Differential Calculus and Differential Equation 2016 – BSc Computer Science Part 1

July 4, 2018July 25, 2018 Lokesh Kumar 0 Comments 1st Year, BSc, Computer Science, Exam paper

Paper code: 13502
1502
B.sc. (Computer Science) (Part 1)
Examination, 2016
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) If $y=\sin mx +\cos mx$ , prove that:

$y_{n}=m^{n} \left [ 1+(-1)^{n} \sin 2mx \right ]^{\frac{1}{2}}$

(b) If $y=a \cos \left ( \log x \right )+ b \sin \left ( \log x \right )$

Show that:

$x^2 y_2+xy_1+y=0$

and $x^2 y_{n+2}+(2n+1)xy_{n+1}+(n^{2}+1) y_n=0$ .

2. (a) State and prove Maclaurin’s theorem.

(b) Expend $\tan^{-1}x$ in powers of $\left ( x-\frac{\pi }{4} \right )$ .

3. (a) If:

$y=x_{n}\log x$

prove that:

$xy_{n+1}=n!$

(b) Evaluate:

$\lim_{x \to 0}\frac{\sin x-x+\frac{x^{3}}{6}}{x^{5}}$

4. For the cardoid $r=a\left ( 1-\cos \theta \right )$ , prove that:

$\phi =\frac {\theta}{2}$
$p=2a\sin^{3}\frac{\theta }{2}$
The Pedal equation is $2ap^{2}=r^{3}$
The Polar sub tangent $= 2a\sin^{2} \frac {\theta}{2} \tan \frac {\theta}{2}$

Section-B

5. (a) Solve:

$\frac {dy}{dx}=\left ( 4x+y+1\right )^{2}$

(b) Solve it:

$x\frac {dy}{dx}=y-x\tan \frac {y}{x}$

6. (a) Solve:

$\frac {dy}{dx}-3y\cot x = \sin 2x$

Given $y=2$ when $x=\frac {\pi}{2}$ .

(b) Solve:

$xdx+ydy+\frac {xdy-ydx}{x^{2}+y^{2}}=0$

7. (a) Solve:

$\left (D^{3}+6D^{2}+11D+6 \right )y=0$

(b) Solve:

$\left (D^{2}+a^{2} \right )y=\sin ax$

8. (a) Solve:

$\left (x^{2} D^{2} + 3xD +1 \right )y=\frac {1}{\left (1-x \right )^{2}}$

Section-C

9. (a) Evaluate:

$\int \frac {5x-2}{1+2x+3x^{2}}dx$

(b) Evaluate:

$\int \sqrt{2-3x-4x^{2}}dx$

10. (a) Evaluate $\int_{a}^{b}x^{2}dx$ by summation.

(b) Evaluate $\int_{0}^{\frac {\pi}{2}}\log \sin x dx$ .

……End……

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