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

July 7, 2023July 8, 2023 Lokesh Kumar 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……

Lokesh Kumar

Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

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