## First Semester | First Year | Tribhuvan University

Computer Science and Information Technology (MTH. 104)

Calculus and Analytical Geometry, Year: 2067

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 a relation and a function from a set into another set. Give suitable example.
- Show that the series converges by using integral test.
- Investigate the convergence of the series
- Find the foci, vertices, center of the ellipse .
- Find the equation for the plane through (-3,0,7) perpendicular to
- Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder 4x
^{2}+ 4y^{2}= 9 in cylindrical coordinates. - Calculate for f(x,y) = 1 – 6x
^{2}y, R : 0 ≤ x ≤ 2, -1 ≤ y ≤ 1. - Define Jacobian determinant for x = g(u, v, w), y = h(u, v, w), z = k(u, v, w).
- What do you mean by local extreme points of f(x,y)? Illustrate the concept by graphs.
- Define partial differential equations of the first index with suitable examples.

**Group B (5×4=20)** - State the mean value theorem for a differentiable function and verify it for the function f(x) = on the interval [-1,1].
- Find the Taylor series and Taylor polynomials generated by the function f(x) = cos x at x = 0.
- Find the length of the cardioid r = 1 – cosθ.
- Define the partial derivative of f(x,y) at a point (x
_{0}, y_{0}) with respect to all variables. Find the derivative of f(x,y) = xe^{y}= cos(x, y) at the point (2, 0) in the direction of A = 3i – 4j. - Find a general solution of the differential equation

**Group C (5×8=40)** - Find the area of the region in the first quadrant that is bounded above by y = √x and below by the x-axis and the line y = x – 2.

**OR**

Investigate the convergence of the integrals

(a) (b) - Calculate the curvature and torsion for the helix

r(t) = (a cos t)i + (a sin t)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 absolute maximum and minimum values of f(x,y) = 2 + 2x + 2y – x
^{2}– y^{2}on the triangular plate in the first quadrant bounded by lines x = 0, y = 0 and x + y = 9.

**OR**

Find the points on the curve xy^{2}= 54 nearest to the origin. How are the Lagrange multipliers defined? - Derive D’ Alembert’s solution satisfying the initials conditions of the one-dimensional wave equation.

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