Elementary Mathematics II — biochemistry Past Questions

University of Ilorin MTH102: 37 questions.

Topics: Functions of a real variable, graphs, limits, and continuity., Derivatives as limits of rate of change, differentiation techniques, maxima and minima.

1. If $f(x) = 16x^2$, find $f(-x)$.
   1. $16x^2$
   2. $16 - x^2$
   3. $-16x^2$
   4. $x^2 - 16$
2. Which of the following is correct about the function $f(a) = a^2 + 4$?
   1. $f(a)$ is an even function
   2. $f(a)$ is an odd function
   3. $f(a)$ is an onto function
   4. $f(a)$ is an into function
3. Find $\\frac{dy}{dx}$ of $y = (2x+1)(1+x)$.
   1. $4x - 3$
   2. $4x^2 - 3x$
   3. $4x^2 - 3x^2$
   4. $4x + 3$
4. 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. $g(r)$ is undefined at $r=0$
5. If $f(x) = (x+1)^2$, find $f(1) + f(a)$.
   1. $a + 2a + 5$
   2. $a^2 + 2a + 5$
   3. $a^2 - 2a + 5$
   4. $a + 2a^2 + 5$
6. 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. $3x^2 - x$
7. 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. $\\frac{5x+2}{3}$
   4. $\\frac{3x-2}{5}$
8. Let $g: X \\to Y$. If $g(x_1) = g(x_2) \\iff x_1 = x_2$, which of the following is true for the function?
   1. $g(x)$ is a one-to-one function
   2. $g(x)$ is a many-to-one function
   3. $g(x)$ is a many-to-many relation
   4. $g(x)$ is a constant function
9. Which of the following represents an identity function?
   1. $g(a) = a$
   2. $g(a) = a + 1$
   3. $g(a) = a^2 + 4$
   4. $g(a) = a + 3$
10. Given $g(a) = 4a^3 - 6a + 2$, find the image of $-1$.
    1. $4$
    2. $6$
    3. $8$
    4. $10$
11. Evaluate $\\lim_{x \\to \\infty} \\frac{5x^5 - 3x + 1}{x^4 + 1}$.
    1. $1$
    2. $\\infty$
    3. $-\\infty$
    4. $0$
12. Evaluate $\\lim_{x \\to 2} \\sqrt{16-x^2}$.
    1. $4$
    2. $4\\sqrt{3}$
    3. $2$
    4. $2\\sqrt{3}$
13. Evaluate $\\lim_{x \\to 2} \\frac{x-2}{x^2+x-6}$.
    1. $\\infty$
    2. $0$
    3. $1/5$
    4. $5$
14. 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. $7x-1$
15. For the limit $\\lim_{x \\to 2} (2x+8) = 12$, find the maximum value of $\\delta$ given that $\\epsilon = 0.5$.
    1. $0.5$
    2. $0.25$
    3. $0.025$
    4. $0.0025$
16. 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. $3a^2 - a + 4$
17. If $f(x) = 7x^3 + 5x^2 - 3x$, find $f(2)$.
    1. $56$
    2. $70$
    3. $76$
    4. $64$
18. Evaluate $\\lim_{x \\to \\infty} (x+1) - \\lim_{x \\to 2} (x-1)$.
    1. $1$
    2. $2$
    3. $0$
    4. $\\infty$
19. Evaluate $\\lim_{x \\to 1} \\frac{x^2 - 2x + 1}{x - 1}$.
    1. $1$
    2. $0$
    3. $\\infty$
    4. $-1$
20. Find the derivative of $y = \\frac{e^x + \\tan x}{\\sin x}$.
    1. $\\frac{e^x \\sin x + \\sec^2 x}{\\sin^2 x}$
    2. $\\frac{e^x \\sin x + \\sin x \\sec^2 x - e^x \\cos x - \\cos x \\tan x}{\\sin^2 x}$
    3. $\\frac{e^x \\cos x - \\sec^2 x}{\\sin^2 x}$
    4. $\\frac{e^x \\sin x - e^x \\cos x - \\cos x \\tan x}{\\sin^2 x}$
21. 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. $4a^3 - a^2 - a + 1$
22. 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. $e^x \\sec^2 x (\\sin x + \\cos x)$
    4. $e^x(1 + \\tan x)$
23. If $f(x) = 6x^2 - 5x$, find $f(-x)$.
    1. $6x^2 - 5x$
    2. $6x^2 + 5x$
    3. $-6x^2 + 5x$
    4. $-6x^2 - 5x$
24. Evaluate the limit $\\lim_{x \\to 2} \\frac{x-2}{x^2+2x-8}$.
    1. $2/3$
    2. $1/6$
    3. $1/3$
    4. $0$
25. If $f(x) = 6x + 4$, find $x$ whose image is 10.
    1. $2$
    2. $3$
    3. $4$
    4. $1$
26. Evaluate $\\lim_{x \\to 4} \\frac{x^2 - 16}{x - 4}$.
    1. $4$
    2. $8$
    3. $0$
    4. $16$
27. 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. $4a^3 + 2a^2 + 11a + 3$
    3. $4a^3 + a^2 - 11a - 3$
    4. $4a^3 + a^2 - 11a + 3$
28. 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. $x e^{x \\sin x}$
29. Evaluate the limit $\\lim_{x \\to -2} \\sqrt{16-4x^2}$.
    1. $0$
    2. $1$
    3. $4$
    4. Undefined
30. Determine the domain of the function $f(x) = \\sqrt{4-x^2}$.
    1. $x = 2$
    2. $[0, \\infty)$
    3. $[-2, 2]$
    4. $(-\\infty, 2]$
31. Find the derivative of $y = x^{5 \\sin x}$.
    1. $5\\sin x \\cdot x^{5 \\sin x - 1}$
    2. $5\\sin x \\cdot x^{5 \\sin x} + 5\\cos x \\ln(x) x^{5 \\sin x}$
    3. $5\\cos x \\ln(x) x^{5 \\sin x}$
    4. $5\\sin x \\cdot x^{5 \\sin x - 1} + 5\\cos x \\ln(x) x^{5 \\sin x}$
32. Evaluate $\\lim_{x \\to 2} \\frac{4}{x-2}$.
    1. $\\infty$
    2. $0$
    3. $1$
    4. $4$
33. Evaluate $\\lim_{x \\to -5} \\frac{x^2 + 5x}{x + 5}$.
    1. $0$
    2. $5$
    3. $-5$
    4. $\\infty$
34. Find the inverse function $F^{-1}(x)$ for $F(x) = \\frac{2x+5}{4x-3}$.
    1. $\\frac{5+3x}{4x+2}$
    2. $\\frac{5+3y}{4y-2}$
    3. $\\frac{3x+5}{4x-2}$
    4. $\\frac{2x-5}{4x+3}$
35. If $g(a) = 4a^3 + 6a + 12$, find $g(2a)$.
    1. $8a^3 + 12a + 12$
    2. $32a^3 + 12a + 24$
    3. $32a^3 + 12a + 12$
    4. $32a^3 + 36a$
36. Find the derivative of $y = (2x+5)\\sin x$.
    1. $2x \\cos x + 5 \\cos x$
    2. $(2x+5)\\cos x$
    3. $2x \\cos x + 5 \\cos x + 2 \\sin x$
    4. $7 \\cos x + 2 \\sin x$
37. Differentiate $y = (3x-7)\\cos x$.
    1. $3x \\sin x - 7x + 3 \\cos x$
    2. $-3x \\sin x + 7 \\sin x + 3 \\cos x$
    3. $(3x-7)\\sin x$
    4. $3 \\cos x$

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