MTH 111 Elementary Mathematics I Past Paper 2024/2025 - Anchor University Lagos PDF

DEPARTMENT OF MATHEMATICS GENERAL OBJECTIVES COURSE CODE: MTH 111 UNIT: 2 COURSE: ELEMENTARY MATHEMATICS I ACADEMIC SESSION: 2024/2025.. ( +1) 1. Prove by induction: 1 + 2 + 3 + ⋯ + = 2. What is the base ca...

DEPARTMENT OF MATHEMATICS GENERAL OBJECTIVES COURSE CODE: MTH 111 UNIT: 2 COURSE: ELEMENTARY MATHEMATICS I ACADEMIC SESSION: 2024/2025.. ( +1) 1. Prove by induction: 1 + 2 + 3 + ⋯ + = 2. What is the base case for = 1? A) 0 B) 1 C) 2 D) n+1 2. If P(k) is true for k = 1 and P(k+1) is true whenever P(k) is true, what can you conclude? A) P(k+1) is false B) P(k) is false C) P(n) is true for all ≥ 1 D) P(n) is false for all ≥ 1 3. The decimal expansion of 22/7 is a. Terminating b. Non-terminating and non-recurring c. Non-terminating and recurring d. None of the above 4. Which of the following option is not irrational? A. (3 + √7) B. (3 – √7) C. (3 + √7) (3 – √7) D. 3√7 5. The sum of a rational number and an irrational number is always a/an: A rational number B. Irrational number c. either rational or irrational D None of the above 6. The product of two irrational numbers is: a. irrational or irrational b. rational number c. rational d. None 7. The largest number that divides 72 and 128, which leaves the remainders 7 and 11, is: 65 b. 15 c. 23 d. 13 8 Which of the following is a valid use of mathematical induction? A) Proving identities B) Solving differential equations C) Finding the derivative of a function D) Solving inequalities 9 For the statement2 > 2 for ≥ 5, what is the induction hypothesis? A) 2 > 2 B) 2 +1 > ( + 1)2 C) 2 > D) 2 < 2 10 Prove 3 + 2 is divisible by 3 for all ≥ 1. What must be shown in the inductive step? A) ( + 1) − ( ) is divisible by 3 B) ( + 1) + ( ) is divisible by C) ( + 1) = ( ) D) ( + 1) ≠ ( ) 11 Find the sum of the first terms of the 2,5,8, … A) 2 + 3 B) /2 ⋅ (2 + ( − 1) ) C) /2 ⋅ ( + ) D) None of these 12 If the 10 ℎ term of an is 25, and the first term is 5, find the common difference. A) 2 B) 3 C) 4 D) 5 13 In an AP, if = 3, = 2, and = 55, find. A) 5 B) 10 C) 11 D) 12 14 The 4th and 8th terms of an AP are 10 and 22, respectively. Find the common difference. A) 2 B) 3 C) 4 D) 5 15 What is the sum of all integers from 1 to 100 that are divisible by 5? A) 1050 B) 1055 C) 1100 D) 1150 16 If the first term of a GP is 2 and the common ratio is 3, find the 5th term. A) 486 B) 54 C) 162 D) 72 17 Find the sum of the infinite GP with = 4 and = 1/2. A) 8 B) 6 C) 10 D) 12 18 The sum of three terms in GP is 21, and their product is 216. Find the terms.  A) 3,6,123, 6, 12 B) 2,6,182, 6, 18 C) 4,8,164, 8, 16 D) 6,6,66, 6, 6 ( − 1) 19 The sum of a finite GP is given by _ = −1 , when > 1. Find _4 if = 2, = 3. A) 80 B) 120 C) 242 D) 26 13 20 If the sum of the first 3 terms of a GP is 2 and their product is 27/8, what are 3 3 3 3 3 3 the terms? A) , , 2 4 8 B) , ,3 4 2 C) 1, 2 , 3 D) None of these 21 Solve 2 − 5 + 6 = 0. A) = 2,3 B) =− 2, − 3 C) = 3, − 3 ) =− 2,3 22 What is the nature of the roots of 2 − 4 + 5 = 0? A) Real and equal B) Real and distinct C) Complex conjugates D) None of these 23 Find the sum of the roots of 2 − 3 + 4 = 0. A) 3/2 B) −3/2 C) 2 D) −2 24 Solve 2 + 7 + 10 ) = − 2, − 5 ) = 2, − 5 ) =− 2,5 ) = 2,5 25 If one root of 2 + + = 0 is 3, find when = − 5. A) − 12 B) − 15 C) 12 D) 15 26 Simplify 35. A) 1 B) –I C) I D) -1 2 27 Express 1 + in the form +. A) 0 + 2 B) 2 + 2 C) 1 + 2 D) −1 + 2 7 28 Find the modulus of 3 − 4. A) 5 B) 7 C) 25 D) 7 29 If = 2 + 3 , find. A) 2 − 3 B) −2 + 3 C) −2 − 3 D) 3 − 2 30 Solve 2 + 1 = 0. A) ± B) ±1 C) 1, − D) None of these ( +1)(2 +1) 31 Prove 12 + 22 + 32 + , … + 2 = 6. What is the base case for = 1? A) 1 B) 2 C) 3 D) 6 32. A __________ is an ordered collection of objects. a Relation b. Function c. Proposition d. Set 33. The set O of odd positive integers less than 10 can be expressed by _____________ a {1, 2, 3} b. {1, 3, 5, 7, 9} c. {1, 2, 5, 9} d. {1, 5, 7, 9, 11} 34. Power set of empty set has exactly _________ subset. a. One b. Two c. Zero d. Three 35. What is the Cartesian product of A = {1, 2} and B = {a, b}? a. {(1, a), (1, b), (2, a), (b, b)} b. {(1, 1), (2, 2), (a, a), (b, b)} c. {(1, a), (2, a), (1, b), (2, b)}d. {(1, 1), (a, a), (2, a), (1, b)} 36. The Cartesian Product B x A is equal to the Cartesian product A x B. a. True b. False c. I don’t Know d. None of the above 37 What is the cardinality of the set of odd positive integers less than 10? a. 10 b. 5 c. 3 d. 20 38 Which of the following two sets are equal? a) A = {1, 2} and B = {1} b. A = {1, 2} and B = {1, 2, 3} c. A = {1, 2, 3} and B = {2, 1, 3}d. A = {1, 2, 4} and B = {1, 2, 3} 39 The set of positive integers is _____________ a. Infinite b. Finite c. Subset d Empty 40. What is the Cardinality of the Power set of the set {0, 1, 2}? a. 8 b. 6 c. 7 d. 9 41. The value of tan 60°/cot 30° is equal to: a. 0 b. 1 c. 2 d. 3 42. 1 – cos2A is equal to: a. sin2A b. tan2A c. 1 – sin2A d. sec2A 43. sin (90° – A) and cos A are: a. Different b. Same c. Not related d. None of the above 44. If cos X = ⅔ then tan X is equal to: a. 5/2 b. √(5/2) c. √5/2 d. 2/√5 45. If cos X = a/b, then sin X is equal to:  a. (b2-a2)/b b. (b-a)/b c. √(b2-a2)/b d.√(b-a)/b : 46 The value of sin 60° cos 30° + sin 30° cos 60° is: a. 0 b. 1 c. 2 d. 4 : 47. 2 tan 30°/(1 + tan230°) = a. sin 60° b. cos 60° c. tan 60° d. sin 30° 48. If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then the order of matrix (5A – 2B) is a. m×3 b. 3 × 3 c. m×n d. 3 × n 2 p  q p  2q   4  3 49. If  =   , then the value of p + q + 2s is  5r  s 4r  3s  11 24  a. 8 b. 10 c. 4 d. -8 1 0 0 50. This matrix 0 1 0 is called   0 0 1 (a) identity matrix (b) symmetric matrix (c) skew symmetric matrix (d) none of these 2x  1 51. is ( x  1)( x  1) a. An equation b. An improper fraction c. An identity d. A proper fraction 52. The identity (5 x  4) 2  25 x 2  40 x  16 is true for? A Two values of x b. All values of x c. None of these d. One value of x x2 53. Partial fractions of are of the form? ( x  1)( x  2) A Bx  C Ax  B C A B a.  b.  c.  x 1 x  2 x 1 x2 x 1 x  2 Ax B d  x 1 x  2 54. The identity ( x  3) 2  x 2  6 x  9 is true for? a. An equation b. An identity c. None of these d. Linear equation 55 For 2 ≥ 2 for ≥ 4, what must be proven in the induction step? A) 2 +1 ≥ + 1 2 B) 2 < 2 C) 2 +1 < 2 D) None of these 56 To prove ( ): ! > 2 for ≥ 4, what is P(k+1) based on? A) + 1 !B) ! + + 1 ! C) ! + 1 > 2 +1 D) ⋅ ( + 1) > 2 +1 57 Prove 3 > 3 for ≥ 4. What is the first inductive step? A) 3 +1 > + 1 3 B) 3 +1 < + 1 3 C) 3 = 3 D) None of these 58 Find the 15 ℎ term of an where = 5 and = 3. A) 47 B) 50 C) 45 D) 48 59 The sum of the first 20 terms of an is 210. If = 5, find d. A) 1 B) 2 C) 3 D) 4 60 How many terms of the 7,10,13, … are needed to make 72? A) 5 B) 6 C) 7 D) 8 61 In an AP, the first term is 4, and the 10th term is 31. Find the sum of the first 10 terms. A) 175 B) 150 C) 180 D) 200 62 The common difference of an is 6, and the 20 ℎ term is 122. Find the first term. A) 2 B) 4 C) 6 D) 8 63 The first term of a GP is 3, and the common ratio is 1/3. Find the sum of 40 41 81 42 the first 4 terms. A) 27 B) 27 C) 2 D) 27 64 If = 2 and = 3, find the 6 ℎ term of the. A) 486 B) 162 C) 243D) 729 65 For an infinite GP, if = 5 and = 1/5 , find the sum. A) 6 B) 5.25 C) 6.25 D) 5.5 1 − 66 The sum of a GP is = 1−. Find 5 if = 2, = 0.  A) 3.94 B) 4.88 C) 4.94 D) 5.12 67 In a GP, = 7 and ∞ = 21. Find r. A) 1/3 B) 2/3 C) ¾ D) None of these 68 Solve 2 + 6 + 9 = 0. A) =− 3, − 3 B) = 3,3 C) =− 3,3 D) None of these 69 The discriminant of 2 − 2 + 7 = 0 is: A) Negative B) Positive C) Zero D) Undefined 70 Find the product of the roots of 2 − 12 + 9 = 0. A) 9/4, 9/4 B) −9/4,-9/4 C) 1212 D) −12-12 71 If α and β are roots of ^2 − 4 + 5 = 0, find + + A) 5 B) 6 C) 4 D) 9 72 Simplify 2 + 3 1 −. A) 5 + 5 + B) 5 − 5 – C) 5 + 3 5 + 3 D) 5 − 3 5 − 3 73 Find the argument of −1 + A) /4 B) 3 /4 C) π/2 D) π 74 If = 1 + 2 , find 2. A) 5 B) 1 C) 2 D) 25 75 Solve 2 = − 4. A) ±2 B) ±2 C) ±4 D) ±1 1 1 1 1 76 Express 1+ in the form +. A) 2 −2 B) 2 +2 C) − 2 + 2 D) None of these 77. Which one is correct? a. log (m + n) = log m + log n (b) log (m – n) = log m – log n c. logb a x loga b = 1 d None of these. 78. Find the value of log b a. logc b. loga c a. abc b. 0 c. 1 d. 4. 79. If log 2 = 0.30103, then the value of log 200 is  a. 18.0103 b. 6.30103 c. 4.30103 d. 2.30103 80. log 144 is equal to a. 2 log 4 + 2 log 2 b 4 log 2 + 2 log 3 c. 3 log 2 + 4 log 3 d. None of these. 81. The relation loge x + loge (1 + x) = 0 is equal to a. x2 + x + e = 2 b. x2 + x – e = 3 c. x2 + x – 1 = 0 d. x2 + x + 4 = 0. 82. If log10 2 = 0.3010, the value of log10 80 is a. 1.9030 b. 1.6020 c. 2.9030 d. 4.9030. 83. Find the value of log3 (27)3 a. 27 b. 9 c. 729 d. 81. 84. If log3 (log2 x) = 1, Find the value of x a. 10 b. 9 c. 3 d. 8