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SPPU First Year Engineering (Semester 1)
Engineering Mathematics-1
May 2017
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

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.
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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