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

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Mathematics IFirst 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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