Calculus and Analytical Geometry Paper 2067 (BSc CSIT)

Tribhuvan University
Institute of Science and Technology
2067

Bachelor Level/ First Year/ First Semeter/ 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 far as practicable.
The figures in the margin indicate full marks.
Attempt all questions.

Group A(10x2=20)

1. Define a relation and a function from a set into another set. Give suitable example.


2. Show that the series  converses by using integral test.


3. Investigate the convergence of the series .


4. Find the foci, vertices, center of the ellipse .


5. Find the equation for the plane through (-3, 0, 7) perpendicular to .


6. Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder  in cylindrical coordinates.


7. Calculate   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(5x4=20)

11. State the mean value theorem for a differentiable function and verify it for the function    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 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) = xe^y + cos(x, y) at the point (2, 0) in the direction of  A = 3i – 4j.


15. Find a general solution of the differential equation .

Group C(5x8=40)

16. Find the area of the region in the first quadrant that is bounded above by    and below by the  x - axis and the line y = x – 2.
    
    OR
    Investigate the convergence of the integrals
    1. 
    2. 
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.
    
    OR
    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.