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RGPV First Year Engineering (Set A) (Semester 1)
Engineering Mathematics -I
May 2013
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) $(\sin^{-1}x)^2=\dfrac {2}{2!}x^2+\dfrac {2.2^2}{4!}x^4+\dfrac {2.2^2.4^2}{6!}x^6+.... \\ and \ hence \ deduce \\\theta^2=2\dfrac {\sin^2\theta}{2!}+2^2\dfrac {2\sin^4\theta}{4!}+2^2.4^2\dfrac {2\sin^6\theta}{6!}+....$
7 M
1 (b) if u(x,y,z)=log(tan x + tan y + tan z), prove that $\sin 2x\dfrac {\partial u}{\partial x}+\sin 2y\dfrac {\partial u}{\partial y}+\sin 2z\dfrac {\partial u}{\partial z}=2$
7 M

2 (a) Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.
7 M
2 (b) Determine the curvature of the parabola y2=2 px at
(i) an arbitary point (x,y).
(ii) the point (P/2, P) and
(iii) the point (0,0)
7 M

Answer any one question from Q3 & Q4
3 (a) Evaluate by expressing the limit of a sum in the form of a definite integral: $\lim_{x\rightarrow \infty} \left [ \left ( 1+\dfrac {1}{n^2} \right )\left (1+ \dfrac {2^2}{n^2} \right ) \left (1+ \dfrac {3^2}{n^2} \right ).... \left (1+ \dfrac {n^2}{n^2} \right )\right ]^{1/n}$
7 M
3 (b) Define B(m,n). Prove that
B(m,n)=B(m+1,n)+B(m,n+1)m,n>0.
7 M

4 (a) Evaluate the following integral by changing the order of integration : $\int^1_0\int^{\sqrt{2-x^2}}_x \dfrac {xdydx}{\sqrt{x^2+y^2}}$
7 M
4 (b) Find the volume cut from the sphere x2 + y2 + z2=a2 by the cylinder x2 + y2=ax.
7 M

Answer any one question from Q5 & Q6
5 (a) Solve (3x2y2 + 2xy)dx + (2x3y3 - x2)dy=0
7 M
5 (b) $Solve \ y-x=x\dfrac {dy}{dx}+\left (\dfrac {dy}{dx} \right )^2$
7 M

6 (a) $Solve \ \dfrac {d^2y}{dx^2}-6 \dfrac {dy}{dx}+13y=8e^{3x}\sin 4x+2^x$
7 M
6 (b) $Solve \ \dfrac {dx}{dt}+4x+3y=t \\ \dfrac {dy}{dt}+2x+5y=e^t$
7 M

Answer any one question from Q6 & Q7
7 (a) Define rank of a matrix. Find the rank of matrix A, where $A=\begin{bmatrix}1^2 &2^2 &3^2 &4^2 \\ 2^2&3^2 &4^2 &5^2 \\ 3^2&4^2 &5^2 &6^2 \\ 4^2 &5^2 &6^2 &7^2 \end{bmatrix}$
7 M
7 (b) Solve completely the system of equation 2w+3x-y-z=0, 4w-6x-2y+2z=0, -6w+12x+3y-4z=0
7 M

8 (a) Determine the eigen values and eigen vectors of the matrix $A=\begin{bmatrix} -2&2 &-3 \\ 2&1 &-6 \\ -1&-2 &0 \end{bmatrix}$
7 M
8 (b) Show that Caley-Hamilton theorem is satisfied by the matrix A. $where \ A=\begin{bmatrix}0 &0 &1 \\ 3& 1&0 \\ -2&1 &4 \end{bmatrix}$ Hence find A-1.
7 M

9 (a) Write the following function into disjunctive normal form of 3 variable x,y,z:
(i) x' + y'
(ii) xy' + x'y
7 M
9 (b) In a Boolean algebra B. Prove that the identity elements 0,1 ? B are unique and prove 0'=1,1'=0
7 M

10 (a) Define the following terms giving example:
(i) Support of fuzzy set.
(ii) Complement of a fuzzy set.
(iii) Union of two fuzzy sets.
(iv) Intersection of two fuzzy sets.
7 M
10 (b) Prove that the number of vertices of odd degree in a graph is always even.
7 M

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