Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

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]
1.
If $\vec{a} = (4,0,3)$ 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]
2.
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]
3.
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]
4.
Integrate

$\int_0^1 x^2 \sqrt{x^3 + 1} dx$
[5]
5.
Find the Maclaurin series expansion of $f(x) = e^x$ at $x = 0$. [5]
6.
Find where the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is increasing and where it is decreasing. [5]
7.
Find y' if $x^3 + y^3 = 6xy$. [5]
8.
Sketch the graph and find the domain and range of the function $f(x) = 2x - 1$. [5]
9.
Determine whether the series converges or diverges

$\sum_{n=1}^{\infty} \frac{n^2}{5n^2+4}$
[5]
10.
If $f(x, y) = x^3 + x^2y^3 - 2y^2$, 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]