Differential Calculus and Differential Equation 2019 – BSc Computer Science Part 1

January 29, 2020 Lokesh Kumar 157 Views 0 Comments 1st Year, BSc, BSc Computer Science, Exam paper

Paper code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2019
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 derivative of:

$\tan^{-1}\frac{2x}{1-x^{2}}$

(b) If $y = x^{2}e^{x}$ , show that:

\frac{d^{n}y}{dx^{n}} = \frac{n}{2}\left ( n-1 \right )\frac{d^{2}y}{dx^{2}}-n\left ( n-2 \right )\frac{dy}{dx}+\frac{1}{2}\left ( n-1 \right )\left ( n-2 \right )y

2. (a) use Maclaurin’s Theorem to find the expansion in ascending power of x of $\log \left ( 1+e^{x} \right )$ upto term containing x⁴.

(b) Use Tailor’s Theorem to prove that

$\tan^{-1}\left ( x+h \right ) = \tan^{-1}x + h\sin z\frac{\sin z}{1}- \left ( h \sin z \right )^{2}\cdot \frac{\sin 2z}{2} + \left ( h \sin z \right )^{3}\cdot \frac{\sin 3z}{3}.....$

3. (a) Find :

$\lim_{x\rightarrow 0}\left ( \frac{\tan x}{x} \right )^{\frac{1}{x}}$

(b) Show that the condition that the curve $ax^{2} + by^{2} = 1$ and $a^{'}x^{2} + b^{'}y^{2} = 1$ should intersect orthogonally is that:

$\frac{1}{a}-\frac{1}{b} = \frac{1}{a^{'}}-\frac{1}{b^{'}}$

4. (a) If $r = a\left ( 1 + \cos\Theta \right )$ , find the polar sub-tangent, polar sub-normal and the lengths of polar tangent and polar normal at the point when $\Theta = \tan^{-1}\left ( \frac{3}{4} \right )$