StudyNp | CSIT First Semester – 2077 Question for Mathematics I

Tribhuwan University

Institute of Science and Technology

2077

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 $f(x) = x^2$ then find

$\frac{f(2+h)-f(2)}{h}$

Dry air is moving upward. If the ground temperature is $20^{\circ}C$ and the temperature at a height of 1km is $10^{\circ}C$, express the temperature T in $^{\circ}C$ as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part(a). What does the slope represent? (c) What is the temperature at a height of 2km? Find the equation of the tangent to the parabola $y = x^2 + x + 1$ at (0, 1). [2.5+5+5]

A farmer has 2000 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? Sketch the curve

$y = \frac{1}{x-3}$

[5+5]

Show that the following integrals converge and diverge respectively.

$\int_1^{\infty} \frac{1}{x^2} dx \text{ and } \int_1^{\infty} \frac{1}{x} dx$

If $f(x, y) = xy/(x^2 + y^2)$, does $f(x, y)$ exist as $(x, y) \to (0, 0)$? A particle moves in a straight line and has acceleration given by $a(t) = 6t^2 + t$. Its initial velocity is 4m/sec and its initial displacement is $s(0) = 5$cm. Find its position function $s(t)$. [2+3+5]

Evaluate

$\int_{3}^{2} \int_{0}^{\frac{\pi}{2}} \left( y + y^{2} \cos x \right) \, dx \, dy$

Find the Maclaurin's series for $\cos x$ and prove that it represents $\cos x$ for all x. [5+5]

Section B

Short Answers Questions

Attempt any Eight questions.

[8*5=40]

If $f(x) = x^2 - 1$, $g(x) = 2x + 1$, find $fog$ and $gof$ and domain of $fog$. [5]

Define continuity of a function at a point $x = a$. Show that the function $f(x) = \sqrt{1-x^2}$ is continuous on the interval $[1, -1]$. [5]

State Rolle's theorem and verify the Rolle's theorem for $f(x) = x^3 - x^2 - 6x + 2$ in [0, 3]. [5]

Find the third approximation $x_3$ to the root of the equation $f(x) = x^3 - 2x - 7$, setting $x_1 = 2$. [5]

Find the derivatives of $r(t) = (1 + t^2)\hat{i} - te^t\hat{j} + \sin 2t\hat{k}$ and find the unit tangent vector at $t=0$. [5]

10.

Find the volume of the solid obtained by rotating about the y-axis the region between $y = x$ and $y = x^2$. [5]

11.

Show that the series converges.

$\sum_{n=0}^{\infty} \frac{1}{1+n^2}$

[5]

12.

Find a vector perpendicular to the plane that passes through the points: $P(1, 4, 6)$, $Q(-2, 5, -1)$ and $R(1, -1, 1)$. [5]

13.

Find the partial derivative of $f(x, y) = x^3 + 2x^3y^3 - 3y^2 + x + y$ at (2,1). [5]

14.

Find the local maximum and minimum values, saddle points of $f(x, y) = x^4 + y^4 - 4xy + 1$. [5]

15.

Solve: $y'' + y = 0$, $y(0) = 5$, $y(\pi/4) = 3$. [5]