Differential Calculus and Differential Equation 2018 – BSc Computer Science Part 1

Paper code: 13502
1502
B.sc. (Computer Science) (Part 1)
Examination, 2018
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 :

\sin^{5}x \cos^{3}x

    (b) If y=e^{a \sin^{-1} x}, find the values of (y_{n})_{0}.

2. (a) Expand 2x^{3}+7x^{2}+x-1 in power of (x-2).

    (b) State and prove Maclaurin’s theorem.

3. (a) Evaluate :

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

    (b) Find the angle between the radius vector and tangent for the curve :

r=a(1+\cos \Theta) at point (r, \Theta)

4. (a) Find the length of polar subtangent of the parabola :

\frac{2a}{r} = 1 + \cos \Theta

(b) Evaluate :

\lim_{x\rightarrow 0}\frac{(1+x)^{1/x}-e}{x}

Section-B

5. (a) Evaluate :

\int_{0}^{\pi/2} \log\sin x dx

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

6. (a) Evaluate :

\int \frac{2x^{2}+3x+4}{x^{2}+6x+10}dx

    (b) Evaluate :

\int \frac{1-4x-2x^{2}}{\sqrt{2x-x^{2}}}dx

7. (a) Solve :

\frac{dy}{dx} = e^{x-y} + x^{2} e^{-y}

    (b) Solve the following :

\frac{dy}{dx} + (2x\tan^{-1}y - x^{3})(1+y^{2})=0

Section-C

8. (a) Solve:

(3x+2)^{2}\frac{d^{2}y}{dx^{2}}+3(3x+2)\frac{dy}{dx}-36y=3x^{2}+4x+1

    (b) Evaluate the following :

\frac{d^{4}y}{dx^{4}}+\frac{d^{2}}{dx^{2}}+y = ax^{2} + be^{-x} \sin 2x

9. (a) Solve :

\frac{d^{2}y}{dx^{2}}+a^{2}y=\tan ax

      (b) Solve :

x^{2} \frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+y= \frac{\log x \sin (\log x)+1}{x}

10. (a) Solve :

\frac{dx}{dt} + wy =0

\frac{dy}{dt} - wx =0

      (b) Solve :

\frac{d^{2}y}{dx^{2}}+a^{2}y=\cos ax

……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.

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