STUPIDSID
Solve any one question from Q.1(a,b,c) & Q.2(a,b,c)
1(a)
Show that system of Linear equations is consistent. Find solution:
x+2y+3z=6
2x+3y=11
4x+y-5z= -3.
x+2y+3z=6
2x+3y=11
4x+y-5z= -3.
4 M
1(b)
Find eign values and eign vectors of the matrix: \[A=\begin{bmatrix}
5 & 4\\
1& 2
\end{bmatrix}\]
4 M
1(c)
Prove that:\[\left ( \cosh x- \sin x\right )^{n}=\cosh nx-\sin h \ \ nx.\]
4 M
2(a)
Are the following vectors are linearly dependent? If so find relation: \[\begin{align*}&X_{1}=(3, 2, 7),\\
&X_{2}=(2, 4, 1), \\
&X_{3}=(1, -2, 6).\end{align*}\]
4 M
2(b)
If α, β roots of equation \( x^{2}-2x+4=0,\)/ prove that:\[ \alpha ^{3}+\beta ^{n}=2^{n+1}\cos \frac{n\pi }{3}.\]
4 M
2(c)
If \( \left ( a+ib \right )^{p}=m^{x+iy},\)/ prove that: \[\frac{y}{x}=\frac{2\tan ^{-1}\frac{b}{a}}{\log \left ( a^{2}+b^{2} \right ).}\]
4 M
Solve any one question from Q.3(a, b, c) Solve any one question from Q.3(a,b,c) &Q.4(a,b,c)
3(a)
Test the covergence of the series \[\begin{align*}& i)\sum \frac{x^{n}}{a+\sqrt{n}}\\
&ii)\frac{1}{\sqrt{5}}-\frac{1}{2\sqrt{6}}+\frac{1}{3\sqrt{7}}\end{align*}\]
4 M
3(b)
Show that: \[\log \left [ \frac{1+e^{2x}}{e^{x}} \right ]=\log 2+\frac{x^{2}}{2}-\frac{x^{4}}{12}+\frac{x^{6}}{45}......\]
4 M
3(c)
Find the nth derivative of: \[y=e^{x}\cos x.\cos 2x.
4 M
Solve any one question from Q.4(a, b, c)
4(a)
\begin{align*}&i)x\overset{lim}{\rightarrow}0\left [ \frac{\pi }{4x}-\frac{\pi }{2x\left ( e^{\pi x}+1 \right )} \right ]\\&ii)x\overset{lim}{\rightarrow}0\left ( \frac{2^{x}+5x+7x}{3} \right )^{1/x}.\end{align*}
4 M
4(b)
Using Taylor's theorem expand \(x^{3}-2x^{2}+3x+1 \)/ in powers of (x-1).
4 M
4(c)
If \( y=e^{a\sin ^{-1}}x,\)/ prove that:\[\left ( 1-x^{2} \right )y_{n+2}-\left ( 2n+1 \right )x\ \ y_{n+1}-\left ( n^{2} +a^{2}\right )y_{n}=0\]
4 M
Solve any two question from Q.5(a,b,c) & Solve any one question from Q.5(a, b,c) &Q.6(a, b, c)
5(a)
Find the value of n for which \(z=t^{n} e^{-r^{2}}/4t \)/ satisfies the partial differential equation:\[\frac{1}{r^{2}}\left [ \frac{\partial }{\partial r}\left ( r^{2} \frac{\partial z}{\partial r}\right ) \right ]=\frac{\partial z}{\partial t}.\]
6 M
5(b)
If \( T =\sin \left ( \frac{xy}{x^{2}+y^{2}} \right )+\sqrt{x^{2}+y^{2}}+\frac{x^{2}y}{x+y},\)/ find the value of \[x\frac{\partial t}{\partial x}+y\frac{\partial t}{\partial y}.\]
7 M
5(c)
If z = f(x, y), where \( \begin{align*}x=u\cos \alpha -v\sin \alpha ,\\
y=u \sin \alpha -v\cos \alpha ,\end{align*}\)/ where α is constant, show that: \[\left ( \frac{\partial z}{\partial x} \right )^{2}+\left ( \frac{\partial z}{\partial y} \right )^{2}=\left ( \frac{\partial z}{\partial u} \right )^{2}+\left ( \frac{\partial z}{\partial v} \right )^{2}.\]
6 M
Solve any two question from Q.6 (a, b, c)
6(a)
If \(\begin{align*}&x^{2}=a\sqrt{u}+b\sqrt{v}\ \ \text{and}\\
&y^{2}=a\sqrt{u}-b\sqrt{v}\end{align*} \)/ when a and b are constants, prove that: \[\begin{align*} \left ( \frac{\partial u}{\partial x} \right )_{y}\left ( \frac{\partial x}{\partial u} \right )_{v}=\frac{1}{2}\left ( \frac{\partial v}{\partial y} \right )_{x}\left ( \frac{\partial y}{\partial v} \right )_{u}.\end{align*}\]
6 M
6(b)
If \( u=\tan ^{-1}\left ( \frac{\sqrt{x^{3}+y^{3}}}{\sqrt{x} +\sqrt{y}}\right ),\\
\text{then show that:}\\
x^{2}\frac{\partial ^{2}u}{\partial x^{2}}+2xy\frac{\partial ^{2}u}{\partial x\partial y}+y^{2}\frac{\partial ^{2}u}{\partial y^{2}}=-2\sin ^{3}u\cos u. \)/
7 M
6(c)
If \( u=x^{2}-y^{2}, v=2xy \ \ \text{and} z= f(u, v),\\
\text{then show that:}\\
x\frac{\partial z}{\partial x}-y\frac{\partial z}{\partial y}=2\sqrt{u^{2}+v^{2}}\frac{\partial z}{\partial u}.\)/
6 M
Solve any one question from Q.7(a, b, c) &Q.8(a, b, c)
7(a)
If \( u+v=x^{2}+y^{2}, u-v=x+2y\\
\text{Find}\ \ \frac{\partial u}{\partial x}\ \ \text{treating y constant}.\)/
4 M
7(b)
Examine for functional dependence:\[u = \frac{x-y}{x+z}, v=\frac{x+z}{y+z}.\]
4 M
7(c)
Find stationary points of : \( f\left ( x,y \right )=x^{3}y^{2}\left ( 1-x-y \right ) \)/ and find fmax where it exists.
5 M
8(a)
If \( x=v^{2}+w^{2}, y=w^{2}+u^{2}, z=u^{2}+v^{2},\\
\text{ prove that JJ'}=1.\)/
4 M
8(b)
Find the percentage error in computing the parallel resistance r of three resistances r1, r2, r3 from the formula: \[\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}} \\
if\ r_{1}, r_{2},r_{3} \ \ \text{are in error by 2% each}\]
4 M
8(c)
Find the stationary points of:\(T(x, y, z)=8x^{2}+4yz-16z+600 \)/ if the condition \( 4x^{2}+y^{2}+4z^{2}=16\)/ is satisfied.
5 M
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