Engineering Mathematics-1 - May 2014

First Year Engg (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

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 marks) 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 marks) 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 marks) 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 marks) 2 (b) Show that $$ \log \left [ \dfrac {\sin (x+iy)}{\sin (x-iy)} \right ]= 2 i \tan ^{-1}(\cot x \tan hy) $$(4 marks) 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 marks)

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 marks) 3 (b) Expand x³+7x²+x-6 in powers of (x-3)(4 marks) 3 (c) $$ If \ y = \log \left ( x+ \sqrt{x^2+1} \right ) $$ prove that
(1+x²)y_n+2+(2n=1)xy_n+1+n²y_n=0.(4 marks) 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 marks) 4 (b) Prove that $$ \log (1+\tan x) = x - \dfrac {x^2}{2}+ \dfrac {2x^3}{3} \cdots \ \cdots $$(4 marks) 4 (c) Find n^th derivative of $$ y = \dfrac {x} {x-1) (x-2)(x-3) }\lt/span\gt\ltspan class='paper-ques-marks'\gt(4 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Solve any two of the following: \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (a)\lt/b\gt Find the value of n such that u=x\ltsup\gtn\lt/sup\gt (3\cos\ltsup\gt2\lt/sup\gty-1) satisfies the partial differential equation. $$ \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 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Answer any one question from Q5 and Q6 \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (b)\lt/b\gt If x=r cos θ, y=r sin kθ then prove that $$ 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 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt5 (c)\lt/b\gt $$ 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 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Solve any two of the following: \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (a)\lt/b\gt $$ 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} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(7 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (b)\lt/b\gt $$ 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 $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt6 (c)\lt/b\gt if f(x,y)=0, ϕ(x,z)=0 then prove that $$ \dfrac {\partial \phi}{\partial x} \dfrac {\partial f}{\partial y} \dfrac {dy}{\partial z} = \dfrac {\partial f}{\partial x}\dfrac {\partial \phi}{\partial z} $$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt -------------- \ltspan class='paper-comments'\gt ### Answer any one question from Q7 and Q8 \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt7 (a)\lt/b\gt if x=e\ltsup\gt4\lt/sup\gt sec u, y=e\ltsup\gt4\lt/sup\gt tan u, find $$ \dfrac {\partial (u,v)} {\partial (x,y)}(4 marks) 7 (b) Examine for functional dependence for $$ u = \dfrac {x}{y-z}, \ v=\dfrac {y}{z-x}, \ w=\dfrac {z}{x-y} $$(4 marks) 7 (c) Find extreme values of f(x,y)=x³+y³-3axy, a>0(5 marks) 8 (a) if x=cos θ - r sin θ, y=sin θ + r cos θ find $$ \dfrac {\partial r} {\partial x} $$(4 marks) 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 marks) 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 marks)