MATHEMATICS I question paper

DEPARTMENT OF MATHEMATICS
MA 2111 ENGINEERING MATHEMATICS -1
MATRICES
Part –A
1. Find the eigen values of the matrix A = ⎥⎦
⎤
⎢⎣
⎡
1 2
5 4
2. Find the constant a and b such that the matrix ⎥⎦
⎤
⎢⎣
⎡
b
a
1
4
has 3 and -2 as its
eigen values.
3. Show that if λ is a characteristic root of the matrix A, then λ + k is a
characteristic root of the matrix A + k I.
4. If λ be an eigen value of a non-singular matrix A, show that
λ
| A | is an
eigen value of adjA.
5. If the characteristic equation of a matrix A is λ2 - 4 λ+3 = 0, find the
characteristic equation of 3A−1 − 4 I
6. Are ⎥⎦
⎤
⎢⎣
⎡−
2
6
and ⎥⎦
⎤
⎢⎣
⎡−
1
6
eigen vectors of ⎥⎦
⎤
⎢⎣
⎡
1 4
2 3
.
7. Let A =
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
3 1 1
1 5 1
1 1 3
If ( 1, 0, -1)T is an eigen vector corresponding to some
eigen value λ of the matrix A. find λ.
8. Discuss the nature of QF 2xy + 2yz + 2zx.
9. Write down the matrices of the following QF 2x2 +3y2+6xy.
10. Write down the QF corresponding to the following matrices
⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢
⎣
⎡
−
− −
−
−
0 0 3 2
2 0 6 3
1 4 0 0
1 1 2 0
DEPARTMENT OF MATHEMATICS
MA 2111 ENGINEERING MATHEMATICS -1
ANALYTICAL GEOMETRY
PART – A
1. Find the equation of the sphere whose centre is ( 2, -3, 4) and radius 5?
2. Show that the spheres x2+y2+z2=25, x2+y2+ z2-18x-24y-40z + 225=0 touch
externally.
3. Find the equation of the sphere on this join ( 2,-3,1) and ( 1,-2,-1) as
diameter.
4. Find the equation of sphere having its centre on the plane 4x-5y-z=3 and
passing through the circle x2+y2+z2 -2x-3y+4z + 8=0,
x2+y2+z2+4x+5y-6z+2 = 0.
5. Prove that the plane x + 2y – z = 4 cuts the sphere x2+y2+z2 –x + z -2 =0 in
a circle of radius unity.
6. Find the equation of the tangent plane to the sphere 3(x2+y2+z2) -2x -3y -
4z -22 = 0 at the point ( 1,2,3).
7. Find the equation to the cone whose vertex is the origin and base the
circle x =a; y2 + z2 = b2.
8. Find the equation of the right circular cone whose vertex is at the origin,
whose axis is the line x = y / 2 =z / 3 and which has semi vertical angle of
30o
9. Find the equation of the quadric cylinder with generators parallel to
x – axis and passing through the curve ax2 + by2 + cz2 = 1, lx+ my+ nz = p.
10. Find the equation of the quadric cylinder whose generators intersect the
curve ax2 + by2 = 2z, lx+ my+ nz = p and are parallel to Z – axis.
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DEPARTMENT OF MATHEMATICS
MA 2111 ENGINEERING MATHEMATICS -1
FUNCTIONS OF SEVERAL VARIABLES
PART – A
1. Find the first order partial derivatives of the function u = yx
2. If Z = log( x2 + xy + y2), prove that = 2.
∂
∂
+
∂
∂
y
y z
x
x z
3. If U = f( y-z, z-x, x-y), prove that = 0
∂
∂
+
∂
∂
+
∂
∂
z
u
y
u
x
u
4. Given U = sin(x/y), x = et and y = t2 find
dt
du
5. If U = x2 + y2 + z2 and x = e2t, y = e2t cos3t, z = e2tsin3t find
dt
du
6. If
x y
w z
z x
V y
y z
U x
−
=
−
=
−
= , , show that 0.
( , , )
( , , ) =
∂
∂
x y z
u v w
7. If U = x (1-y); V = xy prove that JJ’ = 1.
8. Show that
2
log(1 )
x2 ey + x = x + xy − ( approximately)
9. Find the stationary points of xy ( a –x-y).
10. State necessary conditions for a maximum or a minimum.
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DEPARTMENT OF MATHEMATICS
MA 2111 ENGINEERING MATHEMATICS -1
DIFFERENTIAL CALCULUS
PART-A
1. Find the radius of curvature for a circle and straight line.
2. Find the radius of curvature at any point of the catenary y = c cosh (x / c).
3. Find the radius of curvature at any point of y = logsinx.
4. Find ρ at Ө on x = 3a cos Ө - a cos3 Ө, y = 3a sin Ө - a sin3 Ө.
5. Find ρ at ( -2,0) on y2 = x3 + 8.
6. Find the circle of curvature at (0,0) on x + y=x2+y2+x3.
7. Write the method to find the evolute of a given curve y = f(x).
8. State two important properties of the evolute.
9. Find the envelope of the family of straight lines y = mx +a/m.
10. Find the envelope of + = 1
b
y
a
x subject to a + b = c, where c is known
constant
DEPARTMENT OF MATHEMATICS
MA 2111 ENGINEERING MATHEMATICS -1
MULTIPLE INTEGRALS
PART –A
1. Evaluate ∫ ∫
2
1
5
2
xy dxdy
2. Evaluate ∫ ∫
2
0
2
0
π
rdrdθ
3. Evaluate ∫ ∫ x y dxdy + −
3
4
2
1
( ) 2
4. Evaluate ∫ ∫
π θ
θ
0
cos
0
rdrd
5. Shade the region of integration ∫ ∫
−
−
a a x
ax x
dxdy
0
2 2
2
6. Transform the integral ∫ ∫
∞
0 0
y
dxdy to polar coordinates.
7. Change the order of integration ∫ ∫
1 −
0
2
2
( , )
x
x
f x y dydx
8. Express the region x > 0, y > 0, z > 0, x2 + y2 + z2 < 1 by triple integration.
9. Find the area enclosed by the circle x2 + y2 = a2.
10. Find the area of the region bounded by y2 = 4x and x2 = 4y.
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1. Find the eigen values and eigen vectors of the matrix
2
64
2 2 0
2 5 0
0 0 3
3
75
and
hence diagonalize it through orthogonal reduction.
2. Verify Cayley-Hamilton theorem for A =
2
64
2 0 −1
0 2 0
−1 0 2
3
75
and hence
find A−1 and A4.
3. Find the eigen values and eigen vectors of the matrix A =
2
64
2 −2 2
1 1 1
1 3 −1
3
75
.
4. Find the inverse of the matrix A =
2
64
7 2 −2
−6 −1 2
6 2 −1
3
75
by using Cayley-
Hamilton theorem.
5. show that
2
64
0 1 0
1 0 0
0 0 1
3
75
is orthogonal. Find its inverse. Verify that its
eigen values are of unit modulus.
6. Find the characteristic equation of A =
2
64
2 1 1
0 1 0
1 1 2
3
75
and hence express
the matrix A5 in terms of A2,A and I.
7. If (0, 1, 1)T , (2,−1, 1)T , (1, 1,−1)T be the eigen vectors of matrix A
corresponding to the eigen values −1, 1, 4 then find the matrix A.
8. Reduce the quadratic form 8x21
+7x22
+3x23
−12x1x2 +8x2x3 +4x1x3
to the canonical form through orthogonal transformation. Hence show
that it is positive semi-definite.
9. Reduce the quadratic form 6x21
+ 3x22
+ 3x23
− 4x1x2 − 2x2x3 + 4x1x3
into sum of squares by orthogonal transformation. Write also rank,
index and signature.
10. Reduce the quadratic form 17x2 − 30xy + 17y2 to a canonical form
and find the nature of conic 17x2 − 30xy + 17y2 = 128. Find also the
lengths and directions of the principal axes.
QUESTION BANK 1 MATHEMATICS I
11. Find the equation of the sphere which passes through the points (2, 0, 0),
(0, 2, 0) and (0, 0, 2) and which has its radius as small as possible.
12. Find the equations of the tangent planes to the sphere x2+y2+z2+2x−
4y+6z−7 = 0 which passes through the line 6x−3y−23 = 0 = 3z+2.
13. Find the equations of sphere which pass through the circle x+2y+3z =
8, x2 + y2 + z2 − 2x − 4y = 0 and touch the plane 4x + 3y = 25.
14. Find the center and radius of the circle x2+y2+z2−8x+4y+8z−45 = 0
and x − 2y + z − 3 = 0.
15. Show that the spheres x2 + y2 + z2 + 6y + 2z + 8 = 0 and x2 + y2 +
z2 + 6x + 8y + 4z + 20 = 0 cut orthogonally. Find their plane of
intersection. Also prove that this plane is perpendicular to the line
joining the center.
16. Find the equation of the sphere passing through the points (0, 3, 0),
(−2,−1,−4) and cutting orthogonally the two spheres x2 + y2 + z2 +
x − 3z − 2 = 0 and 2x2 + 2y2 + 2z2 + x + 3y + 4 = 0.
17. Find the equation of the cone whose vertex is (1, 2, 3) and guiding
curve is the circle x2 + y2 + z2 = 4, x + y + z = 1.
18. Prove that 9x2 + 9y2 − 4z2 + 12yz − 6zx + 54z − 81 = 0 represents a
cone. Find also its vertex.
19. Find the equation of the cylinder whose axis is x
1 = y
−2 = z
3 and whose
guiding curve is the ellipse x2 + 2y2 = 1, z = 3.
20. Find the equation of the right circular cylinder passing through A(3, 0, 0)
and having the axis x − 2 = z, y = 0.
21. Find , at (a, 0) on y2 = a3−x3
x .
22. Show that the curves y = a
2 (e
x
a + e
−x
a ) and y = a
2 (2 + x2
a2 ) have the
same curvature at their crossing with the Y-axis.
23. Show that at on x = 3a cos − a cos 3 , x = 3a sin − a sin 3 is
3a sin .
24. Show that the line joining any point on x = a( + sin ), y = a(1 −
cos ) to its center of the curvature is bisected by the line y = 2a.
QUESTION BANK 2 MATHEMATICS I
25. For the curve rn = an cos n , prove
= anr1−n
n + 1 .
26. Find the circle of curvature at (0, 0) on x + y = x2 + y2 + x3.
27. Obtain the equation of the evolute of the ellipse x2
a2 + y2
b2 = 1.
28. Show that the evolute of the tractrix x = a(cos t+log tan t
2 ), y = a sin t
is the catenary y = a cosh xa.
29. Given that x
2
3 + y
2
3 = c
2
3 is the envelope of x
a + y
b = 1. Find the
necessary relation between a and b.
30. Find the evolute of x2
a2 + y2
b2 = 1 considering it as the envelope of
normals.
31. Prove that if f(x, y) = p1
y e−(x−a)2
4y , then fxy = fyx.
32. If u = (1 − 2xy + y2)−1
2 prove that
@
@x

(1 − x2)@u
@x

+ @
@y

y2 @u
@y

= 0.
33. Verify Euler’s Theorem for the function
u = sin−1 x
y
+ tan−1 x
y
.
34. If u = log x4+y4
x+y , show that x@u
@x + y @u
@y = 3.
35. By changing the independent variables u and v to x and y by means
of the relations x = u cos − v sin , y = u sin + v cos , show that
@2z
@u2 + @2z
@v2 transforms into @2z
@x2 + @2z
@y2 .
36. If u = xyz, v = xy + yz + zx,w = x + y + z, show that @(u,v,w)
@(x,y,z) =
(x − y)(y − z)(z − x).
37. Expand tan−1( y
x ) in the neighborhood of (1, 1), using Taylor’s series.
38. Discuss the maxima and minima of x3y2(1 − x − y).
QUESTION BANK 3 MATHEMATICS I
39. Show that the rectangular solid of maximum volume that can be inscribed
in a given sphere is a cube.
40. Find the maximum and minimum distances of the point (3, 4, 12) from
the sphere x2 + y2 + z2 = 1.
41. Evaluate
R R
(x+y)2dxdy over the area bounded by the ellipse x2
a2+y2
b2 =
1.
42. Change the order of integration in
I =
Z 1
0
Z 2−x
x2
xydydx
and hence evaluate the same.
43. Evaluate
R 2
1
R 4−x2
0 (x + y)dydx by changing the order of integration.
44. Evaluate
R 1
0
R p
2−x2
x
p x
x2+y2
dydx by changing the order of integration.
45. Evaluate Z 1
0
Z 2−x
x
x
y
dxdy
by changing the order of integration.
46. Evaluate Z 1
0
Z 1−x
0
Z 1−x−y
0
xyzdzdydx.
47. Find the volume of the solid in the positive octant bounded by the
parabolic z = 36 − 4x2 − 9y2.
48. Change
Z pa
2
0
Z p
a2−y2
y
log(x2 − y2)dydx,
a > 0 into polar co-ordinates and hence evaluate it.
49. Evaluate Z a
0
Z p
a2−x2
p
ax−x2
dxdy p
a2 − x2 − y2
using polar co-ordinates.
50. Evaluate
R R
R xydxdy where R is the region in the first quadrant that
lies between the circles x2 + y2 = 4 and x2 + y2 = 25.
QUESTION BANK 4 MATHEMATICS I