Calculus and Analytical Geometry Question Paper 2066 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2066

Bachelor Level/ First Year/ First Semester/ Science                                   Full Marks: 80
Computer Science and Information Technology (MTH 104)                Pass Marks: 32
(Calculus and Analytical Geometry)                                                                Time: 3 hours.

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

Group A (10x2=20)

1. Find the length of the curve  from x = 0  to x = 4.


2. Find the critical points of the function  .


3. Does the following series converge?
   


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 .


7. Evaluate the limit


8. Find  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 (5x4=20)

11. State and prove Rolle’s theorem.


12. Find the length of the cardioid r = 1 + cos θ.


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

Group C (5x8=40)

16. Graph the function .


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  .


20. Show that the solution of the wave equation   is and deduce the result if the velocity is zero.
    OR
    Find a particular integral of the equation    where A, l, m are constants.