Exam Papers

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

 

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.

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