STUPIDSID
1(a)
If cos α cosh \( \beta =\frac{x}{2},\sin \alpha sinh \beta =\frac{y}{2}, \)/ Prove that \( sec\left ( \alpha -i\beta \right )+sec\left ( \alpha +i\beta \right )= \frac{4x}{x^2+y^2}\)/
3 M
1(b)
If \( z=\log \left ( e^x+e^y \right ) \)/, show that rt-s2=0, where \( r = \frac{\partial^2 z}{\partial x^2}, t = \frac{\partial^2 z}{\partial y^2}, s =\frac{\partial^2 z}{\partial x \partial y} \)/
3 M
1(c)
If x = u v, \[y = \frac{u+v}{u-v}. \] Find \[\frac{\partial \left ( u,v \right )}{\partial \left ( x,y \right )}\].
3 M
1(d)
If \( y = 2^{x}\sin ^2x\cos x \)/ find yn
3 M
1(e)
Express the matrix \( A =\begin{bmatrix}
1 & 0& 5& 3\\
-2& 1& 6& 1\\
3 & 2& 7& 1\\
4& -4& 2& 0
\end{bmatrix} \) as the sum of symmetric and skew- symmetric matrices.
4 M
1(f)
Evaluate \[\lim_{x\rightarrow 0}\frac{e^2-\left ( 1+x \right )^2}{x\log \left ( 1+x \right )}\]
4 M
2(a)
Show that the roots of x5=1 can be written as 1,
α,
α2,
α3,
α4. Hence show that \[\left ( 1-\alpha \right )\left ( 1-\alpha ^2 \right )\left ( 1 -\alpha ^3 \right )\left ( 1 -\alpha ^4 \right ) = 5\]
α,
α2,
α3,
α4. Hence show that \[\left ( 1-\alpha \right )\left ( 1-\alpha ^2 \right )\left ( 1 -\alpha ^3 \right )\left ( 1 -\alpha ^4 \right ) = 5\]
6 M
2(b)
Reduce the following matrix to its normal from and hence find its rank \[A = \begin{bmatrix}
3 & -2& 0& 1\\
0& 2& 2& 7\\
1& -2& -3& 2\\
0& 1& 2& 1
\end{bmatrix}\]
6 M
2(c)
Solve the following system of equations by Gauss-Seidel Iterative Method upto four interations.
4x-2y-z=40
x-6y+2z=-28
x-2y+12z = -86
4x-2y-z=40
x-6y+2z=-28
x-2y+12z = -86
8 M
3(a)
Investigate for what values of 'λ' and 'μ' the system of equations \(x+y+z = 6
x+2 y+3 z = 10
x + 2 y + λ z = μ \)/ has i) no solution
ii) a unique solution
iii) an infinite no. of solutions.
x+2 y+3 z = 10
x + 2 y + λ z = μ \)/ has i) no solution
ii) a unique solution
iii) an infinite no. of solutions.
6 M
3(b)
If \(u = x^2+y^2+z^2 \)/, where \( x = e^t, y = e^t \sin t, z = e^t \cos t \)/ Prove that \[\frac{du}{dt} = 4e^{2t}\]
6 M
3(c)(i)
Show that \[\sin \left ( e^x -1 \right ) = x+\frac{x^2}{2}-\frac{5x^4}{24}+.........\]
4 M
3(c)(ii)
Expand 2x3+7x2+x-6 in power of x-2
4 M
4(a)
If x=u+v+w,
y = uv+vw+uw,
z=uvw and φ is a function of x,y and z. Prove that \[x\frac{\partial\phi }{\partial x}+2y\frac{\partial^\phi }{\partial y}+3z\frac{\partial \phi }{\partial z} = u\frac{\partial \phi }{\partial u}+ v\frac{\partial\phi }{\partial v} + \frac{\partial \phi }{\partial w}\]
y = uv+vw+uw,
z=uvw and φ is a function of x,y and z. Prove that \[x\frac{\partial\phi }{\partial x}+2y\frac{\partial^\phi }{\partial y}+3z\frac{\partial \phi }{\partial z} = u\frac{\partial \phi }{\partial u}+ v\frac{\partial\phi }{\partial v} + \frac{\partial \phi }{\partial w}\]
6 M
4(b)
if \( \tan \left ( \theta+i\phi \right )=\tan \alpha + i\sec \alpha \)/, Prove that
i) \[e^{2\phi } = \cot \frac{\alpha }{2}\]
ii) \[2\theta =n\pi +\frac{\pi }{2} + \alpha \]
i) \[e^{2\phi } = \cot \frac{\alpha }{2}\]
ii) \[2\theta =n\pi +\frac{\pi }{2} + \alpha \]
6 M
4(c)
Find the root of the equation x4+x3+7x2-x+5=0 which lies between 2 and 2.1 correct to three places of decimals using Regula Falsi Method.
8 M
5(a)
If \( y = \left ( x+\sqrt{x^2-1} \right )^m \)/, Prove That \[\left ( x^2-1 \right )y_{n+2}+(2n+1)xy_{n+1}+\left ( n^2-m^2 \right ) y _n =0\].
6 M
5(b)
Using the encoding matrix \( \begin{bmatrix}
1 & 1\\
0& 1
\end{bmatrix} \)/, encode and decode the message I* LOVE*MUMBAI*
6 M
5(c)(i)
Consulting only principal values separate into real and imaginary parts \[i^\log \left ( 1+i \right )\]
4 M
5(c)(ii)
Show that \[i\log \left ( \frac{x-i}{x+i} \right ) = \pi -2\tan ^{-1}x\]
4 M
6(a)
Using De Moivre's theorem prove that \[\cos ^6\theta -\sin ^6\theta =\frac{1}{16}\left ( \cos 6\theta +15\cos 2\theta \right )\]
6 M
6(b)
If\( u = sin ^{-1}\left ( \frac{x^\frac{1}{3}+y^\frac{1}{3}}{x^\frac{1}{2}-y^\frac{1}{2}} \right )^\frac{1}{2} \)/, Prove that \[x^2\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x\partial y}+y^2\frac{\partial^2 u}{\partial y^2}=\frac{\tan u}{144}\left ( \tan ^2u +13 \right )\]
6 M
6(c)
Discuss the maxima and minima of \[f\left ( x,y \right )= x^3y^2\left ( 1-x-y \right )\]
8 M
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