LearnCSIT
Tribhuwan University
Institute of Science and Technology
2074
Bachelor Level/ first Semester/ Science
B.Sc.CSIT
Mathematics I
Full Marks: 80
Pass Marks: 32
Time: 3 hours
Candidates are required to give their answer in their own words as far as practicable.
The figures in the margin indicate full marks.
Section A
Attempt any THREE questions
1.
(a) A function is defined by
f(x) = \begin{cases} x + 2, & \text{if } x < 0 \\ 1 - x, & \text{if } x > 0 \end{cases}
,
calculate
f(-1),
f(3),
and sketch the graph. (5)
(b) Prove that the
\lim_{x \to 0} \dfrac{|x|}{x}
does not exist. (5)
2.
(a) Find the derivative of
f(x) = \sqrt{x}
and to state the domain of
f'
. (3+2)
(b) Estimate the area between the curve
y^2 = x
and the lines
x = 0
and
x = 2
. (5)
3.
(a) Find the Maclaurian series for
e^x
and prove that it represents
e^x
for all
x
. (4)
(b) Define initial value problem. Solve that initial value problem of
y' + 5y = 1, y(0) = 2
. (4)
(c) Find the volume of a sphere of radius r. (2)
4.
(a) For what values of x does the series
\displaystyle \sum_{n=1}^{\infty} \dfrac{(x-3)^n}{x}
converge? (5)
(b) Calculate
\displaystyle \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)
Section B
Attempt any TEN questions
5.
If
f(x) = \sqrt{x}
and
g(x) = \sqrt{3 - x}
, find
gof
and
gog
. (5)
6.
Use continuity to evaluate the limit,
\lim_{x \to 4} \dfrac{5+\sqrt{x}}{\sqrt{5+x}}
. (5)
7.
Verify Mean value theorem of
f(x) = x^3 - 3x + 3
for
[-1, 2]
. (5)
8.
Sketch the curve
y = x^3 + x
. (5)
9.
Determine whether the integral
\displaystyle \int_1^\infty \dfrac{1}{x} \, dx
is convergent or divergent. (5)
10.
Find the length of the arc of the semicubical parabola
y^2 = x^3
between the points
(1, 1)
and
(4, 8)
. (5)
11.
Find the solution of
y'' + 6y' + 9 = 0,
y(0) = 2,
y'(0) = 1
. (5)
12.
Test the convergence of the series
\displaystyle \sum_{n=1}^{\infty} \dfrac{n^n}{n!}
. (5)
13.
Define cross product of two vectors. If
a = i + 3j + 4k
and
b = 2i + 7j - 5k
, find the vector
a \times b
and
b \times a
. (1+2+2)
14.
Define limit of a function. Find
\lim_{x \to \infty} (x - \sqrt{x})
. (1+4)
15.
Find the extreme values of
f(x,y) = y^2 - x^2
. (5)