CSIT 1st Semester
Math I Board Question Paper 2080 Old Course

MTH 117-2080 ✡
Tribhuvan University
Institute of Science and Technology
2080
✡
Bachelor Level/First Year/First Semester/Science
Computer Science Information Technology (MTH 117)
(Math I)
(Old Course)
Full Marks:60 Pass Marks:24 Time:3 hours

Candidates are required to give their answers in their own words as for as practicable.
The figures in the margin indicate full marks

Section A
Long Answer Questions
Attempt any Two question.
[2x10=20]
1.

(a) If \(f(x) = 3x^2 - 2x - 5\), then find \(f(1)\), \(f(2)\), \(f(a)\), \(f(-a)\) and \(f(0)\).
(b) Calculate: \(\lim_{x \to 1} \frac{x}{x^2 + x - 2}\).

2.

(a) Find the derivative of \(y = \frac{\sqrt{x}}{x^2 - x}\).
(b) Estimate the area between the parabola \(y^2 = x\) and the line \(y = -x\).

3.

(a) Verify the Mean Value Theorem for the function \(f(x) = x^2 - x\) for \(x \in [-2, 2]\).
(b) Define initial value problem. Solve the equation \(x\dot{y} - y = 4x\), \(y(2) = 6\).

Section B

Attempt any Eight questions

[8x5=40]
4.

(a) Find the local maximum and minimum values and saddle points of \(f(x, y) = x^2 + y^2 - x^2y + 2\).
(b) Where is the function \(f(x) = |x|\) differentiable? Discuss.

5.

Verify Rolle's theorem for \(f(x) = x^2 + 4\) for \(x \in [-2, 2]\).

6.

Find the Maclaurin series expansion of \(e^x\) at \(x = 0\).

7.

If \(f(x) = \sqrt{2x + 1}\) and \(g(x) = \sqrt[3]{x}\), find \((fog)(x)\) and \((gof)(x)\).

8.

Show that the function \(f(x) = 1 + x^2\) is continuous everywhere.

9.

Evaluate: \(\int_0^2 \frac{x \, dx}{\sqrt{1 + 2x^2}}\).

10.

Sketch the curve \(y = 2x^2 + 1\).

11.

Find the solution of \(y'' - 6y' - 7y = 0\).

12.

Test whether the series \(\sum_{n=1}^{\infty} \frac{n^2}{3n^2 + 4}\) diverges or converges.