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VIVEKANANDHA COLLEGE OF TECHNOLOGY FOR WOMEN

ELAYAMPALAYAM, TIRUCHENGODE – 637205.
MATRICES AND CALCULUS
UNIT-1 (MATRICES)
(2021-2022)

I. Answer all the questions 5  2  10

1 1 5
1. Find the Sum and the product of the matrix A = 1 5 1
3 1 1
2. State Cayley Hamilton theorem.
4 1
3. Find the eigenvalues of the matrix  
 3 2
2 1 5
 
4. Find the eigenvalues of the inverse of the matrix A =  0 4 4 
0 0 5
 
5. If 1 and 2 are the eigenvalues of a 2  2 matrix A, what are the eigenvalues of A2 and A-1.

II. Answer all the questions 1  8  8 & 1  16  32

 4 20 10 
 
6. Find the eigenvalues and eigenvectors of  2 10 4  (8)
 6 30 13 
 
 8 8 2 
 
7. Verify Cayley-Hamilton Theorem for the matrix  4 3 2  . Hence find its inverse
 3 4 1 
 
4
and A (16)
2 2 2
8. Reduce the Q.F. 6x +3y +3z -4xy-2yz+4zx into a canonical form and find the nature of
the Q.F. (16)

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VIVEKANANDHA COLLEGE OF TECHNOLOGY FOR WOMEN

ELAYAMPALAYAM, TIRUCHENGODE – 637205.

MATRICES AND CALCULUS

UNIT-2 (DIFFERENTIAL CALCULUS)

(2021-2022)

I. Answer all the questions 5  2  10

x 2 +x-6
1. Evaluate lim+
x 2 x-2
d 
 sinx  
cosx
2. Find
dx 
x+4
3. Find the domain of the function f(x) =
x 2 -9
1
4. Explain why the function is discontinuous at the given number a f(x) = ,a=-2
x+2
5. Find the critical values of the function f(x) = 2x3-3x2-36x

II. Answer all the questions 5  8  40

6. Find an equation of the tangent and normal lines to the given curve at specified point
x 2 -1
f(x) = 2 , (1,0)
x +x+1

..
xx dy
7. (i) Find y’ if y = x (ii) If sin(x+y) = y2cosx, then find
.
dx
8. (i) Prove that equation x3-15x+c = 0 has atmost one real root in the interval [-2, 2].
(ii)If f(1) = 10 and f'(x)  2 for 1  x  4 how small can f(4) possibly be?
9. Find the local maximum and minimum values of function f(x) = x5-5x+3 using both
the first and second derivatives tests.
10. (i) Suppose f and g are continuous functions such that g(2) = 6 and
lim 3f(x)+f(x)g(x)  36. Find f(2)
x 2

(ii) Show that f(x) = 3x2+2x-1 is continuous at x=2.

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VIVEKANANDHA COLLEGE OF TECHNOLOGY FOR WOMEN

ELAYAMPALAYAM, TIRUCHENGODE – 637205.

MATRICES AND CALCULUS

UNIT-3 (FUNCTIOS OF SEVERAL VARIABLES)

(2021-2022)

I. Answer all the questions 5  2  10

xy+5
1. Evaluate lim
x  x 2 +2y 2
y2

dy
2. Find when x 3 +y3 = 3axy
dx
y2 x2  (u, v)
3. Find the domain of the function u = ,v= find
x y  (x,y)
x y z u u u
4. If u = f  , ,  , then prove that x y z 0
y x x x y z
5. Write two properties of jacobians.

II. Answer all the questions 5  8  40

2
     -9
6. If u=log(x +y +z -3xyz), Show that  
3 3 3
  u=
 x y z   x+y+z 
2

7. Show that the functions u = x+y-z, v = x-y+z, w = x2+y2+z2-2yz are dependent. Find
the relation between them.
π
8. Expand the function sin (xy) in powers of x-1 and y- upto second degree terms.
2
4 4 2 2
9. Find the maxima and minima of x +y -2x +4xy-2y
10. Find the maximum volume of the largest rectangular parallelepiped that can be
x 2 y2 z2
inscribed in an ellipsoid + + =1
a 2 b2 c2

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VIVEKANANDHA COLLEGE OF TECHNOLOGY FOR WOMEN

ELAYAMPALAYAM, TIRUCHENGODE – 637205.

MATRICES AND CALCULUS

UNIT-4 (INTEGRAL CALCULUS)

(2021-2022)

I. Answer all the questions 5  2  10

logx
1. Evaluate
x
dx
sin 2 x
2. Evaluate  dx
1  cos 2 x
1

 tan
-1
3. x dx
0

l
4. Evaluate  a 2 -x 2
dx by using trigonometric substitution.


1
5. For what values of p in the integral x
1
p
dx convergent?

II. Answer all the questions 5  8  40

x -2x  dx by using Riemann sum by taking right end points as the sample
2
6. Evaluate
0

points.
π
2
7. Find the reduction formula for  sin n x dx, n  2is an integer and  sin x dx
n

x  2x  4x  1
4 2
8. Evaluate 
x3  x 2  x  1
dx
1
log 1  x 
9. Evaluate  dx
0 1  x 2

  3x  2   x  x  1 dx
2
10. Evaluate

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VIVEKANANDHA COLLEGE OF TECHNOLOGY FOR WOMEN

ELAYAMPALAYAM, TIRUCHENGODE – 637205.

MATRICES AND CALCULUS

UNIT-5 (MULTIPLE INTEGRALS)

(2021-2022)

I. Answer all the questions 5  2  10

3 2

 e
x+y
1. Evaluate dydx
0 0
1 x
2. Sketch roughly the region of integration for   f(x,y)dydx
0 0
π sinθ
3. Find the domain of the function  
0 0
rdrdθ

a a
4. Change the order of integration of   f(x,y)dxdy
0 y

5. Express the region x  0, y  0, z  0, x 2 +y 2 +z 2  1 by triple integration.

II. Answer all the questions 5  8  40

a 2a-x
6. Change the order of integration in  
0 2
xydxdy and hence evaluate the same.
x
a

7. (i) Find the area bounded by the parabolas y2 = 4 - x and y2 = 4 - 4x as a double

integral and evaluate it.
(ii) Find the area of the cardioid r = a 1  cos θ  , using a double integral.
 x  x y
b 1-  c  1- - 
1  a   a b
8. (i) Evaluate    x 2 z dz dy dz
0 0 0
a b c
(ii) Evaluate   x  y 2  z 2  dx dy dz
2

0 0 0

x 2 y2 z2
9. Find the volume of that portion of    1 which lies in the first octant
a 2 b2 c2
using triple integration.
a a
x2
10. Evaluate by changing to polar co-ordinates, the integral 
0 y x 2 +y 2
dxdy

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