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]

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]

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.

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]