## First Semester | First Year | Tribhuvan University

Old Questions, Mathematics I (New), Year: 2073,

Calculus and Analytical Geometry (Old),

Computer Science and Information Technology (MTH.112)

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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 questions.**

**Group A (10×2=20)**

- If f(x) = sin x and g(x) = -x/2. Find f(f(x)) and g(f(x)).
- Define critical point. Find the critical point of f(x) = 2x
^{2}. - Evaluate
- Find the equation of the parabola with vertex at the origin and directrix at x= 7.
- Find a vector parallel to the line of intersection of the planes 3x + 6y – 2z = 5.
- Evaluate
- Find if f(x,y) = x
^{2}+ y^{2} - Evaluate
- Show that y = ax
^{2}+ b is the solution of xy’’ + y’ = 0. - Solve

**Group B (5×4=20)** - Verify Rolle’s theorem for f(x) = x
^{3}, x ∈ [-3,3]. - Find the Taylor series expansion of the case at e
^{x}, at x=0. - Find a Cartesian equivalent of the polar equation r cos (θ-π/3) = 3.
- Evaluate it
- Obtain the general solution of

**Group C (5×8=40)** - Evaluate the integrals and determine whether they converge or diverge

(a) (b)

**OR**

Find the area bounded on the parabola y = 2 – x^{2}and the line y = -x. - Find the curvature of the helix R ⃗(t) = (a cos ωt)i ⃗ + (a sin ωt)j ⃗ + (bt)k ⃗?
- Find the volume enclosed between the surfaces z = x
^{2}+ 3y^{2}and z = 8 – x^{2}– y^{2} - Find the extreme values of the function F(x,y) = xy –x
^{2}–y^{2}-2x -2y + 4

**OR**

Find the extreme values of f(x,y) = xy subject to g(x,y) = x^{2}+ y^{2}– 10 = 0. - Define second order partial differential equation. Define initial boundary value problem. Derive the heat equation or wave equation in one dimension.

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