Question: Engineering Mathematics -I : Dec 2012 - First Year Engineering (Set A) (Semester 1) | Rajiv Gandhi Proudyogiki Vishwavidyalaya (RGPV)
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Engineering Mathematics -I - Dec 2012

First Year Engineering (Set A) (Semester 1)

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-π/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 = xΦ(y/x) + φ(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 xu3 + yu3 = au3(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)