## First Semester | First Year | Tribhuvan University

Computer Science and Information Technology (MTH. 104)

Calculus and Analytical Geometry, Year: 2068

Old Question Collection | Question Bank

**Download Question Paper File**

[File Type: PDF | File Size: 541 KB | Download]

**OR** you can read question paper online below;

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)**

- Define one-to-one and onto functions with suitable examples.
- Show by integral test that the series , converges if p>1.
- Test the convergence of the series

- Find the focus and the directrix of the parabola y
^{2}= 10x. - Find the point where the line X = 8/3 + 2t, y = -2t, z = 1 + t intersects the plane 3x + 2y + 6z = 6.
- Find a spherical coordinate equation for the sphere x
^{2}+ y^{2}+ (z-1)^{2}= 1. - Find the area of the region R bounded by y = x and y = x
^{2}in the first quadrant by using double integrals. - Define Jacobian determinant for X = g(u, v, w) ,y = h(u, v, w), z = k(u, v, w).
- Find the extreme values of f(x,y) = x
^{2}+ y^{2}. - Define partial differential equations of the second order with suitable examples.

**Group B (5×4=20)** - State Rolle’s Theorem for a differential function. Support with examples that the hypothesis of theorem are essential to hold the theorem.
- Test if the following series converges

(a) ( b) - Obtain the polar equations for circles through the origin centered on the x and y axis and radius a.
- Show that the function is continuous at every point except the origin.
- Find the solution of the equation

**Group C (5×8=40)** - Find the area of the region enclosed by the parabola y = 2 – x
^{2}and the line y = -x.

**OR**

Evaluate the integrals

(a) (b) - 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, a^{2}+ b^{2}≠ 0). - Find the volume of the region D enclosed by the surfaces z = x
^{2}+ 3y^{2}and z = 8 – x^{2}– y^{2}. - Find the maximum and minimum values of the function f(x,y) = 3x + 4y on the circle x
^{2}+ y^{2}= 1.

**OR**

State the conditions of second derivative test for local extreme values. Find the local extreme values of the function f(x,y) = x^{2}+ xy + y^{2}+ 3x – 3y + 4. - Define one-dimensional wave equation and one-dimensional heat equations with initial conditions. Derive solution of any of them.

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