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Engineering Mathematics-1 : Question Paper Jun 2015 - First Year Engineering (Semester 1) | Pune University (PU)

## Engineering Mathematics-1 - Jun 2015

### First Year Engg (Semester 1)

TOTAL MARKS:
TOTAL TIME: HOURS

### Answer any one question from Q1 and Q2

1 (a) Find the Eigen values and Eigen vector corresponding to minimum Eigen value for the matrix: $$A= \begin{bmatrix}1 &0 &-1 \\1 &2 &1 \\2 &2 &3 \end{bmatrix}$$(4 marks) 1 (b) Determine the value for λ for which the equations: $$3x_1 + 2x_2 + 4x_3 =3 \\ x_1 + x_2 + x_3= \lambda \\ 5x_1 + 4x_2 + 6x_3 = 15$$ are consistent. Find also the corresponding solution.(4 marks) 1 (c) If z1, z2, z3 are the vertices of an isosceles triangle right angled at z2, prove that: $$z^2_1 + z^2_3 + 2 z^2_2 = 2z_2 (z_1 + z_3)$$(4 marks) 2 (a) If $$2 \cos \theta = x + \dfrac {1} {x} ,$$ Prove that: $$2 \cos r \theta = x^r + \dfrac {1}{x^r}$$(4 marks) 2 (b) if Y-log tan x, prove that: $$i) \ \sin h ny = \dfrac {1}{2} (\tan^n x - \cot ^n x),$$ (ii) 2coshny cosec2x=cosh (n+1) y+cosh(n-1)y.(4 marks) 2 (c) Examine for linear dependence or independence for the given vectors and if dependence, find the relation between Them:
X1 = (1, ?1, 2, 2),
X2 = (2, -3, 4, -1),
X3 = (-1, 2, -2, 3).
(4 marks)

### Answer any one question from Q3 and Q4

3 (a) Test convergence of the series (any one): $$i) \ \ \sum^{\infty}_{n=1} \left ( \dfrac {2n+1}{3n+4} \right )5^n \\ ii) \ \dfrac {1}{1+\sqrt{2}} + \dfrac {2}{1+2\sqrt{3}} + \dfrac {3}{1+3 \sqrt{4}}+ \cdots \ \cdots$$(4 marks) 3 (b) Prove that $$\sin x \cos h \ x= x+ \dfrac {1}{3}x^3 - \dfrac {1}{30}x^5 - \cdots \ \cdots$$(4 marks) 3 (c) Find nth derivative of:
e2 sin h 3x cos 4x.
(4 marks)
4 (a) Solve any one: $$i) \ \ \lim_{x \to a}(x-a)^{(x-a)} \\ ii) \ \lim_{x\to \pi /2} (\sec x - \tan x)$$(4 marks) 4 (b) Using Taylor's theorem expand: 2x3+3x2-8x+7 in powers of x-2.(4 marks) 4 (c) If y=etan-1x, then show that: $$\left ( 1+x^2 \right )y_{n+1} + (2nx-1)y_n + n(n-1)y_{n-1}=0$$(4 marks)

### Solve any two of the following:

5 (a) Find the value of n for which:
z=A e-gx sin (nt - gx),
satisfies the partial differential equation: $$\dfrac {\partial z}{\partial t} = \dfrac {\partial^2 z}{\partial x^2}$$
(6 marks)

### Answer any one question from Q5 and Q6

5 (b) $$If \ u=\dfrac {x^4 + y^4}{x^2y^2}+ x^6 \tan ^{-1} \left [ \dfrac {x^2 + y^2} {x^2 + 2xy} \right ],$$ find the value of: $$x^2 \dfrac {\partial ^2 u}{\partial x^2}+ 2xy \dfrac {\partial ^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^2 u}{\partial y^2}+ \dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}$$ at x=1, y=2.(7 marks) 5 (c) If z=f(u,v) and $$u=\log (x^2 + y^2), \ v=\dfrac {y}{x},$$ show that: $$x \dfrac {\partial y}{\partial y} - y \dfrac {\partial z}{\partial x} = (1+v^2) \dfrac {\partial z}{\partial v}$$(6 marks)

### Solve any two of the following:

6 (a) $$If \ x=\dfrac {\cos \theta}{r}, \ y=\dfrac {\sin \theta}{r},$$ find the value of: $$\left ( \dfrac {\partial x}{\partial r} \right )_\theta \left ( \dfrac {\partial r}{\partial x} \right )_y + \left( \dfrac {\partial y} {\partial r} \right )_\theta \left ( \dfrac {\partial r}{\partial y} \right )_x$$(6 marks) 6 (b) $$If \ u=\sin^{-1}\sqrt{\dfrac {x^2+ y^2}{x+y}}$$ Show that: $$x^2 \dfrac {\partial^2 u}{\partial x^2} + 2xy \dfrac {\partial^2 u}{\partial x \partial y} + y^2 \dfrac {\partial^2 u}{\partial y^2} = \dfrac {1}{4} \tan u [\tan^2 u-1]$$(7 marks) 6 (c) If z=f(u,v) and u=x cos t ? y sin t, v=x sin t + y cos t, where t is a constant, prove that: $$x \dfrac {\partial z}{\partial x} + y \dfrac {\partial z}{\partial y}= u \dfrac {\partial z}{\partial u}+ v \dfrac {\partial z}{\partial v}.$$(6 marks)

### Answer any one question from Q7 and Q8

7 (a) $$If \ u^3 + v^3 = x + y, \ u^2+v^2 = x^3 + y^3, \ find \ \dfrac {\partial (u,v)}{\partial (x, y)}$$(4 marks) 7 (b) Determine whether the following functions are functionally dependent. If functionally dependent, find the relation between them:
u=sin x + sin y, v=sin (x+y).
(4 marks)
7 (c) Examine maxima and minima of the following function and find their extreme values:
(x2+y2+6x+12).
(5 marks)
8 (a) $$If \ x=u+v, \ y=v^2+w^2, \ z=w^3+u^3, \ show \ that: \\ \dfrac {\partial u}{\partial x}= \dfrac {vw}{vw+u^2}$$(4 marks) 8 (b) If ez=sec x cos y and error of magnitude h and -h are made in estimating x and y, where x and y are found to be π/3 and π6 respectively, find the corresponding error in z.(5 marks) 8 (c) Find the minimum distance from origin to the plane:
3x+2y+z=12.
(4 marks)