080030001 MATHEMATICS – I
Mathematics – I
Part-A Questions
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 transformation
UNIT 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 –I MATRICES
1 2
]
0 2
3 1
Find the characteristic polynomial of [
]
−1 2
3 2 −1
Find the characteristic equation of [2 1
0]
4 −1 6
The sum of the Eigen values of a matrix A is equal to
the sum of the elements on its diagonal
If 𝜆1 , 𝜆2 , ……….. 𝜆n are the eigen values of an n x n
1. Find the characteristic equation of [
2.
3.
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 integral
TEXT BOOK:
1. Bali N. P and Manish Goyal, “ Text book of Engineering Mathematics”,
Third
edition, 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).
4.
5.
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
2 0 1
matrix A = [0 2 0]
1 0 2
7. The product of two eigen values of the matrix A =
6 −2 2
[−2 3 −1] is 16.Find the third eigen value
2 −1 3
8. Find the sum and product of the Eigen values of the
matrix[
1
2 −3
]
4 −2
2
A = [1
1
9. Two eigen values of the matrix
are equal to 1 each.Find the eigen value of A
2 1
3 1]
2 2
|𝐴|
19. If λ Eigen values of a matrix A then
6 −2 2
20. Two of the Eigen values of [−2 3 −1]are 2 & 8
2 −1 3
11. Prove that A and 𝐴𝑇 have the same Eigen values.
12. If λ is an Eigen value of A then
λ
Find the third Eigen value.
is an Eigen value of
21. If 3 & 15 are the two Eigen values of A
A-1.
8 −6 2
=[−6 7 −4] Find the third Eigen value.
2 −4 3
13. The product of the Eigen values of a matrix A is equal
to its determinant.
22. If 2,2,3 are the Eigen values of
14. Prove that the Eigen values of a real symmetric matrix
A=
3 10 5
[−2 −3 −4] Find the Eigen value 𝐴𝑇 and 𝐴−1
3
5
7
1 1
23. If the Eigen values of the matrix A=[
] are 2 , - 2
3 −1
must be real
15. The product of the two Eigen values of the matrix
2
A=[
1
is the Eigen
value of the matrix adj A.
-1
10. Give some properties of Eigen values.
1
λ
4
] is 2. Find the third value.
4
then find the Eigen values of 𝐴𝑇
16. Prove that if X is an Eigen value of A corresponding to
2 1
24. Find the Eigen values of A=[0 2
0 0
an Eigen value λ then any non zero scalar multiple of X
is also an Eigen vector of A.
0
1] without using
2
the characteristic equation idea.
17. An Eigen vector of a matrix can not correspond to two
distinct Eigen values.
25. Find the Eigen values of
18. If λ Eigen values of a matrix A then K λ is the Eigen
1 0
A = [0 4
0 0
0
0]
3
26. Find the Eigen values of the inverse of the matrix A =
value of the matrix KA, where K is a non zero scalar.
1 3
[0 2
0 0
2
4
5]
3
27. Two of the Eigen values of
34. If ( 1,1,5) are the Eigen values of
3 −1 1
[−1 5 −1] are 3 and 6 find the Eigen values of
1 −1 3
35. If the Eigen values of a matrix A are 2,3,4 find the
28. Two Eigen values of the matrix
Eigen values of Adj A
A=
1 −2
). Hence
−5 4
1
1] are equal to 1 each find the Eigen values of
2
36. Find the Eigen values of the matrix(
form the matrix whose Eigen values are 1/6 and -1
−1
𝐴
3 5
37. If A = [0 4
0 0
7 4 −4
29. One of the eigen values of A=[4 −8 −1]find other
4 −1 −8
two eigen values
30. Find the Eigen values of 𝐴3 given
A=
1 2
[0 2
0 0
3
6] find the Eigen values of A, 𝐴−1 ,
1
AdjA and 𝐴5
38. Find the sum of the Eigen values of 2A if A =
8 −6 2
[−6 7 −4]
2 −4 3
3
−7]
3
31. If 1 and 2 are the Eigen values of a
39. If the sum of two Eigen values and trace of a 3 x 3
2 x 2 matrix
what are the Eigen values of 𝐴2 and 𝐴−1
32. If 1,1,5 are the Eigen values of
2 2
[1 3
1 2
1
1]
2
find the Eigen values of KA ( K is scalar)
𝐴−1
2 2
[1 3
1 2
2 2
A = [1 3
1 2
A=
matrix are equal find the value of |𝐴|
40. Find the sum of the Eigen values of the inverse of A =
A=
3 0
[8 4
6 2
1
1] find the Eigen values of 5A
2
2
33. Find the Eigen values of A, 𝐴4 ,3A, 𝐴−1 if A = [
0
3
]
5
0
0]
5
41. What is the sum of squares of the eigen values
1 7 5
of [0 2 9]
0 0 5
3
42. . Form the matrix whose eigen values are α-5, β-5, γ-5
where α, β, γ are the eigen values of
−1 −2 −3
A=[ 4
5 −6]
7 −8 9
3 −1
43. If α, and β are the eigen values of [
], form the
−1 5
3
3.
matrix whose eigen values are α and β
44. If the sum of two Eigen values and trace 3 x 3 matrix A
are equal,
find the value of
49. If x= (-1, 0, -1)T is the Eigen vector of the matrix A =
1 1
[1 5
3 1
1
√2
[−1
√2
1]
√2
√2
are
46. Show that
1+𝑖
√2
√2
1+𝑖
,
,
1
the same those of A = [
2
A
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
√2
=[
is also Eigen values of A
2
47. Find the Eigen value of [
0
1
eigen vector [ ]
0
2
]
1
51. If λ1 , λ2 , λ3 , ………. λn , are the Eigen values of A then
1−𝑖
√2
1−𝑖
( −3𝐴−1 ) are
50. Prove that the Eigen values of
45. The Eigen values of the given orthogonal matrix A =
1
3
1] find the corresponding eigen value.
1
0
4
2
]
0
54. Define orthogonal matrix.
3
] corresponding to the
4
3 −1
48. If one of the Eigen vector of A= [−1 5
1 −1
1
[ 0 ] then find the corresponding Eigen value
−1
55. Show that the matrix [
𝑐𝑜𝑠𝜃
−𝑠𝑖𝑛𝜃
𝑠𝑖𝑛𝜃
] is orthogonal.
𝑐𝑜𝑠𝜃
56. If A is orthogonal matrix, show that A-1 is also
orthogonal.
1
−1] is
3
57. If A is an orthogonal matrix prove that A  1
58. Prove that the inverse of orthogonal matrix A is
orthogonal.
59. Write the matrix of the Quadratic form x12 +2x22+x322x1x2+2x2x3
60. Write the matrix of the Quadratic form 2x12 2x22+4x32+2x1x2 -6x1x3+6x2x3
4
3. Find the centre and radius of the sphere2x2+2y2+2z2 -
61. Write the Quadratic form corresponding to the
0 −1 2
symmetric matrix [−1 1 4]
2
4 3
2x+4y+2z+3= 0
4. Find the centre and radius of the sphere 3(x2+y2+z2) -
62. If the matrix of the Q.F. 3x2+2axy+3y2 has Eigen
6x - 12y+18z+9 = 0.
values 2 and 4 ,find the values of a.
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).
11. Find the equation to the sphere of radius 2 units, lying
63. Determine the nature of the Quadratic form f(x1, x2
,x3)= x12 + x22 + x32
64. Find the index, signature and nature of the Quadratic
form x12 + 2 x22 -3 x32
65. Determine the nature of the Quadratic form f(x1, x2
,x3)= x12+ 2x22
66. Determine the nature of the Quadratic form f(x1, x2
,x3)= 2x12-x22
67. Find the nature of the conic
8x2-4xy +5y2=36
2
by reducing the Q.F. 8x -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
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).
1. Find the equation of a sphere whose centre is (2, -3,
13. Find the equation of the tangent plane at the point
4) and radius 3 units.
2. Find the centre and radius of the sphere
(1,-1, 2) to the sphere
x2+y2+z2 -
x2+y2+z2-2x+4y+6z-12 = 0.
6x+8y-10z+25 = 0
14. Find the equation of the normal at the point (2,-1,4)
to the sphere
x2 +y2 +z2 –y-2z-14=0.
5
15. Prove that the plane 2x-y-2+12 = 0 touches the sphere
24. Find the equation of the cone whose vertex is the
x2+y2+z2-2x-4y+2z-3 = 0.
point(α,β, γ) and base
16. Find the equation of a sphere which passes through
25. Find the equation of the cone whose vertex is at the
the point (1,-2, 3) and the circle Z=0, x2+y2+z2-9 = 0.
origin and the guiding curve is
17. Find the equation of the sphere having its centre on
plane XOY is a hyperbola.
circlex2+y2+z2+2x+3y+6= 0; x-2y+4z-9= 0 and the
27. Define Right circular cone & give its equation.
= 0.
28. Show that the equation to the right circular cone
19. Find the equations of the spheres which passes
through the circle
x2+y2+z2=
𝑧2
the section of the cone by a plane parallel to the
18. Find the equation of the sphere through the
centre of the sphere
4
𝑦2
+ 9 + 1 =1,x+y+z=1
origin and base the circle x=a,y2+z2=b2 and show that
= 0, x-2y+z= 8.
x2+y2+z2-2x+4y-6z+5
𝑥2
26. Find the equation to the cone whose vertex is the
the plane 4x-5y-z=3 and passing through the circle
x2+y2+z2-2x-3y+4z+8
ax2 +by2=1,z=0
whose vertex is 0 axis , OX and semi-vertical angle α
5 and x+2y+3z =3 and
y2+z2 = x2 tan2 α
is
touch the plane 4x+3y= 15.
29. Show that the equation of a right circular cone whose
20. Define cone.
vertex is the origin O,axis OZ and semi vertical angle α
21. Find the equation of the cone where vertex is (3, 1, 2)
is x2 +y2 =z2tan2 α.
and base the circle 2x2+3y2 = 1, z = 1.
30. Write the equation of right circular cone whose axis is
22. Find the equation of the cone whose vertex is the
𝑥−α
𝑙
point (1, 1, 0) and whose base is the curve y = 0,
=
𝑦−β
𝑚
=
𝑧−γ
𝑛
31. Find the equation of the cylinder whose generating
x2+z2 = 4.
line have the direction cosines l, m, n and which
23. Find the equation of the cone whose vertex is the
passes through the circumference of the fixed circle
point(α,β, γ) and base y2=4ax,z=0
x2+z 2= a2 in the ZOX plane.
6
6. Find the radius of curvature of the curve y = ex at the
32. Define cylinder.
33. Find the equation of a cylinder whose generating lines
point where it crosses the y-axis.
have the direction cosines(l, m, n) and which passes
7. Find the radius of curvature of the curve√𝑥 + √𝑦 =1 at
through the circle x2+z2 = a2, y = 0.
(1/4 , 1/4)
34. Find equation of the cylinder whose generators are
𝑥
𝑦
parallel to the line 1= −2=
𝑧
3
8. Find the radius of curvature of the curve y = a log sec(
and whose guiding curve
x/a) at any point(x,y)
9. Find the radius of curvature at (0, c) on the curve y = c
is the ellipse x2+2y2 = 1; z = 0.
cos h( x/c)
35. Find the equation of the quadratic cylinder with
10. Find the curvature at any point on the curve S = c log
generators are parallel to x- axis and passing through
(sec 𝛹).
the curve ax2+by2+cz2 = 1, lx + my + nz = p.
11. Find the radius of curvature at x = π/2 on y = 4sinx.
𝜋
12. Find the radius of curvature at x = on the curve y =
36. Find the equation of the quadratic cylinder with
generators are parallel to z-axis and passing through
2
4sinx – sin2x.
the curve ax2+by2 = 2z, lx + my + nz = p.
13. Find the radius of curvature of the curve xy = c2 at
37. Define Right circular cylinder and give its equation.
(c,c).
Unit -III DIFFERENTIAL CALCULUS
14. Determine the radius of curvature of x3+y3 = 2 at (1, 1).
1. Define curvature and radius of curvature.
15. Find the radius of curvature at x = 1 on y =
2. What is the formula for curvature at any point p(x,
16. Find the radius of curvature of the circle
y)on the curve y = f(x).
log 𝑥
𝑥
x2+y 2=
25 at
(3, 4).
3. Prove that radius of curvature of a circle is its radius.
17. Find the curvature of the curve
4. What is the curvature of a 1) circle 2) straight line.
2y+1 =0, at any point on it.
5. What is the curvature of a circle of radius 2 units?
7
2x2+2y 2+5x-
18. Find the radius of curvature of the curve x2+y 2-
29. Find 𝜌 at any point t on the curve x=a(cost +t
6x+4y+6=0.
sint),y=a(sint-t cost)
19. Find the radius of curvature at y = 2a on the curve y2 =
30. Give the radius of curvature of the curve given by x =
3+2 cos 𝜃,
4ax.
20. Find the curvature of the parabola
y2 = 4x at the
formula.
31. Find the centre of curvature of y=x2 at the origin.
vertex (0,0).
21. Find the radius of curvature at (x, y) for the curve a2 y =
32. Write the equation of the circle of curvature..
x3 – a3
33. State any two properties of evolutes.
22. Find the radius of curvature of the curve at (0,0) on y2 =
𝑎3 −𝑥 3
𝑥
y = 4+2sin 𝜃 without using the
34. Find the evolutes of the curve
x2+y2 +4x-6y+3 =
0.
.
𝑐
𝑐
35. If the centre of curvature of a curve is ( 𝑎cos3t , 𝑎sin3t
23. For the curve x2 = 2c(y-x), find the radius of curvature
).Find the evolute of the curve.
at (o, c).
36. If (2+3cosѳ , 3+4sinѳ) is the centre of curvature at the
24. Prove that the radius of curvature of the curve xy2 = a2-
point ѳ ,find the evolute of the curve.
x2 at the point (a, 0) is 3a/2.
37. Given the co-ordinates of the centre of curvature is
25. Find the radius of curvature at (a, a) on the curve x3+y3
given as X=2a+3at2,
=2a3.
38. Y=-2at3, determine the evolute of the curve.
26. Find the radius of curvature of the curve r = a (1+cos𝜃)
39. Define envelope of a family of curves.
at 𝜃 = 𝜋/2..
40. What is the envelope of the family Am2 + Bm2 + c = 0.
27. Find the radius of curvature at the point (r, 𝜃) on the
41. Find the envelope of y = mx + am2, m being the
curve r = a cos 𝜃.
parameter.
28. Find 𝜌 at any point P(at2,2at) on the parabola y2=4ax
8
42. Find the envelope of the family of straight lines y = mx
52. Find the envelope of
a,where α being the parameter.
± √𝑚2 − 1 where m is the parameter.
43. Find the envelope of y = mx + m3,
xcosα + ysinα =
53. Find the envelope of the family of lines
m being the
𝑥
𝑦
cos 𝜃+𝑏 sin 𝜃 = 1. Where 𝜃 is the parameter
𝑎
parameter.
44. Find the envelope of the family of straight lines y = mx
54. Find the envelope of x2+y2-axcos 𝜃-by sin 𝜃=0,where
𝑎
+ 𝑚 where m is a parameter.
𝜃 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟.
55. Find the envelope of the family of straight lines y – 2x
45. Find the envelope of the family of straight lines y = mx
2
=∝
± √𝑚2 + 1 where m is the parameter.
46. Find the envelope of the family given by x = my + 1/m
56. Find the envelope of x + y – ax cos 𝜃 - by sin 𝜃 = 0,
where is a parameter.
where 𝜃 is the parameter.
47. Find the envelope of the family of straight lines y = mx
57. Show that the family of straight lines 2y-4x+𝜆=0,has no
envelope where 𝜆 is the parameter
+√a2 𝑚2 + 𝑏 2 where m is the parameter.
𝑥
48. Find the envelope of the family of circles (x-  )2 + y2 =
58. Find the envelope of the family of lines 𝑡 + yt = 2c,t
4  ,  being the parameter.
being the parameter.
49. Find the envelope of the family of straight lines y = mx
Unit –IV FUNCTIONS OF SEVERAL VARIABLES
+ a√1 + 𝑚2.
50. Find the envelope of y cot2  -x-a cosec2  =0,where
1. If u = x/y + y/z + z/x find x
 being the parameter.
𝑥
𝑦
𝜕𝑢
𝜕𝑥
+y
𝜕𝑢
𝜕𝑢
+z
𝜕𝑦
𝜕𝑢
𝜕𝑧
𝜕𝑢
2. If u = y f(𝑦) + 𝑔(𝑥 ) find x 𝜕𝑥 + 𝑦 𝜕𝑦
51. Find the envelope of the family of straight lines xcosα +
3. If u = (x-y)(y-z)(z-x) show that
ysinα = asecα,where α being the parameter.
𝑥 𝑦 𝑧
4. If u= f (𝑦, 𝑧
9
𝑥,
𝜕𝑢
𝜕𝑢
𝜕𝑥
𝜕𝑢
𝜕𝑢
+ 𝜕𝑦 + 𝜕𝑧 =0
𝜕𝑢
𝜕𝑢
). Prove that x 𝜕𝑥 + 𝑦 𝜕𝑦 +𝑧 𝜕𝑧 = 0.
5. If f(x,y)=log√𝑥 2 + 𝑦 2 ,show that
6. If x = rcos𝜃 y =rsin𝜃. Prove that
𝜕2 𝑓
𝜕x2
𝜕𝑟
𝑥3
17. If u = tan-1 (
𝑥
𝜕2 𝑓
+ 𝜕y2 = 0.
𝜕𝑥
1 𝜕𝑥
𝜕𝜃
= ,
= r 𝜕𝑥
𝜕𝑥 𝜕𝑟 𝑟 𝜕𝜃
+
−
𝜕𝑢
𝜕𝑢
𝑦3
) Prove that x 𝜕𝑥 + 𝑦 𝜕𝑦 =
𝑦
sin2u.
.
18. If u=log(
𝑥 4 +𝑦 4
𝜕𝑟
7. If x = rcos𝜃 ,y =rsin𝜃 find 𝜕𝑥
𝜕𝑢
𝜕𝑢
), show that x 𝜕𝑥 + 𝑦 𝜕𝑦 =3
𝑥+𝑦
𝑦
𝜕2 𝑢
𝜕2 𝑢
𝜕2 𝑢
19. If u =(x – y) f (𝑥 ) find x2 𝜕𝑥 2 + 2xy 𝜕𝑥𝜕𝑦 + y2𝜕𝑦 2
8. If u = f (x, y, z) where x, y, z are functions in t,
𝑦
𝑦
𝜕2 𝑢
𝜕2 𝑢
𝜕2 𝑢
20. If u =x f (𝑥 )+g(𝑥 ) show that x2 𝜕𝑥 2 + 2xy 𝜕𝑥𝜕𝑦 + y2𝜕𝑦 2
𝜕𝑢
then 𝜕𝑡 ?
𝑥
9. If u = 𝑦, , x = et, y = log t find
=0
𝜕𝑢
𝑑𝑦
𝜕𝑡
21. Find 𝑑𝑥 when f (x, y) = log (x2+y2) + tan-1 y/x.
𝜕𝑢
10. If u = x2+y2+3x2y2, find𝜕𝑥 .
22. What is total differential of a function u ?
1
11. If u = xy +yz +zx where x = et, y = e-t and z = 𝑡 . Find
𝑑𝑦
23. Find 𝑑𝑥 when x3 + y3 = 3axy.
𝜕𝑢
.
12. State Euler’s theorem for homogeneous functions.
𝑑𝑦
𝜕𝑡
24. Find 𝑑𝑥 when ysinx=xcosy
𝜕𝑢
13. Verify whether u = ex/y sin(x/y) + ey/x cos (y/x) is
25. If u = x2 + y2 and x = e2t , y = e2t cos3t .Find 𝜕𝑡 as a
homogeneous. If so find its degree.
total derivative.
𝜕𝑢
𝜕𝑢
𝜕𝑢
14. If u = sin-1 (x/y) + tan-1 (y/x). Prove that x 𝜕𝑥 + y 𝜕𝑦 =
26. Ifu = ex siny where x = st2 and y = s2t. Find 𝜕𝑠 and
𝜕𝑢
0.
𝜕𝑡
𝜕𝑢
𝜕𝑢
𝜕𝑧
𝑥 3 +𝑦3
27. Find 𝜕𝑡 when z = xy2 + x2 y, x = at2, y = 2at without
15. Show that x 𝜕𝑥 + y 𝜕𝑦 =2ulogu where log u=3𝑥+4𝑦
𝑥3
16. If u = sin-1 (
𝑥
−
+
.
actual substitution.
𝜕𝑢
𝜕𝑢
𝑦3
) Prove that x 𝜕𝑥 + 𝑦 𝜕𝑦 =
𝑦
28. Define Jacobian of two variables.
2tanu.
29. State the properties of jacobians.
10
𝜕(𝑟,𝜃)
𝜕(𝑥,𝑦)
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
30. If x = rcos 𝜃, y = rsin 𝜃,find 𝜕(𝑥,𝑦) , 𝜕(𝑟,𝜃)
𝜕(𝑥,𝑦)
31. If x = u(1+v) , y = v(1+u) , find 𝜕(𝑢,𝑣) .
32. If u =
33. If u =
y2
𝑥
y2
and v =
,v=
x2 ,
x2
𝑦
variables to have an extremum at a point.
𝜕(𝑥,𝑦)
, find 𝜕(𝑢,𝑣).
45. Define Stationary points?
𝜕(𝑢,𝑣)
find 𝜕(𝑥,𝑦).
46. Define saddle points of a function f (x, y).
34. If x = u(1-v) , y = uv , find the jacobian of the
47. Find the stationary points of f(x,y) = x2-xy + y2 –
transformation.
35. Find
2x+y.
𝜕(𝑥,𝑦)
if x+y = u ,y = uv.
𝜕(𝑢,𝑣)
36. If u = x-y , v = y-z ,w= z-x , find
48. Find the stationary points of
𝜕(𝑢,𝑣,𝑤)
𝜕(𝑥,𝑦,𝑧)
.
f(x,y) = x3+3xy2-
15x2-15y2+72x.
𝜕(𝑥,𝑦,𝑧)
37. If u = x+y+z , y+z = uv, z = uvw ,find J(𝜕(𝑢,𝑣,𝑤)).
38. If u =
𝑦𝑧
𝑥
,v=
𝑧𝑥
𝑦
𝑥𝑦
,w=
𝑧
, find
49. Find the stationary points of
𝜕(𝑢,𝑣,𝑤)
𝜕(𝑥,𝑦,𝑧)
.
.
39. If u = x2 –y2 , v=2xy , and x = rcos 𝜃, y =
rsin 𝜃. Evaluate
𝜕(𝑢,𝑣)
𝜕(𝑟,𝜃)
9 3
f(x,y) = xy +𝑥+𝑦
.
50. A flat circular plate is heated so that the
temperature at any point (𝑥, 𝑦)is u (𝑥, 𝑦)=x2+2y2-x
40. State Maclaurin’s series for a function of two
find the coldest point on the plate
variables x and y.
51. Find the stationary points of f (x, y) = x3+3xy2-15x2-
41. Find the Taylor’s series expansion of xy near the
15y2+72x for extreme values.
point (1,1) upto the first degree term.
52. Examine the extreme of
42. Find Taylor’s series expansion of ex siny near the
point (-1, π/4) upto the first degree terms.
+ xy + y2 + 1/x +1/y.
11
f (x, y) = x2
15. Find the value of ∫0 ∫0
i) f (x, y) = x4 –y4 – 2x2 + 2y2,
√𝑎2 −𝑥 2
𝑎
𝜋/2
17. Evaluate ∫0
𝜋/2
∫0
𝜋
𝑠𝑖𝑛𝜃
18. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∫02 ∫0
Unit – V MUTIPLE OF INTEGRALS
2. Evaluate
3. Evaluate
4.
5.
6.
7.
8.
9.
2 5
∫1 ∫2 𝑥𝑦 𝑑𝑥 𝑑𝑦 .
1 2
∫0 ∫1 𝑥(𝑥 + 𝑦)𝑑𝑥𝑑𝑦 .
𝑎 𝑏
∫0 ∫0 𝑥𝑦 (𝑥 − 𝑦) 𝑑𝑥 𝑑𝑦.
12.
13.
14.
𝑐𝑜𝑠𝜃
𝜋/2
𝜋
𝑎𝑠𝑖𝑛𝜃
𝜋
𝑟 𝑑𝜃 𝑑𝑟 .
𝑟 𝑑𝑟 𝑑𝜃.
𝑎𝑐𝑜𝑠𝜃
∫0
22. Evaluate ∫0 ∫0
𝑟 2 𝑑𝑟 𝑑𝜃.
𝑟 𝑑𝑟 𝑑𝜃 .
𝑎(1−𝑐𝑜𝑠𝜃) 2
23. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∫0 ∫0
𝑟 𝑠𝑖𝑛𝜃 𝑑𝑟 𝑑𝜃.
24. Change the order of integration
𝑎
√𝑎2 −𝑦 2
∫−𝑎 ∫0
𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦.
𝑎
𝑥
25. Change the order of integration in ∫0 ∫0 𝑑𝑦 𝑑𝑥.
∞
𝑦
26. Transform the integration ∫0 ∫0 𝑑𝑥 𝑑𝑦 to polar coordinates.
27. By changing into polar co-ordinate,
𝑥2
10. Evaluate∫0 ∫0 𝑥(𝑥 2 + 𝑦 2 )𝑑𝑥 𝑑𝑦.
11.
𝜋
21. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∫0
1 1
Evaluate ∫0 ∫0 (𝑥 2 +𝑦 2 ) dx dy.
3 2 𝑑𝑥 𝑑𝑦
Evaluate ∫2 ∫1 𝑥𝑦 .
𝑏 𝑎 𝑑𝑥 𝑑𝑦
Evaluate ∫1 ∫1 𝑥𝑦 .
3 2
Evaluate ∫0 ∫0 𝑒 𝑥+𝑦 dy dx.
5 3 𝑑𝑥 𝑑𝑦
Evaluate ∫1 ∫1 𝑥𝑦 .
1 2
Evaluate∫0 ∫0 𝑥𝑦 2 𝑑𝑦 𝑑𝑥.
5
𝑠𝑖𝑛𝜃
20. 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∫0 ∫0
sin(𝑥 + 𝑦) 𝑑𝑥 𝑑𝑦
𝑟 𝑑𝜃 𝑑𝑟.
𝜋/2
19. Evaluate ∫–𝜋/2 ∫0
dx dy
y
𝑑𝑥 𝑑𝑦.
16. Evaluate∫0 ∫0
ii)f (x,y) = x3 + y3 – 12xy
1. Evaluate
y e−y
∞
53. Identify the saddle point and the extreme point of
2
√2𝑥−𝑥 2
evaluate∫0 ∫0
2 𝑥2
Evaluate ∫1 ∫0 𝑥 𝑑𝑦 𝑑𝑥 .
2 𝑥 𝑑𝑥 𝑑𝑦
Evaluate ∫1 ∫0 𝑥 2 +𝑦 2
2 𝑦 𝑦 𝑑𝑥 𝑑𝑦
𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 ∫1 ∫0 𝑥 2 +𝑦 2
2 𝑥
1
Evaluate ∫1 ∫0 𝑥 2 +𝑦 2 𝑑𝑦𝑑𝑥
𝑥
𝑥 2 +𝑦 2
𝑑𝑥 𝑑𝑦.
28. By changing into polar co-ordinate, find the value of
2𝑎
√2𝑎𝑥−𝑥 2
the integral∫0 ∫0
(𝑥 2 + 𝑦 2 )𝑑𝑦 𝑑𝑥 .
𝑎
√𝑎2 −𝑥 2
29. Change in to polar co-ordinates of ∫−𝑎 ∫−√𝑎2 −𝑥 2 𝑑𝑦 𝑑𝑥.
12
𝑎
𝑥2
𝑎
30. Express into polar co − ordinates ∫0 ∫𝑦
(𝑥 2 +𝑦 2 )3/2
dy.
31. Transform into polar co-ordinates the integral
𝑎
𝑎2 −𝑥 2
𝑎2
𝑎
∫0 ∫0
−𝑥 2
𝑓(𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 .
𝑎
𝑥
𝑥+𝑦
∫0 ∫0
1
𝑥
𝑥+𝑦
4
𝑥
𝑥+𝑦
𝑒 𝑥+𝑦+𝑧 dxdy dz .
𝑧 dz dy dx .
𝑧 dx dy dz .
1. Find the Eigen values and Eigen vectors of the matrix
1 1
[
]
3 −1
2. Find the Eigen values and Eigen vectors of
2 2 0
A=[ 2 1 1 ]
(non repeated)
−7 2 −3
3. Find the Eigen values and Eigen vectors of the
1 0 −1
matrix[1 2 1 ] (non repeated)
2 2 3
4. Find the Eigen values and Eigen vectors of
2 2 1
A=[1 3 1] (two repeated non symmetric)
1 2 2
√𝑎2 +𝑥2
𝑎
(𝑏−𝑦)
𝑓(𝑥, 𝑦) dx dy.
∫0 ∫0𝑏
38. Find the limits of integration in the double integral
, 𝑤ℎ𝑒𝑟𝑒 R is in the first quadrant and bounded by
x=1,y=0,y2 = 4x.
39. Find by double integration , the area of the circle
𝑥 2 +𝑦 2 = 𝑎2 ,in polar coordinates.
𝑐
𝑎
Unit - I Matrices
37. Sketch roughly the region of integration of
𝑏
𝜋
Part-B Questions
36. Shade the region of integration ∫0 ∫√𝑎𝑥−𝑥 2 𝑑𝑥 𝑑𝑦 .
𝑎
3
48. Evaluate ∫0 ∫0 ∫0√
𝑥
𝑐
2
47. Evaluate ∫0 ∫0 ∫0√
integral ∫0 ∫0 𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥.
𝑏
1
𝑙𝑜𝑔2
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
𝑎
2
46. Evaluate∫0
34. Find the limits of integration in∬𝑅 𝑓(𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 ,
𝑏
3
2𝜋
33. Sketch roughly the region of integration for the
following
1
2
45. Evaluate ∫0 ∫0 ∫0 𝑟 4 sinφ dr dφ dѳ .
𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦.
double integral
𝑐
44. Evaluate ∫0 ∫0 ∫0 𝑥𝑦𝑧 dx dy dz .
𝑎
32. Sketch roughly the region of integration of
𝑎
𝑏
43. Evaluate ∫0 ∫1 ∫1 𝑥𝑦 2 𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥
∫0 ∫𝑦 𝑓(𝑥, 𝑦) dx dy .
∫0 ∫0
𝑎
42. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥
dx
40. Evaluate ∫0 ∫0 ∫0 (𝑥 + 𝑦 + 𝑧) dz dy dx
41. Evaluate ∫0 ∫0 ∫0 𝑥𝑦𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥
13
5. Find the Eigen values and Eigen vectors of the
1 2 3
matrix[0 2 3] r
0 0 2
6. Find the Eigen values and Eigen vectors of the
2 1 0
matrix[0 2 1] r
0 0 2
7. Find the Eigen values and Eigen vectors of the
7 −2 0
matrix[−2 6 −2]
0 −2 5
8. Find the Eigen values and Eigen vectors of the
1 −1 −1
matrix[−1 1 −1] r
−1 −1 1
9. Find the Eigen values and Eigen vectors of the
0 1 1
matrix[1 0 1]
1 1 0
10. Find the Eigen values and Eigen vectors of the
6 −2 2
matrix[−2 3 −1] (two repeated symmetric)
2 −1 3
11. Find the Eigen values and Eigen vectors of the
13. Find the Eigen values and Eigen vectors of the
3 −4 4
matrix[1 −2 4]
1 −1 3
𝑎
14. Find the constants a and b such that the matrix [
1
4
]
𝑏
has 3 and -2 as its eigen values
15. Using Cayley-Hamilton theorem , find the inverse of
2
the matrix A = [
1
1
]
−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.
1 2
17. If A=[
] find A-1 and A3 using Cayley Hamilton
3 4
theorem and also verify theorem.
1 0
18. If A = [
] , express A3 in terms of A and I using
4 5
Cayley-Hamilton theorem.
19. Using Cayley Hamilton theorem Find A-1 when
8 −6 2
matrix[−6 7 −4]
2 −4 3
1 3 7


A  4 2 3
1 2 1


20. Verify Cayley-Hamilton theorem for thematrix A =
2 −1 1
[−1 2 −1].Hence compute A-1
1 −1 2
12. Find the Eigen values and Eigen vectors of
6
−6
5
A=[14 −13 10] (three repeated)
7
−6
4
14
10 2 5 


27. Reduce the matrix  2 2 3  to diagonal form
 5 3 5 


21. Verify Cayley-Hamilton theorem and hence find A-1 if A
13 −3 5
=[ 0
4
0 ].
−15 −9 −7
1 2 −1
22. Given A = [0 1 −1] find AdjA by using
3 −1 1
Cayley-Hamilton theorem.
23. Verify Cayley-Hamilton theorem and hence find A-1 if
1
2 −2
A = [−1 3
0 ].
0 −2 1
1 0 0


24. If A =  1 0 1  show that
0 1 0


3 1 1 


28. Diagonalise the matrix A=  1 3 1 by means of
 1 1 3 


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+3x22+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=
2
2
2
3x1 +2x2 +3x3 -2x1x2-2x2x3.
34. Reduce the quadratic form x12  2 x2 x3 into a canonical
An  An 2  A2  I for n  3 using Cayley Hamilton
theorem
25. Find the characteristic equation of the matrix A =
2 1 1
[0 1 0] and hence compute A-1
1 1 2
Also find the matrixrepresented by A8 -5A7 +7A6 -3A5 +A4 5A3 +8A2 -2A+I.
2
26. Diagonalise the matrix A = [0
4
orthogonal transformation.
form by means of an orthogonal transformation.
Determine its nature
35. Reduce the quadratic form
x12  5x 22  x 32  2x1x 2  2x 2x 3  6x 3 x1 to
0 4
6 0] by means of an
0 2
Canonical form through an orthogonal transformation
15
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+z22x-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+8z45=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+5y6z+2=0
36. Verify that the eigen vectors of the real symmetric
8 −6 2
matrix A = [−6 7 −4] are in orthogonal pairs.
2 −4 3
37. Reduce the quadratic form 2 x12  6 x22  2 x32  8 x3 x1 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 Geometry
1. Show that the spheres x2+y2+z2=25, x2+y2+z2-18x-24y40z+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
𝑎
the sphere OABC is 𝑥 +
𝑏
𝑦
𝑐
+𝑧 =2
16
15. Prove that the circles x2+y2+z2 -2x+3y+4z5=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.
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 = x2z-3.
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
𝑦
4
=
𝑦−2
1
=
𝑧−1
3
28. Find the equation of the right circular cone whose
𝑥
vertex is at the origin,whose axis the lin
1
𝑦
𝑧
=2 =3
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
𝑥−1
2 and having as axis of the line
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 6x3y-23=0=3z+2
𝑥
𝑥−3
2
=
𝑦−2
1
=
𝑧−3
2
31. Find the equation of the right circular cylinder of radius
𝑥+1
3 and having as axis of the line
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
𝑧
24. The plane 𝑎 + 𝑏 + 𝑐 =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.
17
point(1,2,3)and
𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑖𝑛𝑒𝑠 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙 𝑡𝑜 2, −3,6
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(
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
𝜃
+sin 𝜃) , y= a(1-cos𝜃) is 4acos 2 at any point 𝜃.
11. Find the radius of curvature at any point P(a cos 𝜃,b sin
𝜃) on the ellipse
Unit-III Differential Calculus
3𝑎
𝑎𝑥
2𝜌 2/3
(𝑎)
𝜃),y=a(tan 𝜃 − 𝜃)
at 𝜃
13. Find the centre of curvature at the point (am2,2am) on
𝑥 2
= (𝑦) +
the parabola y2=4ax
𝑦 2
5.
6.
7.
8.
𝑦2
+ =1
𝑎2 𝑏 2
12. Find the radius of curvature of the curve x=a log(sec
3𝑎
1. Find the radius of curvature at the point ( 2 , 2 ) 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 = 𝑎+𝑥 Prove that
𝑥2
(𝑥 ) where ρ is the radius of curvature of the curve.
Find the radius of curvature at any point t on the curve
x=et cost , y=et sint.
Find the radius of curvature of the parabola x=at2 ,
y=2at at t.
Find the radius of curvature at any point (a cos3,a
sin3) on the curve x2/3+y2/3=a2/3
Find the radius of curvature at the origin for the cycloid
x=a( +sin 𝜃) ,
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
18
17. Find the centre and circle of curvature of the curve
√𝑥 + √𝑦 =√𝑎 at
24. Find the equation of the evolute of the ellipse
(a/4, a/4).
𝑥2
𝑎2
𝑦2
+𝑏 2
= 1.
18. For the curve √𝑥 +√𝑦 =1 find the equation of the circle
𝑥2
25. Find the equation of the evolute of the hyperbola𝑎2 -
of curvature
𝑦2
𝑏2
at (1/4,1/4).
= 1.
26. Find the equation of the evolute of the rectangular
19. Find the equation of circle of curvature of the parabola
y2
hyperbola xy=c2
= 12x at the point (3,6).
27. Find the equation of the evolute of the curve
20. Find the equation of circle of curvature at (3,4) on xy
x2/3+y2/3=a2/3.
= 12.
28. Show that the equation of the evolute of the cycloid
21. Find the equation of the circle of curvature at the point
(2,3) on
𝑥2
4
+
𝑦2
9
x=a(𝜃 –sin𝜃) ,
=2
y=a(1-cos𝜃) is another equal cycloid.
22. Find the equation of the evolute of the parabola y2 =
29. Show that the evolute of the curve
4ax.
x = a(cos𝜃+𝜃sin𝜃), y = a(sin𝜃-𝜃cos𝜃) is a circle
23. Find the equation of the evolute of the parabola x2 =
30. Find the evolute of the curve
4ay.
x=a( +sin 𝜃) , y= a(1-cos𝜃).
19
𝑥 𝑦
31. Prove that the evolute of the curve
38. Find the envelope of 𝑎+𝑏 =1 subject to an+bn=cn where
x = ct, y=c/t is (x+y)2/3 – (x-y)2/3 = (4c)2/3.
c is known constant.
𝑎𝑥
32. Find the envelope of the family of straight lines𝑐𝑜𝑠𝜃 𝑏𝑦
𝑠𝑖𝑛𝜃
39. Find the evolute of y2=4ax considering it as the
envelope of normals .
= a2-b2.
40. Find the evolute of x2=4ay considering it as the
𝑥 𝑦
33. Find the envelope of 𝑎+𝑏 =1 subject to a2+b2=c2 ,where
envelope of normals .
c is being constant.
41. Considering the evolute as the envelope of normals
𝑥 𝑦
34. Find the envelope of 𝑎+𝑏 =1 where the parameters a,b
find the evolute of
are related by ab=c2 where c is known
𝑥2 𝑦2
+ =1.
𝑎2 𝑏 2
42. Considering the evolute as the envelope of normals
𝑥 𝑦
35. Find the envelope of 𝑎+𝑏 =1 subject to a+b=c where c is
find the evolute of
known constant.
𝑥2 𝑦2
- =1.
𝑎2 𝑏 2
Unit- IV Functions of Several Variables
𝑥2 𝑦2
36. Find the envelope of 𝑎2 +𝑏2 =1subject to a+b=c where c
1. If u = xy, then show that
is a constant.
𝜕3 𝑢
𝜕x2 ∂y
𝜕3 𝑢
=𝜕𝑥𝜕𝑦𝜕𝑥
2. If u=log(x3+y3+z3-3xyz),show that
𝑥2 𝑦2
37. Find the envelope of 𝑎2 +𝑏2 =1subject to
𝜕
𝜕
𝜕
−9
3. (i)(𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧)2 u=(𝑥+𝑦+𝑧)2
a2+b2=c 2
a. (ii)
where c is a constant.
𝜕2 𝑢
𝜕x2
2
+
𝜕2 𝑢 𝜕2 𝑢
+
𝜕y2 𝜕z2
2 -1/2
4. If u = (x2 + y + z )
𝜕2 𝑢
𝜕x2
20
𝜕2 𝑢
+ 𝜕y2 +
𝜕2 𝑢
𝜕z2
= 0.
𝜕2 𝑢
𝜕2 𝑢 𝜕2 𝑢
−9
+2
+
=
𝜕𝑦𝜕𝑧 𝜕𝑧𝜕𝑥 𝜕𝑥𝜕𝑦 (𝑥+𝑦+𝑧)2
+2
prove that
5. Find the first order partial derivatives of (i)u=tan-1
(
𝑥 2 +𝑦 2
𝑥+𝑦
6. If u =
15. If Z=f(x,y) where x=rcos𝜃 and rsin show that
) (ii) u=cos-1(x/y)
√𝑥− 𝑦
sin-1 𝑥+√𝑦
√ √
𝑥+𝑦
-1
7. If u=cos [
√𝑥+√
𝜕𝑧 2
𝜕𝑢
𝜕𝑧 2
, find x𝜕𝑥 + y 𝜕𝑦 .
x
𝜕𝑥
𝜕𝑢
+𝑦
𝜕𝑦
=-
1
2
cotu
that
𝜕2 𝑔
𝜕𝑢
𝑥+2𝑦+3𝑧
𝜕𝑢
𝜕2 Ψ
),show that
𝜕v2
).
𝜕2 𝑧
𝜕2 𝑧
𝜕2 𝑧
18. If Given transformation u=𝑒 𝑥 cos y and v=𝑒 𝑥 siny and ∅
𝜕𝑢
x 𝜕𝑥 + y 𝜕𝑦 + z 𝜕𝑧 +3tanu=0
is a function of u and also x and y . prove that
10. State and prove Euler’s extension theorem.
𝜕2 ∅
𝑑𝑢
=(𝑢2 + 𝑣 2 ) (𝜕𝑢2 +
11. If u = x logxy where x3 + y3 + 3xy = 1 ,find 𝑑𝑥 .
𝜕2 ∅
𝜕𝑣 2
𝜕2 ∅
𝜕2 ∅
+ 𝜕𝑦 2
𝜕𝑥2
)
19. Find the Jacobian of y1 ,y2 ,y3 with respect to x1, x2, x3 if
𝑥 𝑥
𝑥 𝑥
𝑥 𝑥
y1 = 𝑥2 3 , y2 = 𝑥3 1 , y3 = 𝑥1 2
12. If u = x3y2 + x2y3 where x =at2 ,
y = 2at.Find
𝜕2 Ψ
( 𝜕x2 + 𝜕y2) =
𝜕2 𝑧
sin (x/y) + tan (y/x).
√𝑥 8 +𝑦8 +𝑧 8
𝜕x2
𝜕2 𝑔
+ 𝜕y2 = 4 (x2 + y2) ( 𝜕u2 +
(l2 + m2) (𝜕u2 + 𝜕v2)
-1
9. If u=sin-I(
𝜕𝑧 2
17. If z = f(u,v) where u = lx + my and v = ly-mx. Show
8. Verify Euler’s theorem for
-1
1
16. If g(x,y) = Ψ(u,v) where u = x2 – y2 and v = 2xy .Prove
] prove that
𝑦
that
𝜕𝑢
𝜕𝑧 2
(𝜕𝑥) +(𝜕𝑦) = (𝜕𝑟) +𝑟 2 (𝜕𝜃)
𝜕𝑢
1
2
2
2
3
𝜕(𝑢,𝑣)
20. If v =2xy,u=x -y and x=rcos 𝜃,y=rsin 𝜃 evaluate 𝜕(𝑟,𝜃)
𝑑𝑢
.
𝑑𝑡
13. If z = sin-1(x-y), x = 3t, y = 4t3.
𝜕𝑢
Show that
𝜕𝑢
𝑑𝑧
𝑑𝑡
3
=√1−𝑡 2
21. If u =
𝑥+𝑦
𝑥−𝑦
𝜕𝑢
14. If u = f (x-y, y-z, z-x) find 𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧
𝜕(𝑢,𝑣)
𝜕(𝑥,𝑦)
21
.
and v=tan−1 𝑥 + tan−1 𝑦 find the Jacobian
22. If u = 4x2 + 6xy , v = 2y2 + xy , x = rcosθ , y = rsinθ
.Evaluate
33. Find the extreme values of the function f(𝑥, 𝑦)=x3+y3-
𝜕(𝑢,𝑣)
𝜕(𝑟,𝜃)
3x-12y+20
23. If x=a cosh 𝛼 cos 𝛽,y=a sinh 𝛼 sin 𝛽,then show that
34. Find the extreme values of the function f(𝑥, 𝑦)=x3y2(1-x-
𝜕(𝑥,𝑦) 𝑎2
= (cosh 2 𝛼-cos 2 𝛽)
𝜕(𝛼,𝛽) 2
24. Ifx=sinθ√1 − 𝑐 2 𝑠𝑖𝑛2 ∅, 𝑦 = 𝑐𝑜𝑠θcos∅, 𝑡ℎ𝑒𝑛
𝜕(𝑥,𝑦)
y)
𝜕(θ,∅)
−𝑠𝑖𝑛∅[(1−𝑐 2 )𝑐𝑜𝑠2 θ+𝑐 2 𝑐𝑜𝑠2 ∅]
=
35. Find the maximum and minimum value of x2-xy+y22x+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.
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
√1−𝑐 2 𝑠𝑖𝑛2 ∅
𝜋
25. Expand ex cosy about (0, 2 ) up to the third term using
taylor’s series
26. Expand ex siny around thye point[1,
𝜋
2
] up to the third
term using taylor’s series
𝜋
27. Expand sin xy in powers of (x-1) and (y- 2 ) 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
at 32cc.find the dimensions of the box that requires the
least material for its construction
of third degree
41. A thin closed rectangular box is to have one edge equal
30. Expand xy2+2x-3y in powers of (x+2) and (y-1) upto
to twice the other and constant volume 72m3.Find the
least surface area of the box.
third degree terms.
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
31. Expand f(x,y) = 4x +xy+6y +x-20y+21 in Taylor’s
series about (-1,1)
32. Examine for the extremum values of f(x,y) = x3+ y312x-3y+20.
2
2
22
𝑥2
𝑎2
𝑦2 𝑧 2
𝑎
𝑦 2 ) 𝑑𝑦𝑑𝑥,and hence evaluate it.
5. Change the order of integration
cosA cosB cosC .
4
4
∫0 ∫𝑦
47. The temperature u(x,y,z) at any point in space is u =
𝑥
𝑥 2 +𝑦2
𝑑𝑥𝑑𝑦 and hence evaluate it.
∞
∞ e−y
6. Change the order of integration ∫0 ∫x
400xyz2. Find the highest temperature on the surface of
dy dx
y
and hence evaluate it.
the sphere x2+y2+z2 =1.
3
48. Find the minimum value of x2+y2+z2 subject to the
√4−𝑦
7. Evaluate ∫0 ∫1
(𝑥 + 𝑦)𝑑𝑥 𝑑𝑦, .
By changing the order of integration
8. Change the order of integration in
1
+ 𝑧 =1.
49. Find the extreme values of the functions v= x2+y2+z2
4
3
∫0 ∫04
subject to ax+by+cz = p
√16−𝑥2
𝑥 𝑑𝑥 𝑑𝑦 𝑎𝑛𝑑 ℎ𝑒𝑛𝑐𝑒 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒.
4𝑎
2
2
2√𝑎𝑥
9. Change the order of integration ∫0 ∫𝑥2
2
50. Find the minimum value of x +y +z with the constraint
𝑑𝑦 𝑑𝑥
4𝑎
and hence evaluate it.
10. Change the order of integration in
xy+yz+zx=3a2
51. Find the shortest distance from the origin to the curve
2
𝑎
4. Change the order of integration ∫0 ∫𝑥 (𝑥 2 +
46. In a plane triangle ABC find the maximum value of
𝑦
𝑥
hence evaluate it.
method of constrained maxima and minima.
1
1
3. Change the order of integration ∫0 ∫0 𝑑𝑦 𝑑𝑥 and
(1,2,-1) to the sphere x2+y2+z2 = 24. using Lagrange’s
1
2−𝑥
dy dx.
45. Find the shortest and the longest distances from the point
condition 𝑥 +
1
2. Change the order of integration ∫0 ∫𝑥 2 𝑓(𝑥, 𝑦)
+𝑏2 +𝑐 2 = 1.
𝑎
2𝑎−𝑥
∫0 ∫𝑥2
2
x +8xy+7y =225.
𝑥𝑦 𝑑𝑦 𝑑𝑥 𝑎𝑛𝑑 hence evaluate it.
𝑎
11. Change the order of integration in
Unit-V Multiple Integrals
1
√1+𝑥2
1. Find ∫0 ∫0
𝑎
𝑎+√𝑎2 −𝑦 2
∫0 ∫𝑎−√𝑎2−𝑦2 𝑑𝑦 𝑑𝑥, and hence evaluate it.
𝑑𝑦 𝑑𝑥
1+𝑥 2 +𝑦2
12. Change the order of integration
.
1
2−𝑦
∫0 ∫𝑦 2 𝑥𝑦 𝑑𝑦 𝑑𝑥 ,and hence evaluate it.
23
𝑥2
13. Evaluate
∞ 𝑥
−
∫0 ∫0 𝑥𝑒 𝑦
𝑥2
dy dx,by change the order
𝑎2
24. Find the area between the parabolas 𝑦 2 = 4ax
and 𝑥 2 = 4ay.
25. Find the area of the region bounded by the
parabolas y = 𝑥 2 and x = 𝑦 2 .
26. Find the area bounded by y=x and y=x2
27. Find by double integration,the are between the
parabola 𝑦 2 = 4ax and the line y = x.
28. Find the smaller of the areas bounded by y = 2 –x
and 𝑥 2 + 𝑦 2 =4.
29. Find by double integration ,the area of the
cardiod r = a(1+cosѳ).
30. Evaluate ∬ 𝑟 3 dr 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=4 sin 𝜃
33. Evaluate ∫ ∫ 𝑟 2 sin 𝜃 dr d𝜃 where R is the region
of semicircle r=2acos 𝜃 about the initial line
34. Evaluate ∬ 𝑟 2 dr dѳ,over the area between the
circles r = 2cosѳ and r = 4cosѳ.
of integration.
∞
𝑥
14. Evaluate ∫0 ∫0 𝑥𝑒
−
𝑥
𝑦
dy dx,by change the order
of integration.
15. Change the order of integration in
√2−𝑥 2
1
∫0 ∫𝑥
16. Evaluate
𝑥
dx dy and hence evaluate it.
√𝑥 2 +𝑦 2
2
2
∞ ∞
∫0 ∫0 𝑒 −(𝑥 +𝑦 ) dx
dy by changing to
∞
polar coordinates and hence show that ∫0 𝑒 −𝑥
2
𝜋
dx=√ 2
17. By changing in to polar co-ordinates ,evaluate
𝑎
𝑎
∫0 ∫𝑦
𝑥
𝑥 2 +𝑦 2
𝑦2
+ 𝑏2 = 1.
dx dy.
18. Find the area of a circle of radius a in polar coordinates
19. Evaluate ∬ 𝑥𝑦 dx dy,over the positive quadrant
of the circle 𝑥 2 +𝑦 2 = 1.
20. Find ∬ 𝑑𝑥 𝑑𝑦 , 𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑟𝑒𝑔𝑖𝑜𝑛 bounded by x ≥
0,y ≥ 0,x+y ≤ 1.
21. Find the area enclosed by the curves y = 𝑥 2 and
x+y = 2.
22. Evaluate ∬ 𝑥𝑦 𝑑𝑥 𝑑𝑦 ,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
1
𝜌
2𝜋
35. Evaluate ∫𝜌=0 ∫𝑧=𝜌2 ∫ѳ=0 𝜌 𝑑𝜌 dz dѳ.
36. Transform the integration
5
6
√36−𝑥 2
∫𝑧=0 ∫−6 ∫−√36−𝑥 2 𝑑𝑥 𝑑𝑦 𝑑𝑧.
24
1
√1−𝑥 2
37. Evaluate ∫0 ∫0
𝑑𝑧𝑑𝑦𝑑𝑥
√1−𝑥 2 −𝑦2
∫0
√1−𝑥2 −𝑦 2 −𝑧 2
.
by changing into spherical polar coordinates
38. Express the volume of the sphere 𝑥 2 + 𝑦 2 +𝑧 2 =
𝑎2 , as a volume integral and hence evaluate it
39. Find the volume bounded by
x,y,z ≥ 0 and 𝑥 2 + 𝑦 2 +𝑧 2 ≤1 in triple integration
40. Find the volume bounded by the cylinder 𝑥 2 + 𝑦 2
=4 and the planes y+z = 4 and z = 0.
41. Find the volume of the ellipsoid
𝑥2
𝑦2
𝑧2
+ + = 1.
𝑎2 𝑏 2 𝑐 2
42. Evaluate ∭ 𝑥𝑦𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧, taken over the
positive octant of the sphere 𝑥 2 + 𝑦 2 +𝑧 2 =1
43. Find the volume of the tetrahedron bounded by
𝑥
𝑦
𝑧
the planes x=0,y=0,z=0 and 𝑎 +𝑏 +𝑐 = 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 𝑣 is the finite
region of space(tetrahedron) formed by the
planes x = 0, y = 0, z = 0 and 2x+3y+4z = 12.
46. Evaluate ∭ 𝑥𝑦𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧, taken throughout the
volume for which x,y,z ≥ 0 and 𝑥 2 + 𝑦 2 +𝑧 2 ≤ 9
25
8. What will be the plane section perpendicular to its axis of a
right circular cylinder
9. Find the evolute of the curve 𝑥 2 +𝑦 2+4x-6y+3 = 0
ANNA UNIVERSITY COIMBATORE
1
10. Find the envelope of the family given by x = my + 𝑚 , m being
B.E./B.TECH. DEGREE EXAMINATIONS : JAN-FEB 2009
the parameter
11. 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
REGULATIONS : 2008
FIRST SEMESTER – COMMON TO ALL BRANCHES
08003001 – MATHEMATICS I
PART -A (20 X 2 = 40 Marks)
13. If u = sin-1
ANSWER ALL QUESTIONS
√𝑥−√𝑦
√𝑥+√𝑦
.Find x
𝜕𝑢
𝜕𝑥
𝜕𝑢
𝜕𝑦
𝜕𝑟
𝜕𝑥
2
+y
14. If x = r cosѲ , y = rsinѲ. Find
15. Find the minimum value of F = 𝑥 + 𝑦 2 subject to the
constant x = 1
16. Expand 𝑒 𝑥+𝑦 in power of x-1 and y+1 up to first degree terms
17. Transform into polar co-ordinates the integral
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
4. Find the eigenvector corresponding to the eigenvalue 1 of
2 2 1
the matrix A = [1 3 1]
1 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 𝑥 2 +𝑦 2 +𝑧 2
=9
7. Give the general equation of the cone passes through the
origin
𝑎
𝑎
∫0 ∫𝑦 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
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
𝑎
√𝑎 2 −𝑥 2
double integral ∫0 ∫0
𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
20. Express the volume bounded by x≥0, y≥0,z≥0 and 𝑥 2 +𝑦 2 +
𝑧 2 ≤ 1 in triple integration
PART -B(5 X 1 2 = 60 Marks)
ANSWER ANY FIVE QUESTIONS
26
𝜕
21. a) Using Cayley Hamilton’s theorem find A4 for the matrix A =
2 −1 2
[−1 2 −1] (6)
1 −1 2
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
𝑥 2 +𝑦 2 +𝑧 2 − 4𝑥 + 2𝑦 − 6𝑧 − 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
=
9
(𝑥+𝑦+𝑧)2
3𝑎
(6)
27. a)Find the volume of the ellipsoid
𝑥2
𝑎2
𝑦2 𝑧2
𝑏 𝑐2
+ 2+
= 1 by triple
integration
(6)
b)Change the order of integration and then evaluate
𝑎+√𝑎 2 −𝑦 2
𝑎
∫0 ∫𝑎−√𝑎2 −𝑦2 𝑥𝑦 𝑑𝑥 𝑑𝑦 (6)
28. a)Transform into polar co-ordinates and evaluate
𝑧−3
2
3𝑎
)
2
𝜕
b)Find the minimum value of 𝑥 2 +𝑦 2 +𝑧 2 with the
constraint xy+yz+zx = 3𝑎2
(6)
2
√2𝑥−𝑥 2 𝑥 𝑑𝑦 𝑑𝑥
∫0 ∫0
(6)
23. a)Find the equation of the sphere passing through the circle
𝑥 2 +𝑦 2 +𝑧 2 + 2𝑥 + 3𝑦 + 6 = 0, 𝑥 − 2𝑦 + 4𝑧 =
9 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝ℎ𝑒𝑟𝑒 𝑥 2 +𝑦 2 +𝑧 2 − 2𝑥 + 4𝑦 −
6𝑧 + 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 𝑦 2 +𝑧 2 =
25,
x=4
(6)
24. a)Find the radius of curvature at ( 2 ,
𝜕
26. a) If u = log(𝑥 3 +𝑦 3 +𝑧 3-3xyz) Prove that (𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧)2 u = -
𝑥 2 +𝑦 2
(6)
b)Find the area enclosed by the curves y = 𝑥 2 and x+y-2 = 0
(6)
on 𝑥 3 +𝑦 3 = 3axy
(6)
b)Find the evolute of the parabola 𝑥 2 = 4by
(6)
𝑎 ,𝑎
25. a)Find the circle of curvature at ( 4 4 ) 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
𝑦 2 = 4ax is the parabola 𝑦 2 = 4a(x+a)
(6)
27
28