Questions
Syllabus
Qns. By Syllabus

Semester

Subject

Year

Tribhuwan University

Institute of Science and Technology

2080

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.
Define gradient of vector function of f(x,y,z) and find the derivative of f(x,y,z) = x3−xy2+zx^3 - xy^2 + z at p(1,0,0) in the direction of vecv\\vec{v} = 2veci\\vec{i} - j + veck\\vec{k}. Define Volume of the solid and find the volume of the solid generated by revolving the region bounded by y=xy = \sqrt{x} and the line y = 1, x = 4 about the line y = 1. [5+5]
2.
Evaluate ∫0π4dx1−sin⁡x\int_{0}^{\frac{\pi}{4}} \frac{dx}{1-\sin x}, Evaluate ∫x2sinxdx\int \\ x^2 sin x dx. Solve the differential equation dydx−3yx=x\frac{dy}{dx} - \frac{3y}{x} = x, x > 0. [5+5]
3.
State Rolle's Theorem and show that x3+3x+1=0x^3 + 3x + 1 = 0 has exactly one real solution. Find the area of the region enclosed by the parabola y=2−x2y = 2 - x^2 and the line y = -x. [5+5]
Section B

Short Answers Questions

Attempt any Eight questions.
[8*5=40]
4.
Define absolute value function and Sketch the graph of absolute value. [5]
5.
Find the limit of lim⁡hto∞6h+25−5h2\lim_{h\\to\infty} \frac { \sqrt {6h+25-5} }{h^2}. [5]
6.
State integral test and apply it to test the convergence of the series ∑n=1∞1n2+1\sum_{n=1}^{\infty}\frac{1}{n^2+1}. [5]
7.
Find the Taylors Series generated by f(x)=1xf(x) = \frac{1}{x} at a = 2. where, if anywhere, does the series converge to 1x\frac{1}{x}? [5]
8.
Define implicit differentiation and find the slope of the circle x2+y2=25x^2 + y^2 = 25 at the point (3, -4). [5]
9.
Define partial derivative and find the value of ∂f∂x&∂f∂y\frac{\partial f}{\partial x} \& \frac{\partial f}{\partial y} at the point (4, -5) if f(x,y)=x3+3xy+y−1f(x, y) = x^3 + 3xy + y - 1. [5]
10.
Evaluate ∫0π2(sin⁡2xcos⁡3x+cos⁡2xsin⁡3x)dx\int_0^{\frac{\pi}{2}}(\sin 2x\cos 3x + \cos 2x\sin 3x)dx and ∫0π4dx1−sin⁡x\int_{0}^{\frac{\pi}{4}} \frac{dx}{1 - \sin x}. [5]
11.
Determine the concavity of y = 3 + sin x on [0, 2π\pi]. [5]
12.
Test for convergence of the series ∑n=1∞(1n+1)n\sum_{n=1}^{\infty}\left(\frac{1}{n+1}\right)^n. [5]