Calculus and Analytical Geometry Model Question Paper (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
Bachelor of Science in Computer Science and Information Technology
Model Question Paper
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.
Attempt all questions.

Group A [10x2=20]

1. Verify Rolle’s theorem for the function   on [-1, 1] and hence find the corresponding point.


2. Find the length of the curve   from  x = 2 to x = 3.


3. Test the p-series    for p a real constant.


4. Find the polar equation of the circle x² + (y - 3)² = 9.


5. Find a spherical coordinate equation for x² + y² +z² = 4.


6. Use double integral to find the area of the region bounded by y = x and y = x² in the forst quadrant.


7. Verify the Euler’s theorem for mixed partial derivatives:  w = x sin y + y sin x + xy .


8. Use the chain rule to find the derivative of  w = xy  with respect to t along the path x = cos t, y = sin t.


9. Form a partial differential equation by eliminating the constants a  and b  from the surface (x - a)² + (y - b)² + z² = c².


10. Solve the partial differential equation  p + q = x , where the symbols have their usual meanings.

Group B [5x4=20]

11. State and prove the mean value theorem  for definite integral. Apply the theorem to calculate the average value of f(x) = 4 - x² on [0, 3].


12. Find the area of the region that lies inside the circle r = 1 and outside the cardioid  r = 1 - cos θ.


13. Find the curvature and principal unit normal for the helix r(t) = (a cos t) i + (a sin t) j + (bt)k  with a, b ≥ 0 and  a² + b² ≠ 0, where the symbols have their usual meanings.


14. What do you mean by directional derivative in the plane? Find the derivative of  f(x, y) = xe^y + cos(xy) at the point (2, 0) in the direction of te vector .


15. Find a particular integral of the equation .

Group C [5x8=40]

16. Graph the function .


17. Find the Taylor’s series and the Taylor’s polynomial generated by f(x) = e^ax and g(x) = x cos x at x = 0.


18. Evaluate the double integral   by applying te transformation  and integrating over an appropriate region in the uv-plane.
    OR
    Find the volume of the region D enclosed by z = x² + 3y² and z = 8 - x² - y².


19. Find the local minima, local maxima and saddle points of the function f(x, y) = 2xy - 5x² - 2y² + 4x +4y - 4.
    OR
    Find the maximum and minimum of the function f(x, y) = 3x - y + 6 subject to the constraint x² + y² = 4 and explain its geometry.


20. Show that the solution of the wave equation   is And deduce the result if the initial velocity is zero.