Elementary Mathematics II — andriod Past Questions

University of Ilorin MTH102: 53 questions.

Topics: Derivatives as limits of rate of change, differentiation techniques, maxima and minima., Functions of a real variable, graphs, limits, and continuity., Curve sketching, integration, definite integrals, reduction formulae.

1. Solve for $dy/dx$ given $4xy + 7x + \cos(xy) = 3$.
   1. $\frac{y\sin(xy) - 4y - 7}{x\sin(xy) - 4x}$
   2. $\frac{y\sin(xy) + 4y - 7}{x\sin(xy) + 4x}$
   3. $\frac{y\cos(xy) - 4y - 7}{x\cos(xy) - 4x}$
   4. $\frac{-(y\sin(xy) - 4y - 7)}{x\sin(xy) - 4x}$
2. If $y = \log(\cos x)$ find $dy/dx$.
   1. $-\tan x$
   2. $\tan x$
   3. $-\cot x$
   4. $\sec x$
3. Differentiate $y = e^{-x^2}\sin x^2$.
   1. $2xe^{-x^2}[\cos x^2 - \sin x^2]$
   2. $2xe^{-x^2}[\cos x^2 + \sin x^2]$
   3. $-2xe^{-x^2}[\cos x^2 - \sin x^2]$
   4. $-2xe^{-x^2}[\sin x^2 + \cos x^2]$
4. Find the point of inflexion in the curve $y = 3x^4 - 8x^3 + 6x^2 - 5$.
   1. $1$
   2. $1/3$
   3. $-1$
   4. $3$
5. Find the Maclaurin series of $f(x) = 1/(1-x)$ to the 4th term.
   1. $1 + x + x^2 + x^3$
   2. $1 - x + x^2 - x^3$
   3. $1 + x + x^2/2! + x^3/3!$
   4. $1 + 2x + 3x^2 + 4x^3$
6. Find the stationary point of $y = x^3 + x^2 - x$.
   1. $-1 \text{ or } 1/3$
   2. $1 \text{ or } -1/3$
   3. $0 \text{ or } 1/3$
   4. $-1 \text{ or } 3$
7. Find the stationary point of $y = x^3/3 + x^2 - 3$.
   1. $0 \text{ or } -2$
   2. $0 \text{ or } 2$
   3. $1 \text{ or } -2$
   4. $-1 \text{ or } 2$
8. A function is even if:
   1. $f(-x) = f(x)$
   2. $f(-x) = -f(x)$
   3. $f(x) = -f(-x)$
   4. $f(x) = f(x)^{-1}$
9. The function $|x|$ is continuous at $x = 0$.
   1. True
   2. False
   3. Only from the right
   4. Undefined
10. Differentiate $\log(x^2 + 3x + 5)$.
    1. $\frac{2x + 3}{x^2 + 3x + 5}$
    2. $\frac{1}{x^2 + 3x + 5}$
    3. $\frac{2x}{x^2 + 3x + 5}$
    4. $(2x + 3)\ln(x^2 + 3x + 5)$
11. If $y = \tan^{-1}(5x^2)$ find $dy/dx$.
    1. $\frac{10x}{1 + (5x^2)^2}$
    2. $\frac{5x}{1 + (5x^2)^2}$
    3. $\frac{10x}{\sqrt{1 - (5x^2)^2}}$
    4. $\frac{10x}{1 - (5x^2)^2}$
12. If $y = m/(m^2 + 1)$ and $x = m^2 + 5$ compute the value for $dy/dx$.
    1. $\frac{1 - m^2}{2m(m^2 + 1)^2}$
    2. $\frac{1 + m^2}{2m(m^2 + 1)^2}$
    3. $\frac{1 - m^2}{m(m^2 + 1)^2}$
    4. $\frac{m^2 - 1}{2m(m^2 + 1)^2}$
13. The domain for $f(x) = \sqrt{2-x}$ is:
    1. $(-\infty, 2]$
    2. $[2, \infty)$
    3. $(-\infty, 2)$
    4. $(-\infty, \infty)$
14. Let $f(x) = 5x+4$ for $x<2$, and $f(x) = 3x-2$ for $x \ge 2$. Where $x \to 2$:
    1. $x \to 2 \text{ is not continuous}$
    2. $x \to 2 \text{ is continuous}$
    3. $x \to 2 \text{ is undefined}$
    4. $\text{Limit is } 14$
15. If $g(x) = 5x^2$, $g(x)$ is:
    1. even
    2. odd
    3. neither
    4. constant
16. $f(x) = (x+2)/(x-3)$ what's $f^{-1}(x)$?
    1. $\frac{2 + 3x}{x - 1}$
    2. $\frac{2x + 3}{x - 1}$
    3. $\frac{3x - 2}{x + 1}$
    4. $\frac{2x - 3}{x + 1}$
17. Differentiate $y = 9^{\sin 2x}$.
    1. $9^{\sin 2x}[2\cos 2x \log 9]$
    2. $9^{\sin 2x}[\cos 2x \log 9]$
    3. $9^{\sin 2x}[2\sin 2x \log 9]$
    4. $9^{\sin 2x}[-2\cos 2x \log 9]$
18. Differentiate $f(x) = e^{-x^2}$.
    1. $-2x e^{-x^2}$
    2. $2x e^{-x^2}$
    3. $-x e^{-x^2}$
    4. $e^{-x^2}$
19. Solve $\cos x$ using Maclaurin series.
    1. $1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + \dots$
    2. $x - (x^3/3!) + (x^5/5!) + \dots$
    3. $1 + (x^2/2!) + (x^4/4!) + \dots$
    4. $1 - x + x^2 - x^3 + \dots$
20. Differentiate $y = 4^{(x^2+1)^3}$.
    1. $4^{(x^2+1)^3} 6x(x^2+1)^2 \log 4$
    2. $4^{(x^2+1)^3} 3(x^2+1)^2 \log 4$
    3. $4^{(x^2+1)^3} 6x(x^2+1)^2$
    4. $4^{(x^2+1)^3} (x^2+1)^2 \log 4$
21. Differentiate $f(x) = \tan x + \sqrt{x} + 1/\sqrt{x}$.
    1. $\sec^2 x + 1/(2\sqrt{x}) - 1/(2\sqrt{x^3})$
    2. $\sec^2 x + 1/(2\sqrt{x}) + 1/(2\sqrt{x^3})$
    3. $\sec^2 x - 1/(2\sqrt{x}) - 1/(2\sqrt{x^3})$
    4. $\csc^2 x + 1/(2\sqrt{x}) - 1/(2\sqrt{x^3})$
22. What's the value of delta when $f(x) = x+1$, alpha = 3 and $E=0.01$.
    1. $0.01$
    2. $0.1$
    3. $0.001$
    4. $1.01$
23. Find $f^{-1}(x)$ when $f(x) = 4x-5$.
    1. $\frac{x+5}{4}$
    2. $\frac{x-5}{4}$
    3. $\frac{4}{x+5}$
    4. $4x+5$
24. If $y = 1/\sqrt{2x+3}$ find $dy/dx$.
    1. $\frac{-1}{\sqrt{(2x+3)^3}}$
    2. $\frac{1}{\sqrt{(2x+3)^3}}$
    3. $\frac{-1}{(2x+3)^{3/2}}$
    4. $\frac{-2}{\sqrt{(2x+3)^3}}$
25. Differentiate $x^2 + y^2 = \cos x$.
    1. $\frac{-\sin x - 2x}{2y}$
    2. $\frac{\cos x - 2x}{2y}$
    3. $\frac{\sin x + 2x}{2y}$
    4. $\frac{-\sin x + 2x}{2y}$
26. If $y = \ln(\cos x)$ find $dy/dx$.
    1. $-\tan x$
    2. $\cot x$
    3. $\tan x$
    4. $-\cot x$
27. Given $y = \sin^{-1} x$ what's $dy/dx$.
    1. $\frac{1}{\sqrt{1-x^2}}$
    2. $\frac{-1}{\sqrt{1-x^2}}$
    3. $\frac{1}{1+x^2}$
    4. $\frac{1}{\sqrt{x^2-1}}$
28. Solve $\sin x$ using Maclaurin series.
    1. $(x/1!) - (x^3/3!) + (x^5/5!) - (x^7/7!) + \dots$
    2. $1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + \dots$
    3. $(x/1!) + (x^3/3!) + (x^5/5!) + \dots$
    4. $x - x^2 + x^3 - x^4 + \dots$
29. If $y = |x|$ find the range and domain of y.
    1. $D(R), R[0, \infty)$
    2. $D[0, \infty), R(R)$
    3. $D(R), R(R)$
    4. $D(0, \infty), R[0, \infty)$
30. Find delta when $f(x) = (x^2-1)/(x-1)$, $E=0.01$, alpha = 1.
    1. $0.01$
    2. $0.1$
    3. $0.001$
    4. $1$
31. Given $L = 34$, $E = 0.1$, $f(x) = 5x+4$ find delta.
    1. $0.02$
    2. $0.1$
    3. $0.05$
    4. $0.2$
32. If $y = (2x+3)^{-1/2}$ find $dy/dx$.
    1. $\frac{-1}{\sqrt{(2x+3)^3}}$
    2. $\frac{1}{\sqrt{(2x+3)^3}}$
    3. $\frac{-1}{2x+3}$
    4. $\frac{-2}{\sqrt{(2x+3)^3}}$
33. Let $f(x) = 4x-5$, alpha = 2, $E = 1/2$ find L and delta.
    1. $L=3 \text{ and delta}=0.125$
    2. $L=3 \text{ and delta}=0.25$
    3. $L=2 \text{ and delta}=0.125$
    4. $L=8 \text{ and delta}=0.5$
34. If $f(x) = 2x^2+7$ and $2 \le x < \infty$ find $f(-2)$.
    1. Undefined
    2. $15$
    3. $11$
    4. $-1$
35. For all stationary point, dy/dx=____
    1. $0$
    2. $1$
    3. $\infty$
    4. $-1$
36. Find $dy/dx$ if $x^2+y^2+3y=0$ is at $(1,0)$.
    1. $-2/3$
    2. $2/3$
    3. $-3/2$
    4. $0$
37. If $y = 4x^3 - 3x^2 - 6x + 1$, what's the nature of the curve when $x = -1/2$.
    1. Maximum
    2. Minimum
    3. Inflexion
    4. Undefined
38. Using Taylor series, find $\sqrt{x}$ at $x^\circ=1$.
    1. $1 + (x-1)/2 - (x-1)^2/8 + 3(x-1)^3/48 + \dots$
    2. $1 + (x-1)/2 + (x-1)^2/8 + 3(x-1)^3/48 + \dots$
    3. $1 - (x-1)/2 - (x-1)^2/8 + \dots$
    4. $1 + (x-1) - (x-1)^2/2 + \dots$
39. Differentiate $y = \sin 4x$.
    1. $4\cos 4x$
    2. $-4\cos 4x$
    3. $\cos 4x$
    4. $4\sin 4x$
40. Differentiate $y = \cos 4x$.
    1. $-4\sin 4x$
    2. $4\sin 4x$
    3. $-\sin 4x$
    4. $-4\cos 4x$
41. $1 + \tan^2 x$ is ____
    1. $\sec^2 x$
    2. $\csc^2 x$
    3. $\cot^2 x$
    4. $\cos^2 x$
42. $\tan(x+y) = 4/3$, where $\tan x = 1/2$ find $\tan y\dots$
    1. $1/2$
    2. $1/3$
    3. $2$
    4. $1/4$
43. If $y = \sin^{-1}(2x)$ what's $dy/dx$?
    1. $\frac{2}{\sqrt{1-(2x)^2}}$
    2. $\frac{-2}{\sqrt{1-(2x)^2}}$
    3. $\frac{1}{\sqrt{1-(2x)^2}}$
    4. $\frac{2}{1+(2x)^2}$
44. A binary operation * is defined in R, the set of real numbers $x*y = \sqrt{xy}$, for $x,y \in R\dots$ if $x*(2*18) = 6$ find $x$.
    1. $6$
    2. $36$
    3. $3$
    4. $12$
45. Differentiate $y = \tan^{-1}(5x^2)$ with respect to x.
    1. $\frac{10x}{1+(5x^2)^2}$
    2. $\frac{5x}{1+(5x^2)^2}$
    3. $\frac{10x}{\sqrt{1-(5x^2)^2}}$
    4. $\frac{10x}{1-(5x^2)^2}$
46. If $y = x^x$ find $dy/dx$.
    1. $x^x(1+\log x)$
    2. $x^x \log x$
    3. $x^{x-1}$
    4. $x^x(1-\log x)$
47. If $y = \sqrt{3-x}$ determine the domain of y.
    1. $(-\infty, 3]$
    2. $[3, \infty)$
    3. $(-\infty, 3)$
    4. $[-3, 3]$
48. Solve $f(x) = e^x$ using Maclaurin series.
    1. $1 + x/1! + x^2/2! + x^3/3! + \dots$
    2. $1 - x/1! + x^2/2! - x^3/3! + \dots$
    3. $x/1! + x^2/2! + x^3/3! + \dots$
    4. $1 + x^2/2! + x^4/4! + \dots$
49. $f(x) = x(1-x^2)$ at the point $x = -1/\sqrt{3}$.
    1. minimal
    2. maximal
    3. Inflexional
    4. undefined
50. $4x^3 - 2x^2 - 6x + 1$ what's the nature of the stationary point at $x = 1/2$?
    1. maximal
    2. minimal
    3. Inflexional
    4. undefined
51. Determine the stationary points of $y = x + 9/x$.
    1. $\pm 3$
    2. $\pm 1/3$
    3. $\pm 9$
    4. $3$
52. Find the stationary point of $x^3 - 2x^2 + x + 3$ if the point is maximum.
    1. $1/3$
    2. $1$
    3. $-1/3$
    4. $-1$
53. Find the stationary point of $x^3 - 2x^2 + x + 3$ if the point is minimum.
    1. $1$
    2. $1/3$
    3. $-1$
    4. $3$

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