General Mathematics I — 2024 Past Questions

Lagos State University MAT101: 62 questions.

Topics: Trigonometry Function of Angle, Number System, Addition and Factor Formulae, Theory of Quadratic Equations, Trigonometric Functions of Angles of Any Magnitude, Sequence and Series, Binomial Theorem, Quadratic Equation, Real Sequences and Series, Mathematical Induction, Union, Complements, Venn Diagrams, Intersection

1. Find the set of values of p for which the equation x² + px + 2p − 3 = 0 has no real roots.
   1. 2 < p < 6
   2. −2 < p < 6
   3. −3 < p < 2
   4. 2 < p < 3
2. The sum of the first ten terms of an arithmetic series is 60 and the sum of the first twenty-two terms is 220. Find the common difference and the first term of the series.
   1. Common difference 2/3, first term 3
   2. Common difference 3/2, first term 1
   3. Common difference 1, first term 2/3
   4. Common difference 3, first term 3/2
3. Use mathematical induction to find the sum to n terms of the series 1/(1·2) + 1/(2·3) + ... + 1/[n(n + 1)].
   1. n/(n + 1)
   2. 2n − 1
   3. n(n − 1)
   4. 1/(n + 1)
4. Let N be the universal set of all integers, with P = {x : x is a prime number}, Q = {x : x is a multiple of 4} and R = {x : x is an even number}. Find Q ∩ R.
   1. {4}
   2. {2}
   3. Q
   4. {2, 4}
5. With N the set of all integers, P = {primes}, Q = {multiples of 4} and R = {even numbers}, find P ∩ R ∩ Q′.
   1. {2}
   2. P
   3. {4}
   4. R
6. With N the set of all integers, P = {primes}, Q = {multiples of 4} and R = {even numbers}, find Q ∪ [R ∩ (P′ ∩ Q′)].
   1. {1}
   2. R
   3. Q
   4. R without the element 2 (the even numbers except 2)
7. Give the first four terms of (2 + x)⁵ in ascending powers of x.
   1. 4 + 10x + 10x² + 5x³
   2. 40x³ + 80x² + 80x + 32
   3. 32 + 80x + 80x² + 40x³
   4. 5x³ + 10x² + 10x + 4
8. The roots of the equation x² − 9x + 20 = 0 are α and β, where α > β. Find √(α − β).
   1. 3
   2. 1
   3. √3
   4. 1/3
9. The function f is defined by f(x) = 2x³ − x² − 18x + 9. If f(x) ≡ (x − 3)(ax² + bx + c), find the values of a, b and c.
   1. a = 7, b = 3, c = −3
   2. a = 3, b = 1/2, c = −3
   3. a = −3, b = 1, c = 2
   4. a = 2, b = 5, c = −3
10. k − 6, k and k + 18 are successive terms of a geometric series. Which of the following is/are true about k? (i) k is even (ii) k is odd (iii) k is prime
    1. (i) only
    2. (i) and (iii) only
    3. (ii) only
    4. (ii) and (iii) only
11. For all x except x = 1, (x² + 1)/(x − 1) equals:
    1. x − 1 + 2/(x − 1)
    2. x + 1 + 1/(x − 1)
    3. x + 1 + 2/(x + 1)
    4. x + 1 + 2/(x − 1)
12. Functions f and g are defined by f : x → logₐ x (x ∈ ℝ⁺, a > 1) and g : x → 1/x (x ∈ ℝ⁺). State the range of g.
    1. 0 < y < ∞
    2. −1 ≤ y ≤ 1
    3. 0 < y ≤ 1
    4. 0 ≤ y ≤ 1/2
13. Functions f and g are defined by f : x → logₐ x (x ∈ ℝ⁺, a > 1) and g : x → 1/x (x ∈ ℝ⁺). Find f∘g(x) + f(x).
    1. 1
    2. 0
    3. 10
    4. Cannot be determined
14. Describe the behaviour of each of these sequences: (i) Uₙ = n(n + 1)(n + 2); (ii) Uₙ₊₁ = 1 − 1/(1 + Uₙ) with U₁ = 1; (iii) f : n → cos(600°n), n ∈ ℕ.
    1. Divergent, periodic, convergent
    2. Convergent, divergent, periodic
    3. Convergent, periodic, convergent
    4. Divergent, convergent, periodic
15. Given the matrix A = [[−1, −4], [1, 3]] (rows shown in square brackets), find A³.
    1. [[−1, −64], [1, 27]]
    2. [[−5, −12], [3, 7]]
    3. [[−1, −64], [27, 1]]
    4. [[−9, −20], [5, 11]]
16. Find the solution of x² − 2x − 8 ≤ 0.
    1. 2 ≤ x ≤ 4
    2. x ≤ −2 or x ≥ 4
    3. −2 ≤ x ≤ 4
    4. −4 ≤ x ≤ −2
17. Solve x(x + 1) + 12/[x(x + 1)] = 8.
    1. −3, 2, −2, 1
    2. 3, −2, 2, −1
    3. −3, −2, 2, −1
    4. 3, −2, −2, 1
18. Find the range of values of x for which 2x > 1/x.
    1. −1/√2 < x < 0 or x > 1/√2
    2. x > 1/√2 only
    3. 0 < x < 1/√2
    4. x < −1/√2
19. The function f is defined by f : x → 1/(2 − x), where x ≠ 2. Find the inverse function f⁻¹ and state its domain.
    1. 2 − x, x ∈ ℝ
    2. x/(2x − 4), 0 < x < ∞
    3. (x − 1)/(2x), x ∈ ℝ, x ≠ 0
    4. (2x − 1)/x, x ∈ ℝ, x ≠ 0
20. The three real, distinct, non-zero numbers a, b, c are such that a, b, c are in arithmetic progression and a, c, b are in geometric progression with common ratio r < 0. Find, in terms of a, the 6th term of the arithmetic progression whose first three terms are a, b, c.
    1. 19a/4
    2. −11a/4
    3. −14a/4
    4. 3a/4
21. Use mathematical induction to find the sum to n terms of the series 1³ + 2³ + ... + n³.
    1. n³(n + 1)(n + 2)
    2. (1/3)n²(3n + 1)²
    3. (1/4)n²(n + 1)²
    4. (n/4)(n + 1)(n + 2)(n + 3)
22. Find the first four terms in the expansion of (1 − x)^(−1/2).
    1. 1 + (1/2)x + (3/2)x² + (15/4)x³
    2. 1 − (1/2)x + (3/2)x² + (5/16)x³
    3. −1 + (1/2)x + (5/2)x² + (15/4)x³
    4. 1 + (1/2)x + (3/8)x² + (5/16)x³
23. Out of the 1000 undergraduate students in the Faculty of Science, 650 are offering Algebra and 500 are offering Trigonometry. How many students are offering both Algebra and Trigonometry if only 210 students are offering neither?
    1. 290
    2. 360
    3. 140
    4. 120
24. The equation x² − 2(p − 2)x + 2p − 10 = 0 has real roots. Find the possible values of p when the roots of the equation differ by 6.
    1. −5, −1
    2. 5, 1
    3. 5, −1
    4. −5, 1
25. If α² and β² are the roots of x² − 21x + 4 = 0 and α and β are both positive, find the equation with roots 1/α² and 1/β².
    1. x² − 11x + 4 = 0
    2. 9x² − 11x − 4 = 0
    3. x² − 21x + 4 = 0
    4. 4x² − 21x + 1 = 0
26. Find the range of values of x for which 2|x + 1| > |x − 1|.
    1. x < −3 or x > −1/3
    2. x > −3
    3. −3 < x < −1/3
    4. x < −1/3
27. Given that |x| ≠ 1, find the complete solution for which x/(x − 1) > 1/(x + 1).
    1. −1 < x < 1
    2. x > −1
    3. x > 1
    4. x < −1 or x > 1
28. Find the value of Σ (from r = 1 to ∞) of k/10ʳ.
    1. k/90
    2. k/10
    3. 10/9
    4. k/9
29. Find the term in x⁵ in the expansion of (1 + x)¹⁰.
    1. 120x⁵
    2. 252x⁵
    3. 720x⁵
    4. 144x⁵
30. The roots of the quadratic equation x² + 2x + 3 = 0 are denoted by α and β. Without solving the equation, find the quadratic equation whose roots are α + 1/β and β + 1/α.
    1. x² − 4x + 5 = 0
    2. x² − 4x − 5 = 0
    3. 3x² + 8x + 16 = 0
    4. 3x² − 8x + 1 = 0
31. It is given that 1/(b + c), 1/(c + a), 1/(a + b) are three consecutive terms of an arithmetic series. Which of the following statements is/are true of the series? (i) a², b², c² are also three consecutive terms of an arithmetic series. (ii) The common difference is (c − b)/[(b + a)(c + a)]. (iii) The sum of its four terms is [1 − 3(a − b)]/(b + c).
    1. (i) only
    2. (i) and (ii) only
    3. (i) and (iii) only
    4. (i), (ii) and (iii)
32. The functions f and g are defined by f(x) = x³ − 2x² − 5x + 6 and g(x) = x − 1. Find f∘g(x).
    1. x³ − 2x² − 5x + 5
    2. x³ − 2x² − 5x + 7
    3. x³ − 5x² + 2x + 8
    4. x³ − 5x² − 2x + 7
33. Find the inverse of the matrix A = [[2, 4, 3], [1, −2, −2], [−3, 3, 2]] (rows shown in square brackets).
    1. (1/11)[[2, 1, −2], [4, 13, 7], [−3, −18, −8]]
    2. (1/11)[[2, 4, −3], [1, 13, −18], [−2, 7, −8]]
    3. (1/11)[[2, 1, −2], [4, −2, 3], [3, −2, 2]]
    4. (1/−21)[[2, 4, −3], [1, 13, −18], [−2, 7, −8]]
34. Find, without using tables, the value of cos 80° cos 20° + sin 80° sin 20°.
    1. 1/2
    2. 1/3
    3. 1/4
    4. 1/5
35. If sin θ = 1/√3 and θ is obtuse, find tan θ.
    1. 1/√2
    2. −1/√2
    3. 1/√5
    4. −1/√5
36. If t = tan(θ/2) and 0 ≤ θ < 90°, express √[(1 + sin θ)/(1 − sin θ)] in terms of t.
    1. (1 − t)/(1 + t)
    2. (1 + t²)/(1 − t²)
    3. 2t/(1 + t²)
    4. (1 + t)/(1 − t)
37. The general solution of tan 2x = cot 5x is given by:
    1. x = n(180°/7)
    2. x = (2n + 1)(90°/7)
    3. x = (2n + 1)(180°/7)
    4. x = n(180°) + 90°
38. Solve for θ in the range 0° ≤ θ ≤ 90°, given that 6 sin θ = tan θ.
    1. 0°, 80°24′
    2. 45°, 80°24′
    3. 68°, 80°36′
    4. 60°, 80°36′
39. Given that x = r sin θ cos φ, y = r sin θ sin φ and z = r cos θ, then:
    1. r = x + y + z
    2. x − y = r − z
    3. r = 1/(x² + y² + z²)
    4. r² = x² + y² + z²
40. The general solution of cos x = cos θ (that is, cos x = α with principal angle θ) is given by:
    1. x = n·360° ± θ
    2. x = n·180° ± θ
    3. x = n·180° + (−1)ⁿθ
    4. x = n·270° ± θ
41. The general solution of sin x = sin θ (that is, sin x = α with principal angle θ) is given by:
    1. x = n·360° ± θ
    2. x = n·180° ± θ
    3. x = n·180° + (−1)ⁿθ
    4. x = n·270° ± θ
42. Evaluate cos 75° without using tables.
    1. (√3 − 1)/(2√2)
    2. 1/√3
    3. 2√2/3
    4. 1/4
43. Find the value of tan(θ − φ), given that tan θ = √3 and tan φ = 1/√3.
    1. √3
    2. 1/√3
    3. √2
    4. 1/√2
44. Solve the equation y = 2 cos⁻¹(x/3) for x.
    1. 3 cos(y/2)
    2. cos(3y/2)
    3. 3 cos(2y/3)
    4. cos(y/2)
45. Simplify (sin³x + cos³x)/(sin x + cos x).
    1. 1 − (1/2) sin 2x
    2. 1 + (1/2) cos 2x
    3. cos 3x
    4. sin 2x
46. Simplify sin(θ + 30°) + cos(θ + 60°).
    1. sin θ
    2. (1/2) cos θ
    3. (1/2) sin θ
    4. cos θ
47. Simplify sin²x/(1 + cos x) + sin²x/(1 − cos x).
    1. sin x
    2. cos²x
    3. sin²x
    4. 2
48. If sin θ = x/y and 0° < θ < 90°, find 1/tan θ.
    1. √(y² − x²)/x
    2. x²/√(y² − x²)
    3. x/√(y² − x²)
    4. y/x
49. If sin θ = (m − n)/(m + n), find the value of 1 + tan²θ.
    1. 2(m² + n²)/(m + n)
    2. √[2(m² + n²)]/(m + n)
    3. (m² + n² + 2mn)/(2mn)
    4. (m² + n² + 2mn)/(4mn)
50. Simplify cos(arcsin 3/5).
    1. 4/5
    2. 3/5
    3. 3/4
    4. 5/4
51. Simplify cos(arctan 15/8 − arcsin 4/5).
    1. 297/425
    2. 284/425
    3. 105/425
    4. 84/85
52. Simplify sin(arcsin 12/13 − arcsin 4/5).
    1. 56/65
    2. 20/65
    3. 36/65
    4. 16/65
53. Simplify sin(A + B) sin(A − B).
    1. sin²B − sin²A
    2. sin²A − sin²B
    3. sin²A
    4. sin²(A + B)
54. Simplify sin(2 arctan 3).
    1. 4/5
    2. 3/5
    3. 3/4
    4. 1/2
55. Write the expression cos 4x − cos 2x as a product of trigonometric functions.
    1. cos 3x cos x
    2. −2 sin 3x sin x
    3. sin 6x sin 2x
    4. −cos 4x cos 2x
56. If tan 6x/(1 − tan²6x) = k tan tx, what are the values of k and t?
    1. k = 1 and t = 6
    2. k = 3 and t = 18
    3. k = 1/2 and t = 12
    4. k = 4 and t = 24
57. Given that 5 − 5 cos 8x = a sin²bx, find the values of a and b that make the equality true.
    1. a = 10 and b = 4
    2. a = 5 and b = 2
    3. a = 15 and b = 6
    4. a = 20 and b = 8
58. Simplify (1 + cot θ − cosec θ)(1 + tan θ + sec θ).
    1. 2
    2. 1
    3. 0
    4. sin θ cos θ
59. If t = tan(θ/2), express 3 cos θ + 4 sin θ + 5 in terms of t.
    1. 2(t + 2)²/(1 + t²)
    2. 2(t + 2)²
    3. 1 + t²
    4. 3√t + 4t^(1/3)
60. Simplify 5 sin θ − 20 sin³θ + 16 sin⁵θ.
    1. sin 3θ
    2. cos 5θ
    3. sin 5θ
    4. cos 3θ
61. Simplify sin(270° − θ) + sin(270° + θ).
    1. −2 cos θ
    2. 2 sin 2θ
    3. 2 cos 2θ
    4. sin θ
62. A tree 50 feet in height casts a shadow 60 feet long. What is the angle of elevation of the top of the tree from the end of the shadow, measured with respect to the ground?
    1. 49.8°
    2. 48.9°
    3. 38.9°
    4. 39.8°

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