Engineering Mathematics-1 - Dec 2015

First Year Engg (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

1 (a) Examine for consistency of the system of equations:
2x-3y+5z=1
3x+y-z=2
x+4y-6z=1
if consistent solve it.(4 marks) 1 (b) Find the eigenvalues of matrix: $$ \begin{bmatrix} 1 & 0 &-4 \\0 &5 &4 \\-4 &4 &3 \end{bmatrix} $$ hence find eigenvector corresponding to highest eigenvalue.(4 marks) 1 (c) Find the complex number z if amp$(z+2i) = \dfrac {\pi}{4} \text { and amp}(z-2i) = \dfrac {3\pi }{4} . $(4 marks) 2 (a) Examine for the linear dependence or independence. If dependent, find the relation among the following vectors (1, 1, 1), (1, 2, 3), (2, 3, 8).(4 marks) 2 (b) Find all values of (1+i)^1/4.(4 marks) 2 (c) If sin (α+iβ)=x+iy, then prove that: $$ i) \ \dfrac {x^2}{\cosh^2 \beta} + \dfrac {y^2}{\sinh^2 \beta} = 1, \\ ii) \ \dfrac {x^2}{\sin^2 \alpha} - \dfrac {y^2}{\cos^2 \alpha} = 1 $$(4 marks)

Test convergence of the series (any one):

3 (a) (i) $$ \dfrac {1}{2+3} + \dfrac {2}{2+3^2} + \dfrac {3}{2+3^3}+ \cdots \ \cdots $$(4 marks) 3 (a) (ii) $$ \sum^{\infty}_{n=1} \dfrac {\sqrt{n+1} + \sqrt{n}}{n^3}$$(4 marks) 3 (b) Prove that: $$ e^{x\cos x} = 1 + x+ \dfrac {x^2}{2} - \dfrac {x^3}{3} - \cdots \ \cdots $$(4 marks) 3 (c) Find n^th derivative of $$ \dfrac {x+1}{(x-1)(x+2)(x-3)} $$(4 marks)

Solve any one:

4 (a) (i) $$ \lim_{x\to 0} (\cos x)^{1/x^2}$$(4 marks) 4 (a) (ii) $$ \lim_{x\to 0 } \dfrac {e^{ax} - e^{-ax}} {\log (1+bx)} $$(4 marks) 4 (b) Using Taylor's theorem, expand x⁴-5x³+5x²+x+2 in powers of x-2.(4 marks) 4 (c) if y=cos(m log x), then prove that: $$ x^2 y_{n+2} + (2n+1) xy_{n+1} + (m^2 + n^2) y_n = 0 $$(4 marks)

Solve any two:

5 (a) If u=tan(y+ax)+(y-ax)^3/2, where a is a constant, then show that: $$ \dfrac {\partial^2 u}{\partial x^2} = a^2 \dfrac {\partial ^2 u}{\partial y^2} $$(6 marks) 5 (b) If $$ \displaystyle u=x^3 f\left ( \dfrac {y}{x} \right )+ \dfrac {1}{y^3} \phi \left ( \dfrac {x}{y} \right ), $$ prove that: $$ x^2 \dfrac {\partial ^2 u }{\partial x^2} + 2xy \dfrac {\partial^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^2 u }{\partial y^2}+ x \dfrac {\partial u}{\partial y} + y \dfrac {\partial u}{\partial y} = 9u. $$(7 marks) 5 (c) If u=f(x-y, y-z, z-x), then find the value of: $$ \dfrac {\partial u}{\partial x} + \dfrac {\partial u}{\partial y} + \dfrac {\partial u}{\partial z} $$(6 marks)

Solve any two:

6 (a) If u=mx+ny, v=nx-my, where m, n are constants, then find the value of: $$ \left ( \dfrac {\partial u}{\partial x} \right )_y \cdot \left ( \dfrac {\partial y}{\partial v} \right )_x \cdot \left ( \dfrac {\partial x}{\partial u} \right )_v \cdot \left ( \dfrac {\partial v}{\partial y} \right )_u $$(6 marks) 6 (b) $$ \text{If } u=\tan^{-1} \left [ \dfrac {x^3 + y^3}{x+y} \right ], $$ then prove that: $$ x^2 \dfrac {\partial ^2 u }{\partial x^2} + 2xy \dfrac {\partial ^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^2 u }{\partial y^2} = \sin 2u \left [ 1-4 \sin ^2 u \right ]. $$(7 marks) 6 (c) $$ \text{If } z=f(x,y), \ x=\dfrac {\cos u}{v}, \ y = \dfrac {\sin u}{v},$$, then prove that: $$ v \dfrac {\partial z}{\partial v} - \dfrac {\partial z}{\partial u} = (y-x) \dfrac {\partial z}{\partial x} - ( y +x) \dfrac {\partial z}{\partial y}. $$(6 marks) 7 (a) $$ \text{If } u= \dfrac {y-x}{1+xy}, \ v=\tan^{-1} y -\tan^{-1}x \\ \text{find } \dfrac {\partial (u,v)}{\partial( x, y)} $$(4 marks) 7 (b) Prove that:
u=y+z, v=x+2x², w=x-4yz-2y²,
are functionally dependent and find relation.(5 marks) 7 (c) As dimensions of a triangle ABC are varied, shown that the maximum value of cos A cos B cos C is obtained when the triangle is equilateral.(4 marks) 8 (a) If u+v²=x, v+w²=y, w+u²=z find $\dfrac {\partial u}{\partial x}$.(4 marks) 8 (b) In estimating the cost of a pile of bricks measured 2m×15m×1.2 m, the top of the pile is streched 1% beyond the standard length. If the count is 450 bricks in 1 cubic meter and bricks cost Rs. 450 per thousand, find the approximate error in cost.(5 marks) 8 (c) Find the minimum value of x²+y², subject to the condition ax+by=c.(4 marks)