StudyNp | CSIT First Semester – 2074 Question for Mathematics I

Tribhuwan University

Institute of Science and Technology

2074

Bachelor Level / First Year / First Semester / Science

B.Sc in Computer Science and Information Technology (MTH117)

(Mathematics I)

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 Answers Questions

Attempt any TWO questions.

[2*10=20]

A function is defined by

f(x) = \begin{cases} x+2, & x<0 \\ 1-x, & x>0 \end{cases}

Calculate f(-1), f(3), and sketch the graph. Prove that the limit does not exist.

\lim_{x \to 0} \frac{|x|}{x}

[5+0+5]

Find the derivative of

f(x) = \sqrt{x}

State the domain of f. Estimate the area between the curve and the line x=0 and x=2 where curve is

y^2 = x

[3+2+5]

For what values of x does the series converge?

\sum_{n=1}^{\infty} \frac{(x-3)^n}{x}

Calculate

\iint_{R} f(x, y) dA

, for

f(x, y) = 100 - 6x^2y

, and

R: 0 \leq x \leq 2, -1 \leq y \leq 1

. [5+5]

Find the Maclaurin series for

e^x

and prove that it represents

e^x

for all x. Define initial value problem. Solve that initial value problem of

y' + 5y = 1

y(0) = 2

. Find the volume of a sphere of radius r. [4+4+2]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

f(x) = \sqrt{x}

and

g(x) = \sqrt{3-x}

, find

gof

and

fog

. [5]

Use continuity to evaluate the limit,

\lim_{x \to 4} \frac{5 + \sqrt{x}}{\sqrt{5 + x}}

[5]

Verify Mean value theorem of

f(x) = x^3 - 3x + 3

for [-1,2]. [5]

Sketch the curve

y = x^3 + x

. [5]

Determine whether the integral is convergent or divergent.

\int_1^{\infty} \frac{1}{x} dx

[5]

10.

Find the length f the arc of the semicubical

y^2 = x^2

between the points (1,1) and (4,8). [5]

11.

Test the convergence of the series

\sum_{n=1}^{\infty} \frac{n^n}{n!}

[5]

12.

Define cross product of two vectors. If

\vec{a} = \hat{i} + 3\hat{j} + 4\hat{k}

and

\vec{b} = 2\hat{i} + 7\hat{j} - 5\hat{k}

, find the vector

\vec{a} \times \vec{b}

and

\vec{b} \times \vec{a}

. [5]

13.

Define limit of a function.

\lim_{x \to \infty} \left({x - \sqrt{x}} \right)

[5]

14.

Find the extreme values of

f(x, y) = y^2 - x^2

. [5]

15.

Find the solution of

y'' + 6y' + 9 = 0

y(0) = 2

y'(0) = 1

. [5]