2nd puc question papers of mathematics 1997-2010

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SECOND PUC ANNUAL EXAMINATION APRIL–1997 MATHEMATICS

Time: 3 Hours ] [ Max. Marks: 100

I Answer all questions. 12 x 1 = 12

1. If ab and ba then prove that a = b.

2. If 2 3 2 x 2 4 1

,7 5 y 1 5 7 10

find x and y

3. In the set of all non–zero real numbers, with the operation “” defined by a b ab

,2

find the identity

element.

4 Find the direction cosines of the vector 2i j 2k

5. Find the length of the tangent from (–1,–3) to the circle x2 + y2 – 2x – y – 7=0.

6. Find cos–1[cos100] + sin–1[sin100].

7. If x=cos+isin, find 1

xx

8. If 2

2

1f x x ,

x find f 1

9. Evaluate sinx cosx

dx.sinx cosx

10. Evaluate 5

2

0

1dx.

x 25

11. Find the eccentricity of the hyperbola x2 – y2 = 1. 12. Eliminate c from y = cx + 5 and form the differential equation.

II Answer all questions. 12 x 2 = 24

13. Prove that If p is prime and pab then prove that pa or pb.

14. If 3 1

A ,1 2

find A2 – 6A – 7I

15. Find the area of the parallelogram whose adjacent sides are i j k and 2i j k .

16. Prove that the identity element in a group is unique.

17. If is a binary operation defined on R, the set of reals defined by a b = 2 2a b , show that is

associative. 18. Find k, so that the line 3x + y + k = 0 may be a tangent to the circle x2 + y2 – 2x – 4y – 5 = 0.

19. Find the equation of the ellipse in the form 2 2

2 2

x y1 a b

a b , given that the distance between the foci

is 8 and the distance between the directrices is 32.

20. Find the modulus and amplitude of the complex number 1 + cos + isin . 21. Find the points on the curve x2 + y2 = 25 at which the tangents are parallel to the x–axis.

22. Evaluate e 1 x 1

e x

x edx

x e

23. Differentiate 21 tan xe

w.r.t. x.

24. Show that a. i i a. j j a. k k a

III 25. Find the G.C.D of 506 & 1155. Express it in the form of 506a + 1155b Show that a and b are not unique. 4


26. If

x 2 3 3 1 2 5 3 3

5 y 2 4 2 5 19 5 16 ,

1 1 z 2 0 3 1 3 0

find x, y and z. 4

OR

Prove that

2 2

2 2

2 2

1 b c b c

1 c a c a a b b c c a

1 a b a b

27. With usual notations prove that a b b c c a 2 a b c 4

28. If a b = a + b + 3 for all a, b Z, the set of integers, prove that (Z, ) is an abelian group. IV

29. Derive the condition for two circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2

x + 2f2y + c2 = 0 to cut

orthogonally 4 30. Find the equations of the circles which cut the circles x2 + y2 – 6y + 1 = 0 and x2 + y2 – 4y + 1 = 0 orthogonally and touch the line 3x + 4y + 5 = 0 4

31. Find the condition that the straight line y = mx + c may be a tangent to the hyperbola 2 2

2 2

x y1

a b 4

V

32. Solve for x: sin–1x + sin–1 2

2x .3

4

OR

Find the general solution of the equation 2 3 cos sin 1.

33. Find the cube roots of 1 i 3 and represent them in the Argand diagram. 4

VI 34. Differentiate loge

x with respect to x from first principles. 4 OR

If 2

2 2

1 t 2tx and y ,

1 t 1 t

find

dy

dx in terms of x and y.

35. If y=ex logx, prove that 2

2

d y dyx 2x 1 x 1 y 0

dxdx 4

36. In the curve xmyn = am + n, prove that the subtangent at any point varies as the abscissa of the point. 4 OR

Prove that among all rightangled triangles of a given hypotenuse, the isosceles triangle has the maximum area.

VII

37. Integrate 3cosx sinx

4cosx 3sinx

w.r.t x. 4

38. Prove that a a

o o

f x dx f a x dx and hence prove that 2 2

2 2

0 0

sin x dx cos x dx .4

4

OR Find the area enclosed between the curve y = 11x – 24 – x2 & the line y = x.

39. Solve by the method of separation of variables: 2dy dyy x a y .

dx dx

4

VIII

40. Find the equation of the circle such that the length of the tangent from (2, 3), (4, –4) & (2, –1) are

respectively 6, 5 and 2 units. 4

***********************


SECOND PUC ANNUAL EXAMINATION OCTOBER –1997

MATHEMATICS Time: 3 Hours ] [ Max. Marks: 100

I Answer all questions 12 x 1 = 12 1. Verify whether 48 and 65 are relatively prime or not.

2. Find the value of the determinant:

243 156 300

81 52 100

3 0 4

3. In the group, G = {1, 2, 3, 4} under multiplication modulo 5, find (3 x 4–1)–1.

4. Find the magnitude of 4a 3b where a 2, 3, 5 and b 1, 2, 3 .

5. Find the radius of the circle 4x2 + 4y2 – 16y – 17 = 0

6. Evaluate: 1 7cos tan .

4

7. Find the modulus of 2 i

.1 i

8. Find the derivative of alog xa w. r. t. x.

9. Integrate 1

1 x w.r.t x.

10. Evaluate 2

4

cot x dx.

11. Write the sum of the focal distances of any point on the ellipse 2 2x y

1.16 9

12. Find the order and degree of differential equation

32

2

d y dy1 3 0

dxdx

II Answer all questions. 12 x 2 = 24

13. For all n z show that n(n + 1)(n + 5) is divisible by 6.

14. Solve for x and y : 3 4 x 10

.9 2 y 2

15. Find the projection of b on a if a 3i j k and b i 3j 5k.

16. If every element of a group G is its own inverse, prove that G is abelian.

17. On the set of all rationals, is defined by ab

a b5

. Solve x 3 = 2–1.

18. Find the value of k, if the circles x2 + y2 + 5x – 2y – 3 = 0 and 2x2 + 2y2 + 5y + k = 0 cut orthogonally. 19. If y = 2x + c is tangent to y2 =8x, find c. Also find the point of contact.

20. Express 1 1

i2 2 in polar form.

21. If s = t3 – 6t2 + 9t + 8 with usual notations, find (i) the initial velocity, and (ii) when the body will be at rest momentarily.

22. Evaluate 2

dx

8 6x 9x

23. Solve the differential equation x ydy1 e

dx

.

24. If G is the centroid of a triangle ABC then show that GA GB GC 0

III


25. Define congruence relation. Show that congruence relation is an equivalence relation on the set of integers 4

OR

Solve

x 1 x 2 3

3 x 2 x 1 0

x 1 2 x 3

26. The position vectors of the points A, B, C, D are 3i 2j k, 2i 3j 4k, i j 2k, 4i 5j k respectively.

If the four points lie on a plane, find . 4

27. Show that the set G z z cos isin is an abelian group under multiplication. 4

IV

28. Derive the equation to the radical axis of the circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and

x2 + y2 + 2g2x + 2f2y + c2 = 0. Prove that the radical axis of the circles is perpendicular to the line of centres of the two circles. 4 29. Find the equation of the circle which passes through the point (2, 3), has its centre on the line x + y = 4 and cuts orthogonally the circle x2 + y2 – 4x + 2y – 3 = 0. 4

OR Find the equations of parabolas with latus rectum joining the points (4, 5) and (12, 5). 30. Find the centre, foci and equations to the directrices of the conic 9x2 – 4y2 – 36x–8y–4=0. 4

V

31. Solve for x: 1 1 2tan x 2cot x

3

4

OR

Find the general solution of sin 11xsin 4x + sin 5xsin 2x = 0

32. If cos+cos=sin+sin=0 show that

(i) cos 2 + cos 2 = 2cos ( + + ) (ii) sin 2 + sin 2 = 2sin ( + + ) 4

VI 33. Differentiate ‘sec x‟ w.r.t x from first principles.

OR

Find dy

,dx

if logx xy tanx x .

34. If 1mcos xy e ,

prove that (1 – x2)y2 – xy1 – m2y = 0 4

35. Find the angle of intersection of the curves x2 + y2 = 13 and x2 + y2 – 8x + 3 = 0 at the point of intersection. 4

OR

Find the maximum value of xe–x.

VII

36. Inegrate the following w.r.t. x (i) 2

2x 1

x 3x 2

(ii) x sec2 2x. 4

37. Evaluate 2

0

asinx bcosxdx

sinx cosx

4

OR

Find the area bounded by the curves y2 = 6x and x2 = 6y.

38. Solve: x 2logx 1dy

.dx siny ycosy

4

VIII

39. Prove by vector method that sin (A + B) = sin Acos B + cos Asin B 4

***********************




I Answer any ten questions 10 x 1 = 10

1. The equation 4x 3(mod 8) has no solution. Why?

2. If

1 0 0 x 1

0 y 0 1 0 ,

0 0 1 z 1

find x, y and z.

3. On the set of integers is defined by a b = a + b + 2. Find the inverse of 4.

4. If a i 2j, b j 2k, find a unit vector along the vector 3a 2b.

5. Find the equation to the common tangent of the touching circles

x2 + y2 – 2x + 6y + 6 = 0 and x2 + y2 – 5x + 6y + 15 = 0. 6. Find the eccentricity of the hyperbola x2 – y2 = a2

7. Prove that 1 13 3tan sin .

4 5

8. Find the modulus and amplitude of 1 – i.

9. If y = x tanx find dy

at xdx 4

.

10. If y log cosx, prove that dy 1

tanx.dx 2

11. Evaluate sinx cosx

dx1 sin2x

12. Find 4

0

sec x tanxdx.

PART–II Answer any ten questions. 10 x 2 = 20 13. Find the number of positive divisors of 6615. 14. Solve by Cramer‟s rule: 3x + 4y = 7 and 7x – y = 6.

15. Find the unit vector perpendicular to a and b where a i j 2k and b 2i j k .

16. In a group (G, ), prove that identity element is unique. 17. Write the Multiplication Table of Z = {1, 2, 3, 4} mod 5. 18. Find k so that the circles x2 + y2 – 2x + 22y + 3 = 0 and x2 + y2 + 14x + 6y + k = 0 are orthogonal. 19. Find c so that y = x + c may be a tangent to the parabola y2 = 4x.

20. Find the general solution of sin2x 3 cosx .

21. If 1

x 2cos ,x

prove that one of the values of x is ei

22. The equation of the tangent to the curve y2=ax3+b at (2, 3) is y = 4x – 5. Find the values of „a‟ and „b‟. 23. The sum of two numbers is 48. Find the numbers, when their product is maximum.

24. Integrate with respect to x :

2

2 2

sec x tanx.

sec x tan x

25. Form the differential equation of family of straight lines whose y intercept is equal to slope of the lines.

PART–III

A. Answer any four of the following 4 x 4 = 16 26. Find (506, 1155) and express it in the form 506m + 1155n. Also show that the expression is not unique.

27. Find the inverse of the matrix

cos sin 0

A sin cos 0

0 0 1


28. Show that

2

2 2 2 2

2

a ab ac

ab b bc 4a b c .

ac bc c

29. If a 3, 1,4 , b 1,2,3 c 4,2, 1 , Find a x b x c

30. Q is the set of all rational numbers and is defined by ab

a b2

for all a,bQ. Prove that (Q, ) is an

abelian group.

31. Prove that a non–empty subset H of a group (G, ) is a sub–group of G, if and only if for all

a, b H, a b–1 H.

B. Answer any three questions 3 x 4 = 12 32. Prove that the equation x2 + y2 + 2gx + 2fy + c = 0 represents a circle. Find its centre and radius. What is

the form of equation of a circle passing through the origin? 33. Find the equation of the circle passing through (1, 2) having its centre on the line 2x + y = 1 and cutting orthogonally the circle x2 + y2 – 2x + 1 = 0 34. Find the focus, vertex and equation of directrix of the parabola 9x2 – 6x + 36y + 19 = 0

35. Derive the equation of the ellipse in the form 2 2

2 2

x y1

a b (a > b).

36. Find the equation of the hyperbola in the form 2 2

2 2

x y1

a b given that the distance between foci is 10 and

the length of the latus rectum is9

2.

37. If 1 1 1sin x sin y sin z ,2

prove that x2 + y2 + z2 + 2xyz = 1. 4

OR

Find the general solution of the equation 3 cot x 1 2cosec x.

38. Prove that (cos + i sin )n = cos n + i sin n for integral values of n 4 OR

Solve the equation 3z 1 i 3 and represent the roots in an Argand diagram.

D 39. Differentiate tan–1 x w.r.t. x from first principles. 4

OR

If x 1 y y 1 x 0 where xy, show that

2

dy 1.

dx 1 x

40. If x = a( + sin ), y = a(1 – cos ), prove that 2 4

2

d y sec / 2.

4adx

4

OR

If 1msin xy e ,

prove that (1 – x2)y2 – xy1 – m2y = 0

41. Find the angle between the curves y = 6 + x – x2 and y(x – 1) = (x + 2) at (2, 4).What conclusion can be drawn from this? 4

OR

The volume of a spherical ball is increasing at the rate of 4 cc/sec. Find the rates of increase of radius

and of the surface area when the volume is 288.c.c. E. Answer any three questions 3 x 4 = 12

42. Evaluate 2 2

1dx.

4cos x 9sin x

43. Evaluate 2 2a x dx.

44. Prove that 2

2

0

xsinx.

41 cos x


45. Find the area bounded by the curve y = x2 – 1, the x–axis and ordinates at

x = 0 and x = 2.

46. Solve the equation x y 2 ydye x e

dx

by the method of separation of variables.

PART–IV

Answer all the questions. 47. Eliminate the arbitrary constant and form the differential equation of x2 – y2 = c2. 1

48. The focus of a standard ellipse is (3, 0) and its eccentricity is 3

.5

Find its major axis. 1

49. Prove that 2a 3b a 4b 5 a b . 2

50. Prove that cis cis = cis ( + ) 2 51. Show that, of all the rectangles of a given perimeter, the square has the maximum area. 4

***********************

SECOND PUC ANNUAL EXAMINATION OCTOBER–1998 MATHEMATICS


PART–I I Answer any ten questions 10 x 1 = 10

1. If ab and bc Prove that ac.

2. If 1 2 1 4

A , B ,3 1 3 2

find |AB|.

3. In the set of all non–negative integers is defined by a b = ab. Prove that is not a binary operation.

4. If the vectors ˆ ˆ ˆ ˆ ˆ ˆ2i j 6k and 3i 6j k are orthogonal find .

5. Find the radius of the circle x2 + y2 – 2x cos – 2y sin – 1 = 0.

6. Which conic does the equations x = a sec , y = b tan represent?

7. Prove that 1 1 1cosec x sin ,1 x 1.

x

8. Write the polar form of the no. „1‟.

9. Differentiate 3 x1 2x log sinhx 2 w.r.t x

10. If x = log (log y), find dy

.dx

11. Evaluate:

sinxlogx.cos x dx

x

12. Evaluate: 2

1

x

0

x.e dx.

PART–II Answer any ten questions: 10 x 2 = 20

13. If (a, b) = 1, ac and bc, prove that abc. 14. In a third order determinant, if two rows are identical, prove that the value of the determinant is zero.

15. Find area of the parallelogram whose diagonals are 2i 3j k and i 4j k

16. Prove that in a group (G, ), if a b = a c, then, b = c where a, b, c are elements of the group.

17. On the set of all rational numbers, if ab

a b ,5

prove that is associative.

18. A and B are points (6, 0) and (0, 8). Find the equation of the tangent at the origin O to the circumcircle

of OAB.


19. Prove that the sum of the focal distances of any point on the ellipse 2 2

2 2

x y1

a b is equal to the length of

the major axis.

20. Find the general solution of the equation 3 cosec2 – 8 cosec + 4 = 0

21. Prove that, with the usual notations cis cis = cis ( + ).

22. Differentiate

3 5 / 2

3

x 3 3x 1

2x 5 x 3

w. r. t. x.

23. Show that the curves y = x3, 6y = 7 – x2 cut orthogonally at (1,1). 24. Show that the tangents to the curve y = x3 at x = 1 and x = –1 are parallel to each other.

25. Evaluate cos x dx.

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Find the number of positive divisors and sum of positive divisors of 2304. 27. Solve by Matrix method: 3x + y + 2z = 3, 2x – 3y – z = –3 and x + 2y + z = 4

28. Prove that

2 2

2 2 2 2 2

2 2

a bc ac c

a ab b ac 4a b c .

ab b bc c

29. If a i 2j 3k, b 2i j k, c i 3j 2k, find a unit vector r to a in the same plane as b andc

30. Prove that the set given by

x x x RM

x x x 0 is an abelian group with respect to matrix multiplication.

31. Prove that S = {1, 3, 7, 9} form an abelian group under x mod 10.

B Answer any three questions: 3 x 4 = 12 32. Find analytically equation of the tangent at (x1, y1) on the circle x2 + y2 + 2gx + 2fy + c = 0 33. Find the Radical Centre of the following circles x2 + y2 + 4x + 2y + 1 = 0

2x2 + 2y2 + 8x + 6y – 3 = 0, x2 + y2 + 6x – 2y – 3 = 0 34. Derive the equation of the parabola in the form y2

= 4ax. 35. Find the equations of the tangents to the ellipse 9x2 + 25y2 = 225 at the ends of a latus rectum. 36. Find centre, foci, directrices of the hyperbola x2 – 3y2 – 4x – 6y – 11 = 0 C

37. Prove that 1 1 11 12tan cot 7 2tan

8 5 4

4

OR

Find the general solutions of cos 2 = cos + sin

38. Simplify:

4

5

cos isin

sin icos

4

OR

Find the fourth roots of 3 i

2 2 and represent them in the Argand diagram.

D 39. Differentiate tan–1x from first principles. 4

OR

If sin y = x sin (a + y) prove that 2sin a ydy

.dx sina

40. If loge y = m tan–1 x, prove that 2

2

2

d y dy1 x 2x m 0.

dxdx 4

OR

If x = a sin2t (1 + cos 2t), y = a cos2t (1 – cos 2t), prove that 2

2

d y 1at t .

a 4dx


41. Show that for the curve x5

= ay6, the square of the sub–tangent varies as the cube of the sub–norma 4 OR

The law of motion of a particle is s2 = at2 + 2bt + c. show that acceleration is given by 2a v

A ,s

where

v is the velocity, s is the displacement of the particle. E Answer any three questions: 3 x 4 = 12

42. Evaluate

1

2

tan xdx.

1 x

43. Evaluate: 1

dx.1 tanx

44. Prove that

2

2

0

xdx .

1 sin x 2 2

45. Prove by the method of integration, that the area of the circle (x – h)2 + (y – k)2 = a2 is a2. 46. Solve by the method of seperation of variables :

:cosylog ( sec x + tan x ) dx = cos xlog ( sec y + tan y ) dy. PART–IV

Answer all the questions:

47. Write the modulus and amplitude of

i4e . 1

48. If a b , Prove that a b is perpendicular to a b .

49. Find the points on the parabola y2 = 2x, where the focal distance is 5

.2

2

50. Find the minimum value of the function 2 250x .

x 2

51. If

5 2A ,

3 1 find A–1 using Calyley–Hamilton theorem. 4

***********************


Time: 3 Hours] [ Max. Marks: 100

PART–I (i) Answer any ten questions. 10 x 1 = 10

1. Prove that 7100 1(mod 8)

2. If 1 3

A ,2 0

write adj (A).

3. If is defined by a b = 1 + ab, show that it is a binary operation on the set of rationals.

4. Find the direction cosines of the vector 3i 4j 5k.

5. Find the power of the point (1, 5) w.r.to the circle x2 + y2 + 4x + 2y – 3 = 0. 6. What is the angle between the asymptotes of a rectangular hyperbola?

7. Show that 1 11 42tan tan .

2 3

8. Express 1 i

1 i

in the form a+ib.

9. If

1 tanxy tanh

2, find

dy.

dx

10. Differentiate x2sin–1 x w.r.t x.

PRECIOUS ACADEMY, MYSORE. CET COACHING CLASS-CONTACT: 9964305558/9844204227/9880986967Page | 10

11. Evaluate:

31

2

tan xdx.

1 x

12. Evaluate: x

0

e dx.

PART–II Answer any ten questions. 10 x 2 = 20 13. Find the least positive reminder when 64 x 65 x 66 is divided by 67.

14. Solve for x and y, given 3 4 x 10

.9 2 y 2

15. Find the projection of the vector 2i 3j 4k on the vector 2i 5k.

16. Define a semigroup. Give an example of a semigroup which is not a group.

17. In a group (G, ), prove that (a b)–1 = b–1 a–1.

18. Find the equation of the tangent to the circle x2 + y2 +3x – 7 = 0 at (–3, –2) on it. 19. Find the eccentricity of the hyperbola given that the distance between its foci is 32 and its distance

between the directrices is 8. 20. Find the general solution of cos x + cos 2x = 0.

21. Express i 3 in the polar form.

22. Find the subtangent and subnormal to the cure y = x3 + x2 – 11 at (2, 1). 23. Show that x3 – 6x2 + 12x–3 has neither a maximum nor a minimum at x = 2.

24. Evaluate: 2

cosxdx.

4 sin x

25. Find the area bounded by the x – axis and the curve y = 6x – x2 – 5. PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Prove that prime numbers are infinite.

27. Prove that

2

2

2

1 a bc

1 b ca a b b c c a a b c .

1 c ab

28. Verify Cayley–Hamilton‟s theorem for the matrix 2 1

A .1 2

29. Show that the points A(–6, 3, 2), B(3, –2, 4), C(5, 7, 3) and D(–13, 17, –1) are coplanar.

30. Prove that the set Q of rationals other than 1 forms an Abelian group w.r.t.a binary operation. defined

by a b = a + b + ab. 31. Show that a non–empty subset H of a group G is a sub–group of G, if and only if

(i) a, b H, ab H (ii) a–1 H, a H.

B. Answer any three questions: 3 x 4 = 12 32. A circle passes through the point (2, 5) has its centre on 2x – y – 14 = 0 and cuts orthogonally the

circle x2 + y2 – 5x + 3y + 11 = 0. Find its equation. 33. Obtain the condition for the circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0 may cut orthogonally. 34. Find the equation of the tangent to the parabola y2 =16x which is parallel to the line 2x + 3y = 1.

Find also its point of contact. 35. Derive the equation of an ellipse whose axes are parallel to the co–ordinate axes, centre at (–3, 1) and

eccentricity being 1

2 and passing through (–1, 4).

36. Derive the equation of a hyperbola in the form 2 2

2 2

x y1.

a b

C

37. If 1 1 11 1 1cot cot cot

a b c

, prove that a + b + c = abc. 4

OR


Find the general solution of 3 tan 2sec 1.

38. Prove that mm m

2 2 12nn nm b

a ib a ib 2 a b cos tan .n a

4

OR

Find the fourth roots of 3 i and represent them in an Argand diagram.

D. 39. Differentiate „ elog x ‟ w.r.t. x from first principles. 4

OR

If x = 3 cos – 2 cos3 , y = 3 sin – 2 sin3 then show that

dycot .

dx

40. Differentiate (sinx)x +(x)sinx w.r.t. x. 4

OR

If 1msin xy e ,

prove that (1 – x2)y2 – xy1 – m2y = 0.

41. If the curves 2 2x y

1A B

and 2 2x y

1a b cut orthogonally then show that A – B = a – b 4

OR

Find the maximum area of the rectangle that can be inscribed in a circle of radius r. E. Answer any three questions: 3 x 4 = 12

42. Evaluate the following: (i)

x 1 e 1

x e

e xdx

e x (ii) 2xsec 3xdx.

43. Evaluate: 3sinx 5cosx

dx.sinx 2cosx

44. Show that 2

0

x tanxdx .

sec x cosx 4

45. Find the area of the circle x2+y2=9 by integration. 46. Solve by the method of separation of variables:(ey +1) cos x dx + ey sin x dy = 0

PART–IV

Answer all questions. 47. Form a differential equation for the function xy = c2 by eliminating the constant c. 1

48. Find the amplitude of i / 3

i / 2

e.

e

1

49. Find the sine of the angle between the vectors i 2j 3k and 2i j k. 2

50. Show that 1

xx is maximum at x = e. 2

51. Show that the circles 2 2 2 2x + y + 2x - 4y -4 = 0 and x + y -4x +4y +4 = 0 touch each other. Find also the

point of contact 4

***********************



Time: 3 Hours ] [ Max. Marks:100

PART–I (i) Answer any ten questions.

10 x 1 = 10

1. If ab and ac then prove that ab + c.

2. 1 2 1

A1 0 2

and

2 0 3B ,

3 1 4

find 2A – 3B.

3. Write the identity element in the group, G = {2, 4, 6, 8} under multiplication modulo 10.

4. Find the unit vector in the direction of the vector i j k

5. Define point circle.

6. Find the semi–latus rectum of the ellipse 2 2x y

1.16 12

7. Find the value of 1 11 1sin cos .

2 2

8. Find the amplitude of the complex number i 1.

9. If 2y x y show that

dy 1

dx 1 2y

10. If

1sin xy e , find

dy.

dx

11. Evaluate:

cosx dx

2 3sinx

12. Evaluate / 4

2

0

sec xdx.

PART–II

Answer any ten questions. 10 x 2 = 20

13. If a b modm and x is an integer, prove that a x b x modm

14. Solve by Cramer‟s Rule: 2x + y = 1 and x – 3y = 4

15. Find the volume of the parallelopiped whose co–terminus edges are i 3j 2k, 2i j 3k and i j k.

16. In a group (G,), prove that the inverse of every element is unique. 17. In the set Q1, which is the set of all rational numbers other than 1, find the identity element w.r.t. the

binary operation () defined by a b = a + b – ab, a, b Q1.

18. Find the value of for which the circles x2 + y2

+ 3x – 2y – = 0 and 3x2 + 3y2 – 6x + 9y – 2 = 0 intersect

orthogonally. 19. Find the equation of the parabola with focus at (–3, 2) and vertex (–2, 2). 20. Find the general solution of tan 5x = cot 3x.

21. Simplify:

3

4

cos4 isin4.

cos2 isin2

22 Show that the curves y = x3 + x + 1 and 2y = x3

+ 5x touch each other at the point (1, 3). 23. If the law of motion of a particle is s = t3 – 6t2 + 9t + 8, find its initial velocity.

24. Evaluate sin 2x cos 3x dx.

25. If y = a sec x + b tan x, form the differential equation, eliminating a and b.

PART–III A. Answer any four of the following 4 x 4 = 16 26. Prove that least divisor > 1 of any integer is a prime number. 27. Solve the following by using matrix method : 7x + 6y – 5z = 30, 3x – 4y + z = 0 and x + 2y – 3z = 10


28. Solve for x:

x 1 2 3

1 x 2 3 0

1 2 x 3

29. Prove that, a b b c c a 2 a b c .

30. Prove that the set, {1, 5, 7, 11} form an abelian group under multiplication modulo 12.

31. In a group G, prove that,(i) 1

1a a, a G

(ii) 1 1 1ab b a , a,b G.

B. Answer any three questions: 3 x 4 = 12 32. Derive the equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at (x1, y1) on it. 33. Find the radical centre of the following circles : x2 + y2 + 6x – 4y + 9 = 0, x2 + y2 + 8x – 6y + 7 = 0, x2 + y2 – 4x + 2y + 1 = 0 34. Define a parabola, Derive the equation of a parabola in the form y2 = 4ax, a > 0

35. Find the values of k for which the line x – 2y + k = 0 touches the ellipse, 2 2x y

1.20 11

36. Find the centre, foci and equations of the directricies of the hyperbola, 9x2 – 4y2 – 36x – 8y – 4 = 0. C

37. Solve for x, given 1 1sin x sin 2x .3

4

OR

Find the general solution of sin 2x + sin 4x + sin 6x = 0 38. Find all the cube roots of the complex number, –1 –i, and represent them in an Argand plane. 4

OR

If cos + cos + cos = 0 = sin + sin + sin , prove that

cos 3 + cos 3 + cos 3 = 3 cos ( + + ) and

sin 3 + sin 3 + sin 3 = 3 sin ( + + ) D 39. Differentiate ‘sec3x’, w.r.t. x from first principles. 4

OR

If n

2y x x 1 , prove that 2

2 2 2dyx 1 n y .

dx

40. Find dy

,dx

if 1 1 x 1 xy tan .

1 x 1 x

4

OR

Find 2d y

,dx

given x = et(cos t + sin t) ; y = et(cos t – sin t).

41. For the curve, ay2 =(x + b)3, prove that the square on the subtangent varies as the subnormal. 4 OR

A drop of oil spreads on water in the circular form at the rate of 36 sq.cm/sec. How fast the radius

increasing when the area is 49sq.cms. E. Answer any three questions: 3 x 4 = 12

42. Evaluate the following: (i) 1 sin2x dx (ii) 2

dx.

x 4x 20

43. Evaluate: dx

13 5cosx

44. Prove that

22

0

x tan xdx2

45. Find the area of the ellipse, 4x2 + 9y2 = 36, by the method of integration.

46. Solve: x y 2 ydye x e ,

dx

by separating the variables.

PART–IV


Answer all the questions.

47. Find the degree of the differential equation

12 22

2

dy d y1 4

dx dx

. 1

48. For a point circle, x2 + y2 + 2gx + 2fy + c = 0, prove that g2 + f2 = c 2 49. Prove by vector method that the diagonals of the Rhombus intersect at right angles. 2 50. Find two numbers whose sum is 12 and their product is maximum 2

51. Find the inverse of the matrix, 1 3

A2 1

using Cayley Hamilton theorem. 4

***********************



PART–I (i) Answer any ten questions. 10 x 1 = 10 1. Find the G.C.D of the numbers 65 and 117.

2. If is an imaginary cube root of 1, find the value of

2

2

2

1

1

1

3. If is defined on the set of reals R by 2 2a b a b , find the identity element in R, with respect to .

4. If the vectors ˆ ˆ ˆ3i mj k and ˆ ˆ ˆ2i j 8k are orthogonal, find m.

5. Find the equation of the tangent to the circle x2 + y2 = 10 at the point (3, 1) on it. 6. Find the equation of the directrix of the parabola y2 = –8x.

7. Find the value of 1 1sin sin .

3 2

8. Find the principal value of the amplitude of the complex number 3 i.

9. Define the differential co-efficient of a continuous function y = f(x) with respect to x.

10. If y = x3 2logx, find dy

.dx

11. Evaluate 2 3x sin(x )dx.

12. Evaluate: 4

20

dx.

16 x

PART–II


13. If pab, where p is prime number then Prove that pa or pb.

14. If 2 3

A0 4

and

1 5B ,

2 0 show that (AB)=BA.

15. Find the value of if the vectors 2i j 6kand 3i 6j k are orthogonal.

16. In a group G, if a G a–1 = a, show that G is abelian. 17. Define a subgroup of a group, and give an example of a subgroup. 18. Find the Radical axis of the circles 3x2 + 3y2 – 9x + 6y – 1 = 0 and 2x2 + 2y2 – 8x + 16y – 3 = 0.

19. Find the general solutions of the equation cos2 + cos 2 = 2.

20. If x = cos + isin , y = cos + isin and z = cos + isin , and x + y + z = 0 then, prove that

1 1 1

0.x y z


21. The perimeter of a rectangle is 100mts. Find the length of the sides when the area is maximum. 22. Find the length of the subtangent to curve y = x3 + x2 – 11 at the point (2, 1) on it.

23. Evaluate: 2

logxdx.

x

24. If the distance between the directrices of the ellipse 2 2

2 2

x y1 a b

a b is 9 times the distance between

their foci find its eccentricity.

25. Find the area bounded by the curve y = sin x, with x–axis in between the ordinates at x = 0 & x = 2.

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Find number of positive divisors and sum of positive divisors of the number 22869.

27. State Cayley–Hamilton theorem. Using this theorem, find the inverse of the matrix 3 2

.5 1

28. If a, b, c are all different and

2 3

2 3

2 3

a a 1 a

b b 1 b 0.

c c 1 c

prove that 1 + abc = 0.

29. Find the vector of magnitude 12 units which is perpendicular to the vectors

a 4i j 3k and b 2i j 2k.

30. If Q+ is the set of all positive rationals, prove (Q+ ) is an abelian group, where is defined by

2ab

a b .3

31. Show that the set of all fourth roots of the number 1, form an abelian group under multiplication.

B.Answer any three questions: 3 x 4 = 12 32. Find the equation of the circle which passes through the point (2, 3) and touches the line

2x – 3y – 13 = 0 at the point (2, –3). 33. Obtain the condition for the circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2

= 0 to intersect orthogonally. 34. Show that the equation x2 – 6x – 12y – 15 = 0 represents a parabola. Also find its vertex, focus and

directrix.

35. Derive the equation of the ellipse in the standard form 2 2

2 2

x y1 a b .

a b

36. If the foci of the hyperbola 2 2

2

x y 1

81 25k and the foci of the ellipse

2 2x y1

16 7 coincide,

prove that k2 = 144. C



OR

Find the general solution of the equation tan sec 3.

38. Find the cube roots of the complex number 3 i 3, and represent them in Argand diagram. 4

OR

Prove that

4

5

cos isinsin 4 5 icos 4 5

sin icos

D. 39. Differentiate ‘cotx’ with respect to x from the method of first principles. 4

OR

If x = a( – sin ), y = b(1 – cos ), prove that the value of 2

2 2

d y bat is .

2dx a


40. If y = a cos (log x) + b sin (log x), prove that x2y2 + xy1 + y = 0 4 OR

If 2 1 1 xy sin cot .

1 x

prove that dy 1

.dx 2

41. A spherical balloon is being inflated so that its volume is increasing uniformly at the rate of 30 c.c/mt.

How fast is its surface area increasing when the volume is 36 c.c? 4 OR

If the curves 2 2

2

x y1

4a and y3 = 16x intersect each other orthogonally, then prove that 2 4

a .3

E. answer any three questions: 3 x 4 = 12

42. Evaluate the following: (i)

31 2

dx.

sin x 1 x (ii) sin5x sin11xdx.

43. Evaluate

2

3x 2 dx

4x 4x 5

44. Evaluate: /4

30

dx.

1 cot (2x)

45. Find the area enclosed between the parabola x2 = 4y and the line x = 4y – 2.

46. Solve by the method of separation of variables y 2 2 ydye x x e .

dx

PART–IV Answer all the questions. 47. Express „1‟ in the polar form. 1


322 4

2

2

d y dya 1

dxdx

1

49. Prove that a b c 0 2

50. Prove that the difference of the focal distances of any point on the hyperbola 2 2

2 2

x y1

a b is equal to

length of the traverse axis. 2 51. Find the equation of the circle passing through the point (–2, 3) and having length of the tangent from

(5, 1) as 46 and cuts orthogonally with circle x2 + y2 – 4x + 2y – 23 = 0 4

***********************




PART–I

Answer any ten questions. (10 x 1 = 10) 1. Find the digit in the unit‟s place of 951.

2. For what value of x,

1 6 7

2 x 14 0 ?

0 1 4

3. On the set of Real numbers is defined by a b ab. Is a binaryoperation?

4. If ˆ ˆ ˆ ˆ ˆ ˆ3i mj k and 2i-j-8k are orthogonal then find „m‟

5. Find the radical axis for the circles.2x2 + 2y2 – 4y – 7 = 0 and x2+y2+3x+4y–2=0 6. Find the eccentricity of the hyperbola x2 – y2 = 1.

7. Evaluate cos–1 [cos (1540)].

8. Find the real part of the complex number 2 cis .3

9. If

2

1y

2cos

x

find dy

dx

10. Differentiate cosh [log (tanx)] w.r.t. x.

11. Evaluate sin x

dx.x

12. Evaluate: 3

20

dx.

x 9

PART–II

Answer any ten questions: 10 x 2 = 20 13. If (a, b) = 1 and (a, c) = 1 prove that (a, bc) = 1.

14. If 1 1

2A B0 1

and

0 1A 3B

1 0

find A and B.

15. Find the unit vector perpendicular to both the vectors 3i j 2k and i j 3k.

16. Define a semi–group, give an example of a semi–group which is not a group.

17. In a group (G, ) if a G, a =a–1, prove that G is abelian. 18. Find the value of k so that the line 3x+y+k=0 is a tangent to circle x2+y2 = 10. 19. Find the length of the major axis of the ellipse 9x2 + 16y2 – 36x + 96y +36=0. 20. Find the general solution of tan 2x.tan x = 1.

21. If z = 3 + 2i, evaluate 2 2

z z z z .

22. Find the equation of the normal to the curve, 3

2 xy

2a x

at the point (a, a).

23. Find the maximum value of log x

x if it exists.

24. Evaluate: 2

sinx dx.

4 9cos x

25. Evaluate: / 3

/ 6

dx

1 tanx

.

OCTOBER–2000


PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Prove that relation “Congruence modulo m” is an equivalence relation.

27. Show that, 3

a b c 2a 2a

2b b c a 2b a b c .

2c 2c c a b

28. Find the inverse of 2 3

A2 5

using Cayley Hamilton theorem.

29. If i j k, 4i 2j 9k, 5i j 14k and 3i 2j 7k are the position vectors of four coplanar points, find the

value of .

30. If Q+ is the set of all positive rational numbers and is defined by ab

a b2

a, b Q+,

prove that (Q+ , ) is an abelian group. 31. If a and b are any two elements of a group G, then prove that equations ax = b and ya = b have unique solution in G. B. Answer any three questions: 3 x 4 = 12 32. Define the radical axis. Show that the radical axis of two circles is a straight line perpendicular to the

line of centres. 33. Find the equation of the circle, such that the length of tangents from (–1, 0), (0, 2), (–2, 1) to it are

respectively 3, 10 and 3 3.

34. Find the equation of the tangent to the parabola, y2=8x and parallel to the line 2x – 3y + 1 = 0, also find

the point of contact.


2 2

x y1 a b.

a b

36. Find the centre, eccentricity, foci and directrices of the hyperbola

2 2x 2 y 1

1.16 4

C

37. Prove that, 1 1 14 3 27cos sin tan .

5 1134

4

OR

Find the general solution of sinx 3 cosx 2.

38. For any integer n, prove that, n

1n n2

n1 i 1 i 2 cos .

4

4

OR

If x = cis and y = cis , prove that (i) 2 3

3 2

x y2cos 2 3

y x (ii)

2 3

3 2

x y2isin 2 3 .

y x

D. 39. Differentiate e2x w.r.t x, using first principles. 4

OR

If 1 cosx

y log ,1 cosx

prove that

dy2cosecx.

dx

40. If x2 + 2xy + 3y2 = 1, prove that

2

2 3

d y 2.

dx x 3y

4

OR

If 1y cos mcos x prove that (1 – x2)y2 – xy1 + m2y = 0

41. If x = a( – sin ) and y = a(1 – cos ), show that y is maximum when = 4 OR


The volume of a spherical ball is increasing at the rate of 4 cc/sec. find the rates of increase of radius

and of the surface area, when the volume is 288.c.c. E. Answer any three questions: 3 x 4 = 12

42. Find : (i)

x 2dx.

x 3 x 1

(ii) 1 sin2x dx.

43. Evaluate: 1

dx.5 4cosx

44. Prove that a a

0 0

f(x)dx f(a x) dx, and hence evaluate 2

0

cosxdx.

sinx cosx

45. Find the area enclosed between the curve x2 = 6y and y2 = 6x.

46. Solve by the method of separation of variables, 2 2x 1 y dx y 1 x dy 0.

PART–IV


47. Find the number of incongruent solutions of 6x 12(mod 9). 1 48. Solve x dy + y dx = dx + dy. 1

49. Find the area of the parallelogram when the diagonals are 3i 2j 4k and 4i 2j 2

50. Evaluate 9

3 i using De Moivre‟s theorem. 2

51. Prove that the sum of the focal distances of any point on the ellipse 2 2

2 2

x y1

a b is the constant

equal to 2a. 4

***********************


Time: 3 Hours ] [Max. Marks: 100

PART–I (i) Answer any ten questions. 10 x 1 = 10

1. If 28 a(mod 13), find a.

2. If a b 1 0

A , B ,c d 0 1

find adj (AB).

3. Find the identity element in the set of all positive rationals Q+ defined by ab

a b ,2

a, b Q+.

4. If a 1,0,0 b 1,1,0 ,c 1,0,1 find the magnitude of 2a 3b c

5. Find the equation of the radical axis of the circles x2 + y2 = 5 and x2+y2–3x+4y=1. 6. Find the equation of the axis of the parabola (y – 2)2 = 8(x – 1).

7. Find sin–1 [sin (–600)].

8. Find the amplitude of cis

3 .

cis4

9. Differentiate elog cosecx cot x w.r.t. x

10. If dy

y log tanx, find .dx


11. Evaluate: 1

dx.1 cosx

12. Evaluate: 3

2

0

dx

x 9.

PART–II

Answer any ten questions: 10 x 2 = 20 13. Find the remainder obtained when 135 x 74 x 48 is divided by 7.

14. Find the Eigen values of the matrix 2 3

.0 1

15. The position vectors of A, B and C respectively are ˆ ˆ ˆ ˆ ˆ ˆi j k, 2i+j-k and 3i-2j-k , find the area of the

triangle ABC. 16. G = {2, 4, 6, 8} is a group under multiplication mod 10. Prepare multiplication mod 10 table and hence find the identity element.

17. In a group G, (ab)2 = a2b2, a, b G, prove that G is an abelian group. 18. Find the equation of a circle two of whose diameters are x + y = 6 & x + 2y = 4 and whose radius is 10. 19. Find the eccentricity of the hyperbola x2 – 3y2 – 4x – 6y – 11 = 0.

20. Prove that 1 11 122tan tan .

5 5 2

21. Find the smallest positive integer n such that n

1 i1.

1 i

22. If 2

1

2

1 xy cos ,

1 x

prove that

2

dy 2.

dx 1 x

23. Find the length of subtangent for the curve x5 = ay6.

24. Evaluate:

dx.

x xlogx

25. Solve the differential equation (1 + y2) dx + xdy 0

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. (a) Prove that the number of primes is infinite (b) Find the digit in the unit place of 7123.

27. Prove that

x p q

p x q x p x q x p q .

p q x

28. If

1 2 2

A 0 2 3

3 2 4

and

2

B 2 ,

0

find (A–1 B).

29. Prove that: a b b c c a 2 a b c .

30. Define subgroup of a group. Prove that H = {1,2,4} is a subgroup of the group G = {1, 2, 3, 4, 5, 6}

under multiplication (mod 7).

31. Prove that G = {cos + i sin ‟‟ is real} is an abelian group under multiplication. B. Answer any three questions: 3 x 4 = 12 32. Derive the condition for the line, y = mx + c may be tangent to the circle x2 + y2 = a2. Also find the point

of contact. 33. Find the radical centre of the circles x2 + y2 – 8x + 3y + 21 = 0, x2 + y2 – 7x + 5y + 28 = 0 and x2 + y2 + x + y – 16 = 0. Also find the length of the tangent from this point to the circles. 34. Find the equations of parabolas with (1, 5) and (1, 1) as the end points of latus rectum (LR). 35. Two ends of the major axis of an ellipse are (5, 0) and (–5, 0). If one of the foci lies on the line 3x – 5y = 9, find the equation of the ellipse.

36. Find the equations of tangent and normal to the hyperbola. 2 2

2 2

x y1

a b at a point (x1, y1) on it.


C

37. If cos–1 x + cos–1 y + cos–1 z = prove that x2 + y2 + z2 + 2xyz = 1 4 OR

Find the general solution of 3 1 cosx 3 1 sinx 2.

38. Prove that (cos + i sin )n = cos n + i sin n for integral value of n. 4 OR

Prove that (1 + cos + i sin )n + (1 + cos – i sin )n = n 1 n n2 cos .cos

2 2

39. Differentiate ‘cos2x’ with respect to x, from first principle 4 OR

If x 1 y y 1 x 0 and x y, prove that

2

dy 1.

dx 1 x

40. If x = a(sin – cos ), y = a(cos + sin ), prove that 2

3

2

d y 1cosec .

adx

4

If x = sin , y = sin m prove that (1 – x2)y2 – xy1

+ m2y = 0 41. A ladder 17 ft long leans against a smooth vertical wall. If the lower end is moving at the rate of 9 ft/min, find the rate at which the upper end is moving, when the lower end is 8 ft from the wall. 4

OR

The tangent to the curve y = x2 – 5x + 6 at the point x = 1 intersects the x–axis at A and the y–axis at B.

Show that the area of the triangle 25

AOB sq6

units, O being the origin.

E. Answer any three questions: 3 x 4 = 12

42. Find (i) 2

1dx.

x 4x 9 (ii) 1tan xdx.

43. Find: 3sinx 5cosx

dx.sinx 2cosx

44. Prove that 2

0

xdx

41 x 1 x

45. Find the area enclosed between the parabolas y2 = 4x and x2 = 4y. 46. Solve by separating the variables (y2 + y) dx + (x2 + x) dy = 0.

PART–IV


47. Find the order of the differential equation

32 2

2

dy d y1 7 .

dx dx

1

48. Find the angle between the asymptotes of the hyperbola x2 – y2 = a2. 1

49. Show that

x1

x

has either maxima or minima at 1

x .e

2

50. Find the imaginary part of 5

1 i 3 . 2

51. Show that the angle in a semicircle is a right angle. 4

***********************


SECOND PUC ANNUAL EXAMINATION SEPT–2001 MATHEMATICS

Time: 3 Hours] [Max. Marks:100 I Answer all questions 12 x 1 = 12

1. Find the value of x satisfying 28 x (mod 6)

2. Evaluate :

3 8 9

1 9 3 .

2 10 6

3. In a group of integers Z, an operation is defined by a b = a + b – 1, a, b, Z. Find the identity element.

4 Find a unit vector in the direction of ˆ ˆ ˆ3i 4j 5k.

5. Find the equation of the tangent to the circle x2 + y2 = 5 at the point (2, –1) on it. 6. Find the length of the latus rectum of 9x2 + 4y2 = 36.

7. Evaluate : 1 1

cos 2sin .3

8. If 1 2i

z3 i

then find z .

9. If 1 1 x

y tan1 x

prove that

2

dy 1.

dx 1 x

10. If 1 xy cosh e , find dy

.dx

11. Evaluate : 3tanxsec xdx .

12. Evaluate :

1x

0

xe dx.

II Answer any ten questions. 10 x 2 = 20 13. Verify whether 653 is a prime or not. 14. Solve the equations by Cramer‟s Rule : 3x + 4y =7, 7x – y= 6.

15. Find, if the vectors ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆi j 2k, 2i 3j 4k and i 2j k arecoplanar.

16. In a group (G, ), prove that the inverse of each element is unique.

17. On the set of integers Z, a b = ab + 1, a, b Z. Prove that is not associative. 18. One end of a diameter of the circle x2 + y2 + 4x – 6y + 12 = 0 is (–5, –1). Find the other end.

19. Find the eccentricity and equations of directrices of the ellipse

2 2x 2 y 3

1.32 64

20. Find the general solution of 4sin x cos x = 1.

21. If x = cos + i sin , y = cos + i sin . Then prove that 1

xy 2isin .xy

22. If x = a, a

y ,

show that dy y

0.dx x

23. Show that the curve y = x3 – 6x2 + 12x – 3 has neither maximum nor minimum at x = 2.

24. Evaluate : 3sin xdx.

25. Evaluate : 1

n

0

x 1 x dx.

Part–III A. Answer any four of the following : 4 x 4 = 16 26. Find (1) Unit digit of 7231.


(2) The +ve reminder when 34 x 67 x 1234 is divided by 19

27. Solve for

x 1 x 2 3

x : 3 x 2 x 1 0.

x 1 2 x 3

28. State Cayley–Hamilton theorem and verify it for the matrix 1 3

A .4 7

29. If ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆa 3i j 2k, b i 3j 4k, and c 4i 2j 6k , find a unit vector coplanar with a and b but

perpendicular to c .

30. Prove that G = {1, 2, 3, 4} is an abelian group under multiplication modulo 5.

31. Prove that a non-empty subset H of a group (G, ) is a subgroup of G, if and only if for all

a, b H, a b–1 H.

B. Answer any four of the following : 3 x 4 = 12 32. Find the length of the tangent from the point (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0. Also find the power of the point w.r.t. the circle. 33. Find the equation of the circle through the point (2, –1) and having its centre on the line 2x + y = 1 and

cutting orthogonally the circle x2 + y2 + 2x + 6y + 5 = 0. 34. Derive the condition for the line y = mx + c may be a tangent to the parabola y2 = 4ax. Also write the

(coordinates of ) point of contact. 35. An ellipse has its centre at (3, 1) and passes through (2, 0). If the axes are parallel to coordinate x and y

axes and eccentricity is 1

,2

find the equation of the ellipse.

36. Find the centre, foci and length of latus rectum of the hyperbola 9x2 – 4y2 + 18x – 8y – 31 = 0.

C.

37. Prove that 1 1 1 11 1tan tan cot 18 cot 3 .

7 8

OR

Find the general solution of sec x tanx 3.

38. Find the cube roots of 3 i and represent them on the Argand diagram.

OR

Prove that

n1 sin icos n n

cos n isin n .1 sin icos 2 2

D. 39. Differentiate cosec ax w.r.t. x using first principle method. OR

If 1 1

y sin 1 x 1 x ,2

then show that 2

dy 1.

dx 2 1 x

40. If y y xx e then prove that

2

dy 2 logx.

dx 1 logx

OR

If

1 1

m my y 2x

then prove that 2 22 1x 1 y xy m y.

41. Show that the curves y = x2 –3x + 3 and y = x3 –6x2 + 13x – 9 touch each other at (2, 1). Find the

equations of the common tangent and common normal. OR

A man 6 feet tall moves away from a source of light 20 feet above the ground level his rate of walking being 4 miles per hour. Find the rate at which (i) the length of his shadow increasing, (ii) the tip of his shadow moving. E. Answer any three questions: 3 x 4 = 12


42. Evaluate the following :

(i) 2

x 1dx.

x 2x 3

(ii)

2x 3

dx.x 1 x 2

43. Evaluate : dx

.3 2sinx cosx

44. Prove that 4

0

log 1 tanx dx log2.8

45. Find the area of the circle x2 + y2 = 22 by the method of integration. 46. Solve: (ey + 1) cos x dx + ey sin x dy = 0 by separating the variables.

Part–IV

47. Find the multiplicative inverse of 4 + 3i. 1 48. Define a differential equation. 1

49. Find the projection of 2i 3j 4k on 4i 7j 2k . 2

50. 2x + y – 3 = 0 is the Radical axis of the co–axal system of circles. One of them is x2 + y2 + 8x – 4y = 0 and the other passes through (3, 0). Find its equation. 2 51. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a

is 2a

.3

4

***********************



PART–I


1. Find the number of incongruent solutions of 6x 9(mod 15)

2. Find [A B] if 1 2

A3 1

and

1 0B .

1 0

3. Find the identity element in the set of real numbers where the binary operation defined by

2 2a b a b

4. If ˆ ˆ ˆ ˆ ˆ ˆa i 2j, b j 2k, c i 2k find a b c .

5. Find the length of tangent to x2 + y2 – 3x + y = 0 from A(–1, 1).

6. Give equation of conjugate hyperbola of the curve 2 2

2 2

x y1.

a b

7. Evaluate: sin–1(sin110).

8. Simplify: 16

cos i sin .8 8

9. Differentiate 4

1x 5

x w.r.t x.

10. Find dy

dx if 1 sin x

ey log e .


11. Evaluate: 2

dx.

4 x

12. Find / 2

0

1 cos2x dx .

PART–II

Answer any ten questions. 10 x 2 = 20 13. Find the non negative reminder when 37 x 423 x 625 is divided by 13.

14. Find AA| if 2 3

A .4 5

15. Find value of if i j k, 2i 3j 2k and i j 3k are coplanar.

16. Define a subgroup. Give an example of finite subgroup of a finite group.

17. Verify Associative law if is defined on R as ab=ab+1. a, bR. 18. Find equation of a circle with centre at (1, –2) if it passes through centre of x2 + y2 – 4x + 1 = 0.

19. Find foci of 2 2x y

1.9 4

20. If 1 1tan x tan y ,4

then show that x + y + xy = 1.

21. Find modulus and amplitude of 1 i 3 .

22 Find dy

dx if 1 1 cosx

y tan .1 cosx

23. Find equation of tangent to the curve x2 + 2y2 = 9 at (–1, 2) on it.

24. Evaluate: x

dx.

1 e

25. Evaluate: 2

0

cosxdx

1 sin x

.

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Prove that prime numbers are infinite.

27. If

1 x 1 1

1 1 y 1 0

1 1 1 z

where x0, y0 and z0, then show that 1

1 0 .x

28. Solve the following by matrix method:x + y – 2z = 0 , 2x – y + z = 2 and x + 2y – z = 2.

29. Find a unit vector perpendicular to both a i 2j 4k and b 2i 3j k. Also find angle between a and b .

30. Show that the set of all fourth roots of unity is an abelian group under multiplication.

31. On Q – {1}, (set of all rationals except 1) define as a ,b Q – {1}, a b = a + b – ab, is this structure a group? Justify. B. Answer any three questions: 3 x 4 = 12 32. Obtain the condition for circles x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0 to cut

orthogonally. 33. Find the equation of a circle with centre on 2x + 3y – 7 = 0 given that it cuts x2 + y2 – 10x – 4y + 21 = 0 and x2 + y2 – 4x – 6y + 11 = 0 orthogonally. 34. Derive the equation of parabola in the form of y2 = 4ax. 35. Find centre, focii and length of latus rectum of the curve 4x2 + 9y2 + 16x – 18y – 11 = 0 36. Find equation of hyperbola in standard form if the distance between its foci is 10 units and length of

latus rectum is 9

2 units.

C.

37 Solve for x: 1 1sin x sin 2x .3

4


OR

Find general solution of sin 7 sin 5 = sin 3 sin . 38. Evaluate: (1 + i)8 – (1 – i)8. 4

OR Solve: z7 – z4 – z3 + 1 = 0. Also represent cube roots of unity in Argand‟s diagram. D 39. Differentiate „sin–1 2x’ w.r.t. x from first principles. 4

OR

Find dy

dx , if

x ytan a .

x y

40. If x2–xy+y2=a2 then show that

2 2

2 3

d y 6a.

dx x 2y

4

OR

If y = a cos (log x) + b sin( logx) then show that x2y2 + xy1 + y = 0. 41. Find length of subnormal to the curve x3y2 = a5 at any point on it. Also show that length of sub–tangent varies directly as abscissa at that point.

OR

Show that y = sin x(1 + cos x) is maximum at x .3

Also find its maximum value.


42. Evaluate the following (i) 2

dx

x 3x 2 (ii) 2

2

sec xdx.

2 tan x

43. Integrate 1

2 3sinx cosx w.r.t. x.

44. Evaluate: x

0

e sinx dx .

45. Find area of circle x2 + y2 = 6 by integration.

46. Solve by separating variables: 2 dy1 x xy 5y .

dx

PART–IV Answer all the questions.

47. Prove that (1 + – 2)5 = –32. 1

48. Give equation of pair of asymptotes of 2 2

2 2

x y1.

a b 1

49. If a i j 2k, and b 3i j k, find the cosine of the angle between a and b . 2

50. Find the equation of a circle with (3, 5) and (–2, 2) as the ends of the diameter. 2

51. Find A–1 using Cayley–Hamilton‟s theorem if 2 4

A .7 3

4

***********************


SECOND PUC ANNUAL EXAMINATION SEPT–2002 MATHEMATICS

Time: 3 Hours] [Max. Marks: 100

PART–I Answer any ten questions. 10 x 1 = 10

1. If ab and ac then prove that ab + c.

2. Evaluate 4785 4787

.4789 4791

3. Find the identity element in the set of all +ve integers, If is defined by a b = a + b a, b I+.

4. If a 8, b 1, a b 15 find a b .

5. Find the radius of the circle with centre at (1, 2) and cutting orthogonally the circle x2 + y2 – 6x + 6y – 2 = 0.

6. Which conic does the equation x = a cos and y = b sin represent?

7. Find the value of 1 1a btan tan b.

1 ab

8. Find the amplitude of i 3 1.

9. If y = 2–x, find dy

.dx

10. If y = sin (loge x), prove that 21 ydy

.dx x

11. Evaluate: xe sinx cosx dx.

12. Evaluate: 2

1x

0

xe dx.

PART–II

Answer any ten questions 10 x 2 = 20

13. If a b(mod m) and c d(mod m) prove that ac bd(mod m)

14. If A is square matrix, prove that AA is symmetric.

15. Prove that i j j k k i 0.

16. Prove that in a group (G, ), if b a = c a, then b = c where a, b, c are elements of the group. 17. Write the multiplication table for the set of integers of modulo 5. 18. Find the value of k, for which the circles 2x2 + 2y2 – 3x + 6y + k = 0 and x2 + y2 – 4x + 10y + 16 = 0 cut orthogonally. 19. If the latus rectum of an ellipse is half the minor axis, find its eccentricity.

20. Find 11 4sin cos .

2 5

21. If is an imaginary cube root of unity, prove that (1 – + 2)4 = 16. 22. Find the point on the curve y2 = 3 – 4x, where tangent is parallel to the line 2x + y – 2 = 0.

23. Prove that in the curve x

ay e , the subnormal varies as square of y-ordinate.

24. Evaluate: x

1dx.

1 2e

25. From the differential equation for the family of circles passing through origin and having centers on the

x-axis.

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Find the G.C.D of the numbers a and b where a = 495, b = 243 express it in the form of ax + by. Show

that x and y are not unique.


27. If AX = B where

3 1 2 13 x

A 2 1 1 , B 3 , X y ,

1 3 5 8 z

find the value of x, y, z by using inverse of A.

28. Prove that: 3

a 2b 2c 3b 3c

3a b 2c 2a 3c 4 a b c .

3a 3b c 2a 2b

29. Prove that i x a x i j x a x j k x a x k 2a.

30. If a and b are any two elements of a group G under , then the equation y a = b, has unique solution in G.

31. Prove that the set x x . x R

Mx x .x O

is an abelian group w.r.t matrix multiplication.

B. Answer any three questions: 3 x 4 = 12 32. Find the equation of the tangent at (x1, y1) on the circle. x2 + y2 +2gx +2fy+c=0, using analytical method. 33. Prove that points A(1, 1), B(–2, 2), C(–2, –8) and D(–6, 0) are concyclic. 34. Find the equation of the parabola whose directrix is x–5=0, axis is y + 4 = 0 and latus rectum is 12.

35. Derive the equation of the ellipse in the form

2 2

2 2 2

x y1.

a a 1 e

36. Derive the condition for the line y sin = p – x cos may be tangent to the hyperbola 2 2

2 2

x y1.

a b

C.

37. Solve for x, given 1 1 2sin x sin 2x .

3

4

OR

Find the general solution of sin3 x + sin x. cos x + cos3 x = 1.

38. If and are roots of the equation x2 + 2x + 4 = 0, prove that 6 + 6 = 27 4 OR

Find the cube roots of 3 i.

D. 39. Differentiate ‘sec 3x’ w.r.t. x using first principle method. 4

OR

Differentiate 1 1 xtan

1 x

w.r.t. 1

2

2xtan .

1 x

40. If x = 3 sin 2 + 2 sin 3 and y = 2 cos 3 – 3 cos 2, prove that

32

2

secd y 1 2. .

524dx cos2

4

OR If y = cos (m tan–1 x), prove that (1 + x2)2y2 + 2x(1 + x2)y1 + m2y = 0. 41. Prove that the condition that the curves mx2 + ny2 = 1 and ax2 + by2 = 1, should interest orthogonally is

that 1 1 1 1

.a b m n 4

OR

A right circular cone has a depth of 40 cms and a base radius15cms. Water is poured into it at the rate of 24 c.c. per second.Find the rate of rise of water level and the rate of increase of theradius of the cone when the depth of water level is 16 cms.



42. Find the value of i) 2

cosxdx

16 7cos x ii) 2

1dx.

16 6x x

43. Evaluate:

x 2dx

2x 1 x 2

44. Prove that: a

2 20

1dx .

4x a x

45. Find the area of the ellipse 2 2x y

19 4

by the method of integration.

46. Solve x–1 cos2 y dy + y–1 cos2 x dx = 0 by separating the variables. PART–IV


47. Find the degree of differential equation

452 3

2

d y dy5 0.

dxdx

1

48. Express e(cos + i sin ) in the form a + ib. 1

49. If 5y log sinx find dy

dx 2

50. Find the eigen values of matrix 3 4

A .2 1

2

51. In ABC, prove that 2 2 2b c a

cosA2bc

using vector method. 4

***********************




1. If ab and cd then prove that acbd.

2. For what value of x,

2 3 1

x 2 5 0 ?

1 3 4

3. In a group G = {1, 2, 3, 4} under modulo 5 find (3 x 4–1)–1

4. If ˆ ˆ ˆ ˆ ˆ ˆa 2i j k, b 3i j k find

a b

5. Find the equation of the tangent to the circle x2 + y2 – 6x + 8y + 17 = 0 at (1, –2) on it. 6. Write the eccentricity of the conic represented by the equations x = at2 and y = 2at.

7. Find the value of sin–1 [sin 1550] . 8. Express –3i in the modulus–amplitude form. 9. Find the slope of the tangent to the curve xy + 2x – y = 5 at (2, 1)

10. If

1 sinhx

ey log e then find dy

.dx

11. Evaluate: x xe sin e dx .

12. Evaluate: 5

20

dx.

25 x


PART–II

Answer any ten questions. 10 x 2 = 20 13. Find the sum of all positive divisors of 960. 14. Solve by Cramer‟s rule: x – 3y = 5 3x – y – 7 = 0

15. Show that 4a 3b x 2a 3b 6 a x b .

16. Show that the vectors 2i j k, i 3j 5k and 3i 4j 4k form the sides of a right angled triangle.

17. Show that a group of order 3 is always Abelian. 18. Define the radical centre of three circles and find the radical axis of the circles x2 + y2 + 4x – 6y + 2 = 0

and 3x2 + 3y2 + 6x – 12y + 5 = 0. 19. Find the length of the Transverse axis of the hyperbola 9x2 – 4y2 – 36x – 8y – 8 = 0..

20. Show that 2

1 11 4cos sin .

55

21. If 1

x 2cos ,x

then show that 4

4

1x 2cos4 .

x

22 If 1 cosx

y log ,1 cosx

then show that

dy2cosec x .

dx

23. The perimeter of a rectangle is 80 m. Find the length of the sides when the area is maximum.

24. If d

f x sin5x sinx,dx

then find f(x).

25. Solve the differential equation 2 dy

x ydx

.

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Find (325, 481) and express it in the form 325m + 481n. Also show that this expression is not unique.

27. Show that

2 2

32 2

2 2

bc b bc c bc

a ac ac c ac ab bc ca .

a ab b ab ab

28. State Cayley–Hamilton theorem and find the characteristic- roots of the matrix 1 3

4 2

.

29. Show that the points A(2, 3, –1), B(1, –2, 3), C(3, 4, –2) and D(1, –6, 6) are coplanar. 30. Show that a non–empty subset H of a group G is subgroup of G iff

i) a, b H ab H ii) a–1 H a H.

31. Show that the set G z/ z cos isin , ( is real) is an Abelian group under multiplication.

B. Answer any three questions: 3 x 4 = 12 32. Derive the condition for the line y = mx + c to be a tangent to the circle x2 + y2 = a2. Also find the point of

contact. 33. Find the equation of the circle, which cuts the circles x2 + y2 – 8x + 3y + 21 = 0, x2 + y2 – 7x + 5y + 28 = 0

and x2 + y2 + x + y – 16 = 0 orthogonally.

34. Derive the equation of the hyperbola in the form 2 2

2 2

x y1.

a b

35. Find the equations of the parabolas, whose directrix is x + 2 = 0, axis is y = 3 and the length of the latus

rectum is 8.

36. Any tangent to the ellipse 2 2

2 2

x y1

a b , makes intercepts h and k on the co–ordinate axes.

Show that 2 2

2 2

a b1.

h k


C 37.If cos–1 x + cos–1 y + cos–1 z = , then prove that x2 + y2 + z2 + 2xyz = 1. 4 OR

Find general solution of 2cosec x cot x 3 .

38. State De Moivre‟s Theorem and prove it for integral values of n only. 4 OR

If cos + cos + cos = 0 = sin + sin + sin , then show that

i) cos 3 + cos 3 + cos 3 = 3 cos ( + + ) and

ii) sin 3 + sin 3 + sin 3 = 3 sin ( + + ) . D 39. Differentiate ‘cot 5x’ w.r.t. x from first principles. 4

(OR)

Differentiate 2

1 1 x 1tan

x

w.r.t. 2

1

2

1 xcot .

1 x

40. If t

x a cost log tan2

and y = a sin t, prove that

dytant .

dx

4

OR

If logey=m sin–1x, then prove that (1 – x2)y2 – xy1 – m2y = 0. 41. Show that in the curve ay2 = x3, the square of the subtangent varies as the subnormal. 4

OR

When brakes are applied to a moving car, the car travels a distance s feet in t sec, given by s = 8t – 6t2. when does the car stop? How far does it move before coming to rest? Also find the acceleration of the car. E. Answer any three questions: 3 x 4 = 12

42. Evaluate the following: i) 2

1dx.

1 2x x ii) sin x dx .

43. Evaluate: 2cosx 3sinx

dx .3cosx 5sinx

44. Prove that 2

2

0

x tan xdx .2

45. Find the area enclosed between the curves y2 = 6x and x2 = 6y. 46. Solve (1 –x2) dy + xy dx = xy2 dx by separating the variables.

PART–IV



15 23

2

dy d y1 .

dx dx

1

48. Find the imaginary part of log i. 1

49. If e1 and e2 are the eccentricities of hyperbola and its conjugate hyperbola. Prove that 2 2

1 2

1 11

e e 2

50. Find the area of the parallelogram formed by the diagonals 3i j 2k and 4i 2j k 2

51. Find the equation of the circle passing through the points (1, 1), (–2,–8) and (–6, 0) 4

***********************


SECOND PUC ANNUAL EXAMINATION -OCT–2003 MATHEMATICS



1. The linear congruence 8x 23(mod 24) has no solution. Why?

2. If the matrix 5x 2

10 1

is singular, find the value of x.

3. If is defined on the set of reals by ab3ab

,7

find the identity element in R under .

4. Find the value of so that ˆ ˆ ˆi j k is a unit vector.

5. Show that the point (–6, 1) lies outside the circle x2 + y2 – x + y – 2 = 0 6. If the line y = 4x + k is a tangent to the parabola y2 = 8x, find k.

7. Find the general solution of the equation 4sin2 – 3 = 0.

8. Show that 24

cos isin 1.3 3

9. If y = x3.5x, find dy

dx.

10. If y = loga x, find dy

.dx

11. Find 2cot 5x dx .

12. Evaluate: 10

20

dx.

100 x

PART–II


13. If (a, b) = 1, ac and bc prove that abc.

14. Prove that

2 2 2

2 2 2

2 2 2

5 a b c

5 b c a 0

5 c a b

15. Find the value of if the vectors a 1 i 2j 3k and b 5i j 6k are orthogonal

16. Show that the set of integers I is a semigroup with identity, under multiplication

17. Prove that in any group (G ), the inverse of any element is unique. 18. Find the equation of the circle if the ends of a diameter are (–3, 6) and (1, –2).

19. If the major axis of the ellipse 2 2

2 2

x y1

a b (a > b) is double its minor axis, find the eccentricity of the

ellipse.

20. Find the value of 2 11 3cos cos .

2 5

21. Prove that 1 i 1 i

6 6e e e 3 .

22. If 2

1

2

1 xy cos ,

1 x

prove that 2

dy 2.

dx 1 x

23 Find the equation of the tangent to the curve y = x2 – x + 2 at the point where it crosses the y–axis.

24. Evaluate: 5xxe dx.

25. Evaluate

2

2 2 2 20

dx

a cos x b sin x


PART–II A. Answer any four of the following: 4 x 4 = 16 26. Find the g.c.d o f 1125 and 1225 and express it in the form 1125m + 1225n. show that this expression is

not unique.

27. Prove that:

2

2

2

x x y z

y y z x x y y z z x x y z .

z z x y

28. Find the inverse of the matrix:

1 2 1

1 1 2 .

2 1 1

29. Find the vector of magnitude 10 units perpendicular to the vector a i j k and coplanar with the

vectors b 2i j k and c i 2j k .

30. Show that the set G = { 9n | n I} is an Abelian group under multiplication.

31. Prove that a non–empty subset H of a group (G ) is a sub group of (G ), if and only if a, b H,

a b–1 H. B. Answer any three questions: 3 x 4 = 12 32. Derive the equation of the Radical axis of the circles x2 + y2 + 2g1x + 2f1y + c1 = 0

and x2 + y2 + 2g2x + 2f2y + c2 = 0, and show that it is a straight line perpendicular to the line of centres of the circles. 33. Find the equation of the circle which passes through the point (2, 3), cuts the circle

x2 + y2 – 4x + 2y – 3 = 0 orhtogonally, and the length of the tangent to it from the point (1, 0) is 2.

34. Derive the equation of the hyperbola in the standard form 2 2

2 2

x y1.

a b

35. Find the equation of the parabola whose vertex is (–2, –3) and directrix y=6. 36. Find the centre, eccentricity and the foci of the ellipse 4x2 + 9y2 – 8x – 36y + 4 = 0. C.


prove that x2 + y2 + z2 = 1 – 2xyz. 4

OR

Find the general solution of cosec + cot = 3 .

38. If x = cos + i sin , y = cos + i sin , z = cos + i sin and x + y + z = 0, prove that 1 1 1

0.x y z

Hence show that 2 2 2 3cos cos cos

2

4

Find the cube roots of 3 i

2

and represent them in Argand diagram.

D 39 Differentiate ‘cosec (6x)’ with respect to x by first principles method 4

OR

If x5y3 = (x + y)8, prove that dy y

.dx x

40. If y = a(sin – cos ) and x = a(cos + sin ), prove that 2

2

d y

dx at

8 2is .

4 a

4

OR

If 5

2y x 1 x , prove that (1 + x2)y2 + xy1 – 25y = 0

41. If the curves y2 = 8kx and xy = 2p intersect each other orthogonally, prove that p2 = 128k4. 4


OR

Prove that for the curve (x + 3)3 = 6y2 the square of the sub–tangent varies as the sub–normal at any point on it. E. Answer any three questions: 3 x 4 = 12

42. Evaluate the following: i) 2

2

7x 5dx

x 4

ii) cos 10x cos 3x dx.

43. Evaluate:

2

3x 4 dx.

4x 4x 5

44. Prove that 0

x tanxdx 2 .

sec x tanx 2

45. Find the area enclosed between the parabola y = x2 – 6x + 6 and the line x + y = 0.

46. Solve by separating the variables: dy

3x 4y 6xy 2 .dx

PART–IV

Answer all the questions. 47. Verify whether 667 is a prime number or not. 1 48. Find the multiplicative inverse of 4 + 3i. 1 49. Show that x = 1 is a maximum point for the function f(x) = x5 – 5x4 + 5x3 – 1. 2 50. Show that the distance of any point (h, k) on the parabola y2 = 20x from its focus is (h + 5). 2 51. Show that the line 3x + 4y = 44 touches the circle x2 + y2 – 2x – 8y – 8 = 0. Find the point of contact.4

***********************

SECOND PUC ANNUAL EXAMINATION - APRIL–2004 MATHEMATICS

Time: 3 Hours ] [Max. Marks: 100 Instructions: (i) The question paper has four parts. Answer all the four Parts. (ii) Part–I carries 10 marks, Part–II carries 20 marks, Part–III carries 60 marks and Part–IV is compulsory which carries 10 marks. (iii) Write the question number properly as indicated in the question paper.

PART–I 10 x 1 =10

1. If ax | bx and x0 prove that a | b.

2. If 1 1 3

B A2 3 4

and

2 3 1B A ,

3 4 2

find A.

3. In the group G={1,2,3,4} under multiplication mod 5, find the inverse of 5.

4. If i j k is a Unit vector find ..

5. Find the value of h and k for the equation kx2 + 2hxy + 4y2 – 2x + 3y – 7 = 0 to represent a circle. 6. Find the eccentricity of the ellipse 25x2 + 16y2 = 400.

7. Evaluate : 11 4

sin cos .2 5

8. If is a cube root of unity, then find the value of 33 21 1 .

9. If 1 1 x

y tan ,1 x

find

dy.

dx

10. If the slope at any point on the curve xy = c2 is y

xthen prove that subnormal is proportional to y3.

11. Find xe sec x 1 tanx dx.


12. Evaluate: / 2

2

0

sin x dx.

PART–II Answer any ten questions. 10x2=20

13. If a b(mod m) and c d(mod m) prove that ac bd(mod m). 14. Solve by Cramer‟s Rule: 2 x –3y = 5 and 7x – y = 8.

15. Find the projection of ˆa 5i 2j 3k on ˆ ˆ ˆb 4i 2j 5k.

16. On Q, set of all rational numbers, is defined by a b

a b .2

show that is not associative.

17. In a group G, if every element has its own inverse, show that G is Abelian. 18. Show that the circles x2 + y2 – 4x + 6y + 8 = 0 and x2 + y2 – 10x – 6y + 14 = 0 touch each other.

19. If e1 and e2 are the eccentricities of a hyperbola and its conjugate, prove that 2 2

1 2

1 11.

e e

20. Solve : tan2 – 4 sec + 5 = 0 in (0, 2).

21. Simplify :

2 3

5

cos isin cos2 isin2

cos4 isin4

.

22. If x y

sec a,x y

prove that

dy y.

dx x

23. Find the angle between the curves y2 = 4x and x2 = 2y – 3 at the point (1, 2).

24. Evaluate: 2sin x.cos3xdx.

25. Solve: 2cosx. 1 y dx 2y 1 sinx dy 0

. PART–III

A. Answer any four of the following: 4 x 4 = 16

26. (a) solve for x: 37 2x mod11 (b) If a,c 1 and c ab, then prove that c b

27. Prove that

2 2

2 2 2 2 2

2 2

a bc ac c

a ab b ac 4a b c .

ab b bc c

28. If 2 1 3

A 1 4 2 ,B 1 2 1 ,

0 3 1

find B(adj A).

29. If ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆa i j k, b i 2j k and c i 2j k, find the unit vector perpendicular to both a b and b c.

30. Prove that M of all 2 2 matrices with elements are real numbers is an Abelian group w.r.t. addition. 31. Show that H = {0, 2, 4} is a subgroup of G = {0, 1, 2, 3, 4, 5} under addition mod 6.

B. Answer any three questions: 3 x 4 = 12 32. Derive the equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at (x1, y1) on it. 33. Show that the line 3x + y – 15 = 0 touches the circle x2 + y2 – 2x – 4y – 5 = 0 and also find the point of contact. 34. Find the equations of the parabolas whose directrix is x + 2 = 0, axis is y = 3 and latus rectum is 6.

35. Find the condition so that the line x cos + y sin = p may be a tangent to the ellipse

2 2

2 2

x y1.

a b

36. Find the equation of the hyperbola in the form

2 2

2 2

x y1

a b

given that distance between

vertices = 24 and foci are (–10, 2) and (16, 2).


C 37. Solve : 1 1 2sin x sin 2x

3

. 4

OR

Find the general solution of 1

cos xcos2xcos3x4

38. Show that

4

5

cos isinsin 4 5 icos 4 5 .

sin icos

4

OR

Show that 3n 3n

3n 11 i 3 1 i 3 2 .

D. 39. Differentiate sec ax with respect to x from first principle. 4

OR

If 1

2 2

2 a b xy tan tan ,

a b 2a b

a > b > 0, prove that dy 1

.dx a bcos x

40. If 1 x

sin x 1y x sin x , find dy

.dx

4

OR

If y=sin ht, y = cos hpt, prove that (1 + x2)y2 + xy1 – p2y = 0. 41. The normal to the curve x2 = 4y –12 at (2, 4) cuts the coordinate axes x and y at A and B respectively. Show that the area of triangle AOB = 18 sq. units. 4

OR

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius a is

2a

.3

E. Answer any three questions: 3 x 4 = 12 42. Evaluate the following:

(i)

2

dx

8 6x 9x (ii)

x x

1dx.

e e

43. Evaluate: 2

2x 4dx.

1 2x x

44. Prove that 1

2

0

log 1 xdx log2.

1 x 8

45. Find the area enclosed between the parabolas y2=4ax and x2= 4ay. 46. Solve by the method of separation of variables: (y2–xy2) dx + (x2 + yx2) dy = 0

PART–IV

Answer all questions. 47. Find the imaginary part of log (–1). 1

48. Find the equation of the asymptotes of the hyperbola 2 2x y

1.5 4

1

49. Prove by vector method that the diagonals of a Rhombus are perpendicular. 2

50. If a ib

x iy ,c id

prove that

2 22

2 2

2 2

a bx y .

c d

2

51. State Cayley Hamilton theorem and verify the same for the matrix 7 4

.3 2

4

***********************


SECOND PUC ANNUAL EXAMINATION -OCT–2004 MATHEMATICS


PART–I


1. Verify whether 14x 5(mod 21) has a solution or not? Give reason.

2. Evaluate 4321 4322

.4323 4324

3. In the set of real numbers R if a b = 1 + ab, then show that is commutative.

4. If the direction cosines of a are 2 1

, and n3 3

find n.

5. Find the length of the tangent to the circle 2x2 + 2y2 – x + 3y – 12 = 0 from (6, 4) 6. Find c so that y = x + c may be a tangent to the parabola y2 = 4x.

7. Find the value of cos–1 (cos 350). 8. Express (1 + i) in polar form.

9. Defferentiate 2

5 62

xx w. r. t

x.

10. If y = sin hx + cos hx, find 1 dy

.y dx

11. Integrate 2

x

x 1w.r.t. x.

12. Evaluate: 3

20

dx.

9 x

PART–II


13. If ab, bc and (a, b) = 1 then prove that abc.

14. Solve for x and y 3 2 x 6

.4 3 y 2

15. Find the cosine of the angle between the vectors ˆ ˆ ˆa i 2j 3k and ˆ ˆ ˆb 2i 3j 4k.

16. If the vectors ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2i - 3j + mk, 2i + j - k and 6i - j + 2k are coplanar, find m.

17. Show that if every element of a Group G is its own inverse, then G is Abelian. 18. If one end of a diameter of the circle x2 + y2 – 2x + 6y – 1 = 0 is (–4, 5), find the other end of the

diameter of the circle.

19. Find the equation of the hyperbola in the form 2 2

2 2

x y1.

a b given that distance between foci is 8 and

distance between directrices is 9

.2

20. Show that 1 112 5sin cot .

13 12

21. If 1

x 2cos ,x

then show that n

n

1x 2cosn .

x

22. If 2 2ey log (x x a ) then show that

2 2

dy 1.

dx x a

23. The distance S feet traveled by a particle in time t seconds is given by S = t3– 6t2 + 15t + 2. Find the

velocity when the acceleration is zero.

24. Evaluate 2ex log xdx.


25. Prove that a a

0 0

f x dx f a x dx

PART–III

A. Answer four of the following: 4 x 4 = 16 26. Find the number of positive divisors and the sum of all positive divisors of the number 1920.

27. Prove that

2 2

2 2 2 2 2

2 2

a a ab ab

bc b b bc 4a b c .

ac c ac c

28. State the Cayley–Hamilton theorem and find the inverse of the matrix 1 2

A3 4

using

Cayley–Hamilton theorem.

29. If a 2i j 2k and b 3i 4j k, find the unit vector perpendicular to both a and b . Find also the sine

of angle between the vectors a and b .

30. Prove that the fourth root of unity forms an Abelian group under multiplication. 31. Show that non–empty subset H of a Group G is a subgroup, iff

i) a, b H ab H. ii) a–1 H, a H.

B. Answer any three questions: 3 x 4 = 12 32. Obtain the condition for the two circles x2 + y2 + 2g1x + 2f1y + c1

= 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0,

to cut orthogonally. 33. Find the radical centre of the circles x2 + y2 + 4x + y – 8 = 0, x2 + y2 + 3x – 4 = 0 & x2 + y2 + 2x + 2y = 0.

Find how the radical centre is situated w.r.t. the circles. 34. Find the equation of the tangent to the parabola y2 = 8x that passes through (2, 5). Also find their point of contact.


2 2

x y1 a b .

a b

36. Find the foci, eccentricity, the length of the latus rectum of 9y2–16x2=144. C.

37. Solve 1 1sin x cos x6

(where x > 0) . 4

OR

Find the general solution of tanx sec x 3 .

38. State De Moivre‟s theorem and prove it for integral values of n only. 4 OR

Find the cube roots of 3 i and represent them in the Argand diagram.

D

39. Differentiate sin x w.r.t x from first principles. 4

OR

If xm yn = ( x + y ) m + n, then prove that dy y x m

.dx x y n

40. If 2

2

1 tx

1 t

and

2

2ty .

1 t

Show that

2dy t 1.

dx 2t

4

OR

If y = a cos ( log x ) + b sin ( log x ), then show that x2y2 + xy1 + y = 0. 41. Find the angle of intersection of the curves y(x2 + 1) = x + 3 and y(x – 1) = x2 – 7x at (2, 1). 4

OR

The sum of the sides of a rectangle is constant, if the area is to be maximum, Show that the rectangle is square. E. Answer any three questions: 3 x 4 = 12


42. Evaluate the following: i) 2tanx sec xdx. ii) 2

1dx.

x 4x 1

43. Evaluate dx

.3 2 tanx

44. Show that 2

0

x tanxdx .

sec x cosx 4


125 9

by the method of integration.

46. Solve (ey + 1) cos x dx +ey sin x dy = 0 by separating the variables. PART–IV

Answer all the questions: 47. Form the differential equation given that y = c esinx where c is the parameter. 1

48. Write the modulus and amplitude of i2e .

1

49. Find the area of the triangle formed by the points (1, 3, 2); (2, –1, 1) and (–1, 2, 3) 2 50. Prove that in the curve xmyn = am+n, the subtangent at any point varies as the abscissa of the point. 2 51. Find the equations of the circles which touch both the co-ordinate axes and pass through the point (2, 1). 4

***********************

SECOND PUC ANNUAL EXAMINATION - APRIL–2005

MATHEMATICS


PART–I


1. Solve 5x 4(mod 13).

2. If the matrix

2 x 3

4 1 6

1 2 7

is singular, then find x.

3. In the group of non–zero rational numbers, the binary operation is defined by ab

a b ,5

then

solve 2 x = 5.

4. Find the area of the parallelogram whose adjacent sides are given by the vectors ˆ2i j k & 3i 4j 4k. .

5. Wirte the condition for the circle touching both axes. 6. Find the eccentricity of the ellipse 16x2 + 7y2 = 112.

7. Find the general solution of tan 1.2

8. Find the amplitude of (1 + cos + i sin ).

9. Differentiate 1 xlog tanh e

e w.r.t x.

10. Differentiate tan(x) w.r.t. x.

11. Find log tanxe dx .

12. Evaluate: / 2

2

0

cos x dx .

PART–II



13. Find the number of positive divisors of 17424. 14. Solve by Cramer‟s rule: 2x + 3y + 21 = 0 and 5x + 2y + 3 = 0.

15. Find sine of the angle between the vectors 2i j k and j k 3i

16. If a and b are any two elements of a group (G, ), then show that the equation a x = b has unique

solution in (G, ). 17. Define a subgroup and give an example of a finite subgroup of a finite group. 18. Show that the two circles x2 + y2 – 6x – 2y + 1 = 0 and x2 + y2 + 2x – 8y +13 = 0 touch each other externally. 19. Find the equation of the parabola with vertex at (2, 5) and focus at (4, 5).

20. Show that 1 1

2

2xsin 2tan x.

1 x

21. Show that

31 cos10 isin10 3 1

i .2 21 cos10 isin10

22. If 1 1 cos2xy tan

1 cos2x

, then find dy

dx.

23. The law of motion of a particle at any time t secs is given by s = 4t3 – 2t2 + 3t + 7. Find its velocity and

acceleration at t = 2 secs.

24. Find x logx dx .

25. Find the area bounded by the curve y = 4x –x2 –3 with x axis.

PART–III A. Answer any four of the following: 4 x 4 = 16 26. Prove that the number of primes is infinite.

27. Prove that 3

a 3b 3c 4b 4c

4a b 3c 3a 4c 9 a b c .

4a 4b c 3a 3b

28. If

1 1 1

A 1 2 3 ;

1 3 4

7

B 16 ;

22

x

X y

z

and AX=B, then find the values of x, y and z using the matrix method.

29. If i 2j 3k, i 6j k, 2i 3j k and 3i 4j 2k are the position vectors of 4 coplanar points, then find .

30. Prove that G = {1, 3, 4, 5, 9} forms an Abelian group under multiplication modulo 11.

31. Prove that a non–empty subset H of a group (G, ) is a sub–group of the group (G, ) if and

only if a, b H, a b–1 H. B. Answer any three questions: 3 x 4 = 12 32. Derive the equation of the radical axis of two circles x2 + y2 + 2g1x+2f1y+c1

= 0 and

x2 + y2 + 2g2x + 2f2y + c2 = 0, and show that it is perpendicular to the line joining their centres.

33. Find the equations of the parabolas having the end points of their latus rectum as (2, 5) and (2, –3). 34. Find the equation of the circle passing through the point (1, 2), having its centre on the line 2x + y = 1 and cutting orthogonally the circle, x2 + y2 – 2x +1 = 0.

35. If the tangents drawn from an external point to the ellipse 2 2x y

116 9

are perpendicular to each other,

then prove that the locus of the point represents the circle x2 + y2 = 25.

36. Derive the equation of the hyperbola in the form 2 2

2 2

x y1.

a b

C

37. If sin–1 x + sin–1 y + sin–1 z = 2

, then show that x2 + y2 + z2 = 1 – 2xyz. 4

OR

Find the general solution of 3 1 cosx 3 1 sinx 2 .


38. Prove that pp p

2 2 12qq qp y

x iy x iy 2 x y . cos tan .q x

OR

Find the cube roots of 1 3 i and represent them in the Argand diagram.

D 39. Differentiate ‘sin–1 x’ w.r.t. x from first principles. 4

OR

Find dy

,dx

given that 2 2

1

2 2

1 x 1 xy tan .

1 x 1 x

40 If y = ( tan x) x + (cot x) x, then find dy

.dx

4

OR If y = sin [m tan–1 x], then show that (1 + x2)2y2 + 2x(1 + x2)y1

+ m2y = 0

41. Show that for the curve y = 7x + 2, the sub–tangent is constant and for the curve y 7x 2 , the sub–

normal is constant. 4 OR

The volume of a closed cylinder is given. When its surface area is least, prove that the height of the cylinder is equal to the diameter of its circular base. E. Answer any three question: 3 x 4 = 12

42. i) Find 2

2 2

cosec x. cot xdx.

cosec x cot x ii) Find

x

2

1 x

2 xe . dx

43. Evaluate: dx

13 12cosx

44. Evaluate:

2

21

1dx.

1 x 1 x

45. Find the area enclosed between the curve y2 = 4x and the line y = 2x – 4. 46. Solve by variables separable method: cos y. log (sec x + tan x). dx = cos x. log (sec y + tan y).dy.

PART–IV Answer all the questions. 47. Find the unit digit of 3162. 1 48. Form the differential equation, given that ex + k..ey = 1, where k is the parameter. 1

49. Find the real part of 1 itan

.1 i tan

2

50. Find a unit vector which is perpendicular to i 3j 5k and i 4j 3k 2

51. Find the equation of the tangents to the circle x2 + y2 – 6x – 4y + 5 = 0 which makes an andgle 45 with x axis. 4

***********************


SECOND PUC ANNUAL EXAMINATION – OCT–2005 MATHEMATICS



1. If ab and ac then prove that abc.

2. Find the inverse of a matrix 4 7

A .6 5

3. Find the identity element in the set of rationals except –1 w.r.t. defined by a b = a + b + ab.

4. Find the magnitude of the vector 3i 4j 6k.

5. Find the equation of a circle having (3, 0) and (0, 3) as ends of a diameter.

6. Which conic does the equation x = a cos and y = b sin represent?

7. Evaluate sin–1 (sin 110).

8. Find the amplitude of 1 i 3.

9. Differentiate 4x

3 with respect to x.

10. If y log sec x , then find dy

.dx

11. Find: sin x

dx .x

12. Evaluate: 3

20

dx.

9 x

PART–II

Answer any ten questions. 10 x 2 = 20 13. Prove that the least divisor >1 of any integer is a prime number.

14. If 2 3 1 5

A and B ,0 4 2 0

then prove that (AB)| = B|A|.

15. Find the cosine of the angle between two vectors i 2j 3k and 2i j k

16. In a group G, if (ab)2 = a2b2 a, b G, then prove that G is an Abelian.

17. Prove that (a b)–1 = b–1 a–1 a, b G, where (G, ) is a group. 18. Find k, so that the line 3x + y + k = 0 may be a tangent to the circle x2 + y2 – 2x – 4y – 5 = 0. 19. Find the length of transverse axis of the hyperbola x2 – 4y2 – 2x + 16y – 40 = 0 20. Find the general solution of the equation tan 5x tan 2x = 1.

21. Simplify ( sin 2 + i cos 2)–4.

22. If y log x = x – y, then show that

2

dy logx.

dx 1 logx

23. Find the equation of the tangent to the curve x2 + 3y2 = 4 at (–1, 1).

24. Find 2

sinxdx.

9 4cos x

25. Form the differential equation by eliminating the constants a and b in the equation y = aemx + be–mx.

PART–III

A. Answer any four of the following: 4 x 4 = 16 26. Find the number of positive divisiors and sum of the divisiors of the number 10584.

27. Show that 3

a b 2c a b

c b c 2a b 2 a b c .

c a c a 2b

28. Solve the equations using matrix method:x + y – 3z = 4, 4x + 3y – 2z = 2 and 2x + y – 3z = 1. 29. Show that the following points with coordinates (4, 5, 1), (0, –1, –1), (3, 9, 4) and (–4, 4, 4) are coplanar.


30. Prove that G = {1, 5, 7, 11} is an Abelian group under multiplication modulo 12.

31. Prove that a non–empty subset H of a group (G, ) is a sub–group G iff a, b H a b–1 H. B. Answer any three questions: 3 x 4 = 12 32. Obtain the condition for the two circles x2 + y2 + 2g1x + 2f1y + c1

= 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0, to

cut orthogonally. 33. Find the equation of a circle which passes through the point (2, 3) and has its centre on the line x + y = 4 and cuts orthogonally the circle x2 + y2 – 4x + 2y – 3 = 0. 34. Find the equation of a parabola with latus rectum joining the points (3, 6) and (–5, 6).

35. Find the condtion that the line y = mx + c may touch the ellipse 2 2

2 2

x y1.

a b (a > b).

36. Find the equation of a hyperbola in the standard form 2 2

2 2

x y1.

a b given that the distance between the

foci is 10 and length of a latus rectum is9

.2

C


then prove that x2 + y2 + z2 + 2xyz = 1. 4

OR

Find the general solution of 3 cot x 1 2cosec x.

38. If cos + cos + cos = 0 = sin + sin +sin , then prove that

i) cos 3 + cos 3 + cos 3 = 3 cos ( + + ) ii) sin 3 + sin 3 + sin 3 = 3 sin ( + + ) 4 OR

Show that n n n 1 nn

1 cos isin 1 cos isin 2 cos cos .2 2

D 39. Differentiate ‘tan ax’ w.r.t. x from the 1 st principles. 4

OR

Differentiate 1

2

1sec

2x 1

w.r.t. 21 x .

40. If ey= yx, then show that

2logydy

.dx logy 1

4

OR

If y = sin [ a cosh–1 x ], then show that (x2 – 1)y2 + xy1 + a2y = 0. 41. In the curve x3 y2 = a7, show that the subtangent varies as abscissa of a point. 4

OR The sum of the sides of a rectangle is constant. If the area is to be maximum. show that the rectangle is a square. E. Answer any three questions: 3 x 4 = 12

42. i) Evaluate sin5xsinxdx. ii) Find: 2

dx.

16 6x x

43. Evaluate: 3sinx 5cosx

dx .sinx 2cosx

44. Show that a

2 20

dx.

4x a x

45. Show that the area of the ellipse 2 2

2 2

x y1

a b is ab (By the method of integration).

46. Solve by the method of separation of variables cos y log (sec x + tan x)dx = cos x log (sec y+tan y) dy.


PART–IV



52 24

2

dy d y1 .

dx dx

1

48. Find the real part of sin (x + iy ). 1 49. Define a semi group. Given an example of a semi group which is not a group. 2 50. Prove by vector method that medians of the triangle are concurrent. 2 51. Show that circles x2 + y2 – 2x + 6y + 6 = 0 and x2 + y2 – 5x + 6y + 15 = 0 touch each other. Find the point of contact. 4

***********************



PART–A Answer all the ten questions: 10 x 1 = 10

1. Find the number of incongruent solutions for 6x 3 (mod 15).

2. If the matrix

3 2 x

4 1 1

0 3 4

has no inverse, find x.

3. In a group G = {1, 2, 3, 4} under multiplication modulo 5 find (3 x 4–1)–1.

4. If the vectors a 3i j 2k and b i j 3k are perpendicular find .

5. Find the centre of the circle 4x2 + 4y2 + 4x + 2y + 1 = 0 6. If the line x + y + 2 = 0 touches the parabola y2 = 8x, find the point of contact. 7. Find the value of sec–1 (–2).

8. Express

21 i

3 i

in x + iy form.

9. Differentiate a4log xy a w.r.t. x.

10. Evaluate: 2

2

x 1dx .

x 1

PART–B

Answer any ten questions: 10 x 2 = 20

11. If ab and ac then prove that ab + c. 12. Solve by Cramer‟s Rule: 3x + 2y = 8, 4x –3y = 5. 13. Prove that H = {0, 3} is a sub–group of the group G = {0, 1, 2, 3, 4, 5} under addition modulo 6.

14. Find the volume of the parallelopiped whose coterminous edges are a i 2j 3k, b i 2j k and

c 3i 2j k

15. Find k for which the circle 2x2 + 2y2 – 18x + 6y – 7 = 0 and 3x2 + 3y2 + 4x + ky + 3 = 0 intersect orthogonally. 16. Find the eccentricity of the ellipse if its minor axis is equal to distance between foci.

17. Solve: 1 1 1 1tan x 1 tan x 1 cot .

2

18. Show that n

1 itan 1 i tan(n ).

1 i tan 1 i tan(n )

19. Differentiate 1cos xy x

w.r.t. x.

20. Show that y = sin x(1 + cos x) is maximum when x .3


21. Evaluate: / 2 n

n n0

sin xdx.

cos x sin x

22. Find the order and degree of the differential equation,

32 24

2

dy d y1 k .

dx dx

I. Answer any three questions: 3 x 5 = 15 23. a). Find the number of all positive divisors and the sum of all such positive divisors of 432 3 b). Find the remainder when 71 x 73 x 75 is divided by 23. 2

24. Prove that

2

2 2 2 2

2

a 1 ab ac

ab b 1 bc 1 a b c .

ac bc c 1

5

25. If Q+ is the set of all positive rationals, prove that (Q+, ) is an Abelian group, where is defined by

2ab

a b .3

5

26. a) If a i j k, b i 2j 3k and c 2i j 4k, find the unit vector in the direction of a x b x c . 3

b) If cos , cos and cos are direction cosines of the vector 2i j 2k,

show that cos2 + cos2 + cos2 = 1. 2 II. Answer any two questions: 2 x 5 = 10 27. a) Find the equation of tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at the point (x1, y1) on it. 3 b) Find the equation of the circle two of whose diameters are x + y = 6 and x + 2y = 4 and whose radius is 10 units. 2 28. a) Find the eccentricity and equations to directrices of the ellipse 4x2 + 9y2 – 8x + 36y + 4 = 0. 3 b) Find the equations of the asymptotes of the hyperbola 9x2 – 4y2 = 36. Also find the angle between them. 2

29. a) If 1 1 1sin x sin y sin z ,2

prove that x2 + y2 + z2 + 2xyz = 1 2

b) Find the general solution of tan 3x. tan 2x = 1. 2 III. Answer any three of the following questions: 3 x 5 = 15 30. a) Differentiate ‘sin2x’ w.r.t. x from first principles. 2

b) If

..........xxy x find

dy.

dx 2

31. a) If y = (sinh–1 x)2, prove that (1 + x2)y2 + xy1 – 2 = 0 3

b). If x = a( + sin ), y = a(1 – cos ), prove that dy

tan .dx 2

2

32. a). Show that the curves 2y = x3 + 5x and y = x2 + 2x +1 touch each other at (1, 3). Find the equation to common tangent. 3

b). Evaluate:

x

2

1 xe dx.

2 x

2

33. a). Evaluate: a

a

a xdx.

a x

b). Evaluate: 3x 24 . x dx 5

34. Find the area bounded between the curves x2 = y and y = x + 2. PART–D

Answer any two of the following questions: 2 x 10 = 20

35. a) Define ellipse and derive standard equation to the ellipse 2 2

2 2

x y1.

a b 6

b). State Caley–Hamilton theorem. Verify the Caley–Hamilton

theorem for the matrix 1 2

.3 4

4


36. a). Find the fourth roots of the complex number 1 i 3 and represent them in an Argand diagram. 6

b). Prove that 2

bxc c xa axb a b c .

4

37. a). An inverted circular cone has depth 12 cms and base radius 9 cms. Water is poured into it at the

rate of 1

12

c.c/sec.

Find the rate of rise of water level and the rate of increase of the surface area when depth of water is 4 cm. 6

b). Find the general solution of cos2 3 sin2 1. 4

38. a). Prove that 1

20

log 1 xdx log2 .

81 x

6

b). Solve the differential equation (y2 + y) dx + (x2 + x) dy = 0 4

***********************

SECOND PUC ANNUAL EXAMINATION - JULY–2006 MATHEMATICS


PART–A Answer all the ten questions: 10x1=10 1. Find the digit in the unit place of 312.

2. If

4 34 1 3

A 1 2 , B3 2 3x 5

3 1

and B = A|, find the value of x.

3. If the binary operation on the set of integers Z is defined by a b = a + b+ 5, find the identity element.

4. Find the direction cosines of the vector 3i 6j 2k .

5. Find the radical axis of the circles x2 + y2 + 4x – 7 = 0 and x2 + y2 + 8y + 12 = 0 6. If in the parabola y2 = 8 kx, the length of the latus rectum is 4, find the value of k.


3 2

8. Find the amplitude of 1 3

i .2 2

9. If y = x5.5x, find dy

.dx

10. Evaluate: 1

3

0

3x 1 dx. PART–B


11. If (c, a) = 1 and cab, prove that cb. 12. Solve the equations by Cramer‟s Rule: 5x + 3y = 1 and 3x + 5y = –9

13. Prove that in any group (G, ), (a b)–1 = b–1 a–1, a, b G.

14. Find the volume of the parallelopiped whose coterminus edges are i 3j 4k , 3i j k and 2i 3k .

15. Find the equation of the circle having centre at (6, 1) and touching the straight line 5x + 12y – 3 = 0. 16. Find the length of the latus rectum of an ellipse 4x2 + 9y2 – 8x –36y + 4 = 0

17. If 1 1tna x tan y ,4

prove that x + y + xy =1.


18. If is an imaginary cube root of unity, prove that 5 5

2 21 1 32.

19. If 2

1

2

1 xy cos ,

1 x

prove that 2

dy 2.

dx 1 x

20. Show that in the parabola y2 = 4ax the sub–tangent at any point is twice the abscissa. 21. Product of two numbers is 16.Find the numbers when their sum is minimum.

22. Evaluate: sin x dx .

PART–C I. Answer any three questions: 3 x 5 = 15 23. Find the G.C.D of 252 and 595 and express it in the form 252a + 595b (where a and b are integers ). Also show that this expression is not unique. 5

24. a). If

1 2 2

A 1 3 0

0 2 1

, find Adj A. 3

b). Given 2 1

A1 2

, then using Cayley–Hamilton theorem, prove that A2 – 4A+ 3I = 0. 2

25. Prove that G = { cos + isin | is real } is an Abelian group under multiplication. 5

26. a) Prove that a b b c c a 2 a b c .

3

b) Find the sine of the angle between the vectors a i 2j k b 2i 3j 6k . 2

II. Answer any two questions: 2 x 5 = 10 27. a). Derive the expression for the length of the tangent from the point (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0. 3

b). Find the value of for which the circles. x2 + y2 + 2x – 8y +1 = 0 and 2x2 + 2y2 – 6x + y + = 0 intersect orthogonally. 2 28. a). Derive the condition for the straight line y = mx + c to be a tangent to the parabola y2 = 4ax. 3

b). Find the equation of the ellipse in the from 2 2

2 2

x y1

a b (a > b),

given distance between the directrices = 10 2 and the eccentricity 1

.2

2

29. a). Solve for x, sin–1 x – cos–1 x = sin–1 (3x – 2). 3 b). Find the general solution of tan2x – 4secx + 5 = 0 2 III. Answer any three of the following questions: 3 x 5 = 15 30. a). Differentiate ‘eax’ w.r.t. x from first principles. 3

b). Differentiate sinh–1 x w.r.t. 21 x . 2

31. a). If y = a cos ( log x ) + b sin ( log x ), prove that x2y2 + xy1 + y = 0 3 b). Show that the curves x2 + y2 = 2a2 and xy = a2 touch each other at the point (a, a). 2

32. a). Evaluate: cosx

dx.sinx cosx 3

b). A point moves on a straight line. Its distance s feet from a fixed point on the line at a time t is s = 5 cos 2t. find its acceleration in terms of s. 2

33. a). If ex + ey = ex+y, prove that y xdye .

dx

3

b). Evaluate: x 1 sinxe dx.

1 cosx

2

34. Find the area of the ellipse 2 2

2 2

x y1

a b by the method of integration. 5

PART–D Answer any two of the following questions: 2 x 10 = 20


35. a). Define hyperbola as a locus and derive the standard equation of the hyperbola in the form.

2 2

2 2

x y1.

a b 6

b). Prove that

2

2

2

1 a bc

1 b ca a b b c c a a b c .

1 c ab

4

36. a). State and prove that De Moivre‟s theorem for rational index 6

b). If a i 2j 3k, b 2i j k and c i 3j 2k , find a unit vector perpendicular to a and in the same

plane as b and c . 4

37. a). A right circular cone has depth of 12 cm and a base radius of 9 cm. Water is poured into it at the

rate of 1

12

c.c/sec. Find the rate of rise of water level and rate of increase of water surface when

the depth of water level is 4 cm. 6

b). Find the general solution of 2 cosecx cot x 3. 4

38. a). Prove that a a

0 0

f(x)dx f a x dx and show that 2

0

x dx.

41 x 1 x

6

b). Solve the differential equation x ydy1 6xe .

dx

4

***********************




1. Find an integer x, satisfying 5x 4 ( mod 13 ).

2. If the matrix 6 x 2

3 x

is singular, find x.

3. On the set Z of integers if “o” is defined by a o b = a + b + 1, a, b Z.Find the identity element.

4. If a 2i 3j and b 3i 4j, find the magnitude of a b .

5. Write the condition (in terms of g, f and c) under which x2 + y2 + 2gx + 2fy + c =0 becomes a point circle. 6. Find the equation of the directrix of the parabola y2 = –8x.


2 2

8. Find the modulus of the complex number 2 i

.5i

9. If 2

2

1f x x ,

x find f(1).

10. Evaluate: / 4

3

0

sin xcosxdx .

PART–B Answer any ten questions: 10 x 2 = 20

11. If a b ( mod m ) and n > 1 is a positive divisor of m, prove that a b( mod n ).


12. Evaluate

2

2

2

a ab ac

ab b bc

ac bc c

13. Define the binary operation, on a non–empty set S. Give an example to show that, on Z,

the operation , defined by a b = ab, is not binary.

14. Find the angle between the vectors 2i 2j k and 2i j 2k .

15. Examine whether the point (1, 5) lies outside, inside or on the circle x2 + y2 + 4x + 2y + 3 = 0. 16. The two ends of the major axis of an ellipse are (5, 0) and (–5, 0). If 3x – 5y – 9 = 0 is a focal chord, find

the eccentricity of the ellipse .

17. Prove that 1 11 32tan sin .

2 5 2

18. If x = cos + i sin and y = cos + i sin prove that 3 2

2 3

y x2cos 3 2 .

x y

19. If xy = ax, prove thatdy xloga y

dx xlogx

.

20. Find the length of the subtangent to the curve 2y x x 1 at the point 1, 3 on it.

21. Integrate sin 3x cos x with respect to x. 22. Form the differential equation of the family of straight lines passing through the origin of Cartesian

plane.

PART–C I. Answer any three questions: 3 x 5 = 15 23. Find the G.C.D. of 408 and 1032 using Euclidean algorithm. Express it in two ways in the form 408m + 1032n where m, n are integers, 5

24. a). Find x and y if

x 2 3 3 1 2 5 3 3

5 y 2 4 2 5 19 5 16

1 1 1 2 0 3 1 3 0

3

b). Solve by Cramer‟s rule: 2x – y = 10; x – 2y = 2. 2

25. a). Given that H is a non–empty subset of a set G and (G, ) is a group. If for all a, b H, a b–1 H,

prove that (H, ) is a subgroup of (G, ). 3 b). If, in a group G, every element is its own inverse, prove that G is an Abelian group. 2 26. a). Using vector method, find the area of the triangle whose vertices are (1, 2, 3), (2, –1, 1) and (1, 2, –4). 3

b). Find the volume of the parallelopiped whose co–terminal edges are 2i j k, 3i 2j 2k and

i 3j 3k 2

II. Answer any two questions: 2 x 5 = 10 27. a) Find the equation of the circle which passes through the point (2, 3), has its centre on x + y =4 and cuts orthogonally the circle x2 + y2 – 4x + 2y –3 = 0 3 b) Find the radical centre of the circles x2 + y2 + 2x – 4 = 0, x2 + y2 + 4y – 4 = 0 & x2 + y2 – 2x – 5 = 0.2 28. a) Find the centre and the eccentricity of the hyperbola x2 – 3y2 –4x –6y –11 = 0. 3 b) Find the equation of the parabola with vertex (–4, 2), axis y = 2 & passing through the point (0, 6).2

29. a) If x 0 and y 0, prove that 1 1 1 2 2sin x sin y sin x 1 y y 1 x .

3

b) Find the general solution of the equation cos x – cos 7x = sin 4x 2 III. Answer any three of the following questions: 3 x 5 = 15 30. a) Differentiate ‘ex’ with respect to x from first principles. 3 b) Differentiate log10 (log x) with respect to x. 2 31. a) If y = x coshx, prove that xy2 – 2y1 – xy + 2 coshx = 0. 3

b) Prove that xx function has a minimum value at 1

x .e

2


32. a) Find 2

x 1dx.

x 4x 6

3

b) A stone is thrown up vertically and the height x feet reached by it in time “t” seconds is given by

x = 80t – 16t2. Find the time for the stone to reach its maximum height. Also find the maximum height reached by the stone. 2

33. a) If x = a( + sin ) and y = a(1 – cos ), find dy

dx and

2

2

d y.

dx 3

b) Find

x

2

xedx .

1 x 2

34. Find the area bounded by the curves 4y2 = 9x and 3x2 = 16y 5 PART–D

Answer any two of the following questions: 2 x 10 = 20

35. a)Define ellipse as the locus of a point.Derive the equation of the ellipse in the form 2 2

2 2

x y1

a b (a > b).6

b) Using Caley–Hamilton theorem, find the inverse of the matrix 1 2

.3 1

4

36. a) Find all the cube roots of the complex number 3 i. Represent them in the Argand diagram.

Also find their product. 6 b) Prove by vector method that the medians of a triangle are concurrent. 4 37. a) A man 6 feet tall moves away from a source of light 20 feet above that ground level and his rate of

walking being 4 miles/hour. At what rate, is the length of the shadow changing? At what rate is the tip of the shadow moving? 6

b) Find the general solution of 3 cosx sinx 2 . 4

38. a) Evaluate / 2

0

logsinxdx

6

b) Find the general solution of the differential equation y log x . log y dx + dy = 0 4

***********************

SECOND PUC ANNUAL EXAMINATION - JULY–2007 MATHEMATICS



1. If 3127 x ( mod 10 ), find x.

2. If 4 3 4

A , B ,1 2 3

find AB.

3. In a group (G, ), if a x = e a G, find x.

4. Find the value of j 3k x i j 2k .

5. Find the centre of the circle passing through (0, 0), (3,0) and (0, 5). 6. Find the vertex of parabola (y – 2)2 = – 8x.

7. If cos–1 x – sin–1 x = 0, prove that 1

x .2

8. Find amplitude of 2i – 4.

9. If y = 3–x, find dy

.dx

10. Evaluate: / 2

0

1 cos2x dx.



11. If a b (mod m) and n | m n I, prove that a b (mod n).

12. Without expression, find the value of

2 2

2 2

sin x cos x 1

cos x sin x 1

10 12 2

13. If Q+ is the set of all positive rationals w.r.t. define 2ab

a b3

a, b Q+.

Find a) Identity element b) Inverse of a under .

14. For any vector a, prove that a a. i i a. j j a. k k.

15. Find the length of tangent from the centre of circle x2 + y2 – 8x = 0 to the circle 3x2 + 3y2 = 7 16. Find the centre of ellipse whose vertices are (2, –2) and (2, 4). Also find the length of major axis.

17. If 1 1tan x tan y ,2

prove that xy = 1.

18. If x = cis and y = cis prove that 1 x y

sin .2i y x

19. If y loge x = y – x, prove that

e

2e

2 log xdy.

dx 1 log x

20. Prove that xx is minimum at 1

x .e

21. Evaluate: 3x

1dx.

5e 1

22. Form a differential equation for the equation x2 + y2 + 2ky = 0. PART–C

I. Answer any three questions: 3 x 5 = 15 23. a) Find the G.C.D. of 48 and 18. If 6=48m+18n, find m and n. 3

b) Solve 51 x 32 (mod 7). Write the solution set. 2

24. If

7 6 5 x 30

3 4 1 y 0 .

1 2 3 z 10

find x, y and z using Cramer‟s Rule. 5

25. Prove that the set G = {…5–2, 5–1, 50, 51, 52, …} is an Abelian group under usual mulitiplication. 5 26. a) Find the area of the triangle ABC where position vectors of A, B, C are

i j 2k, 2j k, j 3k respectively 3

b) Prove that ax bxc bx c xa c x axb 0 . 2

II. Answer any two questions: 2 x 5 = 10 27. a) Obtain the condition for two circles x2 + y2 + 2g1x + 2f1y + c1

= 0 x2 + y2 + 2g2x + 2f2y + c2 = 0, to

intersect orthogonally. 3 b) The radical axis of two circles is x – 2y + 6 = 0. The equation of one of the circles is 2x2 + 2y2 – 8x –4y – 22 = 0. if the second circle passes through the point (1, 6), find its equation. 2 28. a) Find the centre and the foci of ellipse 4x2 + 9y2 + 16x – 18y – 11 = 0. 2 b) Find the focal distance of any point (x, y) on the parabola y2 = 4ax. 2

29. a) Prove that 2

1 1

2 2 2

1 2x 1 1 x 2xtan sin cos .

2 21 x 1 x 1 x

3

b) Find the general solution of tan m = tan n 2 III. Answer any three of the following questions: 3 x 5 = 5 30. a) Differentiate ‘cosec 4x’ with respect to x from first principles. 3

b) If 1 2 5tanxy tan ,

5 2tanx

find dy

.dx

2

31. a) If m

2y x 1 x ,

prove that 2

2 2

2

d y dy1 x x m y 0 .

dxdx 3


b) Find a point on the curve y = x3 – 3x, where tangent is parallel to the line joining the point (1, –2) and (2, –5). 2 32. a) A circular blot of ink in a blotting paper increases in area in such a way that the radius r cm at time t

seconds is given by 3

2 tr 2t .

4 Find the rate of increase of area when t = 2. 3

b) Prove that uv dx uv vu dx where du dv

u and v .dx dx

2

33. a) Evaluate: 1

2

xsin xdx.

1 x

3

b) Evaluate: 2

1dx.

1 4x 4x 2

34. Find the area enclosed between the parabolas y2 = 4ax and x2 = 4ay 5 PART–D

Answer any two of the following questions: 2 x 10 = 20 35. a) Define director circle of a hyperbola. Derive the equation of director circle of the hyperbola. 6

b) Using

cosx sinx 0

A x sinx cosx 0

0 0 1

find adj [A (x)]. Prove that adj A (x)] = A (–x). 4

36. a) Find the fourth roots of 3

3 i . Also find their continued product. 6

b) Prove by vector method, sin ( + ) = sin cos + cos sin . 4 37. a) Show that the height of a right circular cylinder of the greatest volume which is inscribed in a sphere

of radius ‘a’ is 2a

.3

Find the radius of the right circular cylinder. 6

b) Find the general solution of sec x tanx 3 0 4

38. a) Prove that 2

2 2 2 20

xdx.

2aba cos x b sin x

6

b) Solve the differential equation 2dytan x y

dx 4

***********************

SECOND PUC ANNUAL EXAMINATION - MARCH–2008 MATHEMATICS

Time: 3 Hours–15 minutes ] [Max. Marks: 100

PART–A

Answer all the ten questions: 10x1=10

1. Find the least positive integer x satisfying 2x + 5 x + 4 (mod 5).

2. If 5 x 2y 8

A0 3

is scalar matrix, find x and y.

3. If 3ab

a b ,7

then prove that is associative.

4. Define co–planar vectors. 5. Write the condition for the circle x2 + y2 + 2gx + 2fy + c = 0 touches both axes. 6. Find the co–ordinates of the end points of length of the latus rectum of the parabola y2 = 12x. 7. Find the value of tan (tan–1 3) + sec–1 {sec(–2)} 8. Write the maultiplicative inverse of i. 9. Define the differential coefficient of a continuous function y = f(x). w.r.t. x.


10. Evaluate: 2

1 cosxdx.

sin x


11. The relation “Congruence modulo m” is an equivalence relation on z or prove that a b (mod m) is an equivalence relation on z.

12. Evaluate: 2001 2004

.2007 2010

13. If in a group (G, ) a G, a–1 = a, then prove that (G, ) is an Abelian group.

14. If the vectors i 2j k and i 3j 2k are orthogonal, find .

15. Find the area of the circle whose parametric equations are x = 3 + 2 cos and y = 1 + 2 sin.

16. Find the equation of the hyperbola in the from 2 2

2 2

x y1

a b . Given that transverse axis = 10, and

eccentricity (e) = 2.

17. Find x if 1 1 11 3 1tan x sin cos .

2 2 2

18. Prove that 1 i / 3 1 i / 3e e e.

19. If n n

x y2,

a b

then find

dy

dx at (a, b).

20. Find the length of the sub–tangent to the curve x3 + xy + y2 = 13 at (1, 3).

21. Evaluate: 2 2

1dx.

sin xcos x

22. Form the differential equation by eliminating the parameter c. sin–1 x + sin–1 y = c.

PART–C I. Answer any three questions: 3 X 5 = 15 23 Find the number of all positive divisors and the sum of all positive divisors of 39744. 5

24. a) Show that

2

2

2

a bc a 1

b ca b 1 2 a b b c c a .

c ab c 1

3

b) Find the values of x and y according to Cramer‟s rule: x + 2y = 7 and 4x – 5y = 2 2

25. a) Prove that set H = {1, 2, 4} 7 is a sub–group of the group G = { 1, 2, 3, 4, 5, 6 } 7 under multiplication modulo 7. 3 b) Prove that the identity element of a group is unique. 2

26. a) If the vectors i j k, 4i 2j 9k , 5i j 14k and 3i 2j 7k are the position vectors of the four

coplanar points, find . 3

b) Find the unit vector in the direction of 2i j 2k. 2

II Answer any two questions: 2 x 5 = 10 27. a) Find the equation of the circle which cuts the two circles x2 + y2 – 6y + 1 = 0 and x2 + y2 – 4y + 1 = 0 orthogonally and whose centre lies on the line 3x + 4y + 5 = 0. 3 b) Find the equation of the circle having (4, 2) and (–5, 7) as end points of the diameter. 2

28. a) Find the condition for the line y = mx + c to be a tangent to the hyperbola 2 2

2 2

x y1

a b . 3

b) Find the focus of the parabola y2 – 8x – 32 = 0 2

29. Prove that 1 1 1a a b c b a b c c a b c

tan tan tanbc ca ab

5

III. Answer any three of the following questions: 3 x 5 = 15 30. a) Differentiate ‘cosec(ax)’ w.r.t. x from the first principle. 3 b) Differentiate sinx with respect to loge x. 2


31. a) If ex + ey = ex + y prove that y xdye .

dx

2

b) If 1 1 tx tan ,

1 t

y = cos–1 (4t3 – 3t), prove that

dy6.

dx 3

32. a) If 2 1 1 xy sin cot ,

1 x

prove that dy 1

.dx 2

3

b) Evaluate: sinx

dx.1 sinx

2

33. a) Evaluate: 2

cosxdx.

2sin x 3sinx 4 3

b) Evaluate: 2

xdx.

x 4 2


125 9

by integration method. 5

PART–D Answer any two of the following questions: 2 x 10 = 20

35. a) Define an ellipse. Derive the equation of the ellipse in the standard form 2 2

2 2

x y1.

a b 6

b) If 2 3

A ,2 5

find A–1 by Cayley–Hamilton theorem. 4

36. a) State and prove D‟Moivre‟s theorem for rational index. 6

b) Prove that the sine rule a b c

sinA sinB sinC by vector method. 4

37. a) Prove that the greatest size rectangle that can be inscribed in a circle of radius „a‟ is a square. 6

b) Find the general solution of 3 1 cos 3 1 sin 2. 4

38. a) Prove that / 2

0

dx 1 2 1log .

sinx cosx 2 2 1

6

b) Solve the differential equation 2dy

x y 1 .dx

4

PART–E Answer any one of the following questions: 1 x 10 = 10 39. a) Find the cube roots of 1+i and represent them in the Argand diagram. 4 b) Find the length of the chord intercepted by the circle x2 + y2 – 8x – 6y = 0 & the line x – 7y – 8 = 0 4 c) Find the digit in the unit place of 7123 2

40. a) If a 13, b 19, a b 24, find a b . 4

b) Find 4tan xdx. 4

c) If y log cosx, find dy

.dx

2

***********************

SECOND PUC ANNUAL EXAMINATION - JUNE–2008 MATHEMATICS

Time: 3 Hours–15 minutes] [Max. Marks: 100 i) The question paper has four Parts – A, B, C, D.and E. Answer all the parts. ii). Part – A carries 10 marks, Part – B carries 20 marks, Part – C carries 40 marks and Part – D carries 20 marks , Part – E carries 10 marks.



1. Find the number of incongruent solutions of 9x 21 (mod 30).

2. Evaluate: 4321 4322

4323 4324.

3. In a group (Z6, + mod 6), find 2 + 64–1 +63

–1. 4. Find the position vector of the point P which is the mid–point of AB where the position vectors of

A and B are i j 2k and 3i 3j 2k.

5. Find the equation to a circle whose centre is (a, 0) and touching the y–axis. 6. Find the equation to directrix of (x +1)2 = –4(y – 3).

7. Find the value of cos–1(sin 330).

8. If 1, , 2 are the cube roots of unity, find the value of (1 + – 2)2

9 If x ey e x , find dy

.dx

10. Evaluate x 1 tanxe dx.

cosx

PART–B Answer any ten questions. 10 x 2 = 20 11. Find the G.C.D. of 352 and 891.

12. Find the characteristic roots of the matrix 1 4

.3 2

13. Prove that a group of order three is Abelian.

14. Find the volume of the parallelopiped whose co–terminus edges are the vectors i 3j 2k, 2i j 3k and

i j k

15. Find the equation to the parabola whose focus is (3, 2) and its directirx is x = 1.

16. Prove that 1 21 xsin 2tan 1 x .

1 x

17. Find the equation of a circle passing through the origin, having its centre on the line y = x and cutting

orthogonally the circle x2 + y2 – 4x – 6y + 10 = 0. 18. Prove that (1 – i)9 = 16 – 16i.

19. If e

1 cosxy log ,

1 cosx

then prove that

dy2cosec x.

dx

20. Find the point on the curve y2 = x the tangent at which makes an angle of 45 with the x–axis.

21. Evaluate 1

7

0

x 1 x dx.

22. Form the differential equation by eliminating the arbitrary constant (y – 2)2 = 4a (x + 1)

PART–C I. Answer any three questions: 3 x 5 = 15 23. a) Find the number of positive divisors and sum of all such positive divisors of 756. 3

b) If abc and (a, b) =1, then prove that ac. 2 24. Solve by matrix method: 3x + y + 2z = 3

2x –3y – z = –3 x + 2y + z = 4 5

25. Prove that the set z of integers is an Abelian group under binary operation defined by

a b = a + b + 3, a, b z. 5

26. a) If a i 2j 3k, b 2i j k and c i 3j 2k, find a unit vector perpendicular to a and in the same

plane on b and c . 3

b) Find the area of the parallelogram whose diagonals are the vectors 2i j k and i 2j 3k. 2


II. Answer any two questions: 2 x 5 = 10 27. a) Find the length of the tangent from the point (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0. 3 b) Find the equations of tangent to the circle x2 + y2 – 2x – 4y – 4 = 0, which are perpendicular to 3x – 4y + 6 = 0. 2 28. a) Find the focus and equation to the directrix of the ellipse 9x2 +5y2 –36x + 10y – 4 = 0 3

b) Find the equation to the hyperbola in the standard form 2 2

2 2

x y1

a b given that

length of latus rectum 14

3 and

4e

3 . 2

29. a) If 1 1 1tan x tan y tan z ,2

prove that xy + yz + zx = 1. 3

b) Find the general solution of 2 5sin cos2 .

4 2

III. Answer any three of the following questions: 3 x 5 = 15 30. a) Differentiate ‘ax’ w.r.t. x from first principles. 3

b) If 1

2

4xy tan ,

4 x

prove that

2

dy 4.

dx 4 x

2

31. a) If 2 2

1 1y sin x cos x , prove that 22 11 x y xy 4 0 3

b) If x = 3 sin 2 + 2 sin 3, and y = 2 cos 3 – 3cos 2, prove that dy

tan .dx 2

2

32. a) Prove that in the curve

x

ay e the subnormal varies as the square of the ordinate and subtangent is

constant. 3

b) Evaluate / 2

40

sinx.cosxdx.

1 sin x

2

33. a) Evaluate: 2 3tanx

dx.1 2tanx

b) Evaluate: x x

1dx.

1 e 1 e 5

34. Find the area of the ellipse 9x2 + 16y2 = 144 by integration. 5 PART–D

Answer any two of the following questions: 2 x 10 = 20 35. a) Define hyperbola as a locus and derive the standard equation of the hyperbola in the form

2 2

2 2

x y1

a b . 6

b) Prove that

2

22 3

2

1 a a

a 1 a a 1 .

a a 1

4

36. a) If cos + cos + cos = 0 = sin + sin + sin , prove that

i) cos 2 + cos 2 + cos 2 = 0 , sin 2 + sin 2 + sin 2 = 0

ii) cos2 + cos2 + cos2

= 3

2 , sin2

+sin2 +sin2

=3

.2

6

b) Prove that 2

axb bxc c xa a b c

4

37. a) The surface area of a sphere is increasing at the rate of 8 sq/sec. Find the rate at which the radius

and the volume of the sphere are increasing when the volume of the sphere is 500

3

c.c. 6

b) Find the general solution of sin + sin 2 + sin 3 = 0 4

38. a) Prove that / 2 2

0

cos xdx .

1 sinxcosx 3 3

6


b) Find the general solution of the differential equation 2

2

2

dy 1 yxy 1 x x

dx 1 x

4

PART–E Answer any one of the following questions: 1 x 10 = 10

39 a) If a b c 0 and a 3, b 5 and c 7, find the angle between a and b. 4

b) Find the cube roots of a complex number 3 i and represent them in argand diagram. 4

c) Find the remainder when 2202 is divided by 11 (least positive remainder). 2 40. a) The sum of the lengths of hypotenuse and another side of a right angled triangle is given. Show that

the area of the triangle is maximum when the angle between these sides is .3

4

b) Evaluate 4cot 3x dx. 4

c) Differentiate w.r.t. x: 25y log 1 x . 2

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SECOND PUC ANNUAL EXAMINATION - MARCH–2009

MATHEMATICS

Time: 3 Hours-15 minutes. Max Marks: 100

Instructions :- 1. The question paper has FIVE parts, A, B, C, D and E. Answer all the parts 2. Part – A carries 10 marks. Part – B carries 20 marks. Part – C carries 40 marks. Part – D carries 20 marks. And Part – E carries 10 marks. 3. Write the question numbers properly as indicated in the question paper.

PART – A Answer all the ten questions: 10 x 1 = 10 1. Find the least positive remainder when 730 is divided by 5.

2. If 4 x 2

2x 3 x 1

is a symmetric matrix, find x.

3. Define a subgroup.

4. Find the direction cosines of the vector 2i 3j 2k

5. If the radius of the circle 2 2x y 4x 2y k 0 is 4 units, then find k

6. Find the equation of the parabola if its focus is (2, 3) and vertex is (4, 3).

7. Find the value of 11sin cos 1

2

8. If 1, , 2 are the cube roots of unity, find the value of (1 – + 2)6.

9. Differentiate x3 sinhx w.r.t. x.

10. Integrate 1 cos2x

w.r.t. x.1 cos2x

PART – B


11. If a b (mod m) and n is a positive divisor of m, prove that a b (mod n)

12. Without actual expansion show that

43 1 6

35 7 4 0

17 3 2

13. Is G = {0, 1, 2, 3}, under modulo 4 a group? Give reason. 14. Find the equation of two circles whose diameters are x + y = 6 and x + 2y = 4 and whose radius is 10 units.


15. Find the area of the parallelogram whose diagonals are given by the vectors 2i j k and 3i 4j k.

16. Find the eccentricity of the ellipse (a > b). If the distance between the directrices is 5 and distance

between the foci is 4

17. Solve cot –1 x + 2 1 5tan x .

6

18. Find the least positive integer n for which n

1 i1

1 i

.

19. If m

2y x 1 x . Prove that 2 2dy1 x m y 0

dx

20. Show that for the curve

x

ay be the subnormal varies as the square of the ordinate y.

21. Evaluate e

e

1

log x dx

22. Find the order and degree of the differential equation

32 22

2

dy d y1

dx dx

PART – C

I. Answer any three questions: 3 X 5 = 15 23 Find the G.C.D of a = 495 and b = 675 using Euclid Algorithm. Express it in the form 495 (x) + 675 (y).

Also show that x and y are not unique where x, y z 5 24. Solve the linear equations by matrix method.

3x y 2z 3

2x 3y z 3

x 2y z 4

25. a) On the set of rational numbers, binary operation is defined by 2 2a b a b , prove tht is both

commutative and associative. Also find the identity element 3

b) If a is an element of the group (G ), then prove that 1

1a a

. 2

26. a) Find the sine of the angle between the vectors i 2j 3k and 2i j k . 3

b) Show that the vectors j 2k, i 3j and i 2j form the vertices of an isosceles triangle. 2

II Answer any two questions: 2 x 5 = 10

27. a) Derive the condition for two circles 2 21 1 1x y 2g x 2f y c 0 and 2 2

2 2 2x y 2g x 2f y c 0 to cut

orthogonally 3

b) Show that the radical axis of the two circles 2 22x 2y 2x 3y 1 0 and 2 2x y 3x y 2 0 is

perpendicular to the line joining the centres of the circles. 2

28. a) Find the ends of latus rectum and directrix of the parabola 2y 4y 10x 14 0 . 3

b) Find the value of k such that the line x 2y k 0 be a tangent to the ellipse 2 2x 2y 12 2

29. a) If 1 1 1tan x tan y tan z , Show that x + y + z – xyz = 0. 3

b) Find the general solution of tan 4 = cot 2. 2 III. Answer any three of the following questions: 3 x 5 = 15 30. a) Differentiate ‘tan x’ w.r.t. x from the first principle. 3

b) If 2

1

2

2 3xy tan

3 2x

. Prove that 4

dy 2x

dx 1 x

2

31. a) If y = cos (p sin–1 x), prove that 2 22 11 x y xy p y 0 3

b) Find the equation of the normal to the curve y = x2 + 7x – 2 at the point where it crosses y-axis. 2


32. a) Integrate 3x 3 tanxe w.r.t x.

cosx

3

b) Find the angle between the curves 3 24y x and y 6 x at (2, 2) 2

33. a) If m nm nx y x y

, prove that dy

x y.dx

3

b) Integrate 2

1

7 6x x w.r.t x. 2

34. Find the area between the curves y2 = 6x and x2 = 6y 5

PART – D Answer any two of the following questions: 2 x 10 = 20

35. a) Define hyperbola is a locus and hence derive the equation of the hyperbola in the form 2 2

2 2

x y1

a b 6

b) Show that

2 2

2 2 2 2 2

2 2

b c ab ac

ba c a bc 4a b c

ca cb a b

4

36. a) If cos + 2 cos + 3 cos = 0, sin + 2 sin + 3 sin = 0, show that

i) cos 3 + 8 cos 3 + 27 cos 3 = 18 cos ( + + )

ii) sin 3 + 8 sin 3 + 27 sin 3 = 18 sin ( + + ) 6

b) Prove that a b b c c a 2 a b c

. 4

37. a) The volume of a sphere is increasing at the rate of 4 c.c/sec. Find the rate of increase of the radius

and its surface area when the volume of the sphere is 288 c.c. 6

b) Find the general solution of 3 tanx 2sec x 1 4

38. a) Show that 4

0

log 1 tanx dx log28

6

b) Solve the differential equation dy

tany sin x y sin x y .dx

4

PART – E

Answer any one of the following questions: 1 x 10 = 10

39. a) Find the cube roots of 3 i 3 and find their continued product. 4

b) Show that 2 22 2

a b a b a b . 4

c) Find the length of the chord of the circle 2 2x y 6x 2y 5 0 intercepted by the line x – y + 1 = 02

40. a) Evaluate 3

0

x 2dx

x 2 5 x

4

b) Show that among all the rectangles of a given perimeter, the square has maximum area. 4

c) Differentiate sec 5x w.r.t. x . 2

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SECOND PUC ANNUAL EXAMINATION - JUNE–2009 MATHEMATICS

Time: 3 Hours 15 minutes Max Marks: 100

Instructions :-


1. The question paper has FIVE parts, A, B, C, D and E. Answer all the parts 2. Part – A carries 10 marks. Part – B carries 20 marks. Part – C carries 40 marks. Part – D carries 20 marks. And Part – E carries 10 marks. 3. Write the question numbers properly as indicated in the question paper. Answer all the ten questions : 10 x 1 = 10

1. 3x 2(mod 6) has no solution why?

2. If direction cosines of a are 1 2

,3 3

and n, find n.

3. On I (the set of all integers) and operation is defined by a b = ab, a, b I. Examine whether is binary or not on I.binary or not on I.

4. A and B are square matrices of the same order and A 4, B 5. Find AB .

5. Given two circles with radii r1, r2 and having d as the distance between their centres. Write the condition for them to touch each other externally. 6. Find the sum of the focal distances of any point on 4x2 + 9y2 = 36

7. Evaluate 1sin sin130

8. Find the least positive integer n for which

n1 i

11 i

9. Given the function f x x . Find L 0f

10. Evaluate 4

3

4

sin x cos x dx.

PART – B Answer any ten questions : 10 x 2 = 10

11. If ca cb (mod m) and c, m are relatively prime then prove that a b(mod m)

12. For the matrix cos sin

A ,sin cos

virify that AA/ is symmetric.

13. Define a semigroup. Examine whether {1, 2, 3, 4} is a semigroup under “addition modulo 5”(+5).

14. On Q+ (set of all positive rationals). an operation is difined by a b = ab

3, a, b Q+. Find the

identity element and a–1 in Q+.

15. If i j 2k , 2i 3j 4k and i 2j k are coplanar. find .

16. Find the equation of the circumcircle of the triangle formed by (0,0), (3, 0) and (0, 4).

17. Solve 1 1 11 1tan x sin cot .

2 3

18. Show that the real and imaginary parts of 1 4

i tan35e

are 3, 4 respectively.

19. If y = 1 1x 1 x 1sin sec ,

x 1 x 1

prove that

dy0.

dx

20. At any point on the curve m n m n,x y a show that the subtangent varies as the abscissa of the point.

21. Evaluate sin logx cos logx dx.

22. Form the differential equation of the family of circles touching y-axis at origin.

PART – C I. Answer any three questions : 3 x 5 = 15 23. a) Define GCD of two integers a and b. Find the GCD of 275 and 726. 3 b) Find the number of positive divisors of 252 by writing it as the product of primes


(prime power factorization). 2 24. Solve by matrix method: 5

2x y 10

x 2y 2

Also, verify that the coefficient matrix of this system satisfies Caley. Hamilton theorem.

25. Prove that a non-empty subset H of a group G, is a subgroup of G. if a, b H,

ab–1 H. Hence prove that. If H and K are subgroups of a group G then H K also, is a subgroup of G. 5

26. a) Given a 2i j k, b i 2j k, find a unit vector perpendicular to a and coplanar

with a and b 3

b) If a b c 0. prove that a b b c c a. 2

II. Answer all the three questions : 3 x 5 = 15 27. a) Derive the condition for the two circles. x2 + y2 + 2g1x + 2f1y + c1 = 0 and x2 + y2 + 2g2x + 2f2y + c2 = 0 to cut each other orthogonally. 3

b) (1, 2) is the radical centre of three circles. One of the circles is x2 + y2 – 2x + 3y = 0. Examine whether the radical centre lies inside or outside all the circles. 2 28. a) Given the equation of the conic 9x2 + 4y2 – 18x + 16y – 11 = 0, find its centre and the area of its auxiliary circle. 3 b) Obtain the equation of the directrix of the parabola x = 2t2, y = 4t 2

29. a) If 1 1 1sin x sin y sin z ,2


b) Find the general solution of tan 2 tan = 1 2 III. Answer all the three questions: 3 x 5 = 15 30. a) Differentiate sin 2x w.r.t. x from first principle. 3

b) Differentiate logx

sinx w.r.t x 2

31. a) Differentiate 1 3cos 4x 3x w.r.t. 1 2cos 1 2x 3

b) Show that curves y = 6 + x – x2 and y(x – 1) = x +2 touch each other at (2, 4) 2

32. a) If 1y sin mcos x prove that 2 22 11 x y xy m y 0 3

b) Evaluate 5

1dx

x x 1 2

33. a) Integrate sinx 18cos x

3sinx 4cos x

w.r.t. x. 2

b) Evaluate 1 x

dx1 x

3

34. Find the area of x2 + y2 = 6 by integration. 5 PART – D

II. Answer any two questions : 2 x 10 = 20

35. a) Derive a condition for y = mx + c to be a tangent to the hyperbola 2 2

2 2

x y1

a b . Also, find the point

of contact. Using the condition derived, find the equations of tangents to 2 2x y

1,16 12

which are

parallel to x – y + 5 = 0 6

b) Prove that

2

2

2

1 a a bc

1 b b ca 2 a b b c c a .

1 c c ab

4


36. a) State De Moivre‟s theorem. Prove it for positive and negative integral indices. Using it prove that

10

10

z 1i tan 5

z 1

if z = cos + i sin . 6

b) Find the general solution of cos2 2 cos sin 4

37. a) The volume of a sphere increases at the rate of 4 c.c/ sec. Find the rates of increase of its radius

and surface area when its volume is 288 c.c. Also find (i) the change in volume in 5 secs , (ii) rate of

increase of volume w.r.t. radius when the volume is 288 c.c 6 b) Obtain the equations of parabolas having (1, 5) and (1, 1) as ends of the latus rectum. 4

38. a) Prove that 2

2 2 2 20

xdx

2aba cos x b sin x

6

b) Find the particular solution of 2 2dyxy 1 x y 1

dx , given that, when x = 1, y = 0.

PART – E

II. Answer any one of the following questions : 1 x 10 = 10

39. a) If a b c a b c . find the angle between a b and c 4

b) Among all right-angled triangles of a given hypotenuse, show that the triangle which is isosceles has maximum area. 4

c) Find the fouth roots of 16cis2

. 2

40. a) If 150 122 3 135 a mod 7 . find the least positive remainder when a is divided by 7. 4

b) Given the circles 2 22 x y 12x 4y 10 0 and 2 2x y 5x 13y 16 0. find the length of

their common chord. 4

c) Evaluate 2

0

xdx

2 x x 2

***********************


Time: 3 Hours 15 minutes Max Marks: 100

PART - A

Answer any ten questions : 10 X 1= 10

1. If ab and bc then prove that ac.

2. If 1 1 3

B A2 3 4

and

2 3 1B A

3 4 2

, find A.

3. Find the identity element in the set of all positive rationals Q+, is defined by

ab

a b a,b Q2

4. If the vectors a 2i j k and b 3i 4j k, find a b .

5. Find the values of h and k for the equation 2 2kx 2hxy 4y 2x 3y 7 0 to represent a

circle. 6. Write the eccentricity of the conic section represented by the parametric equations x = at2 and y = 2at.


7. Evaluate 11 4sin cos

2 5

.

8. Find the amplitude of 1 cos isin

9. Find dy

dx if

1 sinx

ey log e

10. Evaluate 4

2

0

sec x dx

PART - B Answer any ten questions : 10 X 2 = 20

11. Find the sum of all positive divisors of 72.

12. If 1 0 x 0

2 Iy 5 1 2

where I is the identity matrix, find x and y.

13. Write the composition table for G = {2, 4, 6, 8} under multiplication modulo 10 and find the identity element. 14. Find the area of the triangle whose two adjacent sides are determined by the vectors

2i 3j 2k and 4i 5j 3k .

15. Show that the two circles 2 2x y 6x 2y 1 0 and

2 2x y 2x 8y 13 0 touch each

other externally.

16. Find the centre of the hyperbola 2 29x 4y 18x 8y 31 0 .

17. Prove that 1 1 12

2tan tan5 5 2

.

18. If x = cos 4 + isin 4, show that 1

x 2cos2 .x

19. If x = a, a

y ,

show that dy y

0.dx x

20. A particle is travelling in a straight line whose distance is given by s = 4t3 – 6t2 + t – 7 units. Find the velocity of the particle after t = 2 seconds.

21. Evaluate 2

logxdx

x

22. Find the order and degree of the differential equation.

32 22

2

d y dy1

dx dx

PART - C Answer any three questions : 10 X 2 = 20

23. Find the G.C.D of 506 and 1155 and express it in the form of 506(a) + 1155 (b) (Where a and b are integers). Also show that the expression is not unique. 5

24. a) If

1 x 1 1

1 1 y 1 0

1 1 1 z

where, x 0, y 0 and z 0 then show that 1

1 0x

3

b) Solve by Cramer‟s Rule : 2x – 3y = 5, 7x – y = 8. 2 25. Prove that set M of all 2 X 2 matrices with elements of real numbers form an Abelian group with respect to addition of matrices. 5

26. a) Find the vector of magnitude 12 units which is perpendicular to both the vectors

a 4i j 3k and b 2i j 2k. 3

b) If a 5i 2j 3k and b 4i 2j 5k are two vectors find the projection of a on b 2


II Answer any two questions : 2 X 5 = 10

27. a) Derive the condition for the line y = mx + c to be a tangent to the circle 2 2 2x y a in the form

2 2 2c a 1 m . 3

b) Find the radical axis of the circles 2 23x 3y 9x 6y 1 0 and

2 22x 2y 8x 16y 3 0

28. a) Find the equations of the parabolas whose directrix is x + 2 = 0, axis is y = 3 and the length of the latus rectum is 8 units. 3

b) Find the eccentricity of the hyperbola 2 2x 3y 4x 6y 11 0 2

29. a) If 1 1 1cos x cos y cos z , then prove that

2 2 2x y z 2xyz 1 . 3

b) Find the general solution of the equation 2cos cos 2 2 . 2

III Answer any three questions : 3 X 5 = 15

30. a) Differentiate sin ax with respect to x from the first principle. 3

b) Find dy

dx if

1 1 cosxy tan

1 cosx

. 2

31. a) If 1msin xy e ,

prove that 2 2

2 11 x y xy m y 0. 3

b) Find the angle between the curves y2 = 4x and x2 = 2y – 3 at the point (1, 2). 2

32. a) Find the length of sub-normal to the curve x3y2 = a5 at any point on it. Also show that length of subtangent varies directly as abscissa at that point. 3

b) Evaluate

x

sec x 1 tanxdx

e

2

33. a) Evaluate dx

13 12cosx 3

b) Evaluate 2

dx

8 6x 9x 2

34. Find the area of the circle x2 + y2 = 6 by integration method. 5

PART - D Answer any two of the following questions : 2 X 10 = 20

35. a) Define an ellipse as a locus and derive its equation in standard form 2 2

2 2

x y1, a b .

a b 6

b) Find A–1 using Cayley-Hamilton theorem if 2 4

A7 3

4

36. a) State and prove De Moivre‟s theorem for rational index. 6

b) Prove by vector method. sin A B sinAcosB cosAsinB 4

37. a) Show that rectangle of maximum perimeter which can be inscribed in a circle of radius a is a

square of side a 2 6

b) Find the general solution of 3 1 cosx 3 1 sinx 2 4

38. a) Evaluate

0

x tanxdx

sec x tanx

6

b) Solve by the method of separation of variables the differential equation y 2 2 ydye x x e

dx 4

PART - E


Answer any one questions : 1 X 10 = 10

39. a) Find the cube roots of the complex number 3 i and express them in an Argand diagram. 4

b) Find the equation of the chord of the circle 2 2x y 4x 6y 9 0 bisected at (0, 1). 4

c) Find the number of incongruence solution and the incongruence solution of 2x 2 mod6 . 2

40. a) If a b c 0 and a 3, b 5, c 7, then find the angle between the vectors a & b 4

b) Find 3sec 2x dx 4

c) If 5

dyy log sec x , find

dx 2

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2nd puc question papers of mathematics 1997-2010

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