SPPU Engineering Mathematics-1 - December 2013 Exam Question Paper

SPPU First Year Engineering (Semester 1)
Engineering Mathematics-1
December 2013

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 the following system of equations for consistency and solve it, if consistent.
4x-2y+6z=8
x+y-3z=-1, 15x-3y+9z=21

4 M

1 (b) Examine the following vectors for Linear dependence. Find the relation between them, if dependent.
(2, -1, 3, 2), (1, 3, 4, 2) and (3, -5, 2, 2)

4 M

1 (c) If 2 cos ϕ=x+1/x, 2cosψ=y+1/y
prove that, x^{py^q+1/x^py^q=2 cos (pϕ + qψ)}

4 M

2 (a) Use de Moivre's theorem, to solve the equation
x⁷+x⁴+I (x³+1)=0

4 M

2 (b) If (1+ai)(1+bi)=p+iq, then prove that,
i) p tan [tan^-1 + tan^-1b]=q
ii) (1+a²)(1+b²)=p²+q²

4 M

2 (c) Reduce the following matrix A to its normal form and hence find its rank, where

A = | \begin{matrix} 2 & - 3 & 4 & 4 \\ 1 & 1 & 1 & 2 \\ 3 & - 2 & 3 & 6 \end{matrix} |

$A=\begin{vmatrix} 2 &-3 &4 &4 \\1 &1 &1 &2 \\3 &-2 &3 &6 \end{vmatrix}$

4 M

Answer any one question from Q3 and Q4

3 (a) Test convergence of the series (any one)

i) \sum_{n = 1}^{\infty} \frac{2^{n} + 1}{3^{n} + 1} i i) \frac{1 * 2}{3^{2} * 4^{2}} + \frac{3 * 4}{5^{2} * 6^{2}} + \frac{5 * 6}{7^{2} * 8^{2}} + \dots \dots

$i) \ \ \sum^{\infty}_{n=1} \dfrac {2^n + 1}{3^n +1} \\ ii) \ \ \dfrac {1* 2}{3^2 * 4^2} + \dfrac {3 * 4}{5^2 *6^2}+ \dfrac {5* 6}{7^2 * 8^2} + \cdots \ \cdots$

4 M

3 (b) Expand 40+53(x-2)+19(x-2)²+2(x-2)³ in ascending powers of x

4 M

3 (c) If y=x^{n log x then, prove that $Y_{n+1} = \dfrac {n!}{x}$}

4 M

4 (a) Solve any one:

i) E v a l u a t e lim_{x \to \infty} (\cot x)^{\sin x}

$i) \ \ Evaluate \ \lim_{x\to \infty} (\cot x)^{\sin x}$ ii) Find the values of a and b such that,

lim_{x \to 0} \frac{a \cos x - a + b x^{2}}{x^{4}} = \frac{1}{12}

$\lim_{x\to 0}\dfrac {a \cos x - a + b \ x^2}{x^4}= \dfrac {1}{12}$

4 M

4 (b) prove that,

e^{x} \tan x = x + x^{2} + \frac{5 x^{2}}{6} + \frac{x^{4}}{2} + \dots \dots

$e^x \tan x=x +x^2 + \dfrac {5x^2}{6} + \dfrac {x^4}{2}+ \cdots \ \cdots$

4 M

4 (c)

i f Y = \frac{x}{(x + 1)^{4}} f i n d Y_{n}

$if \ Y=\dfrac {x} {(x+1)^4} \ find \ Y_n$

4 M

Solve any two of the following:

5 (a)

V e r i f y \frac{\partial^{2} u}{\partial x \partial y} = \frac{\partial^{2} u}{\partial y \partial x} f o r u = \tan^{- 1} [\frac{y}{x}]

$Verify \ \dfrac {\partial^2 u}{\partial x \partial y} = \dfrac {\partial ^2 u}{\partial y \partial x} \ for \ u =\tan ^{-1}\left [ \dfrac {y}{x} \right ]$

7 M

5 (b) if x=u tan v, y=u sec v prove that

{(\frac{\partial u}{\partial x})}_{y} {(\frac{\partial v}{\partial x})}_{y} = {(\frac{\partial u}{\partial y})}_{x} {(\frac{\partial v}{\partial y})}_{x}

$\left ( \dfrac {\partial u} {\partial x} \right )_y \left ( \dfrac {\partial v}{\partial x} \right )_y = \left ( \dfrac {\partial u}{\partial y} \right )_x \left ( \dfrac {\partial v}{\partial y} \right )_x$

7 M

Answer any one question from Q5 and Q6

5 (c)

I f u = \frac{x^{3} + y^{3}}{y \sqrt{x}} + \frac{1}{x^{7}} \sin^{- 1} (\frac{x^{2} + y^{2}}{2 x y})

$If \ u = \dfrac {x^3 +y^3}{y \sqrt{x}} + \dfrac {1}{x^7} \sin ^{-1} \left ( \dfrac {x^2 + y^2}{2xy} \right )$ Then, find the value of

x^{2} \frac{\partial^{2} u}{\partial x^{2}} + 2 x y \frac{\partial^{2} u}{\partial x \partial y} + y^{2} \frac{\partial^{2} u}{\partial y^{2}} A t p o i n t (1, 1)

$x^2 \dfrac {\partial^2 u}{\partial x^2}+ 2xy \dfrac {\partial ^2 u}{\partial x \partial y} + y^2 \dfrac {\partial ^ 2 u}{\partial y^2} \ At \ point \ (1,1)$

7 M

Solve any two of the following:

6 (a) If u=(x²-y²) f(xy) then show that
u_xx+u_yy=(x⁴-y4) f''(xy)

7 M

6 (b) Verify Euler's theorem for homogeneous functions
F(x,y,z)=3x²yz+5xy2z+4z⁴

7 M

6 (c) If x=u+v+w, y=uv+uw+vw, z=uvw and F is function of x,y,z then prove that,

x \frac{\partial F}{\partial x} + 2 y \frac{\partial F}{\partial y} + 3 z \frac{\partial F}{\partial z} = u \frac{\partial F}{\partial u} + v \frac{\partial F}{\partial v} + w \frac{\partial F}{\partial w}

$x \dfrac {\partial F}{\partial x} + 2y \dfrac {\partial F}{\partial y} +3z \dfrac {\partial F}{\partial z} = u \dfrac {\partial F} {\partial u} + v \dfrac {\partial F}{\partial v}+ w \dfrac {\partial F}{\partial w}$

7 M

Answer any one question from Q7 and Q8

7 (a) If x=v²+w², y=w²+u², z=u²+v² Find

\frac{\partial (u, v, w)}{\partial (x, y, z)}

$\dfrac {\partial (u,v,w)} {\partial (x,y,z)}$

4 M

7 (b) Examine for functional dependence for u=x+y+z, v=x²+y²+z³, w=x³+y³+z³-3xyz.

4 M

7 (c) Find the extreme values of f(x,y)=x³+y-3axy, a>0

5 M

8 (a) If u²+xv²=x+y and v²+yu²=x-y find

\frac{\partial v}{\partial y}

$\dfrac {\partial v} {\partial y}$

4 M

8 (b) The resistance R of a circuit was calculated using the formula I=E/R. If there is an error of 0.1 Amp in reading I and 0.5 Volts in E, find the corresponding percentage error in R when I=15 Amp and E=100 Volts

4 M

8 (c) Divide 24 into three parts such that, the continued product of the first, square of the second and cube of the third may be maximum. Use Lagrange's method.

5 M

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