Differential Calculus and Differential Equation 2017 – BSc Computer Science Part 1

March 30, 2019 Lokesh Kumar 153 Views 0 Comments 1st Year, BSc, Computer Science, Exam paper

Paper code: 13502
1502
B.sc. (Computer Science) (Part 1)
Examination, 2017
Paper No. 1.2
DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION

Time: Three Hours] [Maximum Marks: 50

Note:Attempt five questions in all selecting one question from each Section. All questions carry equal marks.

Section-A

1. (a) Find nth differential coefiicient of $\tan^{-1}\frac{x}{a}$ .

(b) If $y=(x^{2}-1)^{n}$ and $P_{n}(x)=\frac{d^{n}}{dx^{n}}(x^{2}-1)^{n}$ , show that :
$\frac{d}{dx}\left \{ (1-x^{2})\frac{dP_{n}(x)}{dx} \right \}+n(n+1)P_{n}(x)=0$

2. (a) If $e^{x^{y}} = a_{0} + a_{1}x + a_{2}x^{2} +.....+ a_{n}x^{n} +.....$ , prove that :
$a_{n+1} = \frac{1}{n+1}\left \{ a_{n}+\frac{a_{n-1}}{1!} + \frac{a_{n-2}}{2!} +.....+ \frac{a_{0}}{n!} \right \}$

(b) State and prove Taylor’s theorem.

3. (a) Evaluate :

$\lim_{x\rightarrow 0}\frac{x^{1/2}\tan x}{(e^{x}-1)^{3/2}}$

(b) Show that the Pedal equation of the ellipse $\frac{x^2}{a^{2}}+\frac{y^2}{b^{2}}=1$ is:

$\frac{1}{p^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}-\frac{*r^{2}}{a^{2}b^{2}}$

4. (a) Show that the locus of the extremily of the polar subnormal of the curve $r=f(\theta )$ is $r=f'\left ( \theta - \frac{\pi}{2} \right )$ .
(b) Find the polar sub-tangent of the ellipse :