Tribhuvan University

Institute of Science and Technology

2074

Bachelor Level / first-semester / Science

Computer Science and Information Technology( MTH117 )

Mathematics I

Full Marks: 80 + 20

Pass Marks: 32 + 8

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.

**Group A**

**Attempt any three questions**

1 (a)

A function is defined by f(x) = \(\left\{\begin{matrix}

x + 2 \enspace \enspace if \enspace x < 0\\

1 – x \enspace \enspace if \enspace x > 0

\end{matrix}\right.\), Concluate f(-1), f(3) and sketch graph.

b

Prove that \(\lim_{x \to 0} \frac{|x|}{x}\) the does not exist.

2

Estimate the area between the curve y^{2} = x and the lines x = 0 and x = 2.

3 (a)

Find the Maclaurin series for e^{x} and prove that it represents e^{x} for all x.

b

Define Initial Value Problem. Solve that value problem of \(y^2 + 5y = 1\), y(0) = 2

c

Find the volume of a sphere of radius r

4 (a)

For What values of x does the series \(\sum_{n=1}^{∞} \frac{(x – 3)^n}{x}\) converge?

b

Calculate \(\int_R\int f(x,y) dA\) for f(x,y) = 100 – 6x^{2}y and \(R: 0 \leq x \leq 2, -1 \leq y \leq 1 \)

**Group B**

**Attempt any ten questions**

5

If \(f(x) = \sqrt{x}\) and \(g(x) = \sqrt{3-x}\), find gof and gog.

6

Use Continuity to evaluate the limit, \(\lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )\)

7

Verify Mean Value Theorem by f(x) = x^{3} – 3x + 3 for [-1, 2]

8

Sketch the curve y = x^3 + x

9

Determine whether the integral \(\int_{1}^{∞}\left ( \frac{1}{x} \right )dx\) is convergent or divergent

10

Find the length of the arc of the semi cubical parabola y^{2} = x^{2} between the points (1,1) and (4,8)

11

Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1

12

Test the convergence of the series \(\sum_{n=1}^{∞} \left ( \frac{n^n}{n!} \right )\)

13

Define cross product of two vectors. If \(\vec{a} = \vec{i} + 3\vec{j} + 4\vec{k}\) and \(\vec{b} = 2\vec{i} + 7\vec{j} – 5\vec{k}\) find the vector \(\vec{b} \times \vec{a} \enspace and \enspace \vec{a} \times \vec{b}\)

14

Define limit of a function. Find limit \(\lim_{x \to ∞} (x – \sqrt{x}) \)

15

Find the extremes values of f(x, y) = y^{2} – x^{2}

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