1(a)
Prove the following:

\[\dfrac{1}{1- \dfrac{1}{1-\dfrac{1}{1-\cos h^2x}}} = \cos h^2 x\]

\[\dfrac{1}{1- \dfrac{1}{1-\dfrac{1}{1-\cos h^2x}}} = \cos h^2 x\]

3 M

1(b)
If u = log (tanx + tany)

\[2x\dfrac{\partial u}{\partial x}+sin2y\dfrac{\partial u}{\partial y} = 2 \]

\[2x\dfrac{\partial u}{\partial x}+sin2y\dfrac{\partial u}{\partial y} = 2 \]

3 M

1(c)
If the following expression is true,

\[ u=\dfrac{x+y}{1-xy} \ \ , \ v = tan^{-1}x + tna^{-1}y \\ Find \ \ \dfrac{\partial (u,v)}{\partial (x,y)}\]

\[ u=\dfrac{x+y}{1-xy} \ \ , \ v = tan^{-1}x + tna^{-1}y \\ Find \ \ \dfrac{\partial (u,v)}{\partial (x,y)}\]

3 M

1(d)
Expand log (1+sinx) = (x - x

^{2}/2 + x^{3}/6 +...)
3 M

1(e)
Show that every square matrix can be uniquely expressed as P+iQ where P and Q are Hermitian Matrices.

4 M

1(f)
Find n

\[y= \dfrac{x^2+4}{(2x+3)(x-1)^2}\]

^{th}order derivative of\[y= \dfrac{x^2+4}{(2x+3)(x-1)^2}\]

4 M

2(a)
Show that roots of the equation (x+1)

\[-i\cot\Big[ \dfrac{(2k+1)\pi}{12} \Big] \ \ \ , \ k=0,1,2,3,4,5\]

^{6}+ (x-1)^{6}= 0 are given by\[-i\cot\Big[ \dfrac{(2k+1)\pi}{12} \Big] \ \ \ , \ k=0,1,2,3,4,5\]

6 M

2(b)
Reduce the following matrix into normal form and find its rank

\[ \left[ {\begin{array}{cc} 2 & -1 & 1 & 1\\ 1 & 0 & 1 & 2\\ 3 & 3 & 3 & 1\\ 0 & -4 & -1 & 2\\ \end{array} } \right]\]

\[ \left[ {\begin{array}{cc} 2 & -1 & 1 & 1\\ 1 & 0 & 1 & 2\\ 3 & 3 & 3 & 1\\ 0 & -4 & -1 & 2\\ \end{array} } \right]\]

6 M

2(c)
State and prove Euler's theorem for a homogeneous function in two variables. And hence

\[Find \ \ x \dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y} \ \ where \ u=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\]

\[Find \ \ x \dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y} \ \ where \ u=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\]

8 M

3(a)
Test for consistency and solve if consistent -

x

x

4x

x

_{1}-2x_{2}+x_{3}-x_{4}=2;x

_{1}+2x_{2}+2x_{4}=1;4x

_{2}-x_{3}+3x_{4}=-1.
6 M

3(b)
Find all stationary value of x

^{2}+ 3xy - 15x^{2}- 15y^{2}+ 72x.
6 M

3(c)
If tan[(π/4)+iv] = re

(i) r=1

(ii) tanθ = sinh 2v

(iii) tanhv = tan(θ/2)

^{iθ}show that(i) r=1

(ii) tanθ = sinh 2v

(iii) tanhv = tan(θ/2)

8 M

4(a)
If x = u+e

\[Find \ \ \dfrac{\partial u}{\partial y}, \dfrac{\partial v}{\partial x} \ \ using \ \ Jacobian\]

^{(-v)}sin u, and y = v+e^{(-u)}cos u,\[Find \ \ \dfrac{\partial u}{\partial y}, \dfrac{\partial v}{\partial x} \ \ using \ \ Jacobian\]

6 M

4(b)
Considering only the principal value,

if (1 + i tanθ)

if (1 + i tanθ)

^{(1+i tanθ)}is real, prove that its value is (sec?)^{(sec2θ)}.
6 M

4(c)
Solve the system of linear equation by Crout's method

x - y + 2z = 2;

3x + 2y - 3z = 2;

4x - 4y + 2z = 2

x - y + 2z = 2;

3x + 2y - 3z = 2;

4x - 4y + 2z = 2

8 M

5(a)
Expand cos

^{7}θ in a series of cosines of multiple of θ .
6 M

5(b)
Evaluate the following:

\[\displaystyle\lim_{x \to 0}\Big[ \dfrac{1}{x^2} - cot^2x \Big]\]

\[\displaystyle\lim_{x \to 0}\Big[ \dfrac{1}{x^2} - cot^2x \Big]\]

6 M

5(c)
If y = (sin

^{-1}x)^{2}, obtain y_{n}(0).
8 M

6(a)
Show that the vectors are linearly dependent and find the relation between them

X

X

X

X

X

_{1}=[1,2,-1,0],X

_{2}=[1,3,1,2],X

_{3}=[4,2,1,0],X

_{4}=[6,1,0,1].
6 M

6(b)
If the expression is

\[\dfrac{x^2}{(1+u)}+\dfrac{y^2}{(2+u)} +\dfrac{z^2}{(3=1+u)} \]

prove that

\[\Big[\Big(\dfrac{\partial u}{\partial x}\Big)^2+ \Big(\dfrac{\partial u}{\partial y}\Big)^2 +\Big(\dfrac{\partial u}{\partial z}\Big)^2 \Big] = 2\Big[x\dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y} +z\dfrac{\partial u}{\partial z} \Big]\]

\[\dfrac{x^2}{(1+u)}+\dfrac{y^2}{(2+u)} +\dfrac{z^2}{(3=1+u)} \]

prove that

\[\Big[\Big(\dfrac{\partial u}{\partial x}\Big)^2+ \Big(\dfrac{\partial u}{\partial y}\Big)^2 +\Big(\dfrac{\partial u}{\partial z}\Big)^2 \Big] = 2\Big[x\dfrac{\partial u}{\partial x}+ y\dfrac{\partial u}{\partial y} +z\dfrac{\partial u}{\partial z} \Big]\]

6 M

6(c)
Fit a second degree parabolic curve to following data:-

X | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Y | 2 | 6 | 7 | 8 | 10 | 11 | 11 | 10 | 9 |

8 M

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