2 21(1 ) 0x y xy− − = .Hence show that

2 22 1(1 ) (2 1) 0n n nx y n xy n y+ +− − + − =

22) Evaluate 2cos 3sin

4cos 5sin

x xdx

x x

++∫

23) Evaluate

1

20

log(1 )

1

xdx

x

++∫

24) Evaluate

5

01

dx

x+∫ approximately by using the Simpson's rule with n = 4.


10

MODEL QUESTION PAPER – 2013

SECTION-A (10 × 2 = 20)

1) If the equation 2 2 4 6 0x y x y c+ − + + = represents a circle with radius 6, find the value of c .

2) Find the equation of the directrix of the parabola 22 7 0x y+ =

3) Find the length of the latus rectum of the ellipse 2 2

116 8

x y+ =

4) Find the eccentricity of the hyperbola 2 24 4x y− =

5) Find the distance between the two points in a plane whose polar coordinates are 2, , 3,6 4

π π

.

6) If1

2 5y

x=

+, then find ny . 7) Evaluate 1 sin 2xdx+∫

8) Evaluate

1sin

21

xedx

x

−

−∫ 9) Evaluate

42

1

( 1)x x dx−∫

10) State the Simpson’s rule for Numerical Integration of a function ( )f x over the interval [ , ]a b by dividing

[ , ]a b into n sub-intervals.

SECTION-B (5 × 4 = 20)

11) If the line y mx c= + touches the ellipse2 2

2 21

x y

a b+ = , ( )a b> then show that 2 2 2 2c a m b= + .

12) Find the equations of the tangents shown drawn from (–2, 1) to the hyperbola 2 22 3 6x y− = .

13) Transform the polar equation 2cos2

r aθ =

, ( 0)a > origin as pole and the axis as initial line, into

Cartesian form

14) Iflog x

yx

= , then show that1

( 1) 1 1 1log 1

2 3

n

n n

ny x

nx +− ∠ = − − − −

⋯⋯⋯ .

15) Evaluate 6

2

1

1

xdx

x

−+∫

16) Solve 2 2( ) 2x y dx xydy+ = 17) Solve 2 3

2 1

dy x y

dx y x

+ +=+ +

SECTION-C (5 × 7 = 35)

18) Find the equation of the pair of tangents drawn from (3, 2) to the circle 2 2 6 4 2 0x y x y+ − + − = .

19) Find the equation of the circle passing through the points of intersection of the circles2 2 8 6 21 0x y x y+ − − + = , 2 2 2 15 0x y x+ − − = and the point (1, 2).

20) Find the equation of the circle passing through the origin and coaxial with the circles 2 2 6 4 8 0x y x y+ − + − = and 2 2 2 4 0x y x y+ − + + = .

21) Find the pole of the line 2 0x y+ + = with respect to the parabola 2 4 2 3 0y x y+ − − =

22) Evaluate 3sin cos 7

sin cos 1

x xdx

x x

+ ++ +∫

23) Evaluate 1/4

1/2 1

xdx

x +∫

24) Find the area enclosed by the curves 3y x= and 26y x x= − .

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Maths-2B Previous question papers [Intermediate Education, Andhra Pradesh]