Engineering Mathematics - I - Dec 2011

First Year Engineering (Set B) (Semester 2)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.

Answer any one question from Q1 & Q2

1 (a) Expand e^{a sin-1x} in ascending power of x.(7 marks) 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 marks) 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 α=0.4
(A ∪ B)'
iv) A' ∩ B'(7 marks) 10 (b) Prove that the number of vertices of odd degree in a graph in always even.(7 marks) 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 marks) 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 marks)

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 marks) 3 (b) Find the volume common to the cylinders
x²+y²=a², x²+z²=a²(7 marks) 4 (a) Evaluate: $$ \displaystyle \int^\infty_0 \int^{x}_0 xe^{-x^2/y}dy \ dx $$ by changing the order integration.(7 marks) 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 marks)

Answer any one question from Q5 & Q6

5 (a) Solve the equation: $$ (y-x) \dfrac{dy}{dx} = a^2 $$(7 marks) 5 (b) Solve the equation: $$ \dfrac {d^2 y}{dx^2} + 4y = \sec 2x \\ $$ by the method of variation of parameters.(7 marks) 6 (a) Solve the equation: $$ x^2 \dfrac {d^2 y}{dx^2} -2x \dfrac {dy}{dx} - 4y = x^2 + \log x $$(7 marks) 6 (b) Solve the simultaneous equations: $$ \dfrac {dx}{dt} + y = \sin t \\ \dfrac {dy}{dt}+x \cos t $$(7 marks)

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 marks) 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 marks) 8 (a) Text for consistency and solve:
5x+3y+7z=4
3x+26y+2z=9
7x+2y+10z=5(7 marks) 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 marks)

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 marks) 9 (b) Express the following function into disjunctive normal form:
f(x,y,z)=(x+y+z)(x.y+x'.z)'(7 marks)