Answer any one question from Q1 and Q2

1 (a)
Examine for consistency the system of equations

x-y-z=2

x+2y+z=2

4x-7y-5z=2

and solve it consistent.

x-y-z=2

x+2y+z=2

4x-7y-5z=2

and solve it consistent.

4 M

1 (b)
Examine whether the following vectors are linearly dependent or independent, find the relation between them.

x

x

_{1}=[1, -1, 2] x_{2}=[2 3 5] x_{3}=[3 2 1]
4 M

1 (c)
If cosec(x+iy)=u+iv, prove that \[ i) \ \ \dfrac {u^2}{\sin^2 x} - \dfrac {v^2}{\cos^2 x}= (u^2 + v^2)^2 \\ ii) \ \dfrac {u^2}{\cosh^2 y}+ \dfrac {v^2}{\sinh^2 y} = (u^2 + v^2)^2 \]

4 M

2 (a)
A square lies above real axis in Argand diagram and two of its adjacent vertices are the origin and the point 2+3i. Find the complex numbers representing other two vertices.

4 M

2 (b)
\[ If \ arg(z+1) = \dfrac {\pi}{6} \ and \ arg(z-1) = \dfrac {2 \pi}{3} \ then \ find \ z. \]

4 M

2 (c)
Find the Eigen values and Eigen vectors of following matrix. \[ A= \begin{bmatrix}
1&1 &1 \\
0&2 &1 \\0
&0 &3
\end{bmatrix} \]

4 M

Answer any one question from Q3 and Q4

3 (a)
Test convergence of the series (any one) \[ i) \ \ i) \ \ \dfrac {1}{\sqrt{2}}+ \dfrac {1}{\sqrt{9}}+ \dfrac {1}{\sqrt{28}}+ \dfrac {1}{65}+ \cdots \ \cdots \\ ii) \ 1-\dfrac {1}{\sqrt{2}}+ \dfrac {1}{\sqrt{3}}- \dfrac {1}{\sqrt{4}}+ \cdots \cdots \]

4 M

3 (b)
Prove that \[ \log (1+ \sin x) = x - \dfrac {x^2}{2}+ \dfrac {x^3}{6}- \dfrac {x^4}{12}+ \cdots \]

4 M

3 (c)
Find n

^{th}derivative of \[ \dfrac {x^2}{(x-1)(x-2)} \]
4 M

Solve any one:

4 (a)
i) Evaluate \[ \lim_{x\to 0} \ \log_{\tan x} \ \tan 4x \] ii) Find the value of a and b if \[ \lim_{n \to 0} \left [ x^{-3} \sin x + ax^{-2}+b \right ]=0 \]

4 M

4 (b)
Using Taylor's theorem expand 49-69x+42x

^{2}+11x^{3}+x^{4}in powers of (x+2).
4 M

4 (c)
If y=sin log (x

(x+1)

^{2}+2x+1), then prove that(x+1)

^{2}y_{n+2}+(2n+1)(x+1)y_{n+1}+(n^{2}+4)y_{n=0}
4 M

Solve any two of the following:

5 (a)
If u=log (x

^{3}+y^{3}-x^{2}y-xy^{2}) then prove that x^{2}u_{xx}+2_{xy}u_{xy}+ y^{2}u_{yy}=-3.
7 M

Answer any one question from Q5 and Q6

5 (b)
If x=u+v+w, y=uv+vw+wu, z=uvw and ϕ is a function of x,y,z then prove that \[ u \dfrac {\partial \phi}{\partial u} + v \dfrac {\partial \phi}{\partial v}+ w \dfrac {\partial \phi}{\partial w} = x \dfrac {\partial \phi}{\partial x} + 2y \dfrac {\partial \phi}{\partial y}+ 3z \dfrac {\partial \phi}{\partial z} \]

6 M

5 (c)
\[ If \ ux+vy=0 \ and \dfrac {u}{v}+ \dfrac {v}{y}=1, \ then\ prove \ that \ \left ( \dfrac {\partial u}{\partial x} \right )y - \left (\dfrac {\partial v}{\partial y} \right )x = \dfrac {x^2 + y^2}{y^2 - x^2} \]

6 M

Solve any two of the following:

6 (a)
\[ If \ u=\cos \left (\dfrac {xy}{x^2+y^2} \right )+ \sqrt{x^2 + y^2}+ \dfrac {xy^2}{x+y} \] then find the value of xu

_{x}+yu_{y}at (3, 4).
7 M

6 (b)
\[ if \ x=\dfrac {\cos \theta}{u}, \ y=\dfrac {\sin \theta}{u}, \] Then prove that \[ u \dfrac {\partial z}{\partial u}- \dfrac {\partial z}{\partial \theta} = (y-x) \dfrac {\partial z}{\partial x} - (y-x) \dfrac {\partial z}{\partial y} \]

6 M

6 (c)
If u=(x

^{2}-y^{2}) f(xy), then show that u_{xx}-u_{yy}=(x^{4}-y^{4}) f'(xy).
6 M

Answer any one question from Q7 and Q8

7 (a)
\[ If \ x=r \sin \theta \cos \phi, \ y=r \sin \theta \sin \phi, \ z=r \cos \theta \ find \ \dfrac {\partial (x,y,z)}{\partial (r, \theta, \phi)} \]

4 M

7 (b)
Examine for functional dependence \[ u=\sin^{-1}x+\sin^{-1}y, \ v=x\sqrt{1-y^2}+y \sqrt{1-x^2} \] if dependent find the relation between them.

4 M

7 (c)
The area of a triangle ABC is calculated from the formula Δ=1/2 bc sin A. Errors of 1%, 2% and 3% respectively are made in measuring b,c,A. If the correct value of A is 30°, find the percentage error in the calculated value of area of triangle.

5 M

8 (a)
\[ If \ u^2 +xv^2 -uxy=0, \ v^2-xy^2+2uv+u^2=0, \ find \ \dfrac {\partial u}{\partial x} \] by choosing u, v as dependent and x, y as independent variables.

4 M

8 (b)
Show that \[ u = \dfrac {x+y}{1-xy} , \ v=\tan^{-1} x+\tan^{-1}y \] are functionally dependent and find the relation between them.

4 M

8 (c)
Find all the stationary values of the function \[ f(x,y)=x^3 +3 xy^3 - 15 x^2 - 15 y^2 + 72 x.\] Find maximum value of f(x,y) at suitable point.

5 M

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