Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2078

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]
1.
What do you mean by asymptotes? How many types of asymptotes define each? Find horizontal and vertical asymptotes of the following function: $f(x) = -\frac{-8}{x^2-4}$ Does there other asymptotes exist? [4+5+1]
2.
Define area between two curves. Find area of the region enclosed by the parabola $y=2-x^2$ and the line $y= -x$. Define volume integral. Find the volume of solid generated by revolving the region bounded by the curve $y^2=x$ and the line $y=1$, $x=4$ about the line $y=1$. [1+3+6]
3.
Define Newton's-Raphson method with their formula. An open top box is to be made by cutting small congruent squares from the corners of square sheet of tin having length 12 inch. and is bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible? [2+8]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Graph the following functions. Write their symmetricity and specify the interval over which the function is increasing and decreasing. $y = -x^3$ , $y = x^2$ [5]
5.
Find the equations of tangent and normal to the curve $x^3 + y^3 - 9xy = 0$ at the point $(2,4)$. [5]
6.
What is L'Hospital's rule? Using this rule evaluate the following: $\lim_{x \to 0} (\sec x)^{\frac{1}{x^2}}$, $\lim_{x \to 0}\frac{x-\sin x}{x^3}$ [1+4]
7.
Define integration. Evaluate the following integral. $\int \frac{dx}{(x-1)(x-2)}$, $\int_{-1}^{1} 3x^2 \sqrt{x^3 + 1} \,dx$ [1+4]
8.
Define integral test. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2+4}$. [1+4]
9.
Solve the following differential equation: $x \frac{dy}{dx} = x^2 + 3y$, $x > 0$. [5]
10.
Find the derivative of $f(x, y, z) = x^3 - xy^2 - z$ at point $P(1, 1, 0)$ in the direction of $v = 2i - 3j + 6k$. [5]
11.
State Mean value theorem. Verify the mean value theorem if $f(x) = x^2 + 2x - 1$ on $[0,1]$. [1+4]
12.
Find $\frac{dy}{dx}$ if $y^2 - x^2 = \sin x$. Find the slope of circle $x^2 + y^2 = 25$ at the point $(3,4)$. [2.5+2.5]