Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

Model

Bachelor Level / First Year / First Semester / Science

Bachelors in Information Technology (MTH104)

(Basic Mathematics)

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]
1.
In 2000, 100 is invested in a savings account, where it grows by accruing interest that is compounded annually (once a year) at an interest rate of 5.5%\%. Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount AA in the account after xx years have elapsed. Define when the function f(x)f(x) is odd and even. Also, define when a function f(x)f(x) is increasing and decreasing? If y=x2y = x^2 is a given function then determine the interval in which the function is increasing and decreasing and draw the graph of the given function. [5+5]
2.
A rock breaks loose from the top of a tall cliff. Find average speed during the first 2 sec of fall. What is its average speed during the 1sec interval between second 1 and second 2? Find the speed of the falling rock at t=1t = 1 and t=2t = 2. [3+3+4]
3.
Find the positive root of the equation f(x)=x2−2=0f(x) = x^2 - 2 = 0. Find the Taylor series and the Taylor polynomials generated by f(x)=exf(x) = e^x at x=0x = 0. Use the Trapezoidal Rule with n=4n = 4 to estimate ∫12x2dx\int_{1}^{2} x^2 dx. Compare the estimate with the exact value. [3+3+4]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Define horizontal asymptote to a curve y=f(x)y = f(x). Find the horizontal asymptote to the curve f(x)=5x2+8x−33x2+2f(x) = \frac{5x^2 + 8x - 3}{3x^2 + 2} and draw the curve. [2+3]
5.
Find the slope of the curve y=1/xy = 1/x at any point x=ax = a, a≠0a \neq 0. What is the slope at the point x=−1x = -1? Where does the slope equal −1/4-1/4? What happens to the tangent to the curve at the point (a,1/a)(a, 1/a) as aa changes? [2+1.5+1.5]
6.
Water runs into a conical tank at the rate 9 ft3/minutes9\text{ ft}^3/\text{minutes}. The tank stands point down and has a height of 10 ft10\text{ ft} and a base radius of 5 ft5\text{ ft}. How fast is the water level rising when the water is 6 ft6\text{ ft} deep? [5]
7.
Find the absolute maximum and minimum values of f(x)=x2/3f(x) = x^{2/3} on the interval [−2,3][-2, 3]. [5]
8.
Find the area between the curves y=x2−2y = x^2 - 2 and y=2y = 2. [5]
9.
A pyramid 3 m3\text{ m} high has a square base that is 3 m3\text{ m} on a side. The cross section of the pyramid perpendicular to the altitude x mx\text{ m} down from the vertex is a square x mx\text{ m} on a side. Find the volume of the pyramid. [5]
10.
Draw a phase line for the equation dydx=(y+1)(y−2)\frac{dy}{dx} = (y + 1)(y - 2) and use it to sketch solutions to the equation. [5]
11.
Find the second order derivative ∂2f∂x2,∂2f∂y2,∂2f∂x∂y,∂2f∂y∂x\frac{\partial^2 f}{\partial x^2}, \frac{\partial^2 f}{\partial y^2}, \frac{\partial^2 f}{\partial x\partial y}, \frac{\partial^2 f}{\partial y\partial x} of f(x,y)=xcos⁡y+yexf(x,y) = x \cos y + ye^x [5]
12.
Find the local extreme values of the function f(x,y)=xy−x2−y2−2x−2y+4f(x,y) = xy - x^2 - y^2 - 2x - 2y + 4. [5]