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RGPV First Year Engineering (Set A) (Semester 1)
Engineering Mathematics -I
December 2011
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 & Q2
1 (a) Expand ea sin-1x in ascending power of x.
7 M
1 (b) If p= x cos a+y sin a, touches the curve $\left ( \dfrac {x}{a} \right )^{\frac {n}{n-1}} + \left ( \dfrac {y}{b} \right )^{\frac {n}{n-1}}=1$ Prove that: pn=(a cos a)n + (b sin a)n
7 M

2 (a) Show that the radius of curvature at any point of the cycloid $x=a (\theta + \sin \theta ), y=a (1-\cos \theta) \ is \ 4 a \cos \left ( \dfrac {\theta}{2} \right )$
7 M
2 (b) $If \ u=\sin^{-1} \left ( \dfrac {x+y}{\sqrt{x}+ \sqrt{y}} \right ), \ prove \ that : \\ i) \ x\dfrac {\partial u}{\partial x} + y \dfrac {\partial u}{\partial y} = \dfrac {1}{2} \tan u \\ ii) \ x^2 \dfrac {\partial^2 u}{\partial x^2}+ 2xy \dfrac {\partial^2 u}{\partial x \partial y} + y^2 \dfrac {\partial^2 u}{\partial y^2} = - \dfrac {\sin u \cos 2 u}{4 \cos^3 u}$
7 M

Answer any one question from Q3 & Q4
3 (a) Find the limit as n→∞ of the series: $\dfrac {1}{n+1} + \dfrac {1}{n+2}+ \dfrac {1}{n+3} + \cdots \ \cdots + \dfrac {1}{2n}$
7 M
3 (b) Find the volume common to the cylinders
x2+y2=a2, x2+z2=a2
7 M

4 (a) Evaluate: $\displaystyle \int^\infty_0 \int^{x}_0 xe^{-x^2/y}dy \ dx$ by changing the order integration.
7 M
4 (b) Prove that:
$$(i)\ \dfrac{\beta (m+1, n)}{m} = \dfrac{\beta (m,n+1)}{n} = \dfrac{\beta (m,n)}{m+n}\\ (ii)\ \Gamma (m) \Gamma \left(m+\dfrac {1}{2} \right) = \dfrac {\sqrt{\pi}}{2^{2m-1}} \Gamma(2m)$$
7 M

Answer any one question from Q5 & Q6
5 (a) Solve the equation: $(y-x) \dfrac{dy}{dx} = a^2$
7 M
5 (b) Solve the equation: $\dfrac {d^2 y}{dx^2} + 4y = \sec 2x \\$ by the method of variation of parameters.
7 M

6 (a) Solve the equation: $x^2 \dfrac {d^2 y}{dx^2} -2x \dfrac {dy}{dx} - 4y = x^2 + \log x$
7 M
6 (b) Solve the simultaneous equations: $\dfrac {dx}{dt} + y = \sin t \\ \dfrac {dy}{dt}+x \cos t$
7 M

Answer any one question from Q7 & Q8
7 (a) Reduce the matrix: $A= \begin{bmatrix} 2 &3 &4 &5 \\3 &4 &5 &6 \\4 &5 &6 &7 \\9 &10 &11 &12 \end{bmatrix}$ to normal form and find its range.
7 M
7 (b) Find the eigen values and eigen vectors of the matrix: $A=\begin{bmatrix}2 &1 &1 \\1 &2 &1 \\0 &0 &1 \end{bmatrix}$
7 M

8 (a) Text for consistency and solve:
5x+3y+7z=4
3x+26y+2z=9
7x+2y+10z=5
7 M
8 (b) Verify Cayley-Hamilton theorem for the matrix: $A=\begin{bmatrix} 1 &2 &1 \\0 &1 &-1 \\3 &-1 &1 \end{bmatrix}$ and find its inverse.
7 M

Answer any one question from Q9 & Q10
9 (a) Define the following terms with examples:
i) Simple graph
ii) Degree of a vertex
iii) Isomorphic graphs
iv) Spanning tree
7 M
9 (b) Express the following function into disjunctive normal form:
f(x,y,z)=(x+y+z)(x·y+x'·z)'
7 M

10 (a) Let X={a, b, c, d} be a universe of discourse and A, B be the fuzzy sets on X defined by: $A= \left \{ \dfrac {0.3}{a}, \dfrac {0.5}{b}, \dfrac {0.6}{c}, \dfrac {0.4}{d} \right \} \\ B= \left \{ \dfrac {0.2}{a}, \dfrac {0.6}{b}, \dfrac {0.3}{c}, \dfrac {0.7}{d} \right \}$ Find:
i) Height of A ∪ B
ii) α-cut of A ∩ B for \alpha;=0.4
(A ∪ B)'
iv) A' ∩ B'
7 M
10 (b) Prove that the number of vertices of odd degree in a graph in always even.
7 M

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