STUPIDSID
1(a)
If \[ \ \tanh x=\dfrac {2}{3} \]. Find the value of x and then cosh 2x.
3 M
1(b)
\[ if \ u = \tan^{-1} \left ( \dfrac{y}{x} \right ), \] Find these value of \[ \dfrac {\partial ^2 u}{\partial x^2} + \dfrac {\partial ^2 u}{\partial y^2} \]
3 M
1(c)
If x=r cos θ, y=r sin θ Find \[ \dfrac {\partial (x,y)}{\partial (r, \theta)} \]
3 M
1(d)
Prove that \[\log \sec x = \dfrac {1}{2}x^2 + \dfrac {1}{1z}x^2 + \dfrac {1}{45}x^6 \cdots \ \cdots \]
3 M
1(e)
Show that every square matrix can be uniquely expressed as the sum of Hermitian matrix and a skew Hermitian matrix.
4 M
1(f)
Find the nth derivative of y =sin x sin 2x sin 3x.
4 M
2(a)
Solve the equation x6+1=0
6 M
2(b)
Reduce the matrix to normal form and find its rank where, \[ A= \begin{bmatrix}
1 &-1 &3 &6 \\1
&3 &-3 &-4 \\5
&3 &3 &11
\end{bmatrix} \]
6 M
2(c)
State and prove Eulers theorem for a homogeneous function in two variables: Hence verify the Eulers theorem for \[ u = \dfrac {\sqrt{xy}}{\sqrt{x+} \sqrt{y}} \]
8 M
3(a)
Test the consistancy of the following equations and solve them if they are consistent.
2x-y+z=8, 3x-y+z=6
4x-y+2z=7, -x+y-z=4
2x-y+z=8, 3x-y+z=6
4x-y+2z=7, -x+y-z=4
6 M
3(b)
Find the stationary values
x3+3xy2-3x2-3y2+4
x3+3xy2-3x2-3y2+4
6 M
3(c)
Separatic into real and imaginary parts of sin-1 (eio).
8 M
4(a)
\[ if \ x=uv, \ y=\dfrac {u}{v} \] prove that J.J=1
6 M
4(b)
Show that for real values of a and b, \[ e^{2 \ ai \cot ^{-1}b} \left [ \dfrac {bi-1}{bi+1} \right ]^{-2} = 1 \]
6 M
4(c)
Solve the following equations by Gauss-seidal method
27x + 6y-z=85
6x+15y+2z=72
x+y+54z=110
27x + 6y-z=85
6x+15y+2z=72
x+y+54z=110
8 M
5(a)
Expond cos7θ in a series of cosine of multiple of θ
6 M
5(b)
\[ If \lim_{x \to 0} \dfrac {a\sin hx + b \sin x}{x^3} = \dfrac {5}{3}, \] find a and b
6 M
5(c)
\[ If \ y = \dfrac {\sin^{-1}x}{\sqrt{1-x^2}} \] then prove that (1-x2)yn+1 - (2n+1)xyn-n2yn-1=0
8 M
6(a)
Examine whether the vectors
x1= [3,1,1] x2=[2,0,-1]
x3=[4,2,1] are linearly independent.
x1= [3,1,1] x2=[2,0,-1]
x3=[4,2,1] are linearly independent.
6 M
6(b)
If u=f(x-y, y-z, z-x) then show that \[ \dfrac {\partial u}{\partial x}+ \dfrac {\partial u}{\partial y} + \dfrac {\partial u}{\partial z} = 0 \]
6 M
6(c)
Fit a straight line for the following data
| x | 1 | 2 | 3 | 4 | 5 | 6 |
| y | 49 | 54 | 60 | 73 | 80 | 86 |
8 M
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