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Calculus and Analytical Geometry – BSc.CSIT (TU) Question Paper 2067 | First Semester

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calculus question paper 2067First Semester | First Year | Tribhuvan University
Computer Science and Information Technology (MTH. 104)
Calculus and Analytical Geometry, Year: 2067
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 a relation and a function from a set into another set. Give suitable example.
  2. Show that the series  \sum\limits_{n=1}^{\infty} \frac{1}{n^2} converges by using integral test.
  3. Investigate the convergence of the series  \sum\limits_{n=0}^{\infty} \frac{2^n + 5}{3^x}
  4. Find the foci, vertices, center of the ellipse  \frac{x^2}{16} + \frac{y^2}{9} = 1 .
  5. Find the equation for the plane through (-3,0,7) perpendicular to  \vec{n} = \vec{5i} + \vec{2j} - \vec{k}
  6. Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder 4x2 + 4y2 = 9 in cylindrical coordinates.
  7. Calculate  \iint\limits_R \, f(x,y)d4 for f(x,y) = 1 – 6x2y, R : 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.
  8. Define Jacobian determinant for x = g(u, v, w), y = h(u, v, w), z = k(u, v, w).
  9. What do you mean by local extreme points of f(x,y)? Illustrate the concept by graphs.
  10. Define partial differential equations of the first index with suitable examples.
    Group B    (5×4=20)
  11. State the mean value theorem for a differentiable function and verify it for the function f(x) =  \sqrt{1-x^2} on the interval [-1,1].
  12. Find the Taylor series and Taylor polynomials generated by the function f(x) = cos x at x = 0.
  13. Find the length of the cardioid r = 1 – cosθ.
  14. Define the partial derivative of f(x,y) at a point (x0, y0) with respect to all variables. Find the derivative of f(x,y) = xey = cos(x, y) at the point (2, 0) in the direction of A = 3i – 4j.
  15. Find a general solution of the differential equation
     x^2 \frac{\partial z}{\partial x} + y^2 \frac{\partial z}{\partial y} = (x+y)z.
    Group C (5×8=40)
  16. 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.
    Investigate the convergence of the integrals
    (a)  \int\limits_{1}^{0} \frac{1}{1-x} \mathrm{d}x (b)  \int\limits_{0}^{3} \frac{dx}{x-1^{2/3}}
  17. Calculate the curvature and torsion for the helix
    r(t) = (a cos t)i + (a sin t)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 absolute maximum and minimum values of f(x,y) = 2 + 2x + 2y – x2 – y2 on the triangular plate in the first quadrant bounded by lines x = 0, y = 0 and x + y = 9.
    Find the points on the curve xy2 = 54 nearest to the origin. How are the Lagrange multipliers defined?
  20. Derive D’ Alembert’s solution satisfying the initials conditions of the one-dimensional wave equation.
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