mathssnsce.weebly.commathssnsce.weebly.com/.../0/1/25011348/m1_question_bank.docx · web viewunit...

080030001 MATHEMATICS – I

UNIT I MATRICES Characteristic equation – Eigen values and eigen vectors of a real matrix –Properties – Cayley-Hamilton theorem (excluding proof) – Orthogonal transformation of a symmetric matrix to diagonal form – Quadratic form –Reduction of quadratic form to canonical form by orthogonal transformationUNIT II THREE DIMENSIONAL ANALYTICAL GEOMETRY Equation of a sphere – Plane section of a sphere – Tangent Plane – Equation of a cone – Right circular cone – Equation of a cylinder – Right circular cylinder.

UNIT III DIFFERENTIAL CALCULUS Curvature in Cartesian co-ordinates – Centre and radius of curvature – Circle of curvature – Evolutes – Envelopes – Evolutes as envelope of normal.UNIT IV FUNCTIONS OF SEVERAL VARIABLES Partial derivatives – Euler’s theorem for homogenous functions – Total derivatives – Differentiation of implicit functions – Jacobians – Taylor’s expansion– Maxima and Minima – Method of Lagrangian multipliers.UNIT V MULTIPLE INTEGRALS Double integration – Cartesian and polar coordinates – Change of order of integration – Change of variables between Cartesian and polar coordinates –Triple integration in Cartesian co-ordinates – Area as double integral – Volume as triple integralTEXT BOOK:1. Bali N. P and Manish Goyal, “ Text book of Engineering Mathematics”, Thirdedition, Laxmi Publications(p) Ltd.,(2008).REFERENCES:1. Grewal. B.S, “Higher Engineering Mathematics”, 40th Edition,Khanna Publications,Delhi,(2007).2. Ramana B.V, “Higher Engineering Mathematics”, Tata McGrawHill Publishing Company, New Delhi, (2007).3. Glyn James, “Advanced Engineering Mathematics”, 7thEdition,Wiley,India,(2007).4. Jain R.K and Iyengar S.R.K,” Advanced Engineering Mathematics”,3rdEdition, Narosa Publishing House Pvt. Ltd.,(2007).

Mathematics – IPart-A Questions

Unit –I MATRICES

1. Find the characteristic equation of [1 20 2]

2. Find the characteristic polynomial of [ 3 1−1 2]

3. Find the characteristic equation of [3 2 −12 1 04 −1 6 ]

4. The sum of the Eigen values of a matrix A is equal to the sum of the elements on its diagonal

5. If λ1 , λ2 , ……….. λn are the eigen values of an n x n

matrix A , then show that λ1 3 , λ2

3 , ……. λn

3 are the

eigen values of A3

6. Find the sum and product of the eigen values of the

matrix A = [2 0 10 2 01 0 2 ]

7. The product of two eigen values of the matrix A =

[ 6 −2 2−2 3 −12 −1 3 ] is 16.Find the third eigen value

1

8. Find the sum and product of the Eigen values of the

matrix[2 −34 −2]

9. Two eigen values of the matrix A = [2 2 11 3 11 2 2] are

equal to 1 each.Find the eigen value of A-1

10. Give some properties of Eigen values.

11. Prove that A andAT have the same Eigen values.

12. If λ is an Eigen value of A then 1λ is an Eigen value of

A-1.

13. The product of the Eigen values of a matrix A is equal

to its determinant.

14. Prove that the Eigen values of a real symmetric matrix

must be real

15. The product of the two Eigen values of the matrix A=

[2 41 4] is 2. Find the third value.

16. Prove that if X is an Eigen value of A corresponding to

an Eigen value λ then any non zero scalar multiple of X

is also an Eigen vector of A.

17. An Eigen vector of a matrix can not correspond to two

distinct Eigen values.

18. If λ Eigen values of a matrix A then K λ is the Eigen

value of the matrix KA, where K is a non zero scalar.

19. If λ Eigen values of a matrix A then|A|λ

is the Eigen

value of the matrix adj A.

20. Two of the Eigen values of [ 6 −2 2−2 3 −12 −1 3 ]are 2¿8

Find the third Eigen value.

21. If 3¿ 15 are the two Eigen values of A =

[ 8 −6 2−6 7 −42 −4 3 ] Find the third Eigen value.

22. If 2,2,3 are the Eigen values of A =

[ 3 10 5−2 −3 −43 5 7 ] Find the Eigen valueATand A−1

23. If the Eigen values of the matrix A=[1 13 −1] are 2 , - 2

then find the Eigen values of AT

24. Find the Eigen values of A=[2 1 00 2 10 0 2 ]without using the

characteristic equation idea.

2

25. Find the Eigen values of A = [1 0 00 4 00 0 3]

26. Find the Eigen values of the inverse of the matrix A =

[1 3 40 2 50 0 3]

27. Two of the Eigen values of A =

[ 3 −1 1−1 5 −11 −1 3 ] are 3 and 6 find the Eigen values of A−1

28. Two Eigen values of the matrix A = [2 2 11 3 11 2 2]

are equal to 1 each find the Eigen values of A−1

29. One of the eigen values of A=[7 4 −44 −8 −14 −1 −8]find other

two eigen values30. Find the Eigen values of A3 given A =

[1 2 30 2 −70 0 3 ]

31. If 1 and 2 are the Eigen values of a 2 x 2 matrix

what are the Eigen values of A2 and A−1

32. If 1,1,5 are the Eigen values of A = [2 2 11 3 11 2 2]

find the Eigen values of 5A

33. Find the Eigen values of A, A4 ,3A,A−1 if A = [2 30 5 ]

34. If ( 1,1,5) are the Eigen values of A = [2 2 11 3 11 2 2]

find the Eigen values of KA ( K is scalar)

35. If the Eigen values of a matrix A are 2,3,4 find the

Eigen values of Adj A

36. Find the Eigen values of the matrix( 1 −2−5 4 ). Hence

form the matrix whose Eigen values are 1/6 and -1

37. If A = [3 5 30 4 60 0 1] find the Eigen values of A, A−1, AdjA

and A5

38. Find the sum of the Eigen values of 2A if A =

[ 8 −6 2−6 7 −42 −4 3 ]

39. If the sum of two Eigen values and trace of a 3 x 3

matrix are equal find the value of |A|

3

40. Find the sum of the Eigen values of the inverse of A =

[3 0 08 4 06 2 5]

41. What is the sum of squares of the eigen values of

[1 7 50 2 90 0 5 ]

42. . Form the matrix whose eigen values are α-5, β-5, γ-5 where α, β, γ are the eigen values of A=

[−1 −2 −34 5 −67 −8 9 ]

43. If α, and β are the eigen values of [ 3 −1−1 5 ], form the

matrix whose eigen values are α3 and β3.

44. If the sum of two Eigen values and trace 3 x 3 matrix A

are equal, find the value of

45. The Eigen values of the given orthogonal matrix A =

[ 1√2

1√2

−1√2

1√2 ] are

1+ i√2

, 1−i√2

46. Show that √21+ i

, √21−i

is also Eigen values of A

47. Find the Eigen value of [2 30 4] corresponding to the

eigen vector [10 ]

48. If one of the Eigen vector of A=[ 3 −1 1−1 5 −11 −1 3 ] is ¿]

then find the corresponding Eigen value

49. If x= (-1, 0, -1)T is the Eigen vector of the matrix A =

[1 1 31 5 13 1 1] find the corresponding eigen value.

50. Prove that the Eigen values of ( −3 A−1) are

the same those of A = [1 22 1]

51. If λ1 , λ2 , λ3 , ………. λn , are the Eigen values of A then

find the eigen values of the matrix (A-λI)2.

52. State Cayley -Hamilton theorem and give its uses.

53. Verify Cayley-Hamilton theorem for the matrix A =

[0 24 0]

54. Define orthogonal matrix.

55. Show that the matrix [ cosθ sinθ−sinθ cosθ ] is orthogonal.

4

56. If A is orthogonal matrix, show that A-1 is also

orthogonal.

57. If A is an orthogonal matrix prove that

58. Prove that the inverse of orthogonal matrix A is orthogonal.

59. Write the matrix of the Quadratic form x12 +2x2

2+x32-

2x1x2+2x2x3

60. Write the matrix of the Quadratic form 2x12 -

2x22+4x3

2+2x1x2 -6x1x3+6x2x3

61. Write the Quadratic form corresponding to the

symmetric matrix [ 0 −1 2−1 1 42 4 3 ]

62. If the matrix of the Q.F. 3x2+2axy+3y2 has Eigen

values 2 and 4 ,find the values of a.

63. Determine the nature of the Quadratic form f(x1, x2 ,x3)=

x12 + x2

2 + x32

64. Find the index, signature and nature of the Quadratic

form x12 + 2 x2

2 -3 x32

65. Determine the nature of the Quadratic form f(x1, x2 ,x3)=

x12+ 2x2

2

66. Determine the nature of the Quadratic form f(x1, x2 ,x3)= 2x1

2-x22

67. Find the nature of the conic 8x 2-4xy +5y2=36 by reducing the Q.F. 8x2-4xy +5y2 to the form

Ax2+By2(Equations of the transformations are not needed)

68. If the sum of the Eigen values of matrix of Q.F=0, then what will be the nature of Q.F?

Unit -II THREE DIMENSIONAL ANALYTICAL GEOMETRY

1. Find the equation of a sphere whose centre is (2, -3,

4) and radius 3 units.

2. Find the centre and radius of the sphere x2+y2+z2 -

6x+8y-10z+25 = 0

3. Find the centre and radius of the sphere2x2+2y2+2z2 -

2x+4y+2z+3= 0

4. Find the centre and radius of the sphere 3(x2+y2+z2) -

6x - 12y+18z+9 = 0.

5. Find the centre and radius of the sphere a(x2 +y2 +z2)+2ux+2vy+2wz+d=0.

6. Find the centre and radius of the sphere7 x2 +7y2 +7z2

+28x-42y+56z+3=0.7. The point (2,3,4) is one end of the diameter of a

sphere x2 +y2 +z2 -2x-2y+4z-1=0,find the other end.8. Write down the equation of sphere whose diameter is

the line joining (1,1,1) and (-1,-1,-1).9. Find the equation of the sphere on the line AB as

diameter where A is (2,3,0) and B is (1,2,3).10. Find the equation of a sphere which touches the

plane x+2y+2z-1=0 and whose centre is (2,3,4).

5

11. Find the equation to the sphere of radius 2 units, lying

in the first octant and touching the coordinate planes.

12. Find the equation to the sphere whose centre is (3,-5,

4) and which passes through the point (1, -2, 2).

13. Find the equation of the tangent plane at the point

(1,-1, 2) to the sphere

x2+y2+z2-2x+4y+6z-12 = 0.

14. Find the equation of the normal at the point (2,-1,4) to the sphere x2 +y2 +z2 –y-2z-14=0.

15. Prove that the plane 2x-y-2+12 = 0 touches the sphere

x2+y2+z2-2x-4y+2z-3 = 0.

16. Find the equation of a sphere which passes through

the point (1,-2, 3) and the circle Z=0, x2+y2+z2-9 = 0.

17. Find the equation of the sphere having its centre on

the plane 4x-5y-z=3 and passing through the circle

x2+y2+z2-2x-3y+4z+8 = 0, x-2y+z= 8.

18. Find the equation of the sphere through the

circlex2+y2+z2+2x+3y+6= 0; x-2y+4z-9= 0 and the

centre of the sphere x2+y2+z2-2x+4y-6z+5 = 0.

19. Find the equations of the spheres which passes

through the circle x2+y2+z2= 5 and x+2y+3z =3 and

touch the plane 4x+3y= 15.

20. Define cone.

21. Find the equation of the cone where vertex is (3, 1, 2)

and base the circle 2x2+3y2 = 1, z = 1.

22. Find the equation of the cone whose vertex is the

point (1, 1, 0) and whose base is the curve y = 0,

x2+z2 = 4.


point(α,β,γ) and base y2=4ax,z=0


point(α,β,γ) and base ax2 +by2=1,z=0

25. Find the equation of the cone whose vertex is at the

origin and the guiding curve is x2

4 + y2

9 + z

2

1

=1,x+y+z=1

26. Find the equation to the cone whose vertex is the

origin and base the circle x=a,y2+z2=b2 and show that

the section of the cone by a plane parallel to the

plane XOY is a hyperbola.

27. Define Right circular cone & give its equation.

28. Show that the equation to the right circular cone

whose vertex is 0 axis , OX and semi-vertical angle α

is y2+z2 = x2 tan2 α6

29. Show that the equation of a right circular cone whose

vertex is the origin O,axis OZ and semi vertical angle α

is x2 +y2 =z2tan2 α.

30. Write the equation of right circular cone whose axis is

x−αl =

y−βm =

z−γn

31. Find the equation of the cylinder whose generating

line have the direction cosines l, m, n and which

passes through the circumference of the fixed circle

x2+z 2= a2 in the ZOX plane.

32. Define cylinder.

33. Find the equation of a cylinder whose generating lines

have the direction cosines(l, m, n) and which passes

through the circle x2+z2 = a2, y = 0.

34. Find equation of the cylinder whose generators are

parallel to the line x1 =

y−2=

z3 and whose guiding curve

is the ellipse x2+2y2 = 1; z = 0.

35. Find the equation of the quadratic cylinder with

generators are parallel to x- axis and passing through

the curve ax2+by2+cz2 = 1, lx + my + nz = p.

36. Find the equation of the quadratic cylinder with

generators are parallel to z-axis and passing through

the curve ax2+by2 = 2z, lx + my + nz = p.

37. Define Right circular cylinder and give its equation.

Unit -III DIFFERENTIAL CALCULUS

1. Define curvature and radius of curvature.

2. What is the formula for curvature at any point p(x,

y)on the curve y = f(x).

3. Prove that radius of curvature of a circle is its radius.

4. What is the curvature of a 1) circle 2) straight line.

5. What is the curvature of a circle of radius 2 units?

6. Find the radius of curvature of the curve y = ex at the

point where it crosses the y-axis.

7. Find the radius of curvature of the curve√ x + √ y =1 at

(1/4 , 1/4)

8. Find the radius of curvature of the curve y = a log

sec( x/a) at any point(x,y)

9. Find the radius of curvature at (0, c) on the curve y = c

cos h( x/c)

10. Find the curvature at any point on the curve S = c log

(sec Ψ ).

7

11. Find the radius of curvature at x = π/2 on y = 4sinx.

12. Find the radius of curvature at x = π2 on the curve y =

4sinx – sin2x.

13. Find the radius of curvature of the curve xy = c2 at

(c,c).

14. Determine the radius of curvature of x3+y3 = 2 at (1, 1).

15. Find the radius of curvature at x = 1 on y = log xx

16. Find the radius of curvature of the circle x2+y 2= 25 at

(3, 4).

17. Find the curvature of the curve 2x2+2y 2+5x-

2y+1 =0, at any point on it.

18. Find the radius of curvature of the curve x2+y 2-

6x+4y+6=0.

19. Find the radius of curvature at y = 2a on the curve y2 =

4ax.

20. Find the curvature of the parabola y2 = 4x at the

vertex (0,0).

21. Find the radius of curvature at (x, y) for the curve a2 y =

x3 – a3

22. Find the radius of curvature of the curve at (0,0) on y2 =

a3−x3

x.

23. For the curve x2 = 2c(y-x), find the radius of curvature

at (o, c).

24. Prove that the radius of curvature of the curve xy2 = a2-

x2 at the point (a, 0) is 3a/2.

25. Find the radius of curvature at (a, a) on the curve x3+y3

=2a3.

26. Find the radius of curvature of the curve r = a (1+cosθ)

at θ = π/2..

27. Find the radius of curvature at the point (r,θ) on the

curve r = a cosθ.

28. Find ρ at any point P(at2,2at) on the parabola y2=4ax

29. Find ρ at any point t on the curve x=a(cost +t

sint),y=a(sint-t cost)

30. Give the radius of curvature of the curve given by x =

3+2 cosθ, y = 4+2sinθ without using the

formula.

31. Find the centre of curvature of y=x2 at the origin.

32. Write the equation of the circle of curvature..

33. State any two properties of evolutes.

8

34. Find the evolutes of the curve x2+y2 +4x-6y+3 =

0.

35. If the centre of curvature of a curve is (cacos3t ,

ca

sin3t ).Find the evolute of the curve.

36. If (2+3cosѳ , 3+4sinѳ) is the centre of curvature at the

point ѳ ,find the evolute of the curve.

37. Given the co-ordinates of the centre of curvature is

given as X=2a+3at2,

38. Y=-2at3, determine the evolute of the curve.

39. Define envelope of a family of curves.

40. What is the envelope of the family Am2 + Bm2 + c = 0.

41. Find the envelope of y = mx + am2, m being the

parameter.

42. Find the envelope of the family of straight lines y = mx

± √m2−1 where m is the parameter.

43. Find the envelope of y = mx + m3, m being the

parameter.


+ amwhere m is a parameter.


± √m2+1 where m is the parameter.

46. Find the envelope of the family given by x = my + 1/m

where is a parameter.


+√a2m2+b2 where m is the parameter.

48. Find the envelope of the family of circles (x-α )2 + y2 =

4α ,α being the parameter.


+ a√1+m2.

50. Find the envelope of y cot2 α -x-a cosec2α =0,where

α being the parameter.

51. Find the envelope of the family of straight lines xcosα +

ysinα = asecα,where α being the parameter.

52. Find the envelope of xcosα + ysinα =

a,where α being the parameter.

53. Find the envelope of the family of lines

xa cosθ+

yb sinθ=1.Where θis the parameter

54. Find the envelope of x2+y2-axcos θ-by sin θ=0,where

θ is the parameter .

9

55. Find the envelope of the family of straight lines y – 2x

= 2∝

56. Find the envelope of x + y – ax cosθ - by sin θ = 0,

where θ is the parameter.

57. Show that the family of straight lines 2y-4x+λ=0,has no

envelope where λ is the parameter

58. Find the envelope of the family of lines xt + yt = 2c,t

being the parameter.

Unit –IV FUNCTIONS OF SEVERAL VARIABLES

1. If u = x/y + y/z + z/x find x ∂u∂x + y

∂u∂ y + z

∂u∂ z

2. If u = y f(xy¿+g(

yx ) find x

∂u∂x

+ y ∂u∂ y

3. If u = (x-y)(y-z)(z-x) show that ∂u∂x

+ ∂u∂ y +

∂u∂ z=0

4. If u= f (xy ,

yz

zx , ). Prove that x

∂u∂x

+ y ∂u∂ y +z

∂u∂ z

=0.

5. If f(x,y)=log√ x2+ y2,show that ∂2 f∂ x2 + ∂

2 f∂ y2 = 0.

6. If x = rcosθ y =rsinθ. Prove that ∂r∂x =

∂x∂r ,

1r ∂x∂θ =

r ∂θ∂x .

7. If x = rcosθ ,y =rsinθ find ∂r∂x

8. If u = f (x, y, z) where x, y, z are functions in t, then

∂u∂ t ?

9. If u = xy , , x = et, y = log t find

∂u∂ t

10. If u = x2+y2+3x2y2, find∂u∂x .

11. If u = xy +yz +zx where x = et, y = e-t and z = 1t . Find

∂u∂ t .

12. State Euler’s theorem for homogeneous functions.

13. Verify whether u = ex/y sin(x/y) + ey/x cos (y/x) is

homogeneous. If so find its degree.

14. If u = sin-1 (x/y) + tan-1 (y/x). Prove that x ∂u∂x + y

∂u∂ y

= 0.

10

15. Show that x ∂u∂x + y

∂u∂ y =2ulogu where log u=

x3+ y3

3x+4 y

16. If u = sin-1 (x3 −¿ y3

x +¿ y ) Prove that x ∂u∂x

+ y ∂u∂ y =

2tanu.

17. If u = tan-1 (x3 +¿ y3

x −¿ y ) Prove that x ∂u∂x

+ y ∂u∂ y =

sin2u.

18. If u=log( x4+ y4

x+ y ), show that x

∂u∂x

+ y ∂u∂ y =3

19. If u =(x – y) f (yx ) find x2 ∂

2u∂x2 + 2xy ∂2u

∂x ∂ y + y2 ∂

2u∂ y2

20. If u =x f (yx )+g(

yx ) show that x2 ∂

2u∂x2 + 2xy ∂2u

∂x ∂ y +

y2 ∂2u

∂ y2 =0

21. Find dydx when f (x, y) = log (x2+y2) + tan-1 y/x.

22. What is total differential of a function u ?

23. Find dydx when x3 + y3 = 3axy.

24. Find dydx when ysinx=xcosy

25. If u = x2 + y2 and x = e2t , y = e2t cos3t .Find ∂u∂ t as a

total derivative.

26. Ifu = ex siny where x = st2 and y = s2t. Find ∂u∂ s and

∂u∂ t

.

27. Find ∂z∂ t when z = xy2 + x2 y, x = at2, y = 2at without

actual substitution.

28. Define Jacobian of two variables.

29. State the properties of jacobians.

30. If x = rcosθ, y = rsinθ,find ∂ (r ,θ)∂(x , y ) ,

∂(x , y )∂ (r ,θ)

31. If x = u(1+v) , y = v(1+u) , find ∂(x , y )∂(u , v ) .

32. If u = y2

x and v = x

2

y , find

∂(x , y )∂(u , v ) .

33. If u = y2 , v = x2 , find ∂(u , v )∂(x , y ).

34. If x = u(1-v) , y = uv , find the jacobian of the

transformation.

35. Find ∂(x , y )∂(u , v ) if x+y = u ,y = uv.

11

36. If u = x-y , v = y-z ,w= z-x , find ∂(u , v ,w)∂(x , y , z) .

37. If u = x+y+z , y+z = uv, z = uvw ,find J(∂(x , y , z)∂(u , v ,w)).

38. If u = yzx , v=

zxy , w =

xyz , find

∂(u , v ,w)∂(x , y , z) .

39. If u = x2 –y2 , v=2xy , and x = rcosθ, y = rsin

θ .∂(u , v)|∂(r , θ)

.

40. State Maclaurin’s series for a function of two

variables x and y.

41. Find the Taylor’s series expansion of xy near the

point (1,1) upto the first degree term.

42. Find Taylor’s series expansion of ex siny near the point (-1, π/4) upto the first degree terms.

43. Expand ex+y in powers of (x-1) and (y+1)up to the first degree terms.

44. State the sufficient conditions for a function of two

variables to have an extremum at a point.

45. Define Stationary points?

46. Define saddle points of a function f (x, y).

47. Find the stationary points of f(x,y) = x2-xy + y2 –

2x+y.

48. Find the stationary points of f(x,y) = x3+3xy2-

15x2-15y2+72x.

49. Find the stationary points of f(x,y) = xy +9x+

3y .

50. A flat circular plate is heated so that the

temperature at any point ( x , y )is u( x , y )=x2+2y2-x

find the coldest point on the plate

51. Find the stationary points of f (x, y) = x3+3xy2-15x2-

15y2+72x for extreme values.

52. Examine the extreme of f (x, y) = x2 +

xy + y2 + 1/x +1/y.

53. Identify the saddle point and the extreme point of

i) f (x, y) = x4 –y4 – 2x2 + 2y2,

ii)f (x,y) = x3 + y3 – 12xy

Unit – V MUTIPLE OF INTEGRALS

1. Evaluate ∫1

2

∫2

5

xy dx dy .

12

2. Evaluate ∫0

1

∫1

2

x ( x+ y )dxdy .

3. Evaluate ∫0

a

∫0

b

xy (x− y)dx dy.

4. Evaluate ∫0

1

∫0

1

¿¿¿+y2) dx dy.

5. Evaluate ∫2

3

∫1

2 dx dyxy

.

6. Evaluate ∫1

b

∫1

a dx dyxy

.

7. Evaluate ∫0

3

∫0

2

ex+ y dy dx.

8. Evaluate ∫1

5

∫1

3 dx dyxy

.

9. Evaluate∫0

1

∫0

2

xy2dy dx.

10. Evaluate∫0

5

∫0

x2

x (x2+ y2 )dx dy.

11. Evaluate ∫1

2

∫0

x2

x dy dx.

12. Evaluate ∫1

2

∫0

x dx dyx2+ y2

13. ∫1

2

∫0

y y dxdyx2+ y2 |

14. Evaluate∫1

2

∫0

x 1x2+ y2 dydx

15. Find the value of∫0

∞

∫0

y e− y

y dx dy

16. Evaluate∫0

a

∫0

√a2− x2

dxdy .

17. ∫0

π /2

∫0

π /2

sin ( x+ y )dxdy|18. ∫

0

π2

∫0

sinθ

r dθdr .|19. Evaluate ∫

–π /2

π /2

∫0

sinθ

r dθdr .

20. ∫0

π

∫0

cosθ

r drdθ .|21. ∫

0

π /2

∫0

acosθ

r2dr dθ .|22. Evaluate ∫

0

π

∫0

asinθ

r dr dθ .

23. ∫0

π

∫0

a (1−cosθ)

r2 sinθ dr dθ .|24. Change the order of integration ∫

−a

a

∫0

√a2− y2

f ( x , y )dxdy .

13

25. Change the order of integration in ∫0

a

∫0

x

dydx .

26. Transform the integration ∫0

∞

∫0

y

dxdy to polar co-

ordinates.27. By changing into polar co-ordinate, evaluate

∫0

2

∫0

√2x−¿ x2 xx2+ y2 dx dy

¿¿.

28. By changing into polar co-ordinate, find the value of

the integral∫0

2a

∫0

√2ax−¿ x2 (x2+ y2 )dydx

¿¿.

29. Change in to polar co-ordinates of ∫−a

a

∫−√a2− x2

√a2−x2

dy dx.

30. Express into polar co−ordinates∫0

a

∫y

a x2

(x2+ y2)3 /2 dx dy.

31. Transform into polar co-ordinates the integral

∫0

a

∫y

a

f ( x , y ) dx dy .

32. Sketch roughly the region of integration of

∫0

a

∫0

a2−x2

f ( x , y )dx dy .

33. Sketch roughly the region of integration for the following

double integral ∫0

a

∫0

a2−x2

f ( x , y )dx dy .

34. Find the limits of integration in∬R

❑

f ( x , y )dx dy , where

R is the region in the first quadrant bounded by x=0, y=0,x+y = 1.

35. Sketch roughly the region of integration for the double

integral ∫0

1

∫0

x

f ( x , y )dy dx .

36. Shade the region of integration ∫0

a

∫√ax−x2

√a2+ x2

dx dy .

37. Sketch roughly the region of integration of

∫0

b

∫0

ab (b− y)

f (x , y ) dx dy.

38. Find the limits of integration in the double integral ,whereR is in the first quadrant and bounded by x=1,y=0,y2 = 4x.

39. Find by double integration , the area of the circle x2+y2 = a2,in polar coordinates.

40. Evaluate ∫0

a

∫0

b

∫0

c

(x+ y+z ) dz dy dx

41. Evaluate ∫0

a

∫0

b

∫0

c

xyz dz dy dx

14

42. Evaluate ∫0

a

∫0

b

∫0

c

ex+ y+ zdz dydx

43. Evaluate ∫0

2

∫1

3

∫1

2

xy2 zdz dy d x

44. Evaluate ∫0

1

∫0

2

∫0

3

xyz dx dy dz .

45. Evaluate ∫0

2π

∫0

π

∫0

a

r4 sinφ dr dφ dѳ .

46. Evaluate∫0

log 2

∫0

x

∫0

x+ y

❑ex+ y +zdxdy dz .

47. Evaluate ∫0

1

∫0

x

∫0

√ x+ y

z dz dy dx .

48. Evaluate ∫0

4

∫0

x

∫0

√ x+ y

z dx dy dz .

Part-B Questions

Unit - I Matrices

1. Find the Eigen values and Eigen vectors of the matrix

[1 13 −1]

2. Find the Eigen values and Eigen vectors of A=

[ 2 2 02 1 1

−7 2 −3] (non repeated)


[1 0 −11 2 12 2 3 ] (non repeated)


[2 2 11 3 11 2 2] (two repeated non symmetric)


[1 2 30 2 30 0 2] r


[2 1 00 2 10 0 2 ] r


[ 7 −2 0−2 6 −20 −2 5 ]


[ 1 −1 −1−1 1 −1−1 −1 1 ] r


[0 1 11 0 11 1 0 ]

15


[ 6 −2 2−2 3 −12 −1 3 ] (two repeated symmetric)


[ 8 −6 2−6 7 −42 −4 3 ]


[ 6 −6 514 −13 107 −6 4 ] (three repeated)


[3 −4 41 −2 41 −1 3 ]

14. Find the constants a and b such that the matrix [a 41 b ]

has 3 and -2 as its eigen values

15. Using Cayley-Hamilton theorem , find the inverse of

the matrix A=[2 11 −5 ]

16. Show that for a square matrix, (i)There are infinitely many eigen vectors corresponding to a single eigen value.

(ii) Every eigen vector corresponds to a unique eigen value.

17. If A=[1 23 4] find A-1 and A3 using Cayley Hamilton

theorem and also verify theorem.

18. If A=[1 04 5 ] , express A3 in terms of A and I using

Cayley-Hamilton theorem.19. Using Cayley Hamilton theorem Find A-1 when

20. Verify Cayley-Hamilton theorem for thematrix A =

[ 2 −1 1−1 2 −11 −1 2 ].Hence compute A-1

21. Verify Cayley-Hamilton theorem and hence find A-1 if A

= [ 13 −3 50 4 0

−15 −9 −7 ].22. Given A = [1 2 −1

0 1 −13 −1 1 ] find AdjA by using

Cayley-Hamilton theorem.23. Verify Cayley-Hamilton theorem and hence find A-1 if

A = [ 1 2 −2−1 3 00 −2 1 ].

16

24. If A = show that

using Cayley Hamilton theorem

25. Find the characteristic equation of the matrix A =

[2 1 10 1 01 1 2] and hence compute A-1

Also find the matrixrepresented by A8 -5A7 +7A6 -3A5 +A4 -5A3

+8A2 -2A+I.

26. Diagonalise the matrix A = [2 0 40 6 04 0 2 ] by means of an

orthogonal transformation.

27. Reduce the matrix to diagonal form

28. Diagonalise the matrix A= by means of

an orthogonal transformation.

29. Reduce 3x2 +3z2 +4xy+8xz+8yz into canonical form.30. Reduce the quadratic form x2 -4y2 +6z2 +2xy-4xz+2w2 -

6zw into sum of squares.

31. Reduce 8x2 +7y2 +3z2 -12xy+4xz-8yz into canonical form by orthogonal reduction.

32. Reduce 6x12+3x2

2+3x32-4x1x2-2x2x3+4x3x1 into canonical

form by an orthogonal reduction and find the rank ,index ,signature and the nature of the quadratic form.

33. Reduce the quadratic form given below to its normal form by an orthogonal reduction q = 3x1

2+2x22+3x3

2-2x1x2-2x2x3.

34. Reduce the quadratic form into a canonical form by means of an orthogonal transformation. Determine its nature

35. Reduce the quadratic form

to Canonical form through an orthogonal transformation

36. Verify that the eigen vectors of the real symmetric

matrix A = [ 8 −6 2−6 7 −42 −4 3 ] are in orthogonal pairs.

37. Reduce the quadratic form to the

canonical form by an orthogonal transformation

37.Find the matrix A, whose eigen values are 2 ,3 and 6. and the eigen vectors are {1,0,-1}T, {1,1,1}T,{1,-2,1}T .

Unit-II Three Dimensional Analytical Geometry17

1. Show that the spheres x2+y2+z2=25, x2+y2+z2-18x-24y-40z+225=0 touch externally and find their point of contact.

2. Find the equation of the sphere passing through the four points (4,-1,2),(0,-2,3), (1,5,-1)and(2,0,1)

3. Find the equation of the sphere passing through the four points (0,0,0),(0,1,-1), (-1,2,0)and(1,2,3)

4. Find the equation of the sphere passing through the points (1,1,-1), (-5,4,2),(0,2,3)and having its centre on the plane 3x+4y+2z=6

5. A plane passes througha fixed point (a,b,c) and cuts the axes in A,B,C. Show that the locus of the centre of

the sphere OABC is ax +

by +

cz =2

6. A sphere of constant radius k passes through the origin and meets the axes in A,B,C.Prove that the centroid of the triangle ABC lies on the sphere 9( x2

+y2 +z2)=4k2

7. Find the centre radius and area of the circle x2+y2+z2-2x-4y-6z-2=0,x+2y+2z=20

8. Find the centre ,radius and area of the circle which is the intersection of the sphere x2 +y2 +z2 -8x+4y+8z-45=0 and the plane x-2y+2z = 3.

9. Find the centre ,radius and area of the circle in which the sphere x2 +y2 +z2 +2x-2y-4z-19=0 is cut by the plane x+2y+2z+7 = 0

10. Find the equation of the sphere through the circle x2+y2+z2 +2x+3y+6=0,x-2y+4z=9 and the centre of the sphere x2+y2+z2-2x+4y-6z+5=0

11. Find the equation of the sphere having its centre on the plane 4x-5y-z=3 and passing through the circle x2+y2+z2 -2x-3y+4z+8=0,x-2y+z=8

12. Find the equation of the spheres which passes through the circle x2+y2+z2 =5 and x+2y+3z=3 and touch the plane 4x+3y=15

13. Find the equation of the sphere having the circle x2+y2+z2 +10y-4z-8=0,x+y+z=3 as a great circle. Find its centre and radius.

14. Find the equation of the sphere having its centre on the plane 4x-5y-z=3 and passing through the circle with equations x2+y2+z2 -2x-3y+4z+8=0, x2+y2+z2 +4x+5y-6z+2=0

15. Prove that the circles x2+y2+z2 -2x+3y+4z-5=0,5y+6z+1=0; x2+y2+z2 -3x-4y+5z-6=0,x+2y-7z=0 lie on the same sphere and find its equation.

16. Show that the circles x2+y2+z2 -y+2z=0,x-y+z-2=0 and x2+y2+z2 +x-3y+z-5=0,2x-y+4z-1=0 lie on the same sphere and find its equation.

17. Find the equation of the tangent plane to the sphere x2+y2+z2-2x-10y-6z+26=0 at (2,3,5).

18. Find the equation of the tangent plane to the sphere x2+y2+z2-2x+4y+6z-12=0 at (1,-1,2).

19. Show that the plane 2x-2y+z+12=0 touches the sphere x2+y2+z2-2x-4y+2z=3 and find also the point of contact.

18

20. Show that the plane 4x+9y+14z-64=0 touches the sphere 3(x2 +y2+z2)-2x-3y-4z-22=0 and find the point of contact.

21. Find the equation of the tangent planes to the spheres x2+y2+z2=9 which passes through the line x+y-6=0 = x-2z-3.

22. Find the equation of the tangent planes to the sphere x2 +y2 +z2-4x-2y-6z+5=0 which are parallel to the plane x+4y+8z=0 Find their point of contact.

23. Find the equation of the tangent planes to the sphere x2 +y2 +z2+2x-4y+6z-7=0 which intersect in the line 6x-3y-23=0=3z+2

24. The plane xa +

yb +

zc =1 meets the axes in A,B,C.Find

the equation of the cone whose vertex is the origin and the guiding curve is the circle ABC.

25. Find the equation to the right circular cone whose vertex is P(2,-3,5) axis PQ which makes equal angles with the axis and semi vertical angle is 30

26. Find the equation of the right circular cone whose vertex is the point (2,1,-3) whose axis parallel to OY axis and whose semi vertical angel is 45.

27. Find the equation of the right circular cone whose vertex is(3,2,1) semi vertical angle 30 and the axis the

line x−3

4 = y−2

1 = z−1

3

28. Find the equation of the right circular cone whose

vertex is at the origin,whose axis the linx1 =

y2 =

z3

and which has the semi vertical angle 30 .Also find the semi vertical angle 60

29. The axis of the right cone,vertex O,makes equal angles with the co-ordinate axes and the cone passes through the line drawn from O with the direction cosines proportional to 1,-2,2.Find the equation of the cone.

30. Find the equation of the right circular cylinder of radius

2 and having as axis of the line x−1

2 = y−2

1 = z−3

2

31. Find the equation of the right circular cylinder of radius

3 and having as axis of the line x+1

2 = y−3

2 = z−5−1

32. Find the equation of the right circular cylinder whose

axis is the line x−2

2 = y−1

1 = z−0

3 and which passes

through the point(0,0,3)33. Find the equation of the right circular cylinder of radius

2 and having as axis the line line passesthrough the point(1,2,3)and directioncosin es proportional¿2 ,−3,6

34. Find the equation of the right circular cylinder which has the circle x2+y2+z2-2x-4y-4z-1=0,2x-y-2z+13=0 as the guiding curve.

35. Find the equation of the right circular cylinder whose guiding circle is x2+y2+z2=9,x-y+z=3

19

Unit-III Differential Calculus

1. Find the radius of curvature at the point( 3a2

, 3a2 ) on

the curve x3+y3=3axy.2. Find the radius of curvature at the point(a,0) of the

curve xy2=a3-x3

3. In the curve √ x /a+√ y /b=1.show that the radius of curvature at the point (x, y) varies as (ax+by)3/2

4. If y = axa+x Prove that

( 2 ρa )

2/3

=( xy )2

+( yx )2

where ρ is the radius of curvature

of the curve.5. Find the radius of curvature at any point t on the curve

x=et cost , y=et sint. 6. Find the radius of curvature of the parabola x=at2 ,

y=2at at t.7. Find the radius of curvature at any point (a cos3,a

sin3) on the curve x2/3+y2/3=a2/3

8. Find the radius of curvature at the origin for the cycloid x=a( +sin θ) , y= a(1-cosθ).

9. Find the radius of curvature of the curve r = a(1+cosѳ) at the point ѳ = π/2.

10. Show that the radius of curvature of the cycloid x=a(

+sin θ) , y= a(1-cosθ) is 4acos θ2 at any point θ.

11. Find the radius of curvature at any point P(a cos θ,b sin

θ) on the ellipse x2

a2 + y2

b2 =1

12. Find the radius of curvature of the curve x=a log(sec θ

),y=a(tanθ−θ ¿

at θ

13. Find the centre of curvature at the point (am2,2am) on

the parabola y2=4ax

14. Find the centre of curvature of the parabola y2=4ax At

the point (a,2a)

15. Find the centre of curvature of the curve y=3x3+2x2-3

at (0,-3)

16. Find the equation of circle of curvature at (c,c) on xy=c2

17. Find the centre and circle of curvature of the curve

√ x+√ y =√a at (a/4, a/4).

18. For the curve √ x +√ y =1 find the equation of the circle

of curvature

20

at (1/4,1/4).

19. Find the equation of circle of curvature of the parabola

y2 = 12x at the point (3,6).

20. Find the equation of circle of curvature at (3,4) on xy

= 12.

21. Find the equation of the circle of curvature at the point

(2,3) on x2

4+ y2

9=2

22. Find the equation of the evolute of the parabola y2 =

4ax.

23. Find the equation of the evolute of the parabola x2 =

4ay.

24. Find the equation of the evolute of the ellipse x2

a2 +

y2

b2 = 1.

25. Find the equation of the evolute of the hyperbola x2

a2 -

y2

b2 = 1.

26. Find the equation of the evolute of the rectangular

hyperbola xy=c2

27. Find the equation of the evolute of the curve

x2/3+y2/3=a2/3.

28. Show that the equation of the evolute of the cycloid

x=a(θ –sinθ) ,

y=a(1-cosθ) is another equal cycloid.

29. Show that the evolute of the curve

x = a(cosθ+θsinθ), y = a(sinθ-θcosθ) is a circle

30. Find the evolute of the curve

x=a( +sin θ) , y= a(1-cosθ).

31. Prove that the evolute of the curve

x = ct, y=c/t is (x+y)2/3 – (x-y)2/3 = (4c)2/3.

21

32. Find the envelope of the family of straight linesaxcosθ -

bysinθ = a2-b2.

33. Find the envelope of xa+

yb =1 subject to

a2+b2=c2 ,where c is being constant.


yb =1 where the parameters a,b

are related by ab=c2 where c is known


yb =1 subject to a+b=c where c

is known constant.

36. Find the envelope of x2

a2 + y2

b2 =1subject to a+b=c where c

is a constant.

37. Find the envelope of x2

a2 + y2

b2 =1subject to a2+b2=c 2

where c is a constant.


yb =1 subject to an+bn=cn

where c is known constant.

39. Find the evolute of y2=4ax considering it as the

envelope of normals .

40. Find the evolute of x2=4ay considering it as the

envelope of normals .

41. Considering the evolute as the envelope of normals

find the evolute of x2

a2 + y2

b2 =1.

42. Considering the evolute as the envelope of normals

find the evolute of x2

a2 - y2

b2 =1.

Unit- IV Functions of Several Variables

1. If u = xy, then show that ∂3u∂x2∂ y

= ∂3u∂x ∂ y ∂x

2. If u=log(x3+y3+z3-3xyz),show that

3. (i)( ∂∂ x

+ ∂∂ y

+ ∂∂ z

)2

u=−9

(x+ y+z)2

22

a. (ii)∂2u∂x2 + ∂

2u∂ y2 + ∂

2u∂ z2 +2 ∂2u

∂ y∂ z+2 ∂2u

∂z ∂ x+ ∂2u∂x ∂ y

=

−9(x+ y+z)2

4. If u = (x2 + y2 + z2)-1/2 prove that

∂2u∂x2 + ∂

2u∂ y2 + ∂

2u∂ z2 = 0.

5. Find the first order partial derivatives of (i)u=tan-1 (x2+ y2

x+ y) (ii) u=cos-1(x/y)

6. If u = sin-1 √x−√ y√ x+√ y

, find x∂u∂x + y

∂u∂ y .

7. If u=cos-1 [ x+ y√ x+√ y ] prove that

x∂u∂x

+ y ∂u∂ y=-

12 cotu

8. Verify Euler’s theorem for

sin-1 (x/y) + tan-1 (y/x).

9. If u=sin-I(x+2 y+3 z√x8+ y8+z8 ),show that

x ∂u∂x + y

∂u∂ y + z

∂u∂ z +3tanu=0

10. State and prove Euler’s extension theorem.

11. If u = x logxy where x3 + y3 + 3xy = 1 ,find dudx .

12. If u = x3y2 + x2y3 where x =at2 ,

y = 2at.Find dudt .

13. If z = sin-1(x-y), x = 3t, y = 4t3. Show that dzdt =

3√1−t2

14. If u = f (x-y, y-z, z-x) find ∂u∂x

+ ∂u∂ y +

∂u∂ z

15. If Z=f(x,y) where x=rcosθ and rsin show that ( ∂ z∂ x )2

+

( ∂ z∂ y )2

=( ∂ z∂ r )2

+1r2 ( ∂ z∂θ )

2

16. If g(x,y) = Ψ(u,v) where u = x2 – y2 and v = 2xy .Prove

that∂2g

∂ x2 + ∂2 g

∂ y2 = 4 (x2 + y2) (∂2Ψ∂u2 + ∂

2Ψ∂v2 ).

17. If z = f(u,v) where u = lx + my and v = ly-mx. Show

that ( ∂2 z

∂x2 + ∂2 z

∂ y2) =

(l2 + m2) (∂2 z∂u2 + ∂

2 z∂v2 )

18. If Given transformation u=excos y and v=ex siny and ∅

is a function of u and also x and y . prove that ∂2∅∂ x2 +¿

∂2∅∂ y2 =(u2+v2 )( ∂2∅

∂u2 + ∂2∅∂v2 )

23

19. Find the Jacobian of y1 ,y2 ,y3 with respect to x1, x2, x3 if

y1 = x2 x3

x1 , y2 =

x3 x1

x2 , y3 =

x1 x2

x3

20. If v =2xy,u=x2-y2 and x=rcosθ,y=rsin θ evaluate ∂(u , v )∂ (r ,θ)

21. If u =x+ yx− y and v=tan−1x+¿ tan−1 y¿ find the Jacobian

∂(u , v )∂(x , y )

.

22. If u = 4x2 + 6xy , v = 2y2 + xy , x = rcosθ , y =

rsinθ .Evaluate ∂(u , v )∂ (r ,θ)

23. If x=a cosh αcos β,y=a sinh α sin β,then show that ∂(x , y)∂(α ,β )

=a2

2(cosh 2α -cos 2β)

24. Ifx=sinθ√1−c2 si n2∅ , y=cosθcos∅ ,t h en ∂(x , y)∂ (θ ,∅ ) =

−sin∅ [(1−c2 )cos2θ+c2co s2∅ ]

√1−c2 si n2∅

25. Expand ex cosy about (0 , π2 ) up to the third term using

taylor’s series

26. Expand ex siny around thye point[1, π2

¿up to the third

term using taylor’s series

27. Expand sin xy in powers of (x-1) and (y- π2

¿upto the

second degree terms.28. Expand f(x,y) = exy in Taylor’s series at (1,1) upto

second degree.29. .Expand ex log(1+y) in powers of x and y up to the terms

of third degree

30. Expand xy2+2x-3y in powers of (x+2) and (y-1) upto

third degree terms.

31. Expand f(x,y) = 4x2 +xy+6y2+x-20y+21 in Taylor’s series about (-1,1)

32. Examine for the extremum values of f(x,y) = x3+ y3-12x-3y+20.

33. Find the extreme values of the function f( x , y )=x3+y3-3x-

12y+20

34. Find the extreme values of the function f( x , y )=x3y2(1-x-

y)

35. Find the maximum and minimum value of x2-xy+y2-2x+y

36. Find the maximum value of sinx siny sin(x+y) where o<x , y<π.

37. Find the minimum value of sinx + siny + sin(x+y) ,where 0<x,y<π.

24

38. Find the minimum value of F=x2+y2 subject to the constraint x=1

39. Find the minimum value of xy2z2 subject to x+y+z =24.40. A Rectangular box open at the top is to have a volume

at 32cc.find the dimensions of the box that requires the

least material for its construction

41. A thin closed rectangular box is to have one edge equal

to twice the other and constant volume 72m3.Find the

least surface area of the box.

42. Find the maximum value of xmynzp when x+y+z = a.43. Find the maximum values of x2yz3 subject to the

condition 2x+y+3z = a.44. Find the volume of the greatest rectangular

parallelepiped that can be inscribed in the ellipsoid

x2

a2 + y2

b2 + z2

c2 = 1.

45. Find the shortest and the longest distances from the point

(1,2,-1) to the sphere x2+y2+z2 = 24. using Lagrange’s

method of constrained maxima and minima.

46. In a plane triangle ABC find the maximum value of

cosA cosB cosC .

47. The temperature u(x,y,z) at any point in space is u =

400xyz2. Find the highest temperature on the surface of

the sphere x2+y2+z2 =1.

48. Find the minimum value of x2+y2+z2 subject to the

condition 1x +

1y+

1z =1.

49. Find the extreme values of the functions v= x2+y2+z2

subject to ax+by+cz = p

50. Find the minimum value of x2+y2+z2 with the constraint

xy+yz+zx=3a2

51. Find the shortest distance from the origin to the curve

x2+8xy+7y2 =225.

Unit-V Multiple Integrals

1. Find ∫0

1

∫0

√1+ x2

dy dx1+x2+ y2 .

2. Change the order of integration ∫0

1

∫x2

2− x

f (x , y) dy

dx.


1

∫0

x

dydx and

hence evaluate it.4. Change the order of integration

∫0

a

∫x

a

(x¿¿2+ y2)dydx ,¿and hence evaluate it.


4

∫y

4 xx2+ y2 dxdy

and hence evaluate it.

25


∞

∫x

∞ e− y

y dy dx

and hence evaluate it.

7. Evaluate ∫0

3

∫1

√4− y

( x+ y )dx dy , .

By changing the order of integration8. Change the order of integration in

∫0

4

∫0

34 √16− x2

x dx dy∧hence ¿ .¿


4 a

∫x2

4a

2√ax

dy dx and

hence evaluate it.10. Change the order of integration in

∫0

a

∫x2

a

2a− x

xy dydx∧¿¿hence evaluate it.


a

∫a−√a2− y2

a+√a2− y2

dydx

, and hence evaluate it.


1

∫y2

2− y

xy dy dx

,and hence evaluate it.

13. Evaluate ∫0

∞

∫0

x

x e−x2

y dy dx,by change the order of

integration.

14. Evaluate ∫0

∞

∫0

x

x e−xy dy dx,by change the order of

integration.


1

∫x

√2− x2

x√ x2+ y2

dx dy and hence evaluate it.

16. Evaluate ∫0

∞

∫0

∞

e−(x 2+ y2 )dx dy by changing to polar

coordinates and hence show that ∫0

∞

e−x2

dx=√ π2

17. By changing in to polar co-ordinates ,evaluate

∫0

a

∫y

a xx2+ y2 dx dy.

18. Find the area of a circle of radius a in polar co-ordinates

19. Evaluate ∬ xy dx dy,over the positive quadrant of the circle x2+y2=¿ 1.

20. Find ∬dx dy ,the

regionbounded by x ≥ 0,y ≥

0,x+y ≤ 1.21. Find the area enclosed by the curves y = x2 and

x+y = 2.22. Evaluate ∬ xy dxdy ,where R is the domain

bounded by X-axis,ordinate x=2a and the curve x2=4ay

23. Evaluate ∬ ( x+ y ) dx dy,over the positive quadrant of the ellipse

26

x2

a2 + y2

b2 = 1.

24. Find the area between the parabolas y2= 4ax and x2 = 4ay.

25. Find the area of the region bounded by the parabolas y = x2 and x = y2.

26. Find the area bounded by y=x and y=x2

27. Find by double integration,the are between the parabola y2= 4ax and the line y = x.

28. Find the smaller of the areas bounded by y = 2 –x and x2+y2 =4.

29. Find by double integration ,the area of the cardiod r = a(1+cosѳ).

30. Evaluate ∬ r3dr dѳ,over the area bounded between the circles r = 2cosѳ and r = 4cosѳ.

31. Find the area of the region outside the inner circle r = 2cosѳ and inside the outer circle r = 4cosѳ by double integration

32. Calculate∫∫r 3 dr dθ over the area included between the circles r=2 sin θ and r=4sin θ

33. Evaluate ∫∫r 2sin θ dr dθ where R is the region of semicircle r=2acos θ about the initial line

34. Evaluate ∬ r2 dr dѳ,over the area between the circles r = 2cosѳ and r = 4cosѳ.

35. Evaluate ∫ρ=0

1

∫z= ρ2

ρ

∫ѳ=0

2π

ρ dρ dz dѳ.

36. Transform the integration ∫z=0

5

∫−6

6

∫−√36− x2

√36−x2

dxdy dz .

37. Evaluate ∫0

1

∫0

√1− x2

∫0

√1−x2− y2 dzdydx√1−x2− y2−z2

.by changing into spherical polar coordinates

38. Express the volume of the sphere x2+y2+z2 = a2, as a volume integral and hence evaluate it

39. Find the volume bounded by x,y,z ≥ 0 and x2+y2+z2≤1 in triple integration

40. Find the volume bounded by the cylinder x2+y2 =4 and the planes y+z = 4 and z = 0.

41. Find the volume of the ellipsoid

x2

a2 + y2

b2 + z2

c2 = 1.

42. Evaluate ∭ xyz dx dydz , taken over the positive octant of the sphere x2+y2+z2 =1

43. Find the volume of the tetrahedron bounded by

the planes x=0,y=0,z=0 and xa +

yb +

zc = 1.

44. Find the volume in the positive octant bounded by the co-ordinate planes and the plane x+2y+3z = 4 ,by triple integration.

45. Evaluate ∭v

❑

dx dy dz ,where vis the finite region

of space(tetrahedron) formed by the planes x = 0, y = 0, z = 0 and 2x+3y+4z = 12.

27

46. Evaluate ∭ xyz dx dydz , taken throughout the volume for which x,y,z ≥ 0 and x2+y2+z2 ≤ 9

ANNA UNIVERSITY COIMBATORE

B.E./B.TECH. DEGREE EXAMINATIONS : JAN-FEB 2009

REGULATIONS : 2008

FIRST SEMESTER – COMMON TO ALL BRANCHES

08003001 – MATHEMATICS I

PART -A (20 X 2 = 40 Marks)

ANSWER ALL QUESTIONS

1. True or false : “ If A and B are two invertible matrices then AB and BA have the sameeigen values ”

2. If the sum of the eigen values of the matrix of the quadratic form equal to zero,then what will be the nature of the quadratic form?

3. A is a singular matrix of order three, 2 and 3 are the eigenvalues.Find its third eigen value

28

4. Find the eigenvector corresponding to the eigenvalue 1 of

the matrix A = [2 2 11 3 11 2 2]

5. The number of great circles on any sphere is (a) 1 (b) 2 (C)many (d) 0

6. Test whether the plane x = 3 touches the sphere x2+ y2 +z2 = 9

7. Give the general equation of the cone passes through the origin

8. What will be the plane section perpendicular to its axis of a right circular cylinder

9. Find the evolute of the curve x2+ y2+4x-6y+3 = 0

10. Find the envelope of the family given by x = my + 1m , m

being the parameter11. True or False : When the tangent at a point on a curve is

parallel to x-axis then the curvature at the point is same as the second derivative at that point

12. Find the radius of curvature of the curve given by x = 3+2cosѲ , y = 4+2sinѲ

13. If u = sin-1 √x−√ y√ x+√ y

.Find x ∂u∂x + y

∂u∂ y

14. If x = r cosѲ , y = rsinѲ. Find ∂r∂x

15. Find the minimum value of F = x2+ y2 subject to the constant x = 1

16. Expand ex + y in power of x-1 and y+1 up to first degree terms

17. Transform into polar co-ordinates the integral

∫0

a

∫y

a

f ( x , y )dxdy

18. Why do we change the order of integration in multiple integrals? Justify your answer with an example

19. Sketch roughly the region of integration for the following

double integral ∫0

a

∫0

√a2− x2

f ( x , y )dx dy

20. Express the volume bounded by x≥0, y≥0,z≥0 and x2+ y2+z2

≤ 1 in triple integration

PART -B(5 X 1 2 = 60 Marks)

ANSWER ANY FIVE QUESTIONS

21. a) Using Cayley Hamilton’s theorem find A4 for the matrix A =

[ 2 −1 2−1 2 −11 −1 2 ] (6)

b)Obtain an orthogonal transformation which will transform the quadratic form Q = 2x1x2 +2x2x3+2x3x1 into sum of squares (6)

22. a)Find the equation to the tangent planes to the sphere x2+ y2 +z2−4 x+2 y−6 z−11 = 0 which are parallel to the plane x=0 (6)b)Find the equation to the right circular cylinder of radius 2

and whose axis is the line x−1

2= y−2

1= z−3

2

(6)

29

23. a)Find the equation of the sphere passing through the circle

x2+ y2 +z2+2 x+3 y+6=0 , x−2 y+4 z=9∧the centreof the sphere x2+ y2 +z2−2 x+4 y−6 z+5=0 (6)b)Find the equation to the right circular cone whose vertex is at the origin and the guiding curve is the circle y2+ z2 = 25,x = 4 (6)

24. a)Find the radius of curvature at (3a2

, 3a2 ) on x3+ y3 = 3axy

(6)b)Find the evolute of the parabola x2 = 4by (6)

25. a)Find the circle of curvature at (a4, a4 ) on √ x+√ y = √a

(6)b)Show that the envelope of the family of the circles whose diameters are the double ordinates of the parabola

y2 = 4ax is the parabola y2 = 4a(x+a) (6)

26. a) If u = log(x3+ y3 +z3-3xyz) Prove that (∂∂x

+ ∂∂ y

+ ∂∂z )2 u =

-9

(x+ y+z)2 (6)

b)Find the minimum value of x2+ y2 +z2 with the constraint xy+yz+zx = 3a2 (6)

27. a)Find the volume of the ellipsoid x2

a2 +y2

b2 +z2

c2 = 1 by triple

integration (6)

b)Change the order of integration and then evaluate

∫0

a

∫a−√a2− y2

a+√a2− y2

xy dx dy (6)

28. a)Transform into polar co-ordinates and evaluate

∫0

2

∫0

√2x− x2

x dy dxx2+ y2 (6)

b)Find the area enclosed by the curves y = x2 and x+y-2 = 0 (6)

30

mathssnsce.weebly.commathssnsce.weebly.com/.../0/1/25011348/m1_question_bank.docx · web viewunit...

Documents