StudyNp | CSIT First Semester – 2080 Question for Mathematics I

Tribhuwan University

Institute of Science and Technology

2080

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]

\vec{a} = (4,0,3)

and

\vec{b} = (-2,1,5)

, find

|\vec{a}|

3\vec{b}

\vec{a} + \vec{b}

and

2\vec{a} + 5\vec{b}

. Estimate the value of

\lim_{x \to 0} \frac{\sqrt{x^2 + 9} - 3}{x^2}

[5+5]

As dry air moves upward, it expands and cools. If the ground temperature is

20^{\circ}C

and the temperature at height of 1 km is

10^{\circ}C

, express the temperature

T

(in

^{\circ}C

) as a function of the height

h

(in kilometer), assuming that linear model is appropriate. (a) Draw a graph of the function in part (b). What does the slope represent? (c) What is the temperature at a height of 2.5 km? [5+5]

The area of the parabola

y = x^2

from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface. Find the solution of the equation

y^2 dy = x^2 dx

that satisfies the initial condition

y(0) = 2

. [5+5]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

Integrate

\int_0^1 x^2 \sqrt{x^3 + 1} dx

[5]

Find the Maclaurin series expansion of

f(x) = e^x

x = 0

. [5]

Find where the function

f(x) = 3x^4 - 4x^3 - 12x^2 + 5

is increasing and where it is decreasing. [5]

Find y' if

x^3 + y^3 = 6xy

. [5]

Sketch the graph and find the domain and range of the function

f(x) = 2x - 1

. [5]

Determine whether the series converges or diverges

\sum_{n=1}^{\infty} \frac{n^2}{5n^2+4}

[5]

10.

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

, find

f_x(2,1)

and

f_y(2,1)

. [5]

11.

Show that the function

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

is continuous at

x = 4

. [5]

12.

Show that

y = x - \frac{1}{x}

is a solution of the differential equation

xy' + y = 2x

. [5]