STUPIDSID
1 (a)
If log tan x=y then prove that sinh(n+1)y+sinh(n-1)y=2 sinh ny⋅cosec2x
3 M
1 (b)
If z=log (tan x+tan y) then prove that \[ \sin 2x \dfrac {\partial z}{\partial x} + \sin 2y \dfrac {\partial z}{\partial y} = 2 \]
3 M
1 (c)
If x=r sin θ cos φ, y=r sin θ φ, z=r cos θ, then find \[ \dfrac {\partial (r, \theta , \varphi)}{\partial (x,y,z) } \]
3 M
1 (d)
Prove that \[ \log \sec x=\dfrac {x^2}{2}+ \dfrac {x^4}{12}+ \dfrac {x^6}{45}+ \cdots \ \cdots \]
3 M
1 (e)
Find the values of a,b,c and A-1 when \[ A=\dfrac {1}{9} \begin{bmatrix}
-8 &4 &a \\ 1
&4 &b \\4
&7 &c
\end{bmatrix} \] is or orthogonal.
4 M
1 (f)
If y=sinθ+cos θ then prove that \[y_n = r^n \sqrt{1+(-1)^n \sin 2 \theta} \] where θ rx
4 M
2 (a)
If z=-1+i√3 then prove that \[ \left ( \dfrac {z}{2} \right )^n + \left (\dfrac {2}{z} \right )^n = \left\{\begin{matrix}
2 ,& \text{if n is multiple of 3} \ \ \ \ \ \\-1 ,
& \text {if n is not multiple of 3}
\end{matrix}\right.\]
6 M
2 (b)
\[ \text {if} A=\begin{bmatrix}
1 &2 &-2 \\-1
&3 &0 \\0
&-2 &1
\end{bmatrix} \] then find two non-singular matrices P&Q such that PAQ is in normal form also find ρ(A) and A-1.
6 M
2 (c)
State and prove Euler's theorem for functions of two independent variable hence prove that \[ \left ( x \dfrac {\partial u}{\partial x}+ y \dfrac {\partial u} {\partial y} \right ) \left( x \dfrac {\partial v} {\partial x} + y\dfrac {\partial v} {\partial y} \right ) = 0 \] if x=eu \tan v, y=eu, sec v.
8 M
3 (a)
Determine the values of a and b such that system \[ \left\{\begin{matrix}
3x - 2y+z=b \\
5x-8y+9z=3 \\
2x+y+az=-1
\end{matrix}\right. \] has i) no solution, ii) a unique solution, iii) infinite number of solutions
6 M
3 (b)
Discuss the maximum and minimum of f(x,y)=x3+3xy2-15(x2+y2)+72x
6 M
3 (c)
show that \[ \tan^{-1} \left ( \dfrac {x+iy}{x-iy} \right ) = \dfrac {\pi}{4} + \dfrac {i}{2}\log \left ( \dfrac {x+y}{x-y} \right ) \]
8 M
4 (a)
If u=xyz, v=x2+y2+x2, w=x+y+z then prove that \[ \dfrac {\partial x}{\partial u} = \dfrac {1}{(x-y)(x-z)} \]
6 M
4 (b)
\[ \text {if} \sqrt {i}^{\sqrt{i}^{\sqrt{i}\cdots \cdots \infty}} = \alpha + i\beta \] then prove that \[
i) \ \alpha^2 + \beta^2 = e\dfrac {\pi \beta}{2} \\
ii) \ \tan^{-1}\left ( \dfrac {\beta}{\alpha} \right ) = \dfrac {\pi \alpha} {4} \]
6 M
4 (c)
Apply Crout's method to solve \[ \left\{\begin{matrix} x-y + 2z=2 \ \ \\ 3x+2y-3z=2 \\ 4x -4y+2z=2 \end{matrix}\right. \]
8 M
5 (a)
If cos6θ+sin6θ = α cos 4θ+β then prove that α+β=1.
6 M
5 (b)
Find the values of a,b & c such that \[ \lim_{x\to 0} \dfrac {a e^x - be^{-x}+cx}{x-\sin x}=4 \]
6 M
5 (c)
\[ \text {if } x =\cos \Big [ \log \big (y^{1/m} \big) \Big ] \] then prove that
(1-x2)yn+2-(2n+1)xyn+1-(m2+n2)yn = 0
(1-x2)yn+2-(2n+1)xyn+1-(m2+n2)yn = 0
8 M
6 (a)
Define linear dependence and independence of vectors, Examine for linear dependence of following set of vectors and find the relation between them if dependent \[ X_1 = \begin{bmatrix}1\\-1 \\1\end{bmatrix},
X_2 = \begin{bmatrix}2\\1 \\1\end{bmatrix},
X_3 = \begin{bmatrix}3\\0 \\2\end{bmatrix} \]
6 M
6 (b)
If z=f(u,v), u=x2-y2, v=2xy then prove that \[ \dfrac {\partial ^2 z}{\partial x^2} + \dfrac {\partial ^2 z} {\partial y^2} = 4 \sqrt{u^2 + v^2} \left ( \dfrac {\partial ^2 z} {\partial u^2} + \dfrac {\partial ^2 z}{\partial v^2} \right ) \]
6 M
6 (c)
Fit a straight line passing through points (0,1), (1,2), (2,3), (3,4,5), (4,6), (5,7,5).
8 M
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