Elementary Mathematics II — 2026-exam Past Questions

University of Ilorin MTH102: 113 questions.

Topics: Derivatives as limits of rate of change, differentiation techniques, maxima and minima., Curve sketching, integration, definite integrals, reduction formulae., Applications to areas, volumes (Trapezium and Simpson's rules)., Functions of a real variable, graphs, limits, and continuity.

1. If ln y = x, find dy/dx.
   1. y
   2. 1/y
   3. 1
   4. x²
2. The second-derivative condition for a minimum point of a curve (where dy/dx = 0) is:
   1. dy/dx = 0
   2. d²y/dx² < 0
   3. d²y/dx² > 0
   4. d²y/dx² = 0
3. A particle moves so that Y = 20t + (2/3)t³ − (3/2)t² metres. Find its velocity at t = 0 s.
   1. 0 m/s
   2. 20 m/s
   3. −9 m/s
   4. −1 m/s
4. Y = (x − 2)³. Find the stationary (turning) point.
   1. (−2, 2)
   2. (−1, 2)
   3. (3, 3)
   4. (2, 0)
5. Evaluate ∫ x tan⁻¹x dx.
   1. ((x² + 1)/2) tan⁻¹x − x/2 + C
   2. (x²/2) tan⁻¹x − x/2 + C
   3. ((x² − 1)/2) tan⁻¹x + x/2 + C
   4. (x²/2) tan⁻¹x − ½ ln(1 + x²) + C
6. Find the total area enclosed between the curves y = x and y = x³.
   1. 1/6
   2. 1/4
   3. 1/2
   4. 1/3
7. Find the area under the curve y = x³ from x = 1 to x = 2 (that is, between y = 1 and y = 8).
   1. 35/7
   2. 15/4
   3. 45/4
   4. 11
8. If y = cos 3t and x = sin 3t, find dy/dx.
   1. −tan 3t
   2. tan 3t
   3. 3 sin 3t
   4. −cot 3t
9. Evaluate ∫ 3/((x + 1)(2x + 1)) dx
   1. 3 ln|(2x + 1)/(x + 1)| + C
   2. 3 ln|(x + 1)/(2x + 1)| + C
   3. 6 ln|2x + 1| − 3 ln|x + 1| + C
   4. 3 ln|2x + 1| + 3 ln|x + 1| + C
10. If eʸ = x − y, find dy/dx at the point where y = 0.
    1. 1/2
    2. 1
    3. 1/3
    4. 2
11. Find d²y/dx² of y = x eˣ.
    1. eˣ(x + 2)
    2. eˣ(x + 1)
    3. eˣ(x + 3)
    4. x eˣ
12. Differentiate eˣ/x².
    1. eˣ(x − 2)/x³
    2. eˣ(2x)/x²
    3. eˣ(x − 2)/x⁴
    4. (x eˣ − 2)/x³
13. In evaluating ∫ (ln y)/√y dy by parts with u = ln y, find v where dv = dy/√y.
    1. √y
    2. 1/(2√y)
    3. (2/3)√(y³)
    4. 2√y
14. Differentiate with respect to x: y = 10x − 7.
    1. 10
    2. −10
    3. 0
    4. 3
15. Evaluate ∫ (ln x)/x³ dx
    1. (x²/2)(ln x − 1/2) + C
    2. (x³/3) ln x + 1/3 + C
    3. −(1/(2x²))(ln x + 1/2) + C
    4. −(1/(2x²))(ln x − 1/2) + C
16. If y = t² sin t and x = 2t², find dy/dx.
    1. −cos t
    2. (1/8)(sin t − t)
    3. ½ sin t + ¼ + cos t
    4. (1/4)(2 sin t + t cos t)
17. Find the area under the curve y = 3x − x² (above the x-axis).
    1. 3/2
    2. −9/2
    3. 9/5
    4. 9/2
18. Find the area under the curve ∫₁² x³ dx.
    1. 15/4
    2. 17/4
    3. 4
    4. 8/3
19. Find the area enclosed by the curve y = x − 2√x and the x-axis.
    1. 32/3
    2. 8/3
    3. 4/3
    4. 4
20. Differentiate 1/x.
    1. −1/x²
    2. −1/(2x)
    3. x²
    4. −2/x²
21. Find the derivative of y = sin(cos x).
    1. −sin x cos(cos x)
    2. cos x sin(cos x)
    3. −cos x sin(sin x)
    4. sin x sin(cos x)
22. The distance of a particle is x = 20t − (3/2)t² − (2/3)t³ metres. Find its velocity when t = 0.
    1. 20 m/s
    2. 15 m/s
    3. 10 m/s
    4. 25 m/s
23. Find the derivative of (tan⁻¹x)².
    1. 2 tan⁻¹x/(1 + x²)
    2. 2 tan⁻¹x
    3. 2/(1 + x²)
    4. 1/(1 + x²)²
24. Find dy/dz if y = sin(z⁶) + sin⁶z.
    1. 6z⁵ cos(z⁶) + 6 sin⁵z cos z
    2. 6z⁵ cos(z⁶) + 6 sin⁵z
    3. cos(z⁶) + 6 sin⁵z cos z
    4. 6z⁵ cos(z⁶) − 6 sin⁵z cos z
25. Evaluate ∫ 1/(x² − 2) dx
    1. (1/(2√2)) ln|(x − √2)/(x + √2)| + C
    2. (1/√2) ln|(x − √2)/(x + √2)| + C
    3. (1/(2√2)) ln|(x + √2)/(x − √2)| + C
    4. (1/(2√2)) tan⁻¹(x/√2) + C
26. Which rules are suitable for differentiating x² cos x/(x² + 1)?
    1. Product and Quotient
    2. Product and Chain
    3. Quotient and Chain
    4. Sum and Chain
27. Evaluate ∫ 1/√(4 − 9x²) dx
    1. (1/3) sin⁻¹(3x/2) + C
    2. (1/3) tan⁻¹(3x/2) + C
    3. (1/2) sin⁻¹(3x/2) + C
    4. 3 sin⁻¹(3x/2) + C
28. Evaluate ∫ 1/(9 + 16x²) dx
    1. (1/12) tan⁻¹(4x/3) + C
    2. (1/3) tan⁻¹(4x/3) + C
    3. (1/12) sin⁻¹(4x/3) + C
    4. (1/12) tan⁻¹(3x/4) + C
29. The distance of an object is x = 20t − (3/2)t² − (2/3)t³. Find the time t at maximum height.
    1. 2.5 s
    2. 4 s
    3. 5 s
    4. 2 s
30. Evaluate ∫ 1/√(25 − 4x²) dx
    1. (1/5) sin⁻¹(2x/5) + C
    2. (1/2) sin⁻¹(2x/5) + C
    3. (1/2) sin⁻¹(5x/2) + C
    4. (1/5) tan⁻¹(2x/5) + C
31. Differentiate x² ln(2x).
    1. x(2 ln(2x) + 1)
    2. 2x ln(2x) + 1
    3. 2x ln(2x) + x²
    4. x(ln(2x) + 2)
32. Differentiate 1/√(x − 2).
    1. −1/(2(x − 2)^(3/2))
    2. 1/(2(x − 2)^(3/2))
    3. −1/(2√(x − 2))
    4. −2/(x − 2)^(3/2)
33. Evaluate ∫ 2/(x² − 2) dx
    1. (1/√2) ln|(x − √2)/(x + √2)| + C
    2. (1/√2) ln|(x + √2)/(x − √2)| + C
    3. (1/(2√2)) ln|(x − √2)/(x + √2)| + C
    4. (2/√2) ln|x² − 2| + C
34. If x² + y² + 4 = 20, find dy/dx.
    1. −x/y
    2. x/y
    3. −y/x
    4. −2x/y
35. Differentiate tan⁻¹√x.
    1. 1/(2√x(1 + x))
    2. 1/(1 + x)
    3. 1/(2√x)
    4. √x/(1 + x)
36. If x = eᵗ and y = 2 cos 3t, find dy/dx.
    1. −6 sin 3t/eᵗ
    2. 6 sin 3t/eᵗ
    3. −6 cos 3t/eᵗ
    4. −2 sin 3t/eᵗ
37. Evaluate ∫ sin⁻¹x dx
    1. x sin⁻¹x + √(1 − x²) + C
    2. x sin⁻¹x − √(1 − x²) + C
    3. x sin⁻¹x + √(1 + x²) + C
    4. x sin⁻¹x + 1/√(1 − x²) + C
38. Differentiate sin⁻¹(e⁶ᵗ) with respect to t.
    1. 6e⁶ᵗ/√(1 − e¹²ᵗ)
    2. e⁶ᵗ/√(1 − e¹²ᵗ)
    3. 6e⁶ᵗ/(1 − e¹²ᵗ)
    4. 6e⁶ᵗ/√(1 + e¹²ᵗ)
39. Evaluate ∫ (x³ − 3x² − 2/x²) dx
    1. x⁴/4 − x³ + 2/x + C
    2. x⁴/4 − x³ − 2/x + C
    3. 3x² − 6x + C
    4. x⁴/4 − x³ + 2/x² + C
40. Find the derivative of x.
    1. 1
    2. 0
    3. x
    4. 2
41. Evaluate ∫ [ln x]²/x dx
    1. (ln x)³/3 + C
    2. (ln x)²/2 + C
    3. (ln x)³ + C
    4. 2 ln x/x² + C
42. Evaluate ∫₁² x e²ˣ dx
    1. (3e⁴ − e²)/4
    2. e²/4
    3. (3e⁴ + e²)/4
    4. (e⁴ − e²)/2
43. Evaluate ∫₃⁶ (2x² − 5/x²) dx
    1. 751/6
    2. 11/3
    3. 126
    4. 745/6
44. If y = tan⁻¹(5θ), find the differential coefficient dy/dθ at θ = π/180.
    1. 5/(1 + π²/1296)
    2. 5/(1 + π²/1256)
    3. 5/(1 + 25π²)
    4. 5/(1 + π²/32400)
45. Evaluate ∫ (x⁴ + 3x² − 2)(4x³ + 6x) dx
    1. (x⁴ + 3x² − 2)²/2 + C
    2. (x⁴ + 3x² − 2)²/4 + C
    3. (x⁴ + 3x² − 2)³/3 + C
    4. (x⁴ + 3x²)(x⁴ + 3x² − 2) + C
46. If y = 3/(1 − 2x)², find dy/dx.
    1. 12/(1 − 2x)³
    2. −12/(1 − 2x)³
    3. 6/(1 − 2x)³
    4. 3/(1 − 2x)³
47. Evaluate ∫ (6x − 1/(4x²) + 4x − 3) dx
    1. 5x² − 3x + 1/(4x) + C
    2. 5x² − 3x − 1/(4x) + C
    3. 10x² − 3x + C
    4. 5x² − 3x + 1/(4x²) + C
48. If y = (x + 1) sin x, find dy/dx.
    1. sin x + (x + 1) cos x
    2. cos x
    3. sin x − (x + 1) cos x
    4. (x + 1) cos x
49. Find dy/dx if y = x⁴ ln x.
    1. x³(4 ln x + 1)
    2. 4x³ ln x
    3. x³ ln x + 1
    4. 4x³ + x³ ln x
50. Find dy/dx if y = x²/(x + 1).
    1. x(x + 2)/(x + 1)²
    2. x(x − 2)/(x + 1)²
    3. 2x/(x + 1)²
    4. x²/(x + 1)²
51. If x = 3t³ − 2t² + 4t − 1, find the acceleration when t = 1 s.
    1. 4 m/s²
    2. 14 m/s²
    3. 18 m/s²
    4. 22 m/s²
52. Find the turning points of y = x³/3 + x² − 3x + 4.
    1. (1, 7/3) and (−3, 13)
    2. (1, 7/3) and (−3, −5)
    3. (−1, 7/3) and (3, 13)
    4. (1, 13) and (−3, 7/3)
53. If x = 2t − (3/2)t² − (2/3)t³, find the acceleration when t = 0.
    1. −3 m/s²
    2. 3 m/s²
    3. −4 m/s²
    4. 2 m/s²
54. Evaluate ∫ (25 − 4x) dx
    1. 25x − 2x² + C
    2. 25x − 4x² + C
    3. 25 − 2x² + C
    4. 25x² − 2x + C
55. Evaluate ∫ 2x/√(x² + 4) dx
    1. 2√(x² + 4) + C
    2. √(x² + 4) + C
    3. x²/√(x² + 4) + C
    4. 4√(x² + 4) + C
56. Evaluate lim(x→0) (5x² − 2x + 5).
    1. 5
    2. 0
    3. 2
    4. 10
57. Evaluate ∫ 1/(16 + 9x²) dx
    1. (1/12) tan⁻¹(3x/4) + C
    2. (1/12) tan⁻¹(4x/3) + C
    3. (1/4) tan⁻¹(3x/4) + C
    4. (1/12) sin⁻¹(3x/4) + C
58. Evaluate ∫ (sec x tan x)/(sec x + 3) dx
    1. ln|sec x + 3| + C
    2. ln|tan x| + C
    3. sec x + 3 + C
    4. 1/(sec x + 3) + C
59. Evaluate ∫₀^π sin x dx
    1. 2
    2. 0
    3. 1
    4. −2
60. Find dy/dx of y = √(tan 2x).
    1. sec²2x/√(tan 2x)
    2. 1/(2√(tan 2x))
    3. 2 sec²2x √(tan 2x)
    4. sec 2x tan 2x/√(tan 2x)
61. Find dy/dx of x² + xy + y² = 7 at the point (2, 1).
    1. −5/4
    2. 5/4
    3. −4/5
    4. −3/2
62. Find dy/dx of y = e^(3x² + 2x).
    1. (6x + 2) e^(3x² + 2x)
    2. e^(3x² + 2x)
    3. (3x² + 2x) e^(3x² + 2x)
    4. (6x + 2) e^(6x + 2)
63. Find dy/dx of y = csc 2x.
    1. −2 csc 2x cot 2x
    2. −csc 2x cot 2x
    3. 2 csc 2x cot 2x
    4. −2 csc²2x
64. Find dy/dx of y = eˣ/x.
    1. eˣ(x − 1)/x²
    2. eˣ/x²
    3. eˣ(x + 1)/x²
    4. (eˣ − 1)/x
65. Evaluate ∫ x⁴ e^(x⁵) dx
    1. e^(x⁵)/5 + C
    2. 5e^(x⁵) + C
    3. x⁵e^(x⁵)/5 + C
    4. e^(x⁵)/(5x⁴) + C
66. Evaluate ∫₀¹ x²(x³ + 1)² dx
    1. 7/9
    2. 1/3
    3. 7/3
    4. 8/9
67. Evaluate ∫ −x/((2x + 3)(x + 2)) dx
    1. (3/2) ln|2x + 3| − 2 ln|x + 2| + C
    2. 3 ln|2x + 3| − 2 ln|x + 2| + C
    3. (3/2) ln|2x + 3| + 2 ln|x + 2| + C
    4. −(3/2) ln|2x + 3| + 2 ln|x + 2| + C
68. Evaluate ∫ 1/((x + 4)(x − 2)) dx
    1. (1/6) ln|(x − 2)/(x + 4)| + C
    2. 6 ln|(x − 2)/(x + 4)| + C
    3. (1/6) ln|(x + 4)/(x − 2)| + C
    4. (1/2) ln|(x − 2)/(x + 4)| + C
69. Evaluate ∫ 1/(x − 1)² dx
    1. −1/(x − 1) + C
    2. 1/(x − 1) + C
    3. −2/(x − 1)³ + C
    4. ln|x − 1| + C
70. Evaluate ∫ 3x/√(x² + 1) dx
    1. 3√(x² + 1) + C
    2. (3/2)√(x² + 1) + C
    3. 3x²/√(x² + 1) + C
    4. 6√(x² + 1) + C
71. Evaluate ∫ (11 − x)/((x + 4)(x + 1)) dx
    1. 4 ln|x + 1| − 5 ln|x + 4| + C
    2. 5 ln|x + 1| − 4 ln|x + 4| + C
    3. 4 ln|x + 1| + 5 ln|x + 4| + C
    4. −4 ln|x + 1| + 5 ln|x + 4| + C
72. In ∫ x² 3ˣ dx using integration by parts (LIATE), what is u?
    1. x²
    2. 3ˣ
    3. 2x
    4. x³/3
73. Find the second derivative of ln x.
    1. −1/x²
    2. 1/x²
    3. 1/x
    4. −1/x
74. Find the area under the curve y = x^(1/2) between x = 1 and x = 4.
    1. 14/3
    2. 7/3
    3. 16/3
    4. 2
75. Evaluate ∫ dx/(6x² − 7x + 2)
    1. ln|(3x − 2)/(2x − 1)| + C
    2. ln|(2x − 1)/(3x − 2)| + C
    3. 2 ln|2x − 1| − 3 ln|3x − 2| + C
    4. (1/6) ln|(3x − 2)/(2x − 1)| + C
76. What are the values of x at the turning points of y = 4x³ − 3x² − 6x + 1?
    1. x = 1 and x = −1/2
    2. x = −1 and x = 1/2
    3. x = 1 and x = 1/2
    4. x = −1 and x = −1/2
77. Evaluate ∫₁⁴ 1/(x − 2)² dx
    1. −1/3 (naive value)
    2. The integral diverges (does not exist)
    3. 1/3
    4. 2/3
78. Evaluate lim(x→−2) (x² − 7x).
    1. 18
    2. −10
    3. 10
    4. −18
79. Evaluate ∫ (√5 − 2x) dx
    1. √5 x − x² + C
    2. √5 − x² + C
    3. √5 x − 2x² + C
    4. (√5 x)/2 − x² + C
80. Evaluate ∫ 7^(2x + 5) dx
    1. 7^(2x + 5)/(2 ln 7) + C
    2. 7^(2x + 5)/ln 7 + C
    3. 7^(2x + 5)/2 + C
    4. (2x + 5) 7^(2x + 4) + C
81. Evaluate ∫ (4x + 1)³ dx
    1. (4x + 1)⁴/16 + C
    2. (4x + 1)⁴/4 + C
    3. (4x + 1)⁴ + C
    4. 3(4x + 1)² + C
82. In ∫ x³ 3ˣ dx by parts (u = x³, dv = 3ˣ dx), what is v?
    1. 3ˣ ln 3
    2. x⁴/4
    3. 3ˣ/ln 3
    4. 3x²
83. In ∫ x³ 3ˣ dx by parts, what is dv?
    1. x³
    2. 3ˣ dx
    3. 3ˣ
    4. x³ dx
84. Evaluate ∫₂³ 1/√(x − 2) dx
    1. 2
    2. 1
    3. 4
    4. 1/2
85. Differentiate y = x³ ln x.
    1. x²(3 ln x + 1)
    2. 3x² ln x
    3. x³ ln x + 1
    4. 3x² + x³ ln x
86. Evaluate ∫ (3x − 17)/((4x + 1)(3x + 1)) dx
    1. 18 ln|3x + 1| − (71/4) ln|4x + 1| + C
    2. 54 ln|3x + 1| − 71 ln|4x + 1| + C
    3. 18 ln|3x + 1| + (71/4) ln|4x + 1| + C
    4. (71/4) ln|3x + 1| − 18 ln|4x + 1| + C
87. Find dy/dx if x² + 2xy = 7.
    1. −(x + y)/x
    2. −(x + y)/(2x)
    3. (x + y)/x
    4. −x/(x + y)
88. Find dy/dx of y = sin²(3x + π/6) at x = 0.
    1. π/12
    2. π/6
    3. 2√3/3
    4. 3√3/2
89. If y = sin x, find d²y/dx².
    1. −sin x
    2. sin x
    3. cos x
    4. −cos x
90. Evaluate ∫₋₃³ 1/(x² + 9) dx
    1. π/3
    2. π/12
    3. π/6
    4. π/9
91. Find dy/dx if y = (x² + 1)/(x³ + x + 1).
    1. (−x⁴ − 2x² + 2x − 1)/(x³ + x + 1)²
    2. (x⁴ + 2x² − 2x + 1)/(x³ + x + 1)²
    3. (−x⁴ + 2x² + 2x − 1)/(x³ + x + 1)²
    4. 2x/(3x² + 1)
92. Find the derivative of (3x² − 2x)⁴ at x = 1.
    1. 16
    2. −48
    3. 4
    4. 48
93. Evaluate ∫ x sin x dx
    1. −x cos x + sin x + C
    2. x cos x − sin x + C
    3. x sin x + cos x + C
    4. −x sin x − cos x + C
94. Find the area enclosed between the curves y = x² and y = 2x.
    1. 4/3
    2. 8/3
    3. 2/3
    4. 4
95. Evaluate ∫₋₄⁰ (1 − 2x) dx
    1. 20
    2. −20
    3. −12
    4. 12
96. Find the total area enclosed between the curve y = x³ − 2x and the x-axis.
    1. 2
    2. 1
    3. 4
    4. 0
97. Differentiate tan x/x.
    1. (x sec²x − tan x)/x²
    2. (sec²x − tan x)/x²
    3. (x sec²x + tan x)/x²
    4. sec²x/x
98. If dA/dt = t² − 1 and A = 0 when t = 0, find A when t = 1.
    1. −2/3
    2. 2/3
    3. 0
    4. 1/3
99. If f(x) = 15x + 2x² and g(x) = 2x² + x − 3, find f(x) − g(x).
    1. 14x + 3
    2. 16x − 3
    3. 14x − 3
    4. 4x² + 16x − 3
100. Evaluate ∫ x cos x dx
     1. x sin x + cos x + C
     2. x sin x − cos x + C
     3. −x sin x + cos x + C
     4. x cos x + sin x + C

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