Differential Calculus and Differential Equations 2023 – BSc Computer Science Part 1

July 7, 2023 Lokesh Kumar 112 Views 0 Comments 1st Year, BSc, Computer Science, Exam paper

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Paper code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2023
Paper No. 1.2
Differential Calculus and Differential Equations

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 $y = e^{a\sin^{-1}x}$ , then prove that:

$\left ( 1-x^{2} \right ) y_{n+2} - \left ( 2n+1 \right )xy_{n+1} - \left ( n^{2}+a^{2} \right )y_{n} = 0$

hence find the value of $\left ( y_{n} \right )_{0}$

(b) Find $y_{n}$ when $y = \frac{x}{x^{2}+a^{2}}$

2. (a) Apply Maclaurin’s theorem to to prove that:

$\log \sec x = \frac{x^{2}}{2}+\frac{x^{4}}{12}+\frac{x^{6}}{45}+.......$

(b) Use Tailor’s Theorem to prove that

$\int \left ( \frac{x^{2}}{1+x} \right ) = f\left ( x \right ) - \frac{x}{1+x}f^{1}\left ( x \right )+\frac{x^{2}}{\left ( 1+x \right )^{2}} \frac{f^{11}\left ( x \right )}{2!} + ...$

3. (a) Evaluate:

$\lim_{x\rightarrow 0}\left ( \frac{\tan x}{x} \right )^{\frac{1}{x^{2}}}$

(b) If the normal to the curve $x^{2/3} + y^{2/3} = a^{2/3}$ makes an angle Φ with the axis of x, show that its equation is :

$y\cos \phi = x\sin \phi = a\cos 2 \phi$

4. (a) Find the pedal equation of the curve:

$2x = 3a\cos \Theta + a\cos 3\Theta$

$2y = 3a\sin \Theta - a\sin 3\Theta$

(b) Find the polar sub-tangent for the ellipse:

$\frac{1}{r} = 1 + e\cos \Theta$

Section-B

5. (a) Solve the following differential equation:

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

(b) Solve the following differential equation:

$\frac{d^{2}y}{dx^{2}} = \cot x \frac{dy}{dx}-\left ( 1-\cot x \right )y = e^{x}\sin x$

6. (a) Solve the equation:

$D^{2}y - 3Dy + 2y = \cosh x$

(b) Solve the equation:

$p^{3}-4xyp + 8y^{2} = 0$

7. (a) Solve :

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

(b) Solve :

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

Section-C

8. (a) Evaluate :

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

(b) Evaluate :

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

9. (a) Find the indefinite integral :

$\int \left \{ \sin^{2}x + \cos^{2}x + \frac{x^{3}+2x}{x^{1/2}}\right \}dx$

(b) Evaluate :

$\int_{a}^{b}\sin x dx$ as the limit of a sum.

10. (a) Evaluate :

$\int_{a}^{b}x^{2}dx$ by summation.

(b) Evaluate :

$\int_{1}^{2} \frac{x^{3}}{\left(x+1 \right )\left(x^{2}-7x+12 \right)}dx$

……End……

Differential Calculus and Differential Equations 2023 – BSc Computer Science Part 1

Paper code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2023
Paper No. 1.2
Differential Calculus and Differential Equations

Time: Three Hours]

[Maximum Marks: 50

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Lokesh Kumar

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Paper code: 13502 1502 B.Sc. (Computer Science) (Part 1) Examination, 2023 Paper No. 1.2 Differential Calculus and Differential Equations

Time: Three Hours]

[Maximum Marks: 50

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Paper code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2023
Paper No. 1.2
Differential Calculus and Differential Equations