# Mathematics I | BSc.CSIT (TU) Question Paper 2073 | First Semester

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## 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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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)

1. If f(x) = sin x and g(x) = -x/2. Find f(f(x)) and g(f(x)).
2. Define critical point. Find the critical point of f(x) = 2x2.
3. Evaluate $\lim_({n \to \infty}) \frac{a-b^4}{n^4+a}$
4. Find the equation of the parabola with vertex at the origin and directrix at x= 7.
5. Find a vector parallel to the line of intersection of the planes 3x + 6y – 2z = 5.
6. Evaluate $\int_{-1}^{0}\int_{-1}^{1}(x+y+1)dxdy$
7. Find $\frac{dt}{dx} \text{and} \frac{dt}{dy}$ if f(x,y) = x2 + y2
8. Evaluate $\log_{(x,y)\to(0,1)} \frac{x-xy+k}{x^2 y+5xy-y^3}$
9. Show that y = ax2 + b is the solution of xy’’ + y’ = 0.
10. Solve $\frac{d^2 y}{dx^2}-y=0$
Group B (5×4=20)
11. Verify Rolle’s theorem for f(x) = x3, x ∈ [-3,3].
12. Find the Taylor series expansion of the case at ex, at x=0.
13. Find a Cartesian equivalent of the polar equation r cos (θ-π/3) = 3.
14. Evaluate it $(x,y) \to (0,0) \frac{2y^2}{\sqrt{x^2 +xy}}$
15. Obtain the general solution of $(y-z)\frac{dz}{dx} + (x-y)\frac{dz}{dy}=z-x$
Group C (5×8=40)
16. Evaluate the integrals and determine whether they converge or diverge
(a) $\int_{-1}^{\infty}\frac{dx}{x}$ (b) $\int_{-1}^{\infty}\frac{dx}{x^2}$
OR
Find the area bounded on the parabola y = 2 – x2 and the line y = -x.
17. Find the curvature of the helix R ⃗(t) = (a cos ωt)i ⃗ + (a sin ωt)j ⃗ + (bt)k ⃗?
18. Find the volume enclosed between the surfaces z = x2 + 3y2 and z = 8 – x2 – y2
19. Find the extreme values of the function F(x,y) = xy –x2 –y2 -2x -2y + 4
OR
Find the extreme values of f(x,y) = xy subject to g(x,y) = x2 + y2 – 10 = 0.
20. 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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