Differential Calculus and Differential Equation 2014 – BSc Computer Science Part 1

July 2, 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, 2014
Paper No. 1.2
DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION

Time: Three Hours] [Maximum Marks: 50

Note: Attempt all the five questions in all section one question from each Section. All questions carry equal marks.

Section-A

1. (a) If $I_n=\frac{d^n}{{dx}^n}(x^n \log x)$ , prove that:

$I_n=nI_{n-2}+(n-1)!$

(b) If $y=tan^{-1}x$ , find

$(y_n )_0$

2. (a) Apply Maclaurin’s theorem to find the expansion in ascending powers of x of $\log_{e}(1+e^x)$ to the term containing $x^{4}$ .
(b) Expend $\sin x$ in powers of $\left (x-\frac{1}{2}\pi \right)$ .

Section-B

3. (a) Find the equation of the normal at the point ‘t’ on the curve:

$x=a(\sin t)^{3}, x=b(\cos t)^{3}$

(b) Find the length of the polar tangent and polar normal for the curve $r=a(1+\cos\theta )$ .

4. (a) Find the equation of the tangent at the point $(x_1,y_1)$ to the ellipse:

$(\frac{x^2}{a^2} +\frac{y^2}{b^2}=1)$

(b) If $r^m=a^m\cos m\theta$ , prove that:

$\frac{ds}{d\theta }=\frac{a^{m}}{r^{m}-1}$

Section-C

5. (a) Evaluate:

$\lim_{x \to 0}\frac{\log (1-x^{2})}{\log \cos x}$

(b) Evaluate:

$\int \frac{4x-5}{2x^{2}-5x+2}dx$

6. (a) Show that:

$\int_{0}^{\frac{\pi }{2}}\frac{\sin x}{\sin x+\cos x}dx=\frac{\pi }{4}$

(b) Evaluate:

$\lim_{x \to 0}\left [\frac{1}{n+1}+\frac{1}{n+2}+\cdot \cdot \cdot \cdot \cdot \cdot \cdot + \frac{1}{2n} \right ]$

Section-D

7. (a) Solve:

$(xy+x)dy=(yx+y)dx$

(b) Solve:

$\cos (x+y)dy=dx$

8. (a) Solve:

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

(b) Solve:

$\sec xdy = (y+\sin x )dx$

Section-E

9. (a) Solve:

$\frac{d^2y}{dx^2}-4\frac{dy}{dx}+y=e^{2x}$

(b) Solve:

$(D^{2}-4D+3)y=e^{2x}\sin3x$

10. (a) Solve:

$x^{2}\frac{d^2}{dx^2 }+4x \frac{dy}{dx}+2y=e^x$

(b) Solve:

$\frac{dx}{dt}+4x+3y=t$

……………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 Equation 2014 – BSc Computer Science Part 1

Paper code: 13502
1502
B.sc. (Computer Science) (Part 1)
Examination, 2014
Paper No. 1.2
DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION

Time: Three Hours] [Maximum Marks: 50

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Paper code: 13502 1502 B.sc. (Computer Science) (Part 1) Examination, 2014 Paper No. 1.2 DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION

Time: Three Hours] [Maximum Marks: 50

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Paper code: 13502
1502
B.sc. (Computer Science) (Part 1)
Examination, 2014
Paper No. 1.2
DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION