## First Semester | First Year | Tribhuvan University

Computer Science and Information Technology (MTH. 104)

Calculus and Analytical Geometry, Year: 2065

Old Question Collection | Question Bank

**Download Question Paper File**

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**OR** you can read question paper online below;

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)**

- Verify Rolle’s theorem for the function on the interval [-3, 3].
- Obtain the area between two curves y = sec
^{2}x and y = sin x from x = 0 to x = π/4. - Test the convergence of p – series for p > 1.
- Find the eccentricity of the hyperbola 9x
^{2}– 16y^{2}= 144. - Find a vector perpendicular to the plane of P(1, -1, 0), C(2, 1, -1) and R(-1, 1, 2).
- Find the area enclosed by the curve r
^{2}= 4cos2θ. - Obtain the values of and at the point (4, -5) if f(x,y) = x
^{2}+ 3xy + y -1. - Using partial derivatives, find if x
^{2}+ cos y – y^{2}= 0. - Find the partial differential equation of the function (x – a)
^{2}+ (y – b)^{2}+ z^{2}= c^{2}. - Solve the partial differential equation x
^{2}p + q = z^{2}.

Group B (5×4=20) - State and prove the mean value theorem for a differential function.
- Find the length of the Asteroid x = cos
^{3}t, y = sin^{3}t for 0 ≤ t ≥ 2π. - Define a curvature of a curve. Prove that the curvature of a circle of radius a is 1/a.
- What is meant by direction derivative in the plain? Obtain the derivative of the function f(x,y) = x
^{2}+ xy at P(1, 2) in the direction of the unit vector - Find the center of mass of a solid of constant density δ, bounded below by the disk: x
^{2}+ y^{2}= 4 in the plane z = 0 and above by the paraboid z = 4 – x^{2}– y^{2}.

Group C (5×8=40) - Graph the function f(x) = -x
^{3}+ 12x + 5 for -3 ≤x ≤ 3. - 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. - 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 = x
^{2}. - Find the maximum and the minimum values of f(x, y) = 2xy – 2y
^{2}– 5x^{2}+ 4x – 4. Also find the saddle point if it exists.

**OR**

Evaluate the integral - 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 (D^{2}– D^{1})z =2y – x^{2}where .

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