Engineering Mathematics - I - Dec 2012

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 sin x in powers of (x-?/2). Hence. Find the value of sin 91° correct to 4 decimal places.(7 marks) 1 (b) Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.(7 marks) 10 (a) Let (B, +, �, ') be a Boolean algebra and a, b, be any two elements of B. Then prove that
i) (a+b)'=a'�b'
ii) (a�b)'=a'+b'(7 marks) 10 (b) Define the following terms:
i) Support of a fuzzy set.
ii) Complement of a fuzzy set.
iii) Union of two fuzzy set.
iv) Intersection of two fuzzy set.(7 marks) 2 (a) if u = x?(y/x) + ?(y/x), Prove that $$ x^2\dfrac {\partial^2 u}{\partial x^2}+2xy\dfrac {\partial^2u}{\partial x \partial y}+y^2\dfrac {\partial^2u}{\partial y^2}=0 $$(7 marks) 2 (b) Show that the radius of curvature at any point on the cardioid. $$ r=a(1-\cos \theta)\ is \ 2/3 \sqrt{2ar} $$(7 marks)

Answer any one question from Q3 & Q4

3 (a) $$ Evaluate \ \lim_{n\rightarrow \infty}\left \{\dfrac {n!}{n^n} \right \}yn $$(7 marks) 3 (b) Find the whole area of astroid x^u3 + y^u3 = a^u3(7 marks) 4 (a) Find, by triple integration, the volume of the sphere
x² + y² + z² = a²(7 marks) 4 (b) $$ Prove \ That \ \beta(m,n)= \dfrac {\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$$(7 marks)

Answer any one question from Q5 & Q6

5 (a) Solve the differential equation. $$ \dfrac {d^2y}{dx^3}-3\dfrac {d^2y}{dx^2}+4\dfrac {dy}{dx}-2y=e^x+\cos x $$(7 marks) 5 (b) Solve the following differential equation by method of variation of parameters
(D² + a²)y-sec ax.(7 marks) 6 (a) Solve the differential equation. $$ x^2\dfrac {d^2y}{dx^2}+2x\dfrac {dy}{dx}-12y=x^3 \log x $$(7 marks) 6 (b) Solve $$ \dfrac {dx}{dt}-7x + y=0 \\ \dfrac {dy}{dt}-2x-5y=0 $$(7 marks)

Answer any one question from Q7 & Q8

7 (a) Find the normal form of the matrix A and hence find the its rank, where $$ A=\begin{bmatrix}2 &3 &-1 &-1 \\ 1&-1 &-2 &-4 \\ 3&1 &3 &-2 \\ 6&3 &0 &-7 \end{bmatrix} $$(7 marks) 7 (b) For the matix $$ A=\begin{bmatrix}1 &1 &2 \\ 1&2 &3 \\ 0&-1 &-1 \end{bmatrix}$$ Find non-singular matrices P and Q such that PAQ is in the normal form. Also find rank of A.(7 marks) 8 (a) Determine the eigen values and the corresponding eigen vectors of the matrix $$ A=\begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&-4 &3 \end{bmatrix}$$(7 marks) 8 (b) Test the consistency of the following system of equation and solve using matrix methods.
5x + 3y + 7z =4
3x + 26y + 2z = 9
7x + 2y + 10z = 5(7 marks)

Answer any one question from Q9 & Q10

9 (a) Prove that the proposition
P ? (q ? r) ? (p ? q) ? r is a futology.(7 marks) 9 (b) Define the tree and prove that a tree T with n vertices has exactly (n-1) edges.(7 marks)