StudyNp | BIT First Semester – 2081.2 Question | Basic Mathematics (Syllabus Wise)

Semester

Subject

Year

Bachelors Level/First Year/First Semester/Science bit/first semester/basic mathematics/syllabus wise questions

Bachelors In Information Technology

Institute of Science and Technology, TU

Basic Mathematics (MTH104)

Year Asked: 2081.2, syllabus wise question

Application of Definite Integrals

Find the area enclosed by the ellipse

\frac{x^2}{9} + \frac{y^2}{4} = 1

. Evaluate:

\int_0^2 \frac{x}{\sqrt{x^2 + 4}} \, dx

. [5+5]

Differentiation

f(x) = x^2 + 2x - 1

and

g(x) = 2x - 3

, then find

fog(x)

and

gof(x)

. Find the local maxima and local minima of the function

f(x) = 3x^4 - 4x^3 - 12x^2 + 5

. [5+5]

Find a unit vector that has the same direction as the given vector

-3\mathbf{i} + 7\mathbf{j}

. [5]

First Order Differential Equations

Solve:

xy' = y

, when

y(1) = 2

. [5]

Solve:

y'' - y' - 6y = 0

. [5]

Functions and their graphs

Sketch the graph of

f(x) = \sqrt{x}

. Also, find the domain and range. [5]

Infinite Sequence and Series

Determine whether the series

\sum_{n=1}^{\infty} \frac{5}{2n^2 + 4n + 3}

is convergent or divergent. [5]

Find the Maclaurin’s series expansion of

f(x) = \ln x

. [5]

Limits and continuity

Show that the function

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

is continuous on the interval

[-1,1]

. Evaluate:

\lim_{x \to \infty} \sqrt{x}(\sqrt{x-2} - \sqrt{x})

. [5+5]

Test whether the function

f(x) = \begin{cases} \frac{x^2 - 4}{x - 2} & \text{if } x \ne 2 \\ 4 & \text{if } x = 2 \end{cases}

is continuous or discontinuous at

x = 2

. Explain. [5]

Partial Derivatives

Use the chain rule to find

\frac{dz}{dt}

when

z = \cos(x + 4y)

x = 5t^4

y = \frac{1}{t}

. [5]

Find

\frac{\partial z}{\partial x}

and

\frac{\partial z}{\partial y}

x^3 + y^3 + z^3 + 6xyz = 1

. [5]