**Time: 2½ Hours
Maximum Marks: 70**

Q. 1. (a) If z = tan (y + ax) + (y – ax)^{3/2}, find the value of

\frac{\partial ^{2}Z}{\partial X^{2}}-a^{2}\frac{\partial ^{2}z}{\partial y^{2}} .**[6M]**

(b) If T=\sin \left ( \frac{xy}{x^{2}y^{2}} \right )+\sqrt{x^{2}+y^{2}}

by using Euler’s theorem find x\frac{\partial T}{\partial x}+y\frac{\partial T}{\partial y} **[6M]**

(c) If u = x^{2} – y^{2}, v= 2xy and z = f(u,v), then show that

x\frac{\partial T}{\partial x}+y\frac{\partial T}{\partial y} **[6M]**

**OR**

Q. 2. (a) If x = u tan v, y =u sec v, prove that : **[6M]**

(b) If \left ( \frac{\partial u}{\partial x} \right )_{y}\left (\frac{\partial v}{\partial x}\right )_{y}=\left ( \frac{\partial u}{\partial y } \right )_{x}\left ( \frac{\partial v }{\partial y} \right )_{x} , by using Euler’s theorem. prove that :

x^{2}\frac{\partial u^{2}}{\partial x^{2}}+2xy\frac{\partial u}{\partial x\partial y}+y^{2}\frac{\partial ^{2}u}{\partial y^{2}}=\frac{1}{4}\left ( tan^{3}u-tan\, u \right ) **[6M]**

(c) If x=\frac{cos\Theta }{u}, y=\frac{sin\Theta }{u} z= f\left ( x, y \right )

, then show that :

u\frac{\partial z}{\partial u}-\frac{\partial z}{\partial \Theta }=\left ( y-x \right )\frac{\partial z}{\partial x}-\left ( y+x \right )\frac{\partial z}{\partial y} **[6M]**

Q. 3. (a) If u = x + y + z, v = x^{2} + y^{2} + z^{2}, w = xy + yz + zx

find \frac{\left ( u,v,w \right )}{\partial \left ( x,y,z \right )} **[6M]**

(b) Examine whether u=\frac{x-y}{1+xy} v= tan^{-1}-tan^{-1}y are functionally dependent, if so find the relation between them. **[5M]**

(c) Find the extreme values of x^{2}+x^{2}+\frac{2}{x}+\frac{2}{y} · **[6M]**

**OR**

Q. 4. (a) If u = x + y^{2}, v = y + z^{2}, w = z + x^{2}, using Jacobian find \frac{\partial x}{\partial u} · **[6M]**

(b) A power dissipated in a resistor is given by P=\frac{\varepsilon ^{2}}{R}· If errors of 3% and 2% are found in and R respectively, find the percentage error in P. **[5M]**

(c) Using Lagrange’s method find extreme value of xyz if x + y + z = a. **[6M]**

Q. 5. (a) Examine for consistency of the system of linear equations and solve if consistent :**[6M]**

x_{1} + x_{2} + x_{3} = 0

–2x_{1} + 5x_{2 }+ 2x_{3} = 1

8x_{1} + x_{2} + 4x_{3} = –1

(b) Examine for linear dependence or independence the vectors (1, 1, 1, 3), (1, 2, 3, 4), (2, 3, 4, 7). Find the relation between them if dependent. **[6M]**

(c) Determine the values of a, b, c when A is orthogonal where :**[5M]**

A= \begin{bmatrix}0& 2b& c\\a& b& -c\\a& -b& c\end{bmatrix}

**OR**

Q. 6. (a) Investigate for what values of a and b, the system of equations 2x – y + 3z = 2, x + y + 2z = 2, 5x – y + az = b have :

(1) No solution

(2) A unique solution

(3) An infinite number of solutions. **[6M]**

(b) Examine for linear dependence or independence the vectors

x_{1} = (2, 3, 4, –2), x_{2} = (1, 1, 2, –1), x_{3} =(–1/2, – 1, -1, 1/2 ) Find the relation between them if dependent. **[6M]**

(c) Determine the currents in the network given in figure below : **[5M]**

Q. 7. (a) Find the eigen values and the corresponding eigen vectors for the following matrix : **[6M]**

(b) Verify Cayley-Hemilton theorem for A=\begin{bmatrix}1& -1& 0\\2& 3& -2\\-1& 0& 1\end{bmatrix} and use it to find A–1. **[6M]**

(c) Find a matrix P that diagonalizes the matrix

A=\begin{bmatrix}1 & 6 & 1\\1& 2& 0\\0& 0& 3\end{bmatrix} **[6M]**

**OR**

Q. 8. (a) Find the eigen values and the corresponding eigen vectors for the following matrix : **[6M]**

A= \begin{bmatrix}5 & 0& 1\\0& -2 & 0\\1& 0& 5\end{bmatrix}

(b) Verify Cayley-Hamilton theorem for A=\begin{bmatrix}0 & 1& 0\\0 & 0& 1\\1& -3& 3\end{bmatrix} and useit to find A–1. **[6M]**

(c) Reduce the following quadratic form to the sum of the squares form : **[6M]**

Q = 2x^{2} + 9y^{2} + 6z^{2} + 8xy + 8yz + 6xz.

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