RGPV Engineering Mathematics -I - December 2014 Exam Question Paper

RGPV First Year Engineering (Set A) (Semester 1)
Engineering Mathematics -I
December 2014

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

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 M

1 (b) If

u = f (\frac{y}{x})

$u=f \left( \dfrac {y}{x} \right)$ , then show that

x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 0

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

2 M

1 (c) Discuss the maxima and minima of the function x³ + y³ -3axy.

3 M

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 M

1 (e) If

x^{x} y^{y} z^{z} = c

$x^x y^y z^z=c$ then show that

\frac{\partial^{2} z}{\partial x \partial y} = - [x l o g (e x)]^{- 1}

$\dfrac {\partial^2 z}{\partial x \partial y}= - [x log (ex)]^{-1}$

14 M

Solve any one question from Q2 & Q2 (e)

2 (a) Using Gamma function, evaluate

\int_{0}^{\infty} \sqrt{x} e^{- 3 \sqrt{x}} d x

$\displaystyle \int^{\infty}_0 \sqrt{x}e^{-3\sqrt{x}}dx$

2 M

2 (b) Evaluate:

\int_{0}^{2} \int_{0}^{1} (x^{2} + y^{2}) d x d y

$\displaystyle \int^2_0 \int^1_0 (x^2+y^2)dxdy$

2 M

2 (c) Evaluate:

\int_{- 1}^{1} \int_{0}^{z} \int_{x - z}^{x + z} (x + y + z) d x d y d z

$\displaystyle \int^{1}_{-1} \int^{z}_0 \int^{x+z}_{x-z} (x+y+z)dx dy dz$

3 M

2 (d) Evaluate:

{lim}_{n \to \infty} [(1 + \frac{1}{n^{2}}) (1 + \frac{2^{2}}{n^{2}}) (1 + \frac{3^{2}}{n^{2}}) \dots \dots (1 + \frac{n^{2}}{n^{2}})]

$\lim_{n\rightarrow \infty} \left[ \left(1+ \dfrac{1}{n^2} \right) \left(1+ \dfrac {2^2}{n^2} \right) \left(1+ \dfrac {3^2}{n^2} \right ) \cdots \cdots \left( 1+ \dfrac {n^2}{n^2} \right) \right]$

7 M

2 (e) Prove the Legendre's duplication formula

Γ (m) Γ (m + \frac{1}{2}) = \frac{\sqrt{π}}{2^{2 m - 1}} Γ (2 m)

$\Gamma (m) \Gamma \left (m+ \dfrac {1}{2} \right )= \dfrac {\sqrt{\pi}}{2^{2m-1}} \Gamma (2m)$

14 M

Solve any one question from Q3 & Q3 (e)

3 (a) State whether the differential equation (e^y+1) cos x dx+e^y sin x dy=0 is exact differential equation or not.

2 M

3 (b) Solve the differential equation p = sin ( y - x )

2 M

3 (c) Solve the differential equation

\frac{d y}{d x} - \frac{d x}{d y} = \frac{x}{y} - \frac{y}{x}

$\dfrac {dy}{dx}- \dfrac {dx}{dy}= \dfrac {x}{y}- \dfrac {y}{x}$

3 M

3 (d) Solve

x^{2} \frac{d y}{d x} - 3 x \frac{d y}{d x} + 4 y = (1 + x)^{2}

$x^2 \dfrac {dy}{dx}- 3x \dfrac {dy}{dx}+4y=(1+x)^2$

7 M

3 (e) Solve the simultaneous equations:

\frac{d x}{d t} + 5 x + y = e^{r} \frac{d y}{d t} - x + 3 y = e^{2 t}

$\dfrac {dx}{dt} + 5x+y=e^r \dfrac {dy}{dt}-x+3y=e^{2t}$

14 M

Solve any one question from Q4 & Q4 (e)

4 (a) Find one non zero minor of highest order of the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 2 & 3 - 2 & 4 & - 1 - 1 & 2 & 7 \end{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 M

4 (b) Find the sum and product of eigen values of the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 6 & - 2 & 2 - 2 & 3 & 1 2 & - 1 & 3 \end{matrix} ⎤ ⎥ ⎦

$A= \begin{bmatrix}6 &- 2&2 \\-2 &3 &1 \\2 &-1 &3 \end{bmatrix}$ without actually computing them.

2 M

4 (c) Find the characteristic equation of the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 2 & 1 1 & 3 & 1 1 & 2 & 2 \end{matrix} ⎤ ⎥ ⎦

$A= \begin{bmatrix}2 &2 &1 \\1 &3 &1 \\1 &2 &2 \end{bmatrix}$

3 M

4 (d) Find the normal form of the matrix

A = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 2 & 3 & - 1 & - 1 1 & - 1 & - 2 & - 4 3 & 1 & 3 & - 2 6 & 3 & 0 & - 7 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$A=\begin{bmatrix}2 &3 &-1 &-1 \\1 &-1 &-2 &-4 \\3 &1 &3 &-2 \\6 &3 &0 &- 7 \end{bmatrix}$ and hence find its rank.

7 M

4 (e) For what values of λ, the equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ²

14 M

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 M

5 (b) In a Boolean algebra B, prove that a + b = b⇒a, b=a, ∀a, b∈B.

2 M

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 M

5 (d) Prove that a tree with n vertical has (n-1) edges.

7 M

5 (e) If p, q, r are three statement then show that (p⇔q)∧(q⇔r) ⇒(p⇔r) is a tautology.

14 M

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