STUPIDSID
Answer any one question from Q1 and Q2
1 (a)
Show that the system of equations.
3x+4x+5z=α
4x+5y+6z=β
5x+6y+7z=γ
is consistent only of α, β and γ are in arithmetic progression.
3x+4x+5z=α
4x+5y+6z=β
5x+6y+7z=γ
is consistent only of α, β and γ are in arithmetic progression.
4 M
1 (b)
Verify Cayley-Hamilton theorem for A and hence find A-1. \[ A= \begin{bmatrix}
1 &0 &-2 \\2
&2 &4 \\0
&0 &2
\end{bmatrix} \]
4 M
1 (c)
If α=1+i, β=1-i and cot Q=x+1 then prove that \[ \dfrac {(x+\alpha)^n - (x+\beta)^n}{\alpha -\beta} = \sin nQ \cos \ ec^{n}Q \]
4 M
2 (a)
Examine whether the following vectors are linearly dependent or independent. If dependent, find the relation between them.
X1=[1,2,3], X2=[3, -2, 1], X3=[1, -6, -5]
X1=[1,2,3], X2=[3, -2, 1], X3=[1, -6, -5]
4 M
2 (b)
Show that \[ \log \left [ \dfrac {\sin (x+iy)}{\sin (x-iy)} \right ]= 2 i \tan ^{-1}(\cot x \tan hy) \]
4 M
2 (c)
Prove that \[ \tan^{-1} \left [ i \left ( \dfrac {x-a} {x+a}\right ) \right ] = \dfrac {i}{2} \log \left ( \dfrac {a}{x} \right ) \]
4 M
Answer any one question from Q3 and Q4
3 (a)
Test the convergence of the series (any one) \[ i) \ \ \dfrac {2}{1}+ \dfrac {3}{8}+ \dfrac {4}{27}+ \dfrac {5}{64}+ \cdots \ \cdots + \dfrac {n+1}{n^3}+ \cdots \ \cdots \\ ii) \ \ 1-\dfrac {1}{2\sqrt{2}}+ \dfrac {1}{3\sqrt{3}}+ \dfrac {1}{4\sqrt{4}}+ \cdots \ \cdots \]
4 M
3 (b)
Expand x3+7x2+x-6 in powers of (x-3)
4 M
3 (c)
\[ If \ y = \log \left ( x+ \sqrt{x^2+1} \right ) \] prove that
(1+x2)yn+2+(2n=1)xyn+1+n2yn=0.
(1+x2)yn+2+(2n=1)xyn+1+n2yn=0.
4 M
4 (a)
Solve any one: \[ i) \ \ Evaluate \ \lim_{x\to \pi /2} (\cos x)^{\cos x} \\ ii) \ \ If \lim_{x\to 0} \dfrac {\sin 2 x + \rho \sin x} {x^2} \] is finite, find the value of ρ and hence evaluate the limit.
4 M
4 (b)
Prove that \[ \log (1+\tan x) = x - \dfrac {x^2}{2}+ \dfrac {2x^3}{3} \cdots \ \cdots \]
4 M
4 (c)
Find nth derivative of \[ y = \dfrac {x} {x-1) (x-2)(x-3) }
4 M
Solve any two of the following:
5 (a)
Find the value of n such that u=xn (3\cos2y-1) satisfies the partial differential equation. \[ \dfrac {\partial }{\partial x} \left ( x^2 \dfrac {\partial u}{\partial x} \right ) + \dfrac {1}{\sin y} \dfrac {\partial }{\partial y} \left ( \sin y \dfrac {\partial u}{\partial y} \right )=0 \]
6 M
Answer any one question from Q5 and Q6
5 (b)
If x=r cos θ, y=r sin kθ then prove that \[ i) \ \ \left ( \dfrac {\partial y}{\partial r} \right )_x \left ( \dfrac {\partial y}{\partial r} \right )_\theta =1 \\ ii) \ \left ( \dfrac {\partial x}{\partial \theta} \right )_r = r^2 \left ( \dfrac {\partial \theta}{\partial x} \right )_y \]
6 M
5 (c)
\[ If \ u=x^8 \ f \left ( \dfrac {y}{x} \right ) = \dfrac {1}{y^8}\phi \left ( \dfrac {x}{y} \right ) \] then prove that \[ x^2 \dfrac {\partial ^2 u}{\partial x^2} + 2xy \dfrac {\partial^ 2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^ 2 u}{\partial y^2} + x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}= 64 u \]
7 M
Solve any two of the following:
6 (a)
\[ If \ u=\cos^{-1} \left [ \dfrac {x^3 y^2 + 4y^3 x^2}{\sqrt{x^4 + 6 y^4}} \right ] \] find the value of \[ i) \ \ x\dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} \\ ii) \ x^2 \dfrac {\partial^2 u}{\partial x^2} + 2xy \dfrac {\partial ^2 u}{\partial x \partial y}+ y^2 \dfrac {\partial ^2 u}{\partial y^2} \]
7 M
6 (b)
\[ If \ u =f \left ( \dfrac {x}{y}, \dfrac {y}{z}, \dfrac {z}{x} \right ) \] prove that \[ x \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}+ z \dfrac {\partial u}{\partial z} =0 \]
6 M
6 (c)
if f(x,y)=0, ϕ(x,z)=0 then prove that \[ \dfrac {\partial \phi}{\partial x} \dfrac {\partial f}{\partial y} \dfrac {dy}{\partial z} = \dfrac {\partial f}{\partial x}\dfrac {\partial \phi}{\partial z} \]
6 M
Answer any one question from Q7 and Q8
7 (a)
if x=e4 sec u, y=e4 tan u, find \[ \dfrac {\partial (u,v)} {\partial (x,y)}
4 M
7 (b)
Examine for functional dependence for \[ u = \dfrac {x}{y-z}, \ v=\dfrac {y}{z-x}, \ w=\dfrac {z}{x-y} \]
4 M
7 (c)
Find extreme values of f(x,y)=x3+y3-3axy, a>0
5 M
8 (a)
if x=cos θ - r sin θ, y=sin θ + r cos θ find \[ \dfrac {\partial r} {\partial x} \]
4 M
8 (b)
The resonant frequency in a series electrical circuit is given by \[ f=\dfrac {1}{2\pi \sqrt{LC}} \] If the measurement in L and C are in error by 2% and -1% respectively. Find the percentage error in f.
4 M
8 (c)
Use Lagrange's method to find stationary value of \[ u= \dfrac {x^2}{a^3} + \dfrac {y^2}{b^3}+ \dfrac {z^2}{c^3} \ where \ x+y+z=1 \]
5 M
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