### Exam Year

Tribhuvan University

Institute of Science and Technology

2074

Bachelor Level / first-semester / Science

Computer Science and Information Technology( MTH112 )

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 y2 = x and the lines x = 0 and x = 2.

3 (a)

Find the Maclaurin series for ex and prove  that it represents ex 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 – 6x2y 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) = x3 – 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 y2 = x2 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) = y2 – x2