StudyNp | BIT First Semester – 2077 Question for Basic Mathematics

Tribhuwan University

Institute of Science and Technology

2077

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]

What is even and odd function? Give example of each and write their symmetricity. Find the domain and range of the following functions.

f(x) = \sqrt{5x + 10}

f(x) = \frac{x^2 - 3x - 4}{x+1}

[6+4]

Find the Taylor's series generated by

f(x) = \frac{1}{x}

a = 2

. Where, if anywhere, does the series converge to

\frac{1}{x}

? [10]

Sketch the graph of the function

f(x) = x^2

. Shifted vertically up to 1 and -2 units and horizontally up to 3 and -2 units. Find the

\delta

algebraically for the following functions.

\lim_{x \\to 5} \sqrt{x-1}, \text{and } L = 2, \epsilon = 1

\lim_{x \\to 2} (2x - 2), \text{and } L = 6, \epsilon = 0.02

[5+5]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

Find the initial value problem in

\frac{dy}{dx} + 2y = 3

y(0) = 1

. [5]

Determine the convergence or divergence of the series

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

. [5]

Show that

f(x) = \frac{x^2+x-6}{x^2-4}

x \neq 2

has a continuous extension to

x = 2

[5]

Evaluate the following:

\lim_{x \\to \infty} (x-\sqrt{x^2 + 16})

\lim_{x \\to 1} \left(\frac{\sqrt{6x+10}-5}{x^2}\right)

[2.5+2.5]

Find the derivatives of the function

f(x, y) = x^3 - xy^2 + x^2y - y^3

at the point

p_0(5, 5)

in the direction of

\\vec{u} = 4\\vec{i} + 3\\vec{j}

. [5]

Integrate the following:

\int_{0}^\frac{x}{4} \frac{dx}{1-\sin x}

[5]

10.

Find the area of the surface generated by revolving the curve

y = 2\sqrt{x}

1 \leq x \leq 2

, about x-axis. [5]

11.

Determine the concavity and find the inflection point of the function

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

[5]

12.

Find

\frac{dy}{dx}

for:

Y = 2u^3

u = 8x - 1

Y = \sin u

u = x - x\cos x

[5]