StudyNp | CSIT First Semester – 2075 Question for Mathematics I

Tribhuwan University

Institute of Science and Technology

2075

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]

A function is defined by

$f(x) = |x|$

Calculate f(-3), f(4), and sketch the graph. Prove that the limit does not exist.

$\lim_{x \to 2} \frac{|x-2|}{x-2}$

[5+0+5]

Find the domain and sketch the graph of the function

$f(x) = x^2 - 6x$

Estimate the area between the curve $y = x^2$ and the line y = 1 and y = 2. [3+2]

If $f(x, y) = \frac{y}{x}$ $\lim_{(x,y)\to(0,0)} \frac{f(x,y)}{x}$ does not exist, justify. Calculate $\iint_R f(x, y) dA$, for $f(x, y) = 100 - 6x^2y$, and $R: 0 \leq x \leq 2, -1 \leq y \leq 1$. [5+5]

Find the Maclaurin series for $\cos x$ and prove that it represents $\cos x$ for all x. Define initial value problem. Solve that initial value problem of $y' + 2y = 3$, $y(0) = 1$. Find the volume of a sphere of radius $r$. [4+4+2]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

If $f(x) = \sqrt{2-x}$ and $g(x) = \sqrt{x}$, find $fof$ and $fog$. [5]

Define continuity on an interval. Show that the function is continous on the interval [1,-1].

$f(x) = 1 - \sqrt{1 - x^2}$

[5]

Verify Mean value theorem of $f(x) = x^3 - 3x + 2$ for [-1,2]. [5]

Starting with $x_1 = 2$, find the third approximation $x_3$ to the root of the equation $x^3 - 2x - 5 = 0$. [5]

Evaluate

$\int_0^{\infty} x^3 \sqrt{1-x^4} dx$

[5]

10.

Find the volume of the resulting solid which is enclosed by the curve $y = x$ and $y = x^2$, is rotated about the x-axis. [5]

11.

Determine whether the series converges or diverges

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

[5]

12.

If $a = (4,0,3)$ and $b = (-2,1,5)$, find $|a|$, the vector $a-b$ and $2a+b$. [5]

13.

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ if z is defined as a function of x and y by the equation $x^3+y^3+z^3+6xyz=1$. [5]

14.

Find the extreme values of the function $f(x, y) = x^2 + 2y^2$ on the circle $x^2 + y^2 = 1$. [5]

15.

Find the solution of $y'' + 4y' + 4 = 0$. [5]