StudyNp | BIT First Semester – 2081 Question for Basic Mathematics

Credit:

Eli Tamang

Tribhuwan University

Institute of Science and Technology

2081

Bachelor Level / First Year / First Semester / Science

Bachelors in Information Technology (MTH104)

(Basic Mathematics)

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]

Explain the meaning

\lim_{x \to 2} f(x) = 5

. Is it possible for this statement to be true, yet

f(2) = 3

? Explain. Draw a graph of the function

f(x) = x^2 + 4

and find its domain and range. [5+5]

Find the derivative of

y = \dfrac{\tan^{-1} x}{\sqrt{x}}

with respect to

x

. Find the area of the region bounded by

y = -x

and

x = y^2 + 3y

. [5+5]

What is initial value problem? Find the solution of the initial value problem

x \dfrac{dy}{dx} - y = x^2

y(2) = 5

. Evaluate:

\lim_{x \to \infty} \dfrac{3x^2 - 5x + 2}{5x^2 + 8x + 7}

. [1+4+5]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

Evaluate:

\int \sqrt{4 - x^2} \, dx

. [5]

Find the volume of the solid obtained by rotating about the y-axis the region bounded by

y = x

and

y = x^2

. [5]

Evaluate:

\int_0^5 \dfrac{dx}{\sqrt{x - 2}}

, if it exists. [5]

Test whether the series

\sum_{n=2}^{\infty} \dfrac{2}{n^2 - 1}

converges or diverges. [5]

Use Newton's method to find

\sqrt[4]{10}

correct to four decimal places. [5]

Find the partial derivatives

f_x

f_y

and

f_{xy}

f(x,y) = \sqrt{x} y^3 + x^4 y

(-4,1)

. [5]

10.

Verify mean value theorem for the function

f(x) = x^2 + 3x + 1

[-1,1]

. [5]

11.

Test whether the function

f(x) = \begin{cases} \dfrac{x^2 - 2x}{x - 2}, & x \ne 2 \\ 1, & x = 2 \end{cases}

is continuous or discontinuous at

x = 2

. Explain. [5]

12.

Evaluate:

\int_0^{\pi} x \sin x \, dx

. [5]