Choose your answer for the following :-

1 (a) (i)
Then n

\[ (A) \ 2^{n-1}\cos \left(2x+\dfrac {n\pi}{2} \right )\\(B) \ 2^{n}\cos \left(2x+\dfrac {n\pi}{2} \right )\\(C) \ 2^{n-1}\cos \left(2x+n\pi \right )\\(D) \ 2^{n-1}\cos \left(\dfrac {n\pi}{2} \right )\\\]

^{th}derivation of cos^{2}x is\[ (A) \ 2^{n-1}\cos \left(2x+\dfrac {n\pi}{2} \right )\\(B) \ 2^{n}\cos \left(2x+\dfrac {n\pi}{2} \right )\\(C) \ 2^{n-1}\cos \left(2x+n\pi \right )\\(D) \ 2^{n-1}\cos \left(\dfrac {n\pi}{2} \right )\\\]

1 M

1 (a) (ii)
The value of C of the Cauchy mean value theorem for f(x) = e

\[ (A) \ \dfrac {5}{2}\\(B) \ \dfrac {3}{2}\\(C) \ \dfrac {9}{2}\\(D) \ \dfrac {1}{2}\\\]

^{x}and g(x) = e^{-x}in [4, 5] is\[ (A) \ \dfrac {5}{2}\\(B) \ \dfrac {3}{2}\\(C) \ \dfrac {9}{2}\\(D) \ \dfrac {1}{2}\\\]

1 M

1 (a) (iii)
Find the n

\[ (A) \ y_n=\dfrac {(n+1)!}{x}\\(B) \ y_n=\dfrac {n!}{x}\\(C) \ y_n=\dfrac {(n-1)!}{x}\\(D) \ y_n=\dfrac {n!}{x^2}\\\]

^{th}derivative of y=x^{n-1}log x is\[ (A) \ y_n=\dfrac {(n+1)!}{x}\\(B) \ y_n=\dfrac {n!}{x}\\(C) \ y_n=\dfrac {(n-1)!}{x}\\(D) \ y_n=\dfrac {n!}{x^2}\\\]

1 M

1 (a) (iv)
Maclaurin's series expansion of log (1+x) is

\[ (A) \ x+\dfrac {x^2}{2}+\dfrac {x^3}{5}+...\\(B) \ x-\dfrac {x^2}{3}+\dfrac {x^4}{5}-...\\(C) \ x-\dfrac {x^2}{2}+\dfrac {x^3}{3}-\dfrac {x^4}{5}+...\\(D) \ x+\dfrac {x^2}{3}+\dfrac {x^3}{16}+...\\\]

\[ (A) \ x+\dfrac {x^2}{2}+\dfrac {x^3}{5}+...\\(B) \ x-\dfrac {x^2}{3}+\dfrac {x^4}{5}-...\\(C) \ x-\dfrac {x^2}{2}+\dfrac {x^3}{3}-\dfrac {x^4}{5}+...\\(D) \ x+\dfrac {x^2}{3}+\dfrac {x^3}{16}+...\\\]

1 M

1 (b)
By information in two different ways the n

\[ 1+\dfrac {n^2}{1^2}+\dfrac {n^2(n-1)^2}{1^2 \times 2^2}+\dfrac {n^2 (n-1)^2(n-2)^2}{1^2\times 2^2 \times 3^2}+..... =\dfrac {(2n)!}{(n!)^2} \]

^{th}derivation of x^{2n}. Prove that\[ 1+\dfrac {n^2}{1^2}+\dfrac {n^2(n-1)^2}{1^2 \times 2^2}+\dfrac {n^2 (n-1)^2(n-2)^2}{1^2\times 2^2 \times 3^2}+..... =\dfrac {(2n)!}{(n!)^2} \]

6 M

1 (c)
Verify Rolle's theorem for the function \[ f(x)=\dfrac {\sin 2x}{e^{2x}}\ in \ \left [ 0, \dfrac {\pi}{2} \right ] \]

4 M

1 (d)
Using Maclaurin's series expand log sec x upto the term containing x

^{6}.
6 M

Choose your answer for the following :-

2 (a) (i)
\[ The \ value \ of \ \lim_{x\rightarrow \infty} \dfrac {\log x}{x} \ is \]

(A) 0

(B) 1

(C) 2

(D) -2

(A) 0

(B) 1

(C) 2

(D) -2

1 M

2 (a) (ii)
If s is the arc length of the curve x=f(y) then \[ \dfrac {ds}{dy} \ is \ \]

\[ (A) \ \sqrt{1+y^2_1}\\(B) \ \sqrt{1+y_1} \\(C) \ \sqrt{1+ \left ( \dfrac {dx}{dy} \right )^2}\\(D) \ \sqrt{\left ( \dfrac {dy}{dx} \right )^2 + \left (\dfrac {dx}{dy} \right)^2}\\\]

\[ (A) \ \sqrt{1+y^2_1}\\(B) \ \sqrt{1+y_1} \\(C) \ \sqrt{1+ \left ( \dfrac {dx}{dy} \right )^2}\\(D) \ \sqrt{\left ( \dfrac {dy}{dx} \right )^2 + \left (\dfrac {dx}{dy} \right)^2}\\\]

1 M

2 (a) (iii)
Pedal equation of the curve \[ \dfrac {2a}{r}=1 - \cos \theta \ is \ \]

(A) P=ar

(B) P

(C) P

(D) P

(A) P=ar

^{2}(B) P

^{2}=a^{2}r(C) P

^{2}=a^{2}r^{2}(D) P

^{2}=ar
1 M

2 (a) (iv)
The angle between the two curves r=ae

\[ (A)\ \dfrac {\pi}{2}\\(B)\ \dfrac {\pi}{4}\\(C)\ 0\\(D)\ \pi \]

^{?}and re^{?}=b is\[ (A)\ \dfrac {\pi}{2}\\(B)\ \dfrac {\pi}{4}\\(C)\ 0\\(D)\ \pi \]

1 M

2 (b)
For the curve \[ y=\dfrac {ax}{a+x}, \] if ? is the radius of curvature at any point (x, y) show that; \[ \left (\dfrac {2\rho}{2} \right )^{2/3}= \left ( \dfrac {y}{x} \right )^2 + \left ( \dfrac {x}{y} \right )^2 \]

6 M

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

4 M

2 (d)
Find the angle between the curves \[ r= \dfrac {a}{1+\cos \theta}; \ r=\dfrac {b}{1-\cos \theta} \]

6 M

Choose your answer for the following :-

3 (a) (i)
\[ When \ u=y^2 \ \log\left ( \dfrac {x}{y} \right ) \ then \ x\dfrac {\partial u}{\partial x}+ y\dfrac {\partial u}{\partial y} \ is \]

(A) u

(B) u

(C) 2u

(D) 3u

(A) u

(B) u

^{2}(C) 2u

(D) 3u

1 M

3 (a) (ii)
The Taylor's series of f(x, y)=xy at (1, 1) is

(A) 1+[(x-1)+(y-1)]

(B) 1+[(x-1)+(y-1)]+[(x-1)(y-1)]

(C) [(x-1)(y-1)]

(D) None of these

(A) 1+[(x-1)+(y-1)]

(B) 1+[(x-1)+(y-1)]+[(x-1)(y-1)]

(C) [(x-1)(y-1)]

(D) None of these

1 M

3 (a) (iii)
The Jacobian of transformation from the Cartesian to polar coordinates system is

(A) r

(B) r

(C) r

(D) r sin ?

(A) r

^{3}(B) r

(C) r

^{2}(D) r sin ?

1 M

3 (a) (iv)
The rectangle solid of maximum volume which can be inscribed in a sphere is

(A) parallerlogram

(B) rectangle

(C) cube

(D) triangle

(A) parallerlogram

(B) rectangle

(C) cube

(D) triangle

1 M

3 (b)
Examine the function sin x + sin y + sin (x+y) for extreme values.

6 M

3 (c)
Find the possible error in percent in computing the parallel resistance 'r' of two resistances r

_{1}and r_{2}from the formula \[ \dfrac {1}{r}= \dfrac {1}{r_1} + \dfrac {1}{r_2} \] are both in error by 2%.
4 M

3 (d)
\[ If \ z(x+y)=x^2 + y^2 \ show \ that \ \left [ \dfrac {\partial z}{\partial x}- \dfrac {\partial z}{\partial y} \right ]^2 = 4 \left [ 1- \dfrac {\partial z}{\partial x}- \dfrac {\partial z}{\partial y} \right ] \]

6 M

Choose your answer for the following :-

4 (a) (i)
A gradient of the scalar point function ? that is ? ? is

(A) vector function

(B) scalar function

(C) zero

(D) ?

(A) vector function

(B) scalar function

(C) zero

(D) ?

1 M

4 (a) (ii)
The directional derivative of f(x, y, z)=x

\[ (A) \ \dfrac {28}{\sqrt{5}}\\(B) \ \dfrac {30}{\sqrt{4}}\\(C) \ \dfrac {-28}{\sqrt{5}}\\(D) \ \dfrac {20}{\sqrt{6}} \]

^{2}yz+4xz^{2}at the (1, -2, -1) in the direction PQ where P=(1, 2, -1), Q=(-1, 2, 3) is\[ (A) \ \dfrac {28}{\sqrt{5}}\\(B) \ \dfrac {30}{\sqrt{4}}\\(C) \ \dfrac {-28}{\sqrt{5}}\\(D) \ \dfrac {20}{\sqrt{6}} \]

1 M

4 (a) (iii)
If R is the position vector of any point P(x, y, z) then ? . R is

(A) 3

(B) -3

(C) 2

(D) 0

(A) 3

(B) -3

(C) 2

(D) 0

1 M

4 (a) (iv)
\[ If \ \bar{r}=x\widehat{i}+y\widehat{j}+z\widehat{k} \ then \ Curl \ \bar{r}=..... \]

(A) 0

(B) 1

(C) -1

(D) ?

(A) 0

(B) 1

(C) -1

(D) ?

1 M

4 (b)
Find the constant a and b such that F=(axy+z

^{3})i+(3x^{2}-z)j+(bxz^{2}-y)k is irrotational and find scalar potential function ? such that F=??
6 M

4 (c)
\[ Prove \ that \ abla x \left [ \dfrac {ax\bar{r}}{r^n} \right ]=\dfrac {-\bar{a}}{r^3}+\dfrac {3(a.\bar{r})\bar{r}}{r^5} \]

4 M

4 (d)
Prove that the cylindrical coordinates system is orthogonal.

6 M

Choose your answer for the following :-

5 (a) (i)
\[ The \ value \ of \ \int^1_0 x^2 (1-x^2)^{1/2}dx \ is \\(A) \ \dfrac {\pi}{23}\\(B) \ \dfrac {1}{32}\\(C) \ \dfrac {\pi}{32}\\(D) \ \dfrac {\pi}{16} \]

1 M

5 (a) (ii)
The tangent to the curve y

(A) y-axis

(B) x-axis

(C) both x-axis and y-axis

(D) does not exist

^{2}=4ax at origin is(A) y-axis

(B) x-axis

(C) both x-axis and y-axis

(D) does not exist

1 M

5 (a) (iii)
\[ The \ value \ of \ \int^{\pi}_0 \sin^4 \left ( \dfrac {x}{2} \right )dx \ is \\(A) \ \dfrac {3\pi}{18}\\(B) \ \dfrac {3\pi}{8}\\(C) \ \dfrac {3\pi}{16}\\(D) \ \dfrac {3\pi^2}{8} \]

1 M

5 (a) (iv)
The surface area of the sphere of radius 'a' is

(A) 4?a

(B) 4?

(C) 4?a

(D) 2?a

(A) 4?a

^{2}(B) 4?

^{2}a(C) 4?a

(D) 2?a

^{2}
1 M

5 (b)
Obtain the reduction formula for ? sin

^{n}x cos^{n}x dx.
6 M

5 (c)
Evaluate \[ \int^\pi_0 \dfrac {\tan^{-1}(ax)}{x(1+x^2)}dx \] using the method of differentiation under integral sign.

4 M

5 (d)
Find the area of the loop of the curve ay

^{2}=x^{2}(a-x).
6 M

Choose your answer for the following :-

6 (a) (i)
The solution of the differential equation \[ \dfrac {dy}{dx}=e^{x+y} \ is \]

(A) e

(B) e

(C) e

(D) e

(A) e

^{x}+e^{-y}=c(B) e

^{-x}+ e^{-y}=c(C) e

^{x}+ e^{y}=c(D) e

^{x+y}=c
1 M

6 (a) (ii)
If the homogeneous differential equation \[ \dfrac {dy}{dx}=\dfrac {f_1(x, y)}{f_2(x,y)} \] the degree of the homogeneous functions f

(A) different

(B) same

(C) relatively prime

(D) degree of f

_{1}(x,y) and f_{2}(x,y) are(A) different

(B) same

(C) relatively prime

(D) degree of f

_{1}(x,y) > degree of f_{2}(x,y)
1 M

6 (a) (iii)
The integrating factor of the differential equation \[ (1+x^2)\dfrac {dy}{dx}+xy = \sin h^{-1}x\ is \\(A) \ \dfrac {1}{\sqrt{1+x^2}}\\(B) \ \sqrt{1-x^2}\\(C) \ \sqrt{1+x^2}\\(D) \ \dfrac {x}{\sqrt{1+x^2}}\]

1 M

6 (a) (iv)
If replacing \[ \dfrac {dy}{dx}\ by \ -\dfrac {dx}{dy} \] in the differential equation \[ f \left ( x,y \dfrac {dy}{dx} \right )=0 \]

(A) polar trajectory

(B) orthogonal trajectrory

(C) parametric trajectory

(D) parallel trajectory

(A) polar trajectory

(B) orthogonal trajectrory

(C) parametric trajectory

(D) parallel trajectory

1 M

6 (b)
\[ Solve \ (1+xy^2)\dfrac {dy}{dx}=1 \]

6 M

6 (c)
\[ Solve \ \dfrac {dy}{dx}= \dfrac {x(2\log x+)}{\sin y+ y \cos y} \]

4 M

6 (d)
Find the orthogonal trajectory of r

^{n}=a^{n}sin n?
6 M

Choose your answer for the following :-

7 (a) (i)
In a system of linear equations if the rank of the co-efficient matrix=rank of the augmented matrix=n number of unknown then the system has

(A) no solution

(B) unique solution

(C) infinite number of solution

(D) trivial solutions

(A) no solution

(B) unique solution

(C) infinite number of solution

(D) trivial solutions

1 M

7 (a) (ii)
The rank of matrix \[ \begin{bmatrix}2 &-1 &3 &1 \\ 1&4 &-2 &1 \\ 5&2 &4 &3 \end{bmatrix} \ is \]

(A) 3

(B) 4

(C) 2

(D) 5

(A) 3

(B) 4

(C) 2

(D) 5

1 M

7 (a) (iii)
A square matrix in which aij=aji for all I and j then it is called a

(A) unique matrix

(B) symmetric matrix

(C) skew symmetric

(D) triangular matrix

(A) unique matrix

(B) symmetric matrix

(C) skew symmetric

(D) triangular matrix

1 M

7 (a) (iv)
The inverse of the square matrix A is

\[ (A) \ |A| \\(B) \ \dfrac {adj A}{|A|}\\(C) \ adjA \\(D) \ \dfrac {|A|}{adjA}\]

\[ (A) \ |A| \\(B) \ \dfrac {adj A}{|A|}\\(C) \ adjA \\(D) \ \dfrac {|A|}{adjA}\]

1 M

7 (b)
Investigate for what value of ? and ? the simultaneous equation x+y+=6, x+2y+3z=10, x+2y+?z=? have

(A) no solutions

(B) unique solutions

(C) infinite number of solutions

(A) no solutions

(B) unique solutions

(C) infinite number of solutions

6 M

7 (c)
Apply Gauss-elimination method to solve the following equations:

2x-y+3z=1, -3x+4y-5z=0, x+3y-6z=0

2x-y+3z=1, -3x+4y-5z=0, x+3y-6z=0

4 M

7 (d)
Find the rank of \[ \begin{bmatrix}2 &3 &-1 &-1 \\ 1&-1 &-2 &-4 \\ 3&1 &3 &-2 \\ 6&3 &0 &-7 \end{bmatrix} \]

6 M

Choose your answer for the following :-

8 (a) (i)
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) 2,2,8

(C) 8,4,3

(D) 2,-2,8

(A) 2,3,8

(B) 2,2,8

(C) 8,4,3

(D) 2,-2,8

1 M

8 (a) (ii)
A homogeneous expression of the second degree in any number of variables is called a

(A) quadratic form

(B) diagonal form

(C) symmetric form

(D) spectral form

(A) quadratic form

(B) diagonal form

(C) symmetric form

(D) spectral form

1 M

8 (a) (iii)
A square matrix A of order 3 has 3 linearly independent eigen vectors then a matrix P can be found such that P

(A) digonal matrix

(B) symmetric matrix

(C) unit matrix

(D) singular matrix

^{-1}AP is a(A) digonal matrix

(B) symmetric matrix

(C) unit matrix

(D) singular matrix

1 M

8 (a) (iv)
if the eigen vector is (1, 1, 1) then its normalized form is

\[ (A) \ \left (\dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}} \right ) \\(B) \ \left (\dfrac {1}{\sqrt{2}}, 0,-\dfrac {1}{\sqrt{2}} \right )\\(C) \ \left (-\dfrac {1}{\sqrt{3}},\dfrac {1}{\sqrt{2}},-\dfrac {1}{\sqrt{2}} \right )\\(D) \ \left (-\dfrac {1}{\sqrt{3}}, - \dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}} \right ) \]

\[ (A) \ \left (\dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}} \right ) \\(B) \ \left (\dfrac {1}{\sqrt{2}}, 0,-\dfrac {1}{\sqrt{2}} \right )\\(C) \ \left (-\dfrac {1}{\sqrt{3}},\dfrac {1}{\sqrt{2}},-\dfrac {1}{\sqrt{2}} \right )\\(D) \ \left (-\dfrac {1}{\sqrt{3}}, - \dfrac {1}{\sqrt{3}}, \dfrac {1}{\sqrt{3}} \right ) \]

1 M

8 (b)
Reduce 6x

^{2}+ 3y^{2}- 4xy - 2yz + 4zx ito canonical form.
6 M

8 (c)
Find all the eigen values for the matrix, \[ A=\begin{bmatrix}7 &-2 &0 \\ -2&6 &-2 \\ 0&-2 &5 \end{bmatrix} \]

4 M

8 (d)
Reduce the matrix, \[ A=\begin{bmatrix}11 &-4 &7 \\ 7&-2 &-5 \\ 10&-4 &-6 \end{bmatrix} \] into a digonal matrix.

6 M

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