Choose your answer for the following :-

1 (a) (i)
If y=3

(A) (3 log 5)

(B) (5 log 3)

(C) (5 log 3)

(D) (5 log 3)

^{5x}then y_{n}is(A) (3 log 5)

^{n}e^{5x}(B) (5 log 3)

^{n}e^{5x}(C) (5 log 3)

^{-n}e^{5x}(D) (5 log 3)

^{n}e-5x
1 M

1 (a) (ii)
if y=cos

(A) 2

(B) 2

(C) 2

(D) 2

^{2}x then y_{n}is(A) 2

^{n+1}cos(n?/2+2x)(B) 2

^{n-1}cos(n?/2+2x)(C) 2

^{n-1}cos(n?/2-2x)(D) 2

^{2+1}cos(n?/2-2x)
1 M

1 (a) (iii)
The Largrange's mean value theorem for the function f(x)=e

(A) C=0.5413

(B) C=2.3

(C) 0.3

(D) none of these

^{x}in the interval [0, 1] is(A) C=0.5413

(B) C=2.3

(C) 0.3

(D) none of these

1 M

1 (a) (iv)
Expression of log (1+e

\[ (A) \ \log 2-\dfrac {x}{2}+\dfrac {x^2}{8}+\dfrac {x^4}{192}+.... \\(B) \ \log 2+\dfrac {x}{2}+\dfrac {x^2}{8}-\dfrac {x^4}{192}+....\\(C) \ \log 2+\dfrac {x}{2}+\dfrac {x^2}{8}+\dfrac {x^4}{192}+....\\(D) \ \log 2-\dfrac {x}{2}-\dfrac {x^2}{8}-\dfrac {x^4}{192}+....\]

^{x}) in powers of x is _____\[ (A) \ \log 2-\dfrac {x}{2}+\dfrac {x^2}{8}+\dfrac {x^4}{192}+.... \\(B) \ \log 2+\dfrac {x}{2}+\dfrac {x^2}{8}-\dfrac {x^4}{192}+....\\(C) \ \log 2+\dfrac {x}{2}+\dfrac {x^2}{8}+\dfrac {x^4}{192}+....\\(D) \ \log 2-\dfrac {x}{2}-\dfrac {x^2}{8}-\dfrac {x^4}{192}+....\]

1 M

1 (b)
if y

^{1/m}+y^{-1/m}=2x prove that (x^{2}-1) y_{n+2}+ (2n+1) xy_{n+1}+(n^{2}-m^{2}) y_{n}=0
6 M

1 (c)
Verify the Rolle's theorem for the functions: f(x)=e

^{x}(sin x - cos x) in (?/4.5?/4).
6 M

1 (d)
By using Maclaarin's theorem expand log sec x up to the term containing x

^{6}
4 M

Choose your answer for the following :-

2 (a) (i)
The indeterminate form of \[ \lim_{x\rightarrow 0}\dfrac {a^x-b^x}{x} \ is \\(A) \ \log\left ( \dfrac {b}{a} \right ) \\(B) \ \log \left (\dfrac {a}{b} \right ) \\(C) \ 1\\(D) \ -1 \]

1 M

2 (a) (ii)
The angle between radius vector and the tangent for the curves r=a(1-cos ?) is

(A) ?/2

(B) -?/2

(C) ?/2+?

(D) ?/2-?

(A) ?/2

(B) -?/2

(C) ?/2+?

(D) ?/2-?

1 M

2 (a) (iii)
The polar form of a curve is _____

(A) r=f(?)

(B) ?=f(y)

(C) r=f(x)

(D) none of these

(A) r=f(?)

(B) ?=f(y)

(C) r=f(x)

(D) none of these

1 M

2 (a) (iv)
The rate at which the curve is bending called _____

(A) Radius of curvature

(B) Curvature

(C) Circle of curvature

(D) Evaluate

(A) Radius of curvature

(B) Curvature

(C) Circle of curvature

(D) Evaluate

1 M

2 (b)
\[ Evaluate \ \lim_{x \rightarrow 0} \left ( \dfrac {\sin x}{x} \right )^{1/x^2} \]

6 M

2 (c)
Find the angles of intersection of the following pairs of curves, r=a?/(1+?); r=a/(1+?

^{2})
6 M

2 (d)
Find the radius curvature at (3a/2, 3a/2) on x

^{3}+y^{3}=3axy
4 M

Choose your answer for the following :-

3 (a) (i)
If u=x

(A) 2

(B) 0

(C) 2x

(D) 2y

^{2}+y^{2}then \[ \dfrac {(\partial^2u)}{(\partial x \partial y)}\] is equal to(A) 2

(B) 0

(C) 2x

(D) 2y

1 M

3 (a) (ii)
If z=f(x,y) where x=u-v and y=uv then (u+v) \[ \left (\dfrac {\partial z}{\partial x} \right ) \ is \]

\[ (A) \ u\left (\dfrac {\partial z}{\partial u} \right )-v\left ( \dfrac {\partial z}{\partial v} \right )\\(B) \ u\left (\dfrac {\partial z}{\partial u} \right )+v \left (\ \dfrac {\partial z}{\partial v} \right )\\(C) \ \dfrac {\partial z}{\partial u}+\dfrac {\partial z}{\partial v} \\(D) \ \dfrac {\partial z}{\partial u}-\dfrac {\partial z}{\partial v} \]

\[ (A) \ u\left (\dfrac {\partial z}{\partial u} \right )-v\left ( \dfrac {\partial z}{\partial v} \right )\\(B) \ u\left (\dfrac {\partial z}{\partial u} \right )+v \left (\ \dfrac {\partial z}{\partial v} \right )\\(C) \ \dfrac {\partial z}{\partial u}+\dfrac {\partial z}{\partial v} \\(D) \ \dfrac {\partial z}{\partial u}-\dfrac {\partial z}{\partial v} \]

1 M

3 (a) (iii)
If x=r cos ?, y=r sin ? then \[ \dfrac {[\partial (r,\theta)]}{[\partial (x,y)]} \ is \]

(A) r

(B) 1/r

(C) 1

(D) -1

(A) r

(B) 1/r

(C) 1

(D) -1

1 M

3 (a) (iv)
In error and approximations \[ \dfrac {\partial x}{x}, \dfrac {\partial y}{y}, \dfrac {\partial f}{f} \] are called

(A) relative error

(B) percentage error

(C) error in x,y and f

(D) none of these

(A) relative error

(B) percentage error

(C) error in x,y and f

(D) none of these

1 M

3 (b)
If x

^{x}y^{y}z^{z}=c, show that \dfrac {\partial^2z}{\partial x \partial y}=-[x \log ex]^{-1}, when x=y=z
6 M

3 (c)
Obtain the Jacobian of \[ \dfrac {\partial (x.y.z)}{\partial (r.\theta . \phi)} \] for change of coordinate from three dimensional Cartesian coordinates to spherical polar coordinates.

6 M

3 (d)
In estimating the cost of a pile of bricks measured as 2m × 15m × 1.2m, the tape is stretched +1% beyond the standard length. If the count is 450 bricks to I cu.cm and bricks cost of 530 per 1000, find the approximate error in the cost

4 M

Choose your answer for the following :-

4 (a) (i)
If R=xi+yj+zk then div R

(A) 0

(B) 3

(C) -3

(D) 2

(A) 0

(B) 3

(C) -3

(D) 2

1 M

4 (a) (ii)
If F=3x

(A) 0

(B) -2

(C) 2

(D) 3

^{2}i-xyj+(a-3)xzk is solenoidal, then a is equal to(A) 0

(B) -2

(C) 2

(D) 3

1 M

4 (a) (iii)
If F=(x+y+1)i+j-(x+y)k then F. Curl F is _____

(A) 0

(B) x+y

(C) x+y+z

(D) x-y

(A) 0

(B) x+y

(C) x+y+z

(D) x-y

1 M

4 (a) (iv)
The scale factors for cylindrical coordinates system (? ? z) are given by

(A) (?, 1, 1)

(B) (1, ?, 1)

(C) (1, 1, ?)

(D) None of these

(A) (?, 1, 1)

(B) (1, ?, 1)

(C) (1, 1, ?)

(D) None of these

1 M

4 (b)
Prove that curl A=g rad(div A)- ?

^{2}A.
6 M

4 (c)
Find the constant a, b, c such that the vector F=(x+y+az)i+(bx+2y-z)j+(x+cy+2z)k is irrotational

6 M

4 (d)
Derive an expression for ? ? A in orthogonal curvilinear coordinates. Deduce ? ? A is rectangular coordinates.

4 M

Choose your answer for the following :-

5 (a) (i)
The value of \[ \int^{\infty}_0 e^{\alpha x}dx \] is _____

(A) 1/e

(B) -1/e

(C) 1/?

(D) -1/?

(A) 1/e

(B) -1/e

(C) 1/?

(D) -1/?

1 M

5 (a) (ii)
The value of the integral \[ \int^{\pi/2}_{0}\sin^{7} xdx \ is \]

(A) 35/16

(B) 16/35

(C) -16/35

(D) 18/35

(A) 35/16

(B) 16/35

(C) -16/35

(D) 18/35

1 M

5 (a) (iii)
The volume generated by revolving the cardioid r=a(1+ cos ?) about the intial line is

\[ (A) \ \dfrac {(3\pi a^2)}{8} \\(B) \ \dfrac {(3\pi a^3)}{8} \\(C) \ \dfrac {(2\pi a^2)}{9} \\(D) \ None \]

\[ (A) \ \dfrac {(3\pi a^2)}{8} \\(B) \ \dfrac {(3\pi a^3)}{8} \\(C) \ \dfrac {(2\pi a^2)}{9} \\(D) \ None \]

1 M

5 (a) (iv)
The area of the loop of the curve r=a sin 3? is _____

\[ (A) \ \dfrac {a^2}{12} \\(B) \ \dfrac {\pi}{12} \\(C) \ \dfrac {\pi a^2}{12} \\(D) \ none \]

\[ (A) \ \dfrac {a^2}{12} \\(B) \ \dfrac {\pi}{12} \\(C) \ \dfrac {\pi a^2}{12} \\(D) \ none \]

1 M

5 (b)
By applying differential under the integral sign evaluate \[ \int^{\pi/2}_0 \dfrac {\log (1+y \sin^2 x)}{\sin^2 x}dx \]

6 M

5 (c)
Evaluate \[ \int^{\pi/2}_0 \sin^n x \ dx \] where n is any integer.

6 M

5 (d)
Find the length of arch of the cycloid x=a (? - sin?); y=a (1- cos?); 0

4 M

Choose your answer for the following :-

6 (a) (i)
The general solution of the differential equation (dy/dx)=(y/x)+tan(y/x) is

(A) sin (y/x)=c

(B) sin (y/x)=cx

(C) cos (y/x)=cx

(D) cos (y/x)=c

(A) sin (y/x)=c

(B) sin (y/x)=cx

(C) cos (y/x)=cx

(D) cos (y/x)=c

1 M

6 (a) (ii)
An integrating factor for ydx-xdy=0 is

(A) x/y

(B) y/x

(C) 1(x

(D) 1/(x

(A) x/y

(B) y/x

(C) 1(x

^{2}y^{2})(D) 1/(x

^{2}+y^{2})
1 M

6 (a) (iii)
The differential equation satisfying the relation x=A cos (mt-?) is

(A) (dx/dt)=1-x

(B) (d

(C) (d

(D) (dx/dt)=-m

(A) (dx/dt)=1-x

^{2}(B) (d

^{2}x/dt^{2})=-?^{2}x(C) (d

^{2}x/dt^{2})=-m^{2}x(D) (dx/dt)=-m

^{2}x
1 M

6 (a) (iv)
The orthogonal trajectories of the system given by r=a? is

(A) r

(B) r=ke

(C) r

(D) r

(A) r

^{2}=ke^{?}(B) r=ke

^{?}(C) r

^{2}e^{-?2}= k(D) r

^{2}= k e^{-?2}
1 M

6 (b)
Solve (x cos (y/x)+y sin (y/x)) y- (y sin (y/x) -x cos (y/x)) x(dy/dx)=0

6 M

6 (c)
Solve (1+y

^{2})+(x-e^{tan-1y})dy/dx=0
6 M

6 (d)
Prove that the system parabola y

^{2}=4a(x+a) is self orthognal.
4 M

Choose your answer for the following :-

7 (a) (i)
Find the rank of \[ \begin{bmatrix}3 &-1 &2 \\ -6&2 &4 \\ -3&1 &2 \end{bmatrix} \]

(A) 3

(B) 2

(C) 4

(D) 1

(A) 3

(B) 2

(C) 4

(D) 1

1 M

7 (a) (ii)
The exact solution of the system of equation 10x+y+z=12, x+10y+z=12, x+y+10z=12 by inspection is equal to

(A) (-1, 1, 1)

(B) (1, 1, 1)

(C) (-1, -1, -1)

(D) None

(A) (-1, 1, 1)

(B) (1, 1, 1)

(C) (-1, -1, -1)

(D) None

1 M

7 (a) (iii)
If the given system of linear equations in 'n' variables is consistent then the number of linearly independent-solution is given by

(A) n

(B) n-1

(C) r-n

(D) n-r

(A) n

(B) n-1

(C) r-n

(D) n-r

1 M

7 (a) (iv)
The trivial solution for the given system of equations

qx-y+4z=0, 4x-2y+3z=0, 5x+y-6z=0 is

(A) (1, 2, 0)

(B) (0 4 1)

(C) (0 0 0)

(D) (1 -5 0)

qx-y+4z=0, 4x-2y+3z=0, 5x+y-6z=0 is

(A) (1, 2, 0)

(B) (0 4 1)

(C) (0 0 0)

(D) (1 -5 0)

1 M

7 (b)
Using elementary transformation reduce each of following matrices to the normal form \[ \begin{bmatrix}1 &1 &1 &6 \\ 1&-1 &2 &5 \\ 3&1 &1 &8 \\ 2&-2 &3 &7 \end{bmatrix} \]

6 M

7 (c)
Test for consistency and solve the system, 2x+y+z=10, 3x+2y+3z=18, x+4y+9z=16

6 M

7 (d)
Apply Gauss-Jordan method to solve the system of equations, 2x+5y+7z=52, 2x+y-z=0, x+y+z=9

4 M

Choose your answer for the following :-

8 (a) (i)
A square matrix A is called orthogonal if,

(A) A=A

(B) A=A

(C) AA

(D) None

(A) A=A

^{2}(B) A=A

^{-1}(C) AA

^{-1}=1(D) None

1 M

8 (a) (ii)
The eigen values of the matrix \[ \begin{bmatrix}6 &-2 &2 \\ -2&3 &-1 \\ 2&-1 &3 \end{bmatrix} \ are \]

(A) 2,3,8

(B) 2,3,9

(C) 2,2,8

(D) None

(A) 2,3,8

(B) 2,3,9

(C) 2,2,8

(D) None

1 M

8 (a) (iii)
The eigen vector X of the matrix A corresponding to eigen value ? and satisfy the equation.

(A) AX=?X

(B) ?(A-X)=0

(C) XA-A? =0

(D) |A-?|X=0

(A) AX=?X

(B) ?(A-X)=0

(C) XA-A? =0

(D) |A-?|X=0

1 M

8 (a) (iv)
Two square matrices A and B are similar if,

(A) A=B

(B) B=P

(C) A'=B'

(D) A

(A) A=B

(B) B=P

^{-1}AP(C) A'=B'

(D) A

^{-1}=B^{-1}
1 M

8 (b)
Show that the transformation, y

_{1}=2x_{1}-2x_{2}-x_{3}, y_{2}= -4x_{1}+5x_{2}+3x_{3}, y_{3}=x_{1}-x_{2}-x_{3}, is regular and find the inverse transformations.
6 M

8 (c)
Diagonalize the matrix, \[ \begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&-4 &3 \end{bmatrix} \]

6 M

8 (d)
Reduce the quadratic form, \[ x^2_1 +2x^2_2 -7x^2_3-4x_1x_2+8x_2x_3 \] into sum of squares.

4 M

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