## First Semester | First Year | Tribhuvan University

Computer Science and Information Technology (MTH. 104)

Calculus and Analytical Geometry, Year: 2066

Old Question Collection | Question Bank

**Download Question Paper File**

[File Type: PDF | File Size: 548 KB | Download]

**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)**

- Find the length of the curve y = x
^{3/2}from x=0 to x =4. - Find the critical points of the function f(x) = x
^{3/2}(x-4). - Does the following series converge?

- Find the polar equation of the circle (x+2)
^{2}+ y^{2}= 4 - Find the area of the parallelogram where vertices are A(0,0), B(7,3), C(9,8) and D(2,5).
- Evaluate the integral

- Evaluate the limit

- Find if ω = x
^{2}+ y – z + sin t and x + y = t. - Solve the partial differential equation p + q = x.
- Find the general integral of the linear partial differential equation z(xp – yq) = z
^{2}– x^{2}.

**Group B (5×4=20)** - State and prove Rolle’s theorem.
- Find the length of the cardioid r = 1 + cosθ.
- 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.
- 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 – x
^{2}– y^{2}– 2x – 2y + 4. - Find a particular integral of the equation

= 2y – x^{2}

**Group C (5×8=40)** - Graph the function y = x
^{4/3}– 4x^{1/3} - 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.
- Find the volume of the region enclosed by the surface z = x
^{2}+ 3y^{2}and z = 8 – x^{2}– y^{2}. - Obtain the absolute maximum and minimum values of the function.

f(x,y) = 2 + 2x + 2y – x^{2}– y^{2}on the triangular plate in the first quadrant bounded by lines x = 0, y = 0, y = 9 – x.

**OR**

Evaluate the integral - Show that the solution of the wave equation and deduce the result if the velocity is zero.

**OR**

Find a particular integral of the equation (D^{2}– D^{1}) = A cos(lx + my) where A, l, m are constants.

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