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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: 2077, syllabus wise question

Application of Definite Integrals
1.
Find the area of the surface generated by revolving the curve y=2xy = 2\sqrt{x}, 1≤x≤21 \leq x \leq 2, about x-axis. [5]
Application of Derivative
1.
Determine the concavity and find the inflection point of the function f(x)=x3−3x2+2f(x) = x^3 - 3x^2 + 2 [5]
Differentiation
1.
Find dydx\frac{dy}{dx} for: Y=2u3Y = 2u^3, u=8x−1u = 8x - 1, Y=sin⁡uY = \sin u, u=x−xcos⁡xu = x - x\cos x [5]
First Order Differential Equations
1.
Find the initial value problem in dydx+2y=3\frac{dy}{dx} + 2y = 3, y(0)=1y(0) = 1. [5]
Functions and their graphs
1.
What is even and odd function? Give example of each and write their symmetricity. Find the domain and range of the following functions.
f(x)=5x+10f(x) = \sqrt{5x + 10}
f(x)=x2−3x−4x+1f(x) = \frac{x^2 - 3x - 4}{x+1}
[6+4]
2.
Sketch the graph of the function f(x)=x2f(x) = x^2. Shifted vertically up to 1 and -2 units and horizontally up to 3 and -2 units. Find the δ\delta algebraically for the following functions.
lim⁡xto5x−1,and L=2,ϵ=1\lim_{x \\to 5} \sqrt{x-1}, \text{and } L = 2, \epsilon = 1
lim⁡xto2(2x−2),and L=6,ϵ=0.02\lim_{x \\to 2} (2x - 2), \text{and } L = 6, \epsilon = 0.02
[5+5]
Infinite Sequence and Series
1.
Find the Taylor's series generated by f(x)=1xf(x) = \frac{1}{x} at a=2a = 2. Where, if anywhere, does the series converge to 1x\frac{1}{x}? [10]
2.
Determine the convergence or divergence of the series ∑n=1∞n2e−n\sum_{n=1}^{\infty} {n^2}{e^{-n}}. [5]
Integration
1.
Integrate the following: ∫0x4dx1−sin⁡x\int_{0}^\frac{x}{4} \frac{dx}{1-\sin x} [5]
Limits and continuity
1.
Show that f(x)=x2+x−6x2−4f(x) = \frac{x^2+x-6}{x^2-4}, x≠2x \neq 2 has a continuous extension to x=2x = 2 [5]
2.
Evaluate the following: lim⁡xto∞(x−x2+16)\lim_{x \\to \infty} (x-\sqrt{x^2 + 16}) lim⁡xto1(6x+10−5x2)\lim_{x \\to 1} \left(\frac{\sqrt{6x+10}-5}{x^2}\right) [2.5+2.5]
Partial Derivatives
1.
Find the derivatives of the function f(x,y)=x3−xy2+x2y−y3f(x, y) = x^3 - xy^2 + x^2y - y^3 at the point p0(5,5)p_0(5, 5) in the direction of vecu=4veci+3vecj\\vec{u} = 4\\vec{i} + 3\\vec{j}. [5]