DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION - 2016 | B.SC CS 1ST YEAR | MJPRU | EXAM PAPER | My CS Tutorial - My CS Tutorial

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DIFFERENTIAL CALCULUS AND DIFFERENTIAL EQUATION - 2016 | B.SC CS 1ST YEAR | MJPRU | EXAM PAPER | My CS Tutorial


Differential calculas and Differential equation | My CS Tutorial


Paper Code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2016
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) If y=\sin mx +\cos mx, prove that:

y_{n}=m^{n} \left [ 1+(-1)^{n} \sin 2mx \right ]^{\frac{1}{2}}

(b) If y=a \cos \left ( \log x \right )+ b \sin \left ( \log x \right )

Show that:

x^2 y_2+xy_1+y=0

and x^2 y_{n+2}+(2n+1)xy_{n+1}+(n^{2}+1) y_n=0.

2. (a) State and prove Maclaurin’s theorem.

(b) Expend \tan^{-1}x in powers of \left ( x-\frac{\pi }{4} \right ).

3. (a) If:

y=x_{n}\log x

prove that:

xy_{n+1}=n!

(b) Evaluate:

\lim_{x \to 0}\frac{\sin x-x+\frac{x^{3}}{6}}{x^{5}}

4. For the cardoid r=a\left ( 1-\cos \theta \right ), prove that:

  1. \phi =\frac {\theta}{2}
  2. p=2a\sin^{3}\frac{\theta }{2}
  3. The Pedal equation is 2ap^{2}=r^{3}
  4. The Polar sub tangent = 2a\sin^{2} \frac {\theta}{2} \tan \frac {\theta}{2}

Section-B

5. (a) Solve:

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

(b) Solve it:

x\frac {dy}{dx}=y-x\tan \frac {y}{x}

6. (a) Solve:

\frac {dy}{dx}-3y\cot x = \sin 2x

Given y=2 when x=\frac {\pi}{2}.

(b) Solve:

xdx+ydy+\frac {xdy-ydx}{x^{2}+y^{2}}=0

7. (a) Solve:

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

(b) Solve:

\left (D^{2}+a^{2} \right )y=\sin ax

8. (a) Solve:

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

Section-C

9. (a) Evaluate:

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

(b) Evaluate:

\int \sqrt{2-3x-4x^{2}}dx

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

(b) Evaluate \int_{0}^{\frac {\pi}{2}}\log \sin x dx.


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