RGPV Engineering Mathematics - I - December 2012 Exam Question Paper

RGPV First Year Engineering (Set B) (Semester 2)
Engineering Mathematics - I
December 2012

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 sin x in powers of (x-?/2). Hence. Find the value of sin 91° correct to 4 decimal places.

7 M

1 (b) Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.

7 M

2 (a) if u = x?(y/x) + ?(y/x), Prove that

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}} = 0

$x^2\dfrac {\partial^2 u}{\partial x^2}+2xy\dfrac {\partial^2u}{\partial x \partial y}+y^2\dfrac {\partial^2u}{\partial y^2}=0$

7 M

2 (b) Show that the radius of curvature at any point on the cardioid.

r = a (1 - \cos θ) i s 2 / 3 \sqrt{2 a r}

$r=a(1-\cos \theta)\ is \ 2/3 \sqrt{2ar}$

7 M

Answer any one question from Q3 & Q4

3 (a)

E v a l u a t e lim_{n \to \infty} {\frac{n!}{n^{n}}} y n

$Evaluate \ \lim_{n\rightarrow \infty}\left \{\dfrac {n!}{n^n} \right \}yn$

7 M

3 (b) Find the whole area of astroid x^u3 + y^u3 = a^u3

7 M

4 (a) Find, by triple integration, the volume of the sphere
x² + y² + z² = a²

7 M

4 (b)

P r o v e T h a t β (m, n) = \frac{Γ (m) Γ (n)}{Γ (m + n)}

$Prove \ That \ \beta(m,n)= \dfrac {\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$

7 M

Answer any one question from Q5 & Q6

5 (a) Solve the differential equation.

\frac{d^{2} y}{d x^{3}} - 3 \frac{d^{2} y}{d x^{2}} + 4 \frac{d y}{d x} - 2 y = e^{x} + \cos x

$\dfrac {d^2y}{dx^3}-3\dfrac {d^2y}{dx^2}+4\dfrac {dy}{dx}-2y=e^x+\cos x$

7 M

5 (b) Solve the following differential equation by method of variation of parameters
(D² + a²)y-sec ax.

7 M

6 (a) Solve the differential equation.

x^{2} \frac{d^{2} y}{d x^{2}} + 2 x \frac{d y}{d x} - 12 y = x^{3} \log x

$x^2\dfrac {d^2y}{dx^2}+2x\dfrac {dy}{dx}-12y=x^3 \log x$

7 M

6 (b) Solve

\frac{d x}{d t} - 7 x + y = 0 \frac{d y}{d t} - 2 x - 5 y = 0

$\dfrac {dx}{dt}-7x + y=0 \\ \dfrac {dy}{dt}-2x-5y=0$

7 M

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{matrix} 2 & 3 & - 1 & - 1 \\ 1 & - 1 & - 2 & - 4 \\ 3 & 1 & 3 & - 2 \\ 6 & 3 & 0 & - 7 \end{matrix}]

$A=\begin{bmatrix}2 &3 &-1 &-1 \\ 1&-1 &-2 &-4 \\ 3&1 &3 &-2 \\ 6&3 &0 &-7 \end{bmatrix}$

7 M

7 (b) For the matix

A = [\begin{matrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 0 & - 1 & - 1 \end{matrix}]

$A=\begin{bmatrix}1 &1 &2 \\ 1&2 &3 \\ 0&-1 &-1 \end{bmatrix}$ Find non-singular matrices P and Q such that PAQ is in the normal form. Also find rank of A.

7 M

8 (a) Determine the eigen values and the corresponding eigen vectors of the matrix

A = [\begin{matrix} 8 & - 6 & 2 \\ - 6 & 7 & - 4 \\ 2 & - 4 & 3 \end{matrix}]

$A=\begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&-4 &3 \end{bmatrix}$

7 M

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 M

Answer any one question from Q9 & Q10

9 (a) Prove that the proposition
P ? (q ? r) ? (p ? q) ? r is a futology.

7 M

9 (b) Define the tree and prove that a tree T with n vertices has exactly (n-1) edges.

7 M

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 M

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 M

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