Question: Engineering Mathematics - I : Dec 2014 - First Year Engineering (Set B) (Semester 2) | Rajiv Gandhi Proudyogiki Vishwavidyalaya (RGPV)
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## Engineering Mathematics - I - Dec 2014

### 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.

### Solve any one question from Q1 & Q1 (e)

1 (a) Define curvature of a curve at a point and find the radius of curvature at any point (s, ψ) of the curve s=4 a sin ψ.(2 marks) 1 (b) If $u=f \left( \dfrac {y}{x} \right)$, then show that $x \dfrac{\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}=0$(2 marks) 1 (c) Discuss the maxima and minima of the function x3 + y3 -3axy.(3 marks) 1 (d) Compute the approximate value of √11 to four decimal place by taking the first five terms of an approximate Taylor's expansion.(7 marks) 1 (e) If $x^x y^y z^z=c$ then show that $\dfrac {\partial^2 z}{\partial x \partial y}= - [x log (ex)]^{-1}$(14 marks)

### Solve any one question from Q2 & Q2 (e)

2 (a) Using Gamma function, evaluate $\displaystyle \int^{\infty}_0 \sqrt{x}e^{-3\sqrt{x}}dx$(2 marks) 2 (b) Evaluate: $\displaystyle \int^2_0 \int^1_0 (x^2+y^2)dxdy$(2 marks) 2 (c) Evaluate:$\displaystyle \int^{1}_{-1} \int^{z}_0 \int^{x+z}_{x-z} (x+y+z)dx dy dz$(3 marks) 2 (d) Evaluate: $\lim_{n\rightarrow \infty} \left[ \left(1+ \dfrac{1}{n^2} \right) \left(1+ \dfrac {2^2}{n^2} \right) \left(1+ \dfrac {3^2}{n^2} \right ) \cdots \cdots \left( 1+ \dfrac {n^2}{n^2} \right) \right]$(7 marks) 2 (e) Prove the Legendre's duplication formula $\Gamma (m) \Gamma \left (m+ \dfrac {1}{2} \right )= \dfrac {\sqrt{\pi}}{2^{2m-1}} \Gamma (2m)$(14 marks)

### Solve any one question from Q3 & Q3 (e)

3 (a) State whether the differential equation (ey+1) cos x dx+ey sin x dy=0 is exact differential equation or not.(2 marks) 3 (b) Solve the differential equation p = sin ( y - x )(2 marks) 3 (c) Solve the differential equation $\dfrac {dy}{dx}- \dfrac {dx}{dy}= \dfrac {x}{y}- \dfrac {y}{x}$(3 marks) 3 (d) Solve $x^2 \dfrac {dy}{dx}- 3x \dfrac {dy}{dx}+4y=(1+x)^2$(7 marks) 3 (e) Solve the simultaneous equations: $\dfrac {dx}{dt} + 5x+y=e^r \dfrac {dy}{dt}-x+3y=e^{2t}$(14 marks)

### Solve any one question from Q4 & Q4 (e)

4 (a) Find one non zero minor of highest order of the matrix $A= \begin{bmatrix}-1 &- 2 &3 \\\\-2 &4 &-1 \\\\-1 &2 &7 \end{bmatrix}$ and hence find the rank of the matrix A.(2 marks) 4 (b) Find the sum and product of eigen values of the matrix $A= \begin{bmatrix}6 &- 2&2 \\\\-2 &3 &1 \\\\2 &-1 &3 \end{bmatrix}$ without actually computing them.(2 marks) 4 (c) Find the characteristic equation of the matrix $A= \begin{bmatrix}2 &2 &1 \\\\1 &3 &1 \\\\1 &2 &2 \end{bmatrix}$(3 marks) 4 (d) Find the normal form of the matrix $A=\begin{bmatrix}2 &3 &-1 &-1 \\\\1 &-1 &-2 &-4 \\\\3 &1 &3 &-2 \\\\6 &3 &0 &- 7 \end{bmatrix}$ and hence find its rank.(7 marks) 4 (e) For what values of λ, the equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2(14 marks)

### Solve any one question from Q5 & Q5 (e)

5 (a) Let p ≡ Raju is tall, q ≡ Raju is handsome and r ≡ People like Raju then write the following statements in language.
i) (p⇒q)∨(p⇒r)
ii) p⇒(q∨r)
iii) ∼ p∨∼q
iv) ∼ (∼p∨∼q)
(2 marks)
5 (b) In a Boolean algebra B, prove that a + b = b⇒a, b=a, ∀a, b∈B.(2 marks) 5 (c) Draw the switching circuit for the following functions and replace it by simpler one:
F(x,y,z)=x,y,z+(x+y),(x+z)
(3 marks)
5 (d) Prove that a tree with n vertical has (n-1) edges.(7 marks) 5 (e) If p, q, r are three statement then show that (p⇔q)∧(q⇔r) ⇒(p⇔r) is a tautology.(14 marks)