StudyNp | CSIT First Semester – 2079 Question for Mathematics I

Tribhuwan University

Institute of Science and Technology

2079

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]

If a function is defined by

f(x) = \begin{cases} 1 + x, & x \leq -1 \\ x^2, & x > -1 \end{cases}

, evaluate

f(-3)

f(-1)

and

f(0)

and sketch the graph. Prove that

\lim_{x \to 0} \frac{|x|}{x}

does not exist. [10+0]

Sketch the curve

y = x^2 + 1

with the guidelines of sketching. If

z = xy^2 + y^3

x = \sin t

y = \cos t

, find

\frac{dz}{dt}

t = 0

. [10+0]

Estimate the area between the curve

y = x^2

and the lines

x = 0

and

x = 1

, using rectangle method, with four sub intervals. A particle moves a line so that its velocity

v

at time

t

is (1) Find the displacement of the particle during the time period

1 \leq t \leq 4

(2) Find the distance travelled during this time period. [10+0]

Define initial value problem. Solve:

y'' + 4y' - 6y = 0

y(0) = 1

y'(0) = 0

Find the Taylor's series expansion for

\cos x

x = 0

. [10+0]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

Dry air is moving upward. If the ground temperature is

20^\circ

and the temperature at a height of 2km is

10^\circ

, express the temperature

T

^\circ

C as a function of the height

h

(in km), assuming that a linear model is appropriate. (b) Draw the graph of the function and find the slope. Hence, give the meaning of slope. (c) What is the temperature at a height of 2km? [5]

Find the equation of the tangent at (1,3) to the curve

y = x^2 + 1

. [5]

State Rolle's theorem and verify the theorem for

f(x) = x^2 - 9

x \in [-3,3]

. [5]

Starting with

x_1 = 1

, find the third approximate

x_3

to the root of the equation

x^3 - x - 5 = 0

. [5]

Show the integral coverages

\int_0^3 \frac{dx}{x-1}

. [5]

10.

Use Trapezoidal rule to approximate the integral

\int_1^2 \frac{dx}{x}

, with n=5. [5]

11.

Find the derivative of

\mathbf{r}(t) = t^2\mathbf{i} - te^{-t}\mathbf{j} + \sin(2t)\mathbf{k}

and find the unit tangent vector at

t = 0

. [5]

12.

What is sequence? Is the sequence

a_n = \frac{n}{\sqrt{5+n}}

convergent? [5]

13.

Find the angle between the vectors

a = (2, 2, -1)

and

b = (1, 3, 2)

. [5]

14.

Find the partial derivative

f_{xx}

and

f_{yy}

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

at (1,2). [5]

15.

Solve

\int_0^3 \int_1^2 x^2y \, dx \, dy

[5]