Calculus and Analytical Geometry Question Paper 2065 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2065
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. Verify Rolle’s theorem for the function  on the interval [-3, 3].


2. Obtain the area between two curves y = sec²x and y = sin x from x = 0 to .


3. Test the convergence of p – series  for p > 1.


4. Find the eccentricity of the hyperbola 9x² - 16y² = 144.


5. Find a vector perpendicular to the plane of P(1, -1, 0), C(2, 1, -1) and R(-1, 1, 2).


6. Find the area enclosed by the curve r² = 4 cos2θ.


7. Obtain the values of  and   at the point (4, -5) if  f(x, y) = x² + 3xy + y - 1.


8. Using partial derivatives, find   if   x² + cosy - y² = 0.


9. Find the partial differential equation of the function (x - a)² + (y - b)² + z² = c².


10. Solve the partial differential equation x²p + q = z².

Group B [5x4=20]

11. State and prove the mean value theorem for a differential function.


12. Find the length of the Astroid   x = cos³t, y = sin³t  for 0 ≤ t ≥ 2π.


13. Define a curvature of a curve. Prove that the curvature of a circle of radius a is 1/a.


14. What is meant by direction derivative in the plain? Obtain the derivative of the function f(x, y) = x² + xy at P(1, 2) in the direction of the unit vector .


15. Find the center of mass of a solid of constant density δ, bounded below by the disk: x² + y² = 4 in the plane z = 0 and above by the paraboloid z = 4 - x² - y².

Group C[5x8=40]

16. Graph the function f(x) = - x3 + 12x + 5 for  -3 ≤ x ≤ 3.


17. Define Taylor’s polynomial of order  n. Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) = e^x at x = 0.


18. Obtain the centroid and the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x².


19. Find the maximum and the minimum values of f(x, y) = 2xy – 2y² – 5x² + 4x – 4. Also find the saddle point if it exists.
    
    OR
    Evaluate the integral   .


20. What do you mean by d’ Alembert’s solution of the one-dimensional wave equation? Derive it.
    
    OR
    Find the particular integral of the equation (D2 – D1)z = 2y – x2 where