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Tribhuvan University: Institute of Science and Technology

1. The document is a past exam paper for a Calculus and Analytical Geometry course at Tribhuvan University Institute of Science and Technology. 2. The exam is 3 hours long and consists of 3 sections - Section A has 10 multiple choice questions worth 2 marks each, Section B has 5 questions worth 4 marks each, and Section C has 5 questions worth 8 marks each. 3. Candidates are required to answer all questions in their own words as far as possible. The questions cover topics like relations and functions, series tests, derivatives, integrals, differential equations, and optimization.

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233 views2 pages

Tribhuvan University: Institute of Science and Technology

1. The document is a past exam paper for a Calculus and Analytical Geometry course at Tribhuvan University Institute of Science and Technology. 2. The exam is 3 hours long and consists of 3 sections - Section A has 10 multiple choice questions worth 2 marks each, Section B has 5 questions worth 4 marks each, and Section C has 5 questions worth 8 marks each. 3. Candidates are required to answer all questions in their own words as far as possible. The questions cover topics like relations and functions, series tests, derivatives, integrals, differential equations, and optimization.

Uploaded by

vikash
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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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.

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Candidates are required to give their answers in their own words as far as practicable.

The figures in the margin indicate full marks.

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Attempt all questions.

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Group A (10x2=20)

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

2. Show that the series ∑ g


converses by using integral test.
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3. Investigate the convergence of the series ∑ .
.b

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


g
lo

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


itb

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

7. Calculate ( ) for f(x, y) = 1 – 6x2y R:0≤x≤ - ≤y≤ .


cc

8. Define Jacobian determinant for x = g(u, v, w), y = h(u, v, w), z = k(u, v, w).
bs

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.

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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) = x + cos(x, y) at the point (2, 0) in the direction of A = 3i – 4j.

.c
15. Find a general solution of the differential equation

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( ) .

sp
Group C (5x8=40)
g
16. Find the area of the region in the first quadrant that is bounded above by √ and below by
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the x - axis and the line y = x – 2.
OR
.b

Investigate the convergence of the integrals


(a) (b)
g

⁄
lo

17. Calculate the curvature and torsion for the helix


r(t) = (a cos t)i + (a sin t)j + b t k, a, b ≥ 0 a2 + b2 ≠ 0.
itb

18. Find the volume of the region D enclosed by the surfaces z = x2 + 3y2 and z = 8 – x2 – y2.
s

19. Find the absolute maximum and minimum values of


cc

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
bs

Find the points on the curve xy = 54 nearest to the origin. How are the Lagrange multipliers
2

defined?

20. Derive Alembert’s solution satisfying the initials conditions of the one-dimensional wave
equation.

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