Calculus and Analytical Geometry – BSc.CSIT (TU) Question Paper 2066 | First Semester

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calculus question paper 2066First Semester | First Year | Tribhuvan University
Computer Science and Information Technology (MTH. 104)
Calculus and Analytical Geometry, Year: 2066
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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. Find the length of the curve y = x3/2 from x=0 to x =4.
  2. Find the critical points of the function f(x) = x3/2 (x-4).
  3. Does the following series converge?
     \sum\limits_{n=1}^{\infty} \dfrac {1}{n^2} = 1 + \dfrac{1}{2^2} + \dfrac{1}{3^2} + ..............
  4. Find the polar equation of the circle (x+2)2 + y2 = 4
  5. Find the area of the parallelogram where vertices are A(0,0), B(7,3), C(9,8) and D(2,5).
  6. Evaluate the integral
     \int\limits_{t}^{2t} \int\limits_{0}^{1} (sin x + cos y) \mathrm{d}x \mathrm{d}y
  7. Evaluate the limit
     \lim_{(x,y) \to (0,0)} \frac{x^2 - xy}{\sqrt{x} - \sqrt{y}}
  8. Find  \left( \frac{\partial w}{\partial x} \right)_{y,z} if ω = x2 + y – z + sin t and x + y = t.
  9. Solve the partial differential equation p + q = x.
  10. Find the general integral of the linear partial differential equation z(xp – yq) = z2 – x2.
    Group B     (5×4=20)
  11. State and prove Rolle’s theorem.
  12. Find the length of the cardioid r = 1 + cosθ.
  13. Define unit tangent vector of a differentiable curve. Find the unit tangent vector of the curve r(t) = (cos t + t sin t)i + (sin t – t cos t)j, t > 0.
  14. What do you mean by critical point of a function f(x,y) in a region? Find local extreme values of the function f(x,y) = xy – x2 – y2 – 2x – 2y + 4.
  15. Find a particular integral of the equation
     \frac{\partial^2 z}{\partial x^2} - \frac{\partial z}{\partial y} = 2y – x2
    Group C     (5×8=40)
  16. Graph the function y = x4/3 – 4x1/3
  17. What do you mean by Taylor’s polynomial of order n? Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) =cos x at x =0.
  18. Find the volume of the region enclosed by the surface z = x2 + 3y2 and z = 8 – x2 – y2.
  19. Obtain the absolute maximum and minimum values of the function.
    f(x,y) = 2 + 2x + 2y – x2 – y2 on the triangular plate in the first quadrant bounded by lines x = 0, y = 0, y = 9 – x.
    OR
    Evaluate the integral  \int\limits_{0}^{1} \int\limits_{0}^{3-3x} \int\limits_{0}^{3-3x-y} \mathrm{d}z \mathrm{d}y \mathrm{d}x
  20. Show that the solution of the wave equation  \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, c^2 = \frac{T}{p} is, u(x,t) = \frac{1}{2} [f(x+ct) + f(x-ct)] + \frac{1}{2c} \int\limits_{x-ct}^{x+ct} g(s) \mathrm{d}s  and deduce the result if the velocity is zero.
    OR
    Find a particular integral of the equation (D2 – D1) = A cos(lx + my) where A, l, m are constants.
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