## Engineering Mathematics - I - May 2013

### 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) ** $$ (\sin^{-1}x)^2=\dfrac {2}{2!}x^2+\dfrac {2.2^2}{4!}x^4+\dfrac {2.2^2.4^2}{6!}x^6+.... \\ and \ hence \ deduce \\\theta^2=2\dfrac {\sin^2\theta}{2!}+2^2\dfrac {2\sin^4\theta}{4!}+2^2.4^2\dfrac {2\sin^6\theta}{6!}+.... $$(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 y^{2}=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 x^{2} + y^{2} + z^{2}=a^{2} by the cylinder x^{2} + y^{2}=ax.(7 marks)

### Answer any one question from Q5 & Q6

**5 (a)** Solve (3x^{2}y^{2} + 2xy)dx + (2x^{3}y^{3} - x^{2})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)