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SPPU First Year Engineering (Semester 1)
Engineering Mathematics-1
May 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) 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.
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]
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.
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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