StudyNp | BIT First Semester – 2079 Question for Basic Mathematics

Tribhuwan University

Institute of Science and Technology

2079

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]

If a function is defined by

f(x) = \begin{cases} 1 + x & \text{if } x \leq -1 \\\ x^2 & \text{if } x > -1 \end{cases}

Evaluate

f(-3)

f(-1)

, and

f(0)

and sketch the graph. Define different types of discontinuity at a point. At what points the function becomes continuous of the function

f(x) = \frac{x-2}{x^2-7x+10}

[5+5]

Define Gradient vector and directional derivative.Find the direction in which

f(x, y) = \frac{x^2}{2} + \frac{y^2}{2}

increases and decreases most rapidly at the point

(1,1)

. What is the direction of zero change in

f

(1,1)

? Derivative of

f(x, y)

at the point

(1,1)

in the direction

v = 3i - 4j

. [5+2+3]

Define the concavity of the function. The graph of the function is then

f(x) = x^4 - 4x^3 + 10

. Find the intervals on which

f

is increasing and on which

f

is decreasing. Find where the graph of

f

is concave up and where it is concave down. Find the local maximum or local minimum value of function if exist. [2+3+3+2]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

Find the domain and range of the function

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

. Draw the graph of the function

y = x^2

shifted up by 1 unit, down by 2 units, also shift 3 units to left, and 2 units right with new position of function. [2+3]

Define horizontal and vertical asymptotes. Find the appropriate asymptotes to the function:

f(x) = x - \sqrt{x^2 + 16}

. [2+3]

Find the area of the region between the x-axis and the graph of

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

1 \leq x \leq 2

. [5]

Define arc-length of the curve. Find the length of the curve

y = (\frac{x}{2})^{\frac{2}{3}}

from

x = 0

x = 2

. [1+4]

Evaluate the following integral.

\int_{0}^{\frac{\pi}{4}} \sqrt{1 + \cos x}dx

\int \frac{3x^2-7x+1}{3x} , dx

[5]

What is a first order linear differential equation? Solve the initial value problem:

t \frac{dy}{dt} + 2y = t^3

t > 0

y(2) = 1

. [1+4]

10.

Determine whether the following series are convergence or divergence

\sum_{n=1}^{\infty} \frac{5}{5n-1}

\sum_{n=0}^{\infty} \frac{1}{n!}

. [5]

11.

What is chain rule for function

w = f(x, y)

? To use this rule, find the derivative of

w = xy

w.r.t.

t

along the path

x = \cos t

y = \sin t

. Also, find derivative of

w

t = \frac{\pi}{2}

. [1+4]

12.

Find

\frac{dy}{dx}

of the following.

y^2 - x^2 = \cos(xy)

x^2 = \frac{9}{y^2}

. [5]