Question: Engineering Mathematics -I : May 2013 - First Year Engineering (Set A) (Semester 1) | Rajiv Gandhi Proudyogiki Vishwavidyalaya (RGPV)
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## Engineering Mathematics -I - May 2013

### 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) $$(\sin^{-1}x)^2=\dfrac {2}{2!}x^2+\dfrac {2.2^2}{4!}x^4+\dfrac {2.2^2.4^2}{6!}x^6+.... \\ and \ hence \ deduce \\\theta^2=2\dfrac {\sin^2\theta}{2!}+2^2\dfrac {2\sin^4\theta}{4!}+2^2.4^2\dfrac {2\sin^6\theta}{6!}+....$$(7 marks) 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 marks) 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 marks)
10 (b) Prove that the number of vertices of odd degree in a graph is always even.(7 marks) 2 (a) Prove that if the perimeter of a triangle is constant, its area is maximum when the triangle is equilateral.(7 marks) 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 marks)

### 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 marks) 3 (b) Define B(m,n). Prove that
B(m,n)=B(m+1,n)+B(m,n+1)m,n>0.
(7 marks)
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 marks) 4 (b) Find the volume cut from the sphere x2 + y2 + z2=a2 by the cylinder x2 + y2=ax.(7 marks)

### Answer any one question from Q5 & Q6

5 (a) Solve (3x2y2 + 2xy)dx + (2x3y3 - x2)dy=0(7 marks) 5 (b) $$Solve \ y-x=x\dfrac {dy}{dx}+\left (\dfrac {dy}{dx} \right )^2$$(7 marks) 6 (a) $$Solve \ \dfrac {d^2y}{dx^2}-6 \dfrac {dy}{dx}+13y=8e^{3x}\sin 4x+2^x$$(7 marks) 6 (b) $$Solve \ \dfrac {dx}{dt}+4x+3y=t \\ \dfrac {dy}{dt}+2x+5y=e^t$$(7 marks)

### 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 marks) 7 (b) Solve completely the system of equation 2w+3x-y-z=0, 4w-6x-2y+2z=0, -6w+12x+3y-4z=0(7 marks) 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 marks) 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 marks) 9 (a) Write the following function into disjunctive normal form of 3 variable x,y,z:
(i) x' + y'
(ii) xy' + x'y
(7 marks)
9 (b) In a Boolean algebra B. Prove that the identity elements 0,1 ? B are unique and prove 0'=1,1'=0(7 marks)