# Calculus and Analytical Geometry – BSc.CSIT (TU) Question Paper 2068 | First Semester

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## First Semester | First Year | Tribhuvan University Computer Science and Information Technology (MTH. 104) Calculus and Analytical Geometry, Year: 2068 Old Question Collection | Question Bank

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Full Marks: 80 | Pass Marks: 32 | Time: 3 hours.

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.

Attempt all the questions:

Group A     (10×2=20)

1. Define one-to-one and onto functions with suitable examples.
2. Show by integral test that the series $\sum\limits_{n=1}^{\infty} \frac{1}{x^p}$, converges if p>1.
3. Test the convergence of the series
$\sum\limits_{n=1}^{\infty} (-1)^{x+1} \frac{1}{x^2}$
4. Find the focus and the directrix of the parabola y2 = 10x.
5. Find the point where the line X = 8/3 + 2t, y = -2t, z = 1 + t intersects the plane 3x + 2y + 6z = 6.
6. Find a spherical coordinate equation for the sphere x2 + y2 + (z-1)2 = 1.
7. Find the area of the region R bounded by y = x and y = x2 in the first quadrant by using double integrals.
8. Define Jacobian determinant for X = g(u, v, w) ,y = h(u, v, w), z = k(u, v, w).
9. Find the extreme values of f(x,y) = x2 + y2.
10. Define partial differential equations of the second order with suitable examples.
Group B     (5×4=20)
11. State Rolle’s Theorem for a differential function. Support with examples that the hypothesis of theorem are essential to hold the theorem.
12. Test if the following series converges
(a) $\sum\limits_{n=1}^{\infty} \frac{x^2}{2^x}$       (  b) $\sum\limits_{n=1}^{\infty} \frac{2^x}{x^2}$
13. Obtain the polar equations for circles through the origin centered on the x and y axis and radius a.
14. Show that the function $f(x) = \begin{cases} \frac{2xy}{x^2 + y^2}, (x,y) \neq (0,0) \\ 0, \quad \quad (x,y) = 0 \end{cases}$  is continuous at every point except the origin.
15. Find the solution of the equation $\frac{\partial^2 y}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = x - y.$
Group C    (5×8=40)
16. Find the area of the region enclosed by the parabola y = 2 – x2 and the line y = -x.
OR
Evaluate the integrals
(a) $\int\limits_{0}^{3} \frac{dx}{(x-1)^{2/3}}$  (b)$\int\limits_{-\infty}^{\infty} \frac{dx}{1 + x^2}$
17. Define a curvature of a space curve. Find the curvature for the helix
r(t) = (a cost)i + (a sint)j + btk(a,b ≥ 0, a2 + b2 ≠ 0).
18. Find the volume of the region D enclosed by the surfaces z = x2 + 3y2 and z = 8 – x2 – y2.
19. Find the maximum and minimum values of the function f(x,y) = 3x + 4y on the circle x2 + y2 = 1.
OR
State the conditions of second derivative test for local extreme values. Find the local extreme values of the function f(x,y) = x2 + xy + y2 + 3x – 3y + 4.
20. Define one-dimensional wave equation and one-dimensional heat equations with initial conditions. Derive solution of any of them.
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