STUPIDSID
1(a)
If cos α cosh β=x2,sinαsinhβ=y2,/ Prove that sec(α−iβ)+sec(α+iβ)=4xx2+y2/
3 M
1(b)
If z=log(ex+ey)/, show that rt-s2=0, where r=∂2z∂x2,t=∂2z∂y2,s=∂2z∂x∂y/
3 M
1(c)
If x = u v, y=u+vu−v. Find ∂(u,v)∂(x,y).
3 M
1(d)
If y=2xsin2xcosx/ find yn
3 M
1(e)
Express the matrix A=[1053−216132714−420] as the sum of symmetric and skew- symmetric matrices.
4 M
1(f)
Evaluate lim
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.
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
More question papers from Applied Mathematics 1