Elementary Mathematics II — 2023 Past Questions

University of Ilorin MTH102: 70 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. If $f(x)$ and $g(x)$ are continuous and $g(x)$ is not equal to zero, what can be said about the quotient $f(x)/g(x)$?
    1. It is discontinuous
    2. It is continuous
    3. It is commutative
    4. It is non-commutative
  2. A stationary point on a curve is also commonly referred to as a?
    1. Identify point
    2. Common point
    3. Turning point
    4. Undefined point
  3. Evaluate $\lim_{x \to 0} \frac{7x^2-3}{5x^2+3}$.
    1. $3$
    2. $-1$
    3. $4$
    4. $0$
  4. Find $\frac{dy}{dx}$ of the implicit function $x^2 + y^2 + x + 3y = 0$ at the point $(0,0)$.
    1. $-1/3$
    2. $1/3$
    3. $3$
    4. $4$
  5. Find $\frac{dy}{dx}$ if $x^2 + y^2 = \cos x$.
    1. $\frac{\cos x - 2x}{2x}$
    2. $\frac{\cos x - 2x}{2y}$
    3. $\frac{-\sin x - 2x}{2y}$
    4. $\frac{\sin x - 2x}{2y}$
  6. Determine $\lim_{x \to \infty} \frac{x^3 - x^2 + 4x}{2x^3 + 2x^2 - 7}$.
    1. $0.5$
    2. $2$
    3. $\infty$
    4. $0$
  7. Evaluate the integral $\int \frac{x^3 - 3x^2 + 2x + 5}{x} dx$.
    1. $\frac{x^3}{3} - \frac{3x^2}{2} + 2x + 5 \ln x + C$
    2. $\frac{x^4}{4} - x^3 + x^2 + 5x + C$
    3. $x^2 - 3x + 2 + \frac{5}{x} + C$
    4. $\frac{x^3}{3} - 3x^2 + 2x + 5 \ln x + C$
  8. The sum of two positive numbers is 20. Find their difference if their product is a maximum.
    1. $4$
    2. $2$
    3. $0$
    4. $3$
  9. Find the derivative of $y = \ln(\sqrt{x-1})$.
    1. $\frac{2}{x-1}$
    2. $\frac{1}{x-1}$
    3. $\frac{1}{2(x-1)}$
    4. $\frac{-1}{x-1}$
  10. Determine the domain of the function $f(x) = \frac{x}{x^2+4}$.
    1. Set of all negative numbers
    2. Set of all real numbers
    3. Set of all integers
    4. Set of all positive numbers
  11. Compute the limit: $\lim_{x \to 6} \frac{x-6}{x^2-36}$.
    1. $\infty$
    2. $0$
    3. $\frac{1}{12}$
    4. $12$
  12. Find $\frac{dy}{dx}$ if $y = \sin x - e^2 \cos x$.
    1. $\cos x - e^2 \sin x$
    2. $\cos x + e^2 \sin x$
    3. $\cos x - 2e \cos x$
    4. $\cos x + 2e \sin x$
  13. Integrate $\int e^{-5x+4} dx$.
    1. $-5e^{-5x+4} + C$
    2. $\frac{-1}{5}e^{-5x+4} + C$
    3. $\frac{1}{5}e^{-5x+4} + C$
    4. $5e^{-5x+4} + C$
  14. Find the integral $\int \frac{x^2}{x^{1/2}} dx$.
    1. $\frac{2}{5}x^{5/2} + C$
    2. $\frac{5}{2}x^{5/2} + C$
    3. $\frac{2}{3}x^{3/2} + C$
    4. $\frac{3}{2}x^{3/2} + C$
  15. Find the domain of the function $y = \sqrt{x-4}$.
    1. $(-\infty, 4]$
    2. $[4, \infty)$
    3. $(4, \infty)$
    4. $(-\infty, \infty)$
  16. Evaluate the integral $\int (2t+3)(t+1)(t-2) dt$.
    1. $\frac{t^4}{2} + \frac{t^3}{3} - \frac{7t^2}{2} - 6t + C$
    2. $\frac{t^4}{4} + \frac{t^3}{3} - \frac{7t^2}{2} - 6t + C$
    3. $\frac{t^4}{2} - \frac{t^3}{3} - \frac{7t^2}{2} + 6t + C$
    4. $2t^4 + t^3 - 7t^2 - 6t + C$
  17. If $f(x) = 3x^2 + 6x$, find $f(-x)$.
    1. $3x^2 + 6x$
    2. $-3x^2 - 6x$
    3. $-3x^2 + 6x$
    4. $3x^2 - 6x$
  18. Find $\frac{dy}{dx}$ where $y = x^6 \sin x$.
    1. $6x^5 \sin x - x^6 \cos x$
    2. $x^6 \cos x + 6x^5 \sin x$
    3. $6x^5 \cos x + x^6 \sin x$
    4. $6x^5 \cos x$
  19. Differentiate implicitly to find $\frac{dy}{dx}$ for $x + xy + y = 4$.
    1. $\frac{-(y+1)}{x+1}$
    2. $\frac{y-1}{x+1}$
    3. $\frac{-(x+1)}{y+1}$
    4. $\frac{y+1}{x+1}$
  20. Given $F(x) = \frac{6x^2 - 3x + 2}{3x^2}$, evaluate $F(6)$.
    1. $\frac{50}{27}$
    2. $\frac{200}{108}$
    3. $\frac{25}{27}$
    4. $\frac{50}{9}$
  21. Find the derivative of $y = (1+x^3)^5$.
    1. $15x^2(1+x^3)^4$
    2. $5(1+x^3)^4$
    3. $15x(1+x^3)^4$
    4. $3x^2(1+x^3)^5$
  22. Differentiate $y = x \sqrt{x-1}$.
    1. $\frac{3x-2}{2\sqrt{x-1}}$
    2. $\frac{2x-1}{2\sqrt{x-1}}$
    3. $\frac{3x-2}{\sqrt{x-1}}$
    4. $\frac{x-1}{2\sqrt{x-1}}$
  23. Evaluate the definite integral $\int_{2}^{5} 6x^3 dx$.
    1. $912.5$
    2. $913.5$
    3. $914.5$
    4. $915.5$
  24. If $y = (3x^2 - \cos x)^2$, find $\frac{dy}{dx}$.
    1. $2(3x^2 - \cos x)(6x + \sin x)$
    2. $2(3x - \cos x)(6x + \sin x)$
    3. $2(3 - \cos x)(6 + \sin x)$
    4. $2(6x + \sin x)(3x - \sin x)$
  25. Evaluate $\lim_{x \to 0} \frac{\sin 2x}{4x}$.
    1. $0$
    2. $1/2$
    3. $1/4$
    4. $4$
  26. Find $\frac{dy}{dx}$ of $2x^4 - 7x^3 + 4x^2 + 3x - 4$.
    1. $8x^4 - 21x + 8x + 3$
    2. $8x^3 - 21x^2 + 8x + 3$
    3. $6x^2 - 21x^2 + 8x$
    4. $6x^3 - 21x + 8x + 3$
  27. Evaluate the integral $\int 4 dt$.
    1. $4x + C$
    2. $4t + C$
    3. $4t^2 + C$
    4. $4x^2 + C$
  28. If $y = (4x - 3)^6$, find the second derivative $\frac{d^2y}{dx^2}$.
    1. $24(4x - 3)^6$
    2. $6(4x - 3)^5$
    3. $24(4x - 3)^4$
    4. $480(4x - 3)^4$
  29. Find $\frac{dy}{dx}$ if $y = \cos 2x$.
    1. $-2\sin 2x$
    2. $2\sin 2x$
    3. $\cot 2x$
    4. $2\cos 2x$
  30. Evaluate the integral $\int \sin 8x dx$.
    1. $\frac{1}{8} \cos 8x + C$
    2. $-\frac{1}{8} \cos 8x + C$
    3. $\frac{1}{8} \csc 8x + C$
    4. $\frac{1}{8} \sin 8x + C$
  31. Evaluate $\int_{0}^{1} (3x^2 + 4x + 7) dx$.
    1. $10$
    2. $3$
    3. $6$
    4. $-3$
  32. If $S(t) = t^2 + t - 6$, find the third derivative $\frac{d^3S}{dt^3}$.
    1. $2$
    2. $2t + 7$
    3. $0$
    4. $-2$
  33. If $f(x) = 8x^2 + \tan x$, find $\frac{dy}{dx}$.
    1. $8x + \tan^2 x$
    2. $8x + \csc^2 x$
    3. $16x + \sec^2 x$
    4. $8 + \sec^2 x$
  34. If $y = t^2 \cos t + \sin 4t$, find $\frac{dy}{dt}$.
    1. $2t \cos t - t^2 \sin t + 4 \cos 4t$
    2. $2t \cos t + t^2 \sin t + 4 \cos 4t$
    3. $2t \cos t - t^2 \sin t - 4 \cos 4t$
    4. $2t \sin t - t^2 \cos t + 4 \cos 4t$
  35. If $y = \ln(\cos 4x)$, find $\frac{dy}{dx}$.
    1. $4 \cos 4x$
    2. $4 \cot 4x$
    3. $4 \tan 4x$
    4. $-4 \tan 4x$
  36. If the displacement is given by $s = 2t + 4t^2$, find the velocity at $t = 2$ seconds.
    1. $16 \text{ m/s}$
    2. $18 \text{ m/s}$
    3. $14 \text{ m/s}$
    4. $12 \text{ m/s}$
  37. Find the area under the curve for $\int_{0}^{2} (-3x^2 + 4x + 6) dx$.
    1. $10$
    2. $12$
    3. $-6$
    4. $18$
  38. Evaluate $\int_{1}^{4} 4x^2 dx$.
    1. $84$
    2. $83$
    3. $82$
    4. $81$
  39. Evaluate the integral $\int_{1}^{2} (1 + 3x^2) dx$.
    1. $8$
    2. $18$
    3. $100$
    4. $10$
  40. If $f(x) = \frac{4x}{2x^2 - 1}$, find $f'(x)$.
    1. $\frac{-8x^2 - 4}{(2x^2 - 1)^2}$
    2. $\frac{8x^2 - 4}{(2x^2 - 1)^2}$
    3. $\frac{8x^2 + 4}{(2x^2 - 1)^2}$
    4. $\frac{-8x^2 + 4}{(2x^2 - 1)^2}$
  41. If $y = x^4 + x^3$, find $\frac{dy}{dx}$.
    1. $4x + 3x$
    2. $4x^3 + 3x^2$
    3. $4x^2 + 3x$
    4. $4x^3 + 3x$
  42. If $y = \sin^{-1}(x^3)$, find $\frac{dy}{dx}$.
    1. $\frac{x^3}{\sqrt{1-x^6}}$
    2. $\cos^{-1}(x^3)$
    3. $\frac{3x^2}{\sqrt{1-x^6}}$
    4. $\frac{3x}{\sqrt{1-x^2}}$
  43. If $x = 2\sin t$ and $y = 3\cos 2t$, find $\frac{dy}{dx}$.
    1. $6\cos t$
    2. $-6\sin t$
    3. $6\cos t \sin t$
    4. $-6\cos t$
  44. Differentiate $4 \ln(5y)$ implicitly with respect to $x$.
    1. $\frac{4}{y} \frac{dy}{dx}$
    2. $4 \frac{dy}{dx}$
    3. $\frac{4}{5y} \frac{dy}{dx}$
    4. None
  45. Find the 4th term of the Maclaurin series expansion of $e^{3x}$.
    1. $3x$
    2. $1x$
    3. $\frac{27x^3}{3!}$
    4. $\frac{x^3}{3!}$
  46. Find $\frac{dy}{dx}$ of $y = (2x+1)(1+x)$.
    1. $4x - 3$
    2. $4x^2 - 3x$
    3. $4x^2 - 3x^2$
    4. $4x + 3$
  47. Find $\frac{dy}{dx}$ if $y = e^x \sec x$.
    1. $e^x \sec x \tan x$
    2. $e^x \sec x (1 + \tan x)$
    3. $e^x \tan x (1 + \sec x)$
    4. $e^x(1 + \tan x)$
  48. Evaluate $\lim_{x \to 1} \frac{x^2 - 2x + 1}{x - 1}$.
    1. $1$
    2. $0$
    3. $\infty$
    4. $-1$
  49. Find $\frac{dy}{dx}$ of $y = \frac{\cot x}{\cos x}$.
    1. $\sec x(\cot x \tan x - \csc^2 x)$
    2. $\sec^2 x(\cot x \tan x + \csc^2 x)$
    3. $\sec x(\cot^2 x \tan x - \csc^2 x)$
    4. $\sec x(\cot x \tan x + \csc^2 x)$
  50. Evaluate the integral $\int x \cos x dx$.
    1. $x \sin x + \cos x + C$
    2. $x \sin x - \cos x + C$
    3. $\sin x + C$
    4. $x \sin x \cos x + C$
  51. Evaluate the integral $\int \ln x dx$.
    1. $x(\ln x) + C$
    2. $x \ln x + 1/x + C$
    3. $x(\ln x - 1) + C$
    4. $\ln x - x + C$
  52. Evaluate the integral $\int \frac{2}{2x+3} dx$.
    1. $\ln(2x+3) + C$
    2. $2\ln(2x+3) + C$
    3. $\frac{1}{2}\ln(2x+3) + C$
    4. $\frac{-2}{(2x+3)^2} + C$
  53. If $f(x) = 16x^2$, find $f(-x)$.
    1. $16x^2$
    2. $16 - x^2$
    3. $-16x^2$
    4. All of the above
  54. Which of the following statements is true regarding the function $g(r) = 4r^3 - r$?
    1. $g(r)$ is an odd function
    2. $g(r)$ is an even function
    3. $g(r)$ is a constant function
    4. None of the above
  55. Given $f(x) = x^2 - 3x - 2$, find $f(2x) - f(x)$.
    1. $3x(x+1)$
    2. $3x(x^2-1)$
    3. $3x(x-1)$
    4. All of the above
  56. Find the inverse function $F^{-1}(x)$ for $F(x) = \frac{5x-3}{2}$.
    1. $\frac{2x-3}{5}$
    2. $\frac{2x+3}{5}$
    3. $\frac{2y+3}{5}$
    4. $\frac{2y-3}{5}$
  57. Given $g(a) = 4a^3 - 6a + 2$, find the image of $-1$.
    1. $4$
    2. $6$
    3. $8$
    4. $10$
  58. Evaluate $\lim_{x \to \infty} \frac{5x^5 - 3x + 1}{x^4 + 1}$.
    1. $1$
    2. $\infty$
    3. $-\infty$
    4. $0$
  59. Evaluate $\lim_{x \to 2} \sqrt{16-x^2}$.
    1. $4$
    2. $4\sqrt{3}$
    3. $2$
    4. $2\sqrt{3}$
  60. Evaluate $\lim_{x \to 2} \frac{x-2}{x^2+x-6}$.
    1. $\infty$
    2. $0$
    3. $1/5$
    4. $5$
  61. If $f(x) = 7x - 1$, find its inverse $f^{-1}(x)$.
    1. $\frac{x-1}{7}$
    2. $\frac{x+1}{7}$
    3. $7x+1$
    4. $7x-1$
  62. If $f(a) = 3a^2 - a + 5$ and $h(a) = 2a + 1$, evaluate $f(a) - h(a)$.
    1. $3a^2 - 3a + 4$
    2. $3a^2 - 3a + 10$
    3. $3a^2 + 3a - 4$
    4. $3a^2 - a + 4$
  63. If $f(x) = 7x^3 + 5x^2 - 3x$, find $f(2)$.
    1. $56$
    2. $70$
    3. $76$
    4. None of the above
  64. Find the derivative of $y = \frac{e^x + \tan x}{\sin x}$.
    1. $\frac{e^x \sin x + \sec^2 x}{\sin x}$
    2. $\frac{\sin x(e^x + \sec^2 x) - \cos x(e^x + \tan x)}{\sin^2 x}$
    3. $\frac{e^x \cos x - \sec^2 x}{\sin^2 x}$
    4. $\frac{e^x \sin x + \sin x \sec^2 x - e^x \cos x - \cos x \tan x}{\sin^2 x}$
  65. Given $f(a) = 4a^3 - 6a + 2$ and $g(a) = a^2 - 5a + 1$, find $f(a) - g(a)$.
    1. $4a^3 - a^2 + 11a + 3$
    2. $4a^3 - a^2 - 11a + 3$
    3. $4a^3 - a^2 + a + 1$
    4. $4a^3 - a^2 - a + 1$
  66. Evaluate the limit $\lim_{x \to 2} \frac{x-2}{x^2+2x-8}$.
    1. $2/3$
    2. $1/6$
    3. $1/3$
    4. $0$
  67. If $f(x) = 6x + 4$, find $x$ whose image is 10.
    1. $2$
    2. $3$
    3. $4$
    4. $1$
  68. Find the derivative of $y = e^{x \sin x}$.
    1. $e^{x \sin x}$
    2. $x \sin x \cdot e^{x \sin x}$
    3. $(\sin x + x \cos x)e^{x \sin x}$
    4. $x e^{x \sin x}$
  69. Evaluate the limit $\lim_{x \to -2} \sqrt{16-4x^2}$.
    1. $0$
    2. $1$
    3. $4$
    4. Undefined
  70. Determine the domain of the function $f(x) = \sqrt{4-x^2}$.
    1. $x = 2$
    2. $[0, \infty)$
    3. $[-2, 2]$
    4. $(-\infty, 2]$

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