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.

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.

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  

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.