Calculus and Analytical Geometry – BSc.CSIT (TU) Question Paper 2065 | First Semester First Semester | First Year | Tribhuvan University Computer Science and Information Technology (MTH. 104) Calculus and Analytical Geometry, Year: 2065 Old Question Collection | Question Bank

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Full Marks: 80 | Pass Marks: 32 | 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 the questions:

Group A     (10×2=20)

1. Verify Rolle’s theorem for the function $f(x) = \dfrac{x^3}{3} - 3x$ on the interval [-3, 3].
2. Obtain the area between two curves y = sec2x and y = sin x from x = 0 to x = π/4.
3. Test the convergence of p – series $\sum\limits_{n=1}^\infty \dfrac{1}{n^p}$ 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 = 4cos2θ.
7. Obtain the values of $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ at the point (4, -5) if f(x,y) = x2 + 3xy + y -1.
8. Using partial derivatives, find $\dfrac{dy}{dx}$ if x2 + cos y – 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      (5×4=20)
11. State and prove the mean value theorem for a differential function.
12. Find the length of the Asteroid 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 $v = \left(\dfrac{1}{\sqrt{2}} \right)i + \left(\dfrac{1}{\sqrt{2}} \right)j$
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 paraboid z = 4 – x2 – y2.
Group C    (5×8=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) = ex 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 $\int\limits_{0}^{\sqrt{2}} \int\limits_{0}^{3y} \int\limits_{x^2-3y^2}^{6-x^2-y^2} \mathrm{d}z \mathrm{d}x \mathrm{d}y$
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 $D = \dfrac{\partial}{\partial x} , D' = \dfrac{\partial}{\partial y}$.
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