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SPPU First Year Engineering (Semester 1)
Engineering Mathematics-1
December 2014
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

Answer any one question from Q1 and Q2
1 (a) Examine for consistency the system of equations
x-y-z=2
x+2y+z=2
4x-7y-5z=2
and solve it consistent.
4 M
1 (b) Examine whether the following vectors are linearly dependent or independent, find the relation between them.
x1=[1, -1, 2] x2=[2 3 5] x3=[3 2 1]
4 M
1 (c) If cosec(x+iy)=u+iv, prove that $i) \ \ \dfrac {u^2}{\sin^2 x} - \dfrac {v^2}{\cos^2 x}= (u^2 + v^2)^2 \\ ii) \ \dfrac {u^2}{\cosh^2 y}+ \dfrac {v^2}{\sinh^2 y} = (u^2 + v^2)^2$
4 M

2 (a) A square lies above real axis in Argand diagram and two of its adjacent vertices are the origin and the point 2+3i. Find the complex numbers representing other two vertices.
4 M
2 (b) $If \ arg(z+1) = \dfrac {\pi}{6} \ and \ arg(z-1) = \dfrac {2 \pi}{3} \ then \ find \ z.$
4 M
2 (c) Find the Eigen values and Eigen vectors of following matrix. $A= \begin{bmatrix} 1&1 &1 \\ 0&2 &1 \\0 &0 &3 \end{bmatrix}$
4 M

Answer any one question from Q3 and Q4
3 (a) Test convergence of the series (any one) $i) \ \ i) \ \ \dfrac {1}{\sqrt{2}}+ \dfrac {1}{\sqrt{9}}+ \dfrac {1}{\sqrt{28}}+ \dfrac {1}{65}+ \cdots \ \cdots \\ ii) \ 1-\dfrac {1}{\sqrt{2}}+ \dfrac {1}{\sqrt{3}}- \dfrac {1}{\sqrt{4}}+ \cdots \cdots$
4 M
3 (b) Prove that $\log (1+ \sin x) = x - \dfrac {x^2}{2}+ \dfrac {x^3}{6}- \dfrac {x^4}{12}+ \cdots$
4 M
3 (c) Find nth derivative of $\dfrac {x^2}{(x-1)(x-2)}$
4 M

Solve any one:
4 (a) i) Evaluate $\lim_{x\to 0} \ \log_{\tan x} \ \tan 4x$ ii) Find the value of a and b if $\lim_{n \to 0} \left [ x^{-3} \sin x + ax^{-2}+b \right ]=0$
4 M
4 (b) Using Taylor's theorem expand 49-69x+42x2+11x3+x4 in powers of (x+2).
4 M
4 (c) If y=sin log (x2+2x+1), then prove that
(x+1)2yn+2+(2n+1)(x+1)yn+1+(n2+4)yn=0
4 M

Solve any two of the following:
5 (a) If u=log (x3+y3-x2y-xy2) then prove that x2 uxx+2xy uxy+ y2uyy=-3.
7 M
Answer any one question from Q5 and Q6
5 (b) If x=u+v+w, y=uv+vw+wu, z=uvw and ϕ is a function of x,y,z then prove that $u \dfrac {\partial \phi}{\partial u} + v \dfrac {\partial \phi}{\partial v}+ w \dfrac {\partial \phi}{\partial w} = x \dfrac {\partial \phi}{\partial x} + 2y \dfrac {\partial \phi}{\partial y}+ 3z \dfrac {\partial \phi}{\partial z}$
6 M
5 (c) $If \ ux+vy=0 \ and \dfrac {u}{v}+ \dfrac {v}{y}=1, \ then\ prove \ that \ \left ( \dfrac {\partial u}{\partial x} \right )y - \left (\dfrac {\partial v}{\partial y} \right )x = \dfrac {x^2 + y^2}{y^2 - x^2}$
6 M

Solve any two of the following:
6 (a) $If \ u=\cos \left (\dfrac {xy}{x^2+y^2} \right )+ \sqrt{x^2 + y^2}+ \dfrac {xy^2}{x+y}$ then find the value of xux+yuy at (3, 4).
7 M
6 (b) $if \ x=\dfrac {\cos \theta}{u}, \ y=\dfrac {\sin \theta}{u},$ Then prove that $u \dfrac {\partial z}{\partial u}- \dfrac {\partial z}{\partial \theta} = (y-x) \dfrac {\partial z}{\partial x} - (y-x) \dfrac {\partial z}{\partial y}$
6 M
6 (c) If u=(x2-y2) f(xy), then show that uxx-uyy=(x4-y4) f'(xy).
6 M

Answer any one question from Q7 and Q8
7 (a) $If \ x=r \sin \theta \cos \phi, \ y=r \sin \theta \sin \phi, \ z=r \cos \theta \ find \ \dfrac {\partial (x,y,z)}{\partial (r, \theta, \phi)}$
4 M
7 (b) Examine for functional dependence $u=\sin^{-1}x+\sin^{-1}y, \ v=x\sqrt{1-y^2}+y \sqrt{1-x^2}$ if dependent find the relation between them.
4 M
7 (c) The area of a triangle ABC is calculated from the formula Δ=1/2 bc sin A. Errors of 1%, 2% and 3% respectively are made in measuring b,c,A. If the correct value of A is 30°, find the percentage error in the calculated value of area of triangle.
5 M

8 (a) $If \ u^2 +xv^2 -uxy=0, \ v^2-xy^2+2uv+u^2=0, \ find \ \dfrac {\partial u}{\partial x}$ by choosing u, v as dependent and x, y as independent variables.
4 M
8 (b) Show that $u = \dfrac {x+y}{1-xy} , \ v=\tan^{-1} x+\tan^{-1}y$ are functionally dependent and find the relation between them.
4 M
8 (c) Find all the stationary values of the function $f(x,y)=x^3 +3 xy^3 - 15 x^2 - 15 y^2 + 72 x.$ Find maximum value of f(x,y) at suitable point.
5 M

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