Pre-Calculus I
Introduction
Pre-Calculus I develops the mathematical language and algebraic skills needed for calculus and higher mathematics. Students practise sets, functions, equations, matrices and trigonometry.
Recommended Background
No formal prerequisite.
Topic-Wise Course Outline
Mathematics and Number Systems
A brief history of mathematics; mathematical applications; natural, integer, rational, real and complex numbers.
Sets and Functions
Types of sets; set operations; complex numbers and their polar representation; functions, inverse functions and graphs.
Matrices and Determinants
Types and operations of matrices; 2×2 and 3×3 determinants; elementary row and column operations; echelon forms; linear systems; Cramer’s rule.
Quadratic Equations and Polynomials
Quadratic formula and factorization; polynomial functions; remainder theorem; worked algebraic problems.
Partial Fractions
Proper and improper rational expressions; division of polynomials; decomposition into partial fractions.
Mathematical Induction and Binomial Theorem
Proof by mathematical induction; binomial coefficients and expansions; applications.
Fundamentals of Trigonometry
Degrees and radians; trigonometric ratios and identities; standard angles; compound and half-angle formulas; periodic functions.
How to Study This Subject
Start with the definitions and core concepts in each topic. Work through examples, check every step of a solution, and then practise questions to test understanding.
Begin with the MCQs and worked examples below, then review each topic in the outline. More practice material will be added progressively.
Foundational MCQs and Step-by-Step Practice
Ten introductory topic-wise questions with explanations, followed by two fully worked examples. This is a starter set; further coverage is planned.
Number Systems
Q1. Which number is irrational?
- A. 3/7
- B. √2
- C. −4
- D. 0.25
Show correct answer and step-by-step explanation
Correct answer: B. √2
√2 cannot be expressed as a ratio of two integers.
Sets
Q2. If A={1,2,3} and B={3,4}, what is A ∩ B?
- A. {1,2,3,4}
- B. {1,2}
- C. {3}
- D. ∅
Show correct answer and step-by-step explanation
Correct answer: C. {3}
The intersection contains the common elements.
Functions
Q3. For f(x)=2x+3, find f(4).
- A. 7
- B. 8
- C. 11
- D. 14
Show correct answer and step-by-step explanation
Correct answer: C. 11
Substitute x=4: 2×4+3=11.
Inverse Functions
Q4. Find the inverse of f(x)=3x−6.
- A. (x+6)/3
- B. 3x+6
- C. (x−6)/3
- D. x/3−6
Show correct answer and step-by-step explanation
Correct answer: A. (x+6)/3
Solve y=3x−6 for x, obtaining x=(y+6)/3.
Matrices
Q5. Find det([[2,1],[3,4]]).
- A. 11
- B. 5
- C. −5
- D. 8
Show correct answer and step-by-step explanation
Correct answer: B. 5
For a 2×2 matrix, determinant is ad−bc=8−3=5.
Quadratic Equations
Q6. What are the roots of x²−5x+6=0?
- A. −2 and −3
- B. 1 and 6
- C. 2 and 3
- D. −1 and −6
Show correct answer and step-by-step explanation
Correct answer: C. 2 and 3
Factor as (x−2)(x−3)=0.
Polynomial Remainders
Q7. What is the remainder of x²+2x+1 divided by x−1?
- A. 0
- B. 2
- C. 3
- D. 4
Show correct answer and step-by-step explanation
Correct answer: D. 4
The remainder theorem gives p(1)=1+2+1=4.
Partial Fractions
Q8. For x≠0,−1, what equals 1/[x(x+1)]?
- A. 1/x−1/(x+1)
- B. 1/x+1/(x+1)
- C. 1/(x+1)−1/x
- D. 1/x²
Show correct answer and step-by-step explanation
Correct answer: A. 1/x−1/(x+1)
Combining 1/x−1/(x+1) gives 1/[x(x+1)].
Binomial Theorem
Q9. What is the coefficient of x² in (1+x)⁴?
- A. 4
- B. 6
- C. 8
- D. 12
Show correct answer and step-by-step explanation
Correct answer: B. 6
The coefficient equals C(4,2)=6.
Trigonometry
Q10. Convert 180 degrees to radians.
- A. π/2
- B. 2π
- C. π
- D. π/4
Show correct answer and step-by-step explanation
Correct answer: C. π
Multiply 180 by π/180 to obtain π radians.
Solved Examples with Step-by-Step Answers
Example 1: Quadratic equation
Problem: Solve x²−7x+12=0.
- Find two numbers whose sum is 7 and product is 12: 3 and 4.
- Factor as (x−3)(x−4)=0.
- Use the zero-product property.
Final answer: x=3 or x=4.
Example 2: Determinant
Problem: Find det([[3,2],[5,4]]).
- Use determinant = ad−bc.
- Calculate 3×4−2×5=12−10.
Final answer: The determinant is 2.
Additional Topic-Wise MCQs — Questions 11–20
Additional concept and calculation questions. Select a question’s expandable answer to review the reasoning.
Complex Numbers
Q11. What is i², where i is the imaginary unit?
- A. 1
- B. −1
- C. i
- D. 0
Show correct answer and explanation
Correct answer: B. −1
The imaginary unit satisfies i²=−1.
Q12. What is the modulus of 3+4i?
- A. 3
- B. 4
- C. 5
- D. 7
Show correct answer and explanation
Correct answer: C. 5
The modulus is √(3²+4²)=5.
Sets
Q13. How many subsets does a set with four elements have?
- A. 4
- B. 8
- C. 16
- D. 24
Show correct answer and explanation
Correct answer: C. 16
A set of n elements has 2ⁿ subsets, so 2⁴=16.
Matrices
Q14. What is the rank of [[1,2],[2,4]]?
- A. 0
- B. 1
- C. 2
- D. 4
Show correct answer and explanation
Correct answer: B. 1
The second row is twice the first, so only one row is independent.
Linear Systems
Q15. Solve x+y=7 and x−y=3. Find x.
- A. 2
- B. 3
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: D. 5
Adding gives 2x=10, so x=5.
Polynomials
Q16. If p(x)=x³−2x+1, what is p(−1)?
- A. −2
- B. 0
- C. 2
- D. 4
Show correct answer and explanation
Correct answer: C. 2
p(−1)=−1+2+1=2.
Partial Fractions
Q17. For x≠1,−1, which decomposition equals 2/(x²−1)?
- A. 1/(x−1)−1/(x+1)
- B. 1/(x−1)+1/(x+1)
- C. 2/(x−1)
- D. 2/(x+1)
Show correct answer and explanation
Correct answer: A. 1/(x−1)−1/(x+1)
Subtracting gives [(x+1)−(x−1)]/[(x−1)(x+1)]=2/(x²−1).
Induction
Q18. Which statement is the inductive step in a proof for all integers n≥1?
- A. Prove only n=1
- B. Assume P(k) and prove P(k+1)
- C. Assume P(k+1) without proof
- D. Check n=1,2,3 only
Show correct answer and explanation
Correct answer: B. Assume P(k) and prove P(k+1)
Induction proves a base case and the implication P(k)⇒P(k+1).
Trigonometry
Q19. What is sin(π/6)?
- A. 0
- B. 1
- C. 1/2
- D. √3/2
Show correct answer and explanation
Correct answer: C. 1/2
The standard sine value at 30 degrees is 1/2.
Q20. What is cos²θ+sin²θ for every real θ?
- A. 0
- B. 1
- C. 2
- D. cos 2θ
Show correct answer and explanation
Correct answer: B. 1
The Pythagorean trigonometric identity equals 1.
Additional Practice MCQs — Questions 21–30
Further foundational and intermediate practice, with four answer options and explanations.
Complex Numbers
Q21. What is the conjugate of 2−5i?
- A. 2+5i
- B. 2−5i
- C. −2+5i
- D. −2−5i
Show correct answer and explanation
Correct answer: A. 2+5i
The conjugate reverses the sign of the imaginary part.
Q22. What is (1+i)(1−i)?
- A. 0
- B. 1
- C. 2
- D. 2i
Show correct answer and explanation
Correct answer: C. 2
By difference of squares, 1−i²=2.
Functions
Q23. What is the domain of f(x)=1/(x−3)?
- A. All real numbers
- B. All real numbers except 3
- C. Only x>3
- D. Only x≥3
Show correct answer and explanation
Correct answer: B. All real numbers except 3
The denominator must not equal zero.
Sets
Q24. For A={1,2} and B={2,3}, find A∪B.
- A. {2}
- B. {1,3}
- C. {1,2,2,3}
- D. {1,2,3}
Show correct answer and explanation
Correct answer: D. {1,2,3}
The union contains distinct elements from either set.
Matrices
Q25. For [[1,2],[0,1]] × [[2,0],[3,4]], what is the top-left entry?
- A. 2
- B. 5
- C. 8
- D. 10
Show correct answer and explanation
Correct answer: C. 8
The first row dot first column is 1×2+2×3=8.
Quadratics
Q26. Find the discriminant of 2x²−4x+3=0.
- A. −8
- B. 8
- C. 16
- D. −4
Show correct answer and explanation
Correct answer: A. −8
b²−4ac=16−24=−8.
Polynomials
Q27. If x+2 divides p(x), which statement holds?
- A. p(2)=0
- B. p(−2)=0
- C. p(0)=2
- D. p'(2)=0
Show correct answer and explanation
Correct answer: B. p(−2)=0
The factor theorem requires p(−2)=0.
Binomial Theorem
Q28. Find the coefficient of x³ in (1+x)⁵.
- A. 5
- B. 6
- C. 10
- D. 20
Show correct answer and explanation
Correct answer: C. 10
C(5,3)=10.
Trigonometry
Q29. What is tan(π/4)?
- A. 0
- B. 1/2
- C. √3
- D. 1
Show correct answer and explanation
Correct answer: D. 1
The tangent at 45° is 1.
Induction
Q30. To prove P(n) for all n≥1 by induction, what base case is needed?
- A. P(1)
- B. P(2) only
- C. P(100)
- D. P(n+2)
Show correct answer and explanation
Correct answer: A. P(1)
The base statement P(1) starts induction.
Additional Step-by-Step Solved Example
Complex-number division
Problem: Simplify (3+2i)/(1−i).
- Multiply numerator and denominator by 1+i.
- Expand numerator: (3+2i)(1+i)=1+5i.
- Evaluate denominator: (1−i)(1+i)=2.
Final answer: The result is (1+5i)/2.
Additional Practice MCQs — Questions 31–150
Q31. Evaluate f(1) for f(x)=2x+3.
- A. 4
- B. 8
- C. 5
- D. 6
Show correct answer and explanation
Correct answer: C. 5
Substitute x=1: 2×1+3=5.
Q32. Evaluate f(2) for f(x)=3x+4.
- A. 13
- B. 10
- C. 11
- D. 9
Show correct answer and explanation
Correct answer: B. 10
Substitute x=2: 3×2+4=10.
Q33. Evaluate f(3) for f(x)=4x+5.
- A. 17
- B. 18
- C. 16
- D. 20
Show correct answer and explanation
Correct answer: A. 17
Substitute x=3: 4×3+5=17.
Q34. Evaluate f(4) for f(x)=5x+6.
- A. 27
- B. 25
- C. 29
- D. 26
Show correct answer and explanation
Correct answer: D. 26
Substitute x=4: 5×4+6=26.
Q35. Evaluate f(5) for f(x)=6x+7.
- A. 36
- B. 40
- C. 37
- D. 38
Show correct answer and explanation
Correct answer: C. 37
Substitute x=5: 6×5+7=37.
Q36. Evaluate f(6) for f(x)=7x+8.
- A. 53
- B. 50
- C. 51
- D. 49
Show correct answer and explanation
Correct answer: B. 50
Substitute x=6: 7×6+8=50.
Q37. Evaluate f(7) for f(x)=8x+9.
- A. 65
- B. 66
- C. 64
- D. 68
Show correct answer and explanation
Correct answer: A. 65
Substitute x=7: 8×7+9=65.
Q38. Evaluate f(8) for f(x)=9x+10.
- A. 83
- B. 81
- C. 85
- D. 82
Show correct answer and explanation
Correct answer: D. 82
Substitute x=8: 9×8+10=82.
Q39. Evaluate f(9) for f(x)=10x+11.
- A. 100
- B. 104
- C. 101
- D. 102
Show correct answer and explanation
Correct answer: C. 101
Substitute x=9: 10×9+11=101.
Q40. Evaluate f(10) for f(x)=11x+12.
- A. 125
- B. 122
- C. 123
- D. 121
Show correct answer and explanation
Correct answer: B. 122
Substitute x=10: 11×10+12=122.
Q41. Find the sum of the roots of (x−1)(x−4)=0.
- A. 5
- B. 6
- C. 4
- D. 8
Show correct answer and explanation
Correct answer: A. 5
The roots are 1 and 4; their sum is 5.
Q42. Find the sum of the roots of (x−2)(x−5)=0.
- A. 8
- B. 6
- C. 10
- D. 7
Show correct answer and explanation
Correct answer: D. 7
The roots are 2 and 5; their sum is 7.
Q43. Find the sum of the roots of (x−3)(x−6)=0.
- A. 8
- B. 12
- C. 9
- D. 10
Show correct answer and explanation
Correct answer: C. 9
The roots are 3 and 6; their sum is 9.
Q44. Find the sum of the roots of (x−4)(x−7)=0.
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
The roots are 4 and 7; their sum is 11.
Q45. Find the sum of the roots of (x−5)(x−8)=0.
- A. 13
- B. 14
- C. 12
- D. 16
Show correct answer and explanation
Correct answer: A. 13
The roots are 5 and 8; their sum is 13.
Q46. Find the sum of the roots of (x−6)(x−9)=0.
- A. 16
- B. 14
- C. 18
- D. 15
Show correct answer and explanation
Correct answer: D. 15
The roots are 6 and 9; their sum is 15.
Q47. Find the sum of the roots of (x−7)(x−10)=0.
- A. 16
- B. 20
- C. 17
- D. 18
Show correct answer and explanation
Correct answer: C. 17
The roots are 7 and 10; their sum is 17.
Q48. Find the sum of the roots of (x−8)(x−11)=0.
- A. 22
- B. 19
- C. 20
- D. 18
Show correct answer and explanation
Correct answer: B. 19
The roots are 8 and 11; their sum is 19.
Q49. Find the sum of the roots of (x−9)(x−12)=0.
- A. 21
- B. 22
- C. 20
- D. 24
Show correct answer and explanation
Correct answer: A. 21
The roots are 9 and 12; their sum is 21.
Q50. Find the sum of the roots of (x−10)(x−13)=0.
- A. 24
- B. 22
- C. 26
- D. 23
Show correct answer and explanation
Correct answer: D. 23
The roots are 10 and 13; their sum is 23.
Q51. Compute det([[2,1],[2,5]]).
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
Use ad−bc: 2×5−1×2=8.
Q52. Compute det([[3,2],[2,6]]).
- A. 17
- B. 14
- C. 15
- D. 13
Show correct answer and explanation
Correct answer: B. 14
Use ad−bc: 3×6−2×2=14.
Q53. Compute det([[4,3],[2,7]]).
- A. 22
- B. 23
- C. 21
- D. 25
Show correct answer and explanation
Correct answer: A. 22
Use ad−bc: 4×7−3×2=22.
Q54. Compute det([[5,4],[2,8]]).
- A. 33
- B. 31
- C. 35
- D. 32
Show correct answer and explanation
Correct answer: D. 32
Use ad−bc: 5×8−4×2=32.
Q55. Compute det([[6,5],[2,9]]).
- A. 43
- B. 47
- C. 44
- D. 45
Show correct answer and explanation
Correct answer: C. 44
Use ad−bc: 6×9−5×2=44.
Q56. Compute det([[7,6],[2,10]]).
- A. 61
- B. 58
- C. 59
- D. 57
Show correct answer and explanation
Correct answer: B. 58
Use ad−bc: 7×10−6×2=58.
Q57. Compute det([[8,7],[2,11]]).
- A. 74
- B. 75
- C. 73
- D. 77
Show correct answer and explanation
Correct answer: A. 74
Use ad−bc: 8×11−7×2=74.
Q58. Compute det([[9,8],[2,12]]).
- A. 93
- B. 91
- C. 95
- D. 92
Show correct answer and explanation
Correct answer: D. 92
Use ad−bc: 9×12−8×2=92.
Q59. Compute det([[10,9],[2,13]]).
- A. 111
- B. 115
- C. 112
- D. 113
Show correct answer and explanation
Correct answer: C. 112
Use ad−bc: 10×13−9×2=112.
Q60. Compute det([[11,10],[2,14]]).
- A. 137
- B. 134
- C. 135
- D. 133
Show correct answer and explanation
Correct answer: B. 134
Use ad−bc: 11×14−10×2=134.
Q61. How many subsets are there in a set containing 3 distinct elements?
- A. 8
- B. 9
- C. 7
- D. 11
Show correct answer and explanation
Correct answer: A. 8
A set of n elements has 2^n subsets, giving 2^3=8.
Q62. How many subsets are there in a set containing 4 distinct elements?
- A. 17
- B. 15
- C. 19
- D. 16
Show correct answer and explanation
Correct answer: D. 16
A set of n elements has 2^n subsets, giving 2^4=16.
Q63. How many subsets are there in a set containing 5 distinct elements?
- A. 31
- B. 35
- C. 32
- D. 33
Show correct answer and explanation
Correct answer: C. 32
A set of n elements has 2^n subsets, giving 2^5=32.
Q64. How many subsets are there in a set containing 6 distinct elements?
- A. 67
- B. 64
- C. 65
- D. 63
Show correct answer and explanation
Correct answer: B. 64
A set of n elements has 2^n subsets, giving 2^6=64.
Q65. How many subsets are there in a set containing 7 distinct elements?
- A. 128
- B. 129
- C. 127
- D. 131
Show correct answer and explanation
Correct answer: A. 128
A set of n elements has 2^n subsets, giving 2^7=128.
Q66. How many subsets are there in a set containing 8 distinct elements?
- A. 257
- B. 255
- C. 259
- D. 256
Show correct answer and explanation
Correct answer: D. 256
A set of n elements has 2^n subsets, giving 2^8=256.
Q67. How many subsets are there in a set containing 9 distinct elements?
- A. 511
- B. 515
- C. 512
- D. 513
Show correct answer and explanation
Correct answer: C. 512
A set of n elements has 2^n subsets, giving 2^9=512.
Q68. How many subsets are there in a set containing 10 distinct elements?
- A. 1027
- B. 1024
- C. 1025
- D. 1023
Show correct answer and explanation
Correct answer: B. 1024
A set of n elements has 2^n subsets, giving 2^10=1024.
Q69. How many subsets are there in a set containing 11 distinct elements?
- A. 2048
- B. 2049
- C. 2047
- D. 2051
Show correct answer and explanation
Correct answer: A. 2048
A set of n elements has 2^n subsets, giving 2^11=2048.
Q70. How many subsets are there in a set containing 12 distinct elements?
- A. 4097
- B. 4095
- C. 4099
- D. 4096
Show correct answer and explanation
Correct answer: D. 4096
A set of n elements has 2^n subsets, giving 2^12=4096.
Q71. What is the coefficient of x² in (1+x)^4?
- A. 5
- B. 9
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: C. 6
The binomial coefficient C(4,2)=6.
Q72. What is the coefficient of x² in (1+x)^5?
- A. 13
- B. 10
- C. 11
- D. 9
Show correct answer and explanation
Correct answer: B. 10
The binomial coefficient C(5,2)=10.
Q73. What is the coefficient of x² in (1+x)^6?
- A. 15
- B. 16
- C. 14
- D. 18
Show correct answer and explanation
Correct answer: A. 15
The binomial coefficient C(6,2)=15.
Q74. What is the coefficient of x² in (1+x)^7?
- A. 22
- B. 20
- C. 24
- D. 21
Show correct answer and explanation
Correct answer: D. 21
The binomial coefficient C(7,2)=21.
Q75. What is the coefficient of x² in (1+x)^8?
- A. 27
- B. 31
- C. 28
- D. 29
Show correct answer and explanation
Correct answer: C. 28
The binomial coefficient C(8,2)=28.
Q76. What is the coefficient of x² in (1+x)^9?
- A. 39
- B. 36
- C. 37
- D. 35
Show correct answer and explanation
Correct answer: B. 36
The binomial coefficient C(9,2)=36.
Q77. What is the coefficient of x² in (1+x)^10?
- A. 45
- B. 46
- C. 44
- D. 48
Show correct answer and explanation
Correct answer: A. 45
The binomial coefficient C(10,2)=45.
Q78. What is the coefficient of x² in (1+x)^11?
- A. 56
- B. 54
- C. 58
- D. 55
Show correct answer and explanation
Correct answer: D. 55
The binomial coefficient C(11,2)=55.
Q79. What is the coefficient of x² in (1+x)^12?
- A. 65
- B. 69
- C. 66
- D. 67
Show correct answer and explanation
Correct answer: C. 66
The binomial coefficient C(12,2)=66.
Q80. What is the coefficient of x² in (1+x)^13?
- A. 81
- B. 78
- C. 79
- D. 77
Show correct answer and explanation
Correct answer: B. 78
The binomial coefficient C(13,2)=78.
Q81. An arithmetic sequence has first term 2 and common difference 3. Find term 5.
- A. 14
- B. 15
- C. 13
- D. 17
Show correct answer and explanation
Correct answer: A. 14
Use a_n=a_1+(n−1)d=2+4×3=14.
Q82. An arithmetic sequence has first term 3 and common difference 4. Find term 6.
- A. 24
- B. 22
- C. 26
- D. 23
Show correct answer and explanation
Correct answer: D. 23
Use a_n=a_1+(n−1)d=3+5×4=23.
Q83. An arithmetic sequence has first term 4 and common difference 5. Find term 7.
- A. 33
- B. 37
- C. 34
- D. 35
Show correct answer and explanation
Correct answer: C. 34
Use a_n=a_1+(n−1)d=4+6×5=34.
Q84. An arithmetic sequence has first term 5 and common difference 6. Find term 8.
- A. 50
- B. 47
- C. 48
- D. 46
Show correct answer and explanation
Correct answer: B. 47
Use a_n=a_1+(n−1)d=5+7×6=47.
Q85. An arithmetic sequence has first term 6 and common difference 7. Find term 9.
- A. 62
- B. 63
- C. 61
- D. 65
Show correct answer and explanation
Correct answer: A. 62
Use a_n=a_1+(n−1)d=6+8×7=62.
Q86. An arithmetic sequence has first term 7 and common difference 8. Find term 10.
- A. 80
- B. 78
- C. 82
- D. 79
Show correct answer and explanation
Correct answer: D. 79
Use a_n=a_1+(n−1)d=7+9×8=79.
Q87. An arithmetic sequence has first term 8 and common difference 9. Find term 11.
- A. 97
- B. 101
- C. 98
- D. 99
Show correct answer and explanation
Correct answer: C. 98
Use a_n=a_1+(n−1)d=8+10×9=98.
Q88. An arithmetic sequence has first term 9 and common difference 10. Find term 12.
- A. 122
- B. 119
- C. 120
- D. 118
Show correct answer and explanation
Correct answer: B. 119
Use a_n=a_1+(n−1)d=9+11×10=119.
Q89. An arithmetic sequence has first term 10 and common difference 11. Find term 13.
- A. 142
- B. 143
- C. 141
- D. 145
Show correct answer and explanation
Correct answer: A. 142
Use a_n=a_1+(n−1)d=10+12×11=142.
Q90. An arithmetic sequence has first term 11 and common difference 12. Find term 14.
- A. 168
- B. 166
- C. 170
- D. 167
Show correct answer and explanation
Correct answer: D. 167
Use a_n=a_1+(n−1)d=11+13×12=167.
Q91. A geometric sequence starts at 2 with common ratio 2. Find term 3.
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
Use a_n=a_1 r^(n−1)=2×2^2=8.
Q92. A geometric sequence starts at 3 with common ratio 2. Find term 4.
- A. 27
- B. 24
- C. 25
- D. 23
Show correct answer and explanation
Correct answer: B. 24
Use a_n=a_1 r^(n−1)=3×2^3=24.
Q93. A geometric sequence starts at 4 with common ratio 2. Find term 5.
- A. 64
- B. 65
- C. 63
- D. 67
Show correct answer and explanation
Correct answer: A. 64
Use a_n=a_1 r^(n−1)=4×2^4=64.
Q94. A geometric sequence starts at 5 with common ratio 2. Find term 6.
- A. 161
- B. 159
- C. 163
- D. 160
Show correct answer and explanation
Correct answer: D. 160
Use a_n=a_1 r^(n−1)=5×2^5=160.
Q95. A geometric sequence starts at 6 with common ratio 2. Find term 7.
- A. 383
- B. 387
- C. 384
- D. 385
Show correct answer and explanation
Correct answer: C. 384
Use a_n=a_1 r^(n−1)=6×2^6=384.
Q96. A geometric sequence starts at 7 with common ratio 2. Find term 8.
- A. 899
- B. 896
- C. 897
- D. 895
Show correct answer and explanation
Correct answer: B. 896
Use a_n=a_1 r^(n−1)=7×2^7=896.
Q97. A geometric sequence starts at 8 with common ratio 2. Find term 9.
- A. 2048
- B. 2049
- C. 2047
- D. 2051
Show correct answer and explanation
Correct answer: A. 2048
Use a_n=a_1 r^(n−1)=8×2^8=2048.
Q98. A geometric sequence starts at 9 with common ratio 2. Find term 10.
- A. 4609
- B. 4607
- C. 4611
- D. 4608
Show correct answer and explanation
Correct answer: D. 4608
Use a_n=a_1 r^(n−1)=9×2^9=4608.
Q99. A geometric sequence starts at 10 with common ratio 2. Find term 11.
- A. 10239
- B. 10243
- C. 10240
- D. 10241
Show correct answer and explanation
Correct answer: C. 10240
Use a_n=a_1 r^(n−1)=10×2^10=10240.
Q100. A geometric sequence starts at 11 with common ratio 2. Find term 12.
- A. 22531
- B. 22528
- C. 22529
- D. 22527
Show correct answer and explanation
Correct answer: B. 22528
Use a_n=a_1 r^(n−1)=11×2^11=22528.
Q101. For finite sets with |A|=8, |B|=6 and |A∩B|=2, find |A∪B|.
- A. 12
- B. 13
- C. 11
- D. 15
Show correct answer and explanation
Correct answer: A. 12
Inclusion-exclusion gives 8+6−2=12.
Q102. For finite sets with |A|=9, |B|=7 and |A∩B|=3, find |A∪B|.
- A. 14
- B. 12
- C. 16
- D. 13
Show correct answer and explanation
Correct answer: D. 13
Inclusion-exclusion gives 9+7−3=13.
Q103. For finite sets with |A|=10, |B|=8 and |A∩B|=4, find |A∪B|.
- A. 13
- B. 17
- C. 14
- D. 15
Show correct answer and explanation
Correct answer: C. 14
Inclusion-exclusion gives 10+8−4=14.
Q104. For finite sets with |A|=11, |B|=9 and |A∩B|=5, find |A∪B|.
- A. 18
- B. 15
- C. 16
- D. 14
Show correct answer and explanation
Correct answer: B. 15
Inclusion-exclusion gives 11+9−5=15.
Q105. For finite sets with |A|=12, |B|=10 and |A∩B|=6, find |A∪B|.
- A. 16
- B. 17
- C. 15
- D. 19
Show correct answer and explanation
Correct answer: A. 16
Inclusion-exclusion gives 12+10−6=16.
Q106. For finite sets with |A|=13, |B|=11 and |A∩B|=7, find |A∪B|.
- A. 18
- B. 16
- C. 20
- D. 17
Show correct answer and explanation
Correct answer: D. 17
Inclusion-exclusion gives 13+11−7=17.
Q107. For finite sets with |A|=14, |B|=12 and |A∩B|=8, find |A∪B|.
- A. 17
- B. 21
- C. 18
- D. 19
Show correct answer and explanation
Correct answer: C. 18
Inclusion-exclusion gives 14+12−8=18.
Q108. For finite sets with |A|=15, |B|=13 and |A∩B|=9, find |A∪B|.
- A. 22
- B. 19
- C. 20
- D. 18
Show correct answer and explanation
Correct answer: B. 19
Inclusion-exclusion gives 15+13−9=19.
Q109. For finite sets with |A|=16, |B|=14 and |A∩B|=10, find |A∪B|.
- A. 20
- B. 21
- C. 19
- D. 23
Show correct answer and explanation
Correct answer: A. 20
Inclusion-exclusion gives 16+14−10=20.
Q110. For finite sets with |A|=17, |B|=15 and |A∩B|=11, find |A∪B|.
- A. 22
- B. 20
- C. 24
- D. 21
Show correct answer and explanation
Correct answer: D. 21
Inclusion-exclusion gives 17+15−11=21.
Q111. Which formula gives the inverse of f(x)=2x+1?
- A. 2x−1
- B. (x−2)/1
- C. (x−1)/2
- D. (x+1)/2
Show correct answer and explanation
Correct answer: C. (x−1)/2
Solve y=2x+1 for x: x=(y−1)/2.
Q112. Which formula gives the inverse of f(x)=3x+2?
- A. (x−3)/2
- B. (x−2)/3
- C. (x+2)/3
- D. 3x−2
Show correct answer and explanation
Correct answer: B. (x−2)/3
Solve y=3x+2 for x: x=(y−2)/3.
Q113. Which formula gives the inverse of f(x)=4x+3?
- A. (x−3)/4
- B. (x+3)/4
- C. 4x−3
- D. (x−4)/3
Show correct answer and explanation
Correct answer: A. (x−3)/4
Solve y=4x+3 for x: x=(y−3)/4.
Q114. Which formula gives the inverse of f(x)=5x+4?
- A. (x+4)/5
- B. 5x−4
- C. (x−5)/4
- D. (x−4)/5
Show correct answer and explanation
Correct answer: D. (x−4)/5
Solve y=5x+4 for x: x=(y−4)/5.
Q115. Which formula gives the inverse of f(x)=6x+5?
- A. 6x−5
- B. (x−6)/5
- C. (x−5)/6
- D. (x+5)/6
Show correct answer and explanation
Correct answer: C. (x−5)/6
Solve y=6x+5 for x: x=(y−5)/6.
Q116. Which formula gives the inverse of f(x)=7x+6?
- A. (x−7)/6
- B. (x−6)/7
- C. (x+6)/7
- D. 7x−6
Show correct answer and explanation
Correct answer: B. (x−6)/7
Solve y=7x+6 for x: x=(y−6)/7.
Q117. Which formula gives the inverse of f(x)=8x+7?
- A. (x−7)/8
- B. (x+7)/8
- C. 8x−7
- D. (x−8)/7
Show correct answer and explanation
Correct answer: A. (x−7)/8
Solve y=8x+7 for x: x=(y−7)/8.
Q118. Which formula gives the inverse of f(x)=9x+8?
- A. (x+8)/9
- B. 9x−8
- C. (x−9)/8
- D. (x−8)/9
Show correct answer and explanation
Correct answer: D. (x−8)/9
Solve y=9x+8 for x: x=(y−8)/9.
Q119. Which formula gives the inverse of f(x)=10x+9?
- A. 10x−9
- B. (x−10)/9
- C. (x−9)/10
- D. (x+9)/10
Show correct answer and explanation
Correct answer: C. (x−9)/10
Solve y=10x+9 for x: x=(y−9)/10.
Q120. Which formula gives the inverse of f(x)=11x+10?
- A. (x−11)/10
- B. (x−10)/11
- C. (x+10)/11
- D. 11x−10
Show correct answer and explanation
Correct answer: B. (x−10)/11
Solve y=11x+10 for x: x=(y−10)/11.
Q121. Evaluate (3+3)²−4×9.
- A. 0
- B. 1
- C. -1
- D. 3
Show correct answer and explanation
Correct answer: A. 0
The expression is (2×3)²−4×3²=0.
Q122. Evaluate (4+4)²−4×16.
- A. 1
- B. -1
- C. 3
- D. 0
Show correct answer and explanation
Correct answer: D. 0
The expression is (2×4)²−4×4²=0.
Q123. Evaluate (5+5)²−4×25.
- A. -1
- B. 3
- C. 0
- D. 1
Show correct answer and explanation
Correct answer: C. 0
The expression is (2×5)²−4×5²=0.
Q124. Evaluate (6+6)²−4×36.
- A. 3
- B. 0
- C. 1
- D. -1
Show correct answer and explanation
Correct answer: B. 0
The expression is (2×6)²−4×6²=0.
Q125. Evaluate (7+7)²−4×49.
- A. 0
- B. 1
- C. -1
- D. 3
Show correct answer and explanation
Correct answer: A. 0
The expression is (2×7)²−4×7²=0.
Q126. Evaluate (8+8)²−4×64.
- A. 1
- B. -1
- C. 3
- D. 0
Show correct answer and explanation
Correct answer: D. 0
The expression is (2×8)²−4×8²=0.
Q127. Evaluate (9+9)²−4×81.
- A. -1
- B. 3
- C. 0
- D. 1
Show correct answer and explanation
Correct answer: C. 0
The expression is (2×9)²−4×9²=0.
Q128. Evaluate (10+10)²−4×100.
- A. 3
- B. 0
- C. 1
- D. -1
Show correct answer and explanation
Correct answer: B. 0
The expression is (2×10)²−4×10²=0.
Q129. Evaluate (11+11)²−4×121.
- A. 0
- B. 1
- C. -1
- D. 3
Show correct answer and explanation
Correct answer: A. 0
The expression is (2×11)²−4×11²=0.
Q130. Evaluate (12+12)²−4×144.
- A. 1
- B. -1
- C. 3
- D. 0
Show correct answer and explanation
Correct answer: D. 0
The expression is (2×12)²−4×12²=0.
Q131. Find the modulus of the complex number 6+8i.
- A. 9
- B. 13
- C. 10
- D. 11
Show correct answer and explanation
Correct answer: C. 10
The modulus is sqrt((6)²+(8)²)=10.
Q132. Find the modulus of the complex number 9+12i.
- A. 18
- B. 15
- C. 16
- D. 14
Show correct answer and explanation
Correct answer: B. 15
The modulus is sqrt((9)²+(12)²)=15.
Q133. Find the modulus of the complex number 12+16i.
- A. 20
- B. 21
- C. 19
- D. 23
Show correct answer and explanation
Correct answer: A. 20
The modulus is sqrt((12)²+(16)²)=20.
Q134. Find the modulus of the complex number 15+20i.
- A. 26
- B. 24
- C. 28
- D. 25
Show correct answer and explanation
Correct answer: D. 25
The modulus is sqrt((15)²+(20)²)=25.
Q135. Find the modulus of the complex number 18+24i.
- A. 29
- B. 33
- C. 30
- D. 31
Show correct answer and explanation
Correct answer: C. 30
The modulus is sqrt((18)²+(24)²)=30.
Q136. Find the modulus of the complex number 21+28i.
- A. 38
- B. 35
- C. 36
- D. 34
Show correct answer and explanation
Correct answer: B. 35
The modulus is sqrt((21)²+(28)²)=35.
Q137. Find the modulus of the complex number 24+32i.
- A. 40
- B. 41
- C. 39
- D. 43
Show correct answer and explanation
Correct answer: A. 40
The modulus is sqrt((24)²+(32)²)=40.
Q138. Find the modulus of the complex number 27+36i.
- A. 46
- B. 44
- C. 48
- D. 45
Show correct answer and explanation
Correct answer: D. 45
The modulus is sqrt((27)²+(36)²)=45.
Q139. Find the modulus of the complex number 30+40i.
- A. 49
- B. 53
- C. 50
- D. 51
Show correct answer and explanation
Correct answer: C. 50
The modulus is sqrt((30)²+(40)²)=50.
Q140. Find the modulus of the complex number 33+44i.
- A. 58
- B. 55
- C. 56
- D. 54
Show correct answer and explanation
Correct answer: B. 55
The modulus is sqrt((33)²+(44)²)=55.
Q141. How many ordered pairs are in A×B if |A|=3 and |B|=5?
- A. 15
- B. 16
- C. 14
- D. 18
Show correct answer and explanation
Correct answer: A. 15
The Cartesian product has 3×5=15 pairs.
Q142. How many ordered pairs are in A×B if |A|=4 and |B|=6?
- A. 25
- B. 23
- C. 27
- D. 24
Show correct answer and explanation
Correct answer: D. 24
The Cartesian product has 4×6=24 pairs.
Q143. How many ordered pairs are in A×B if |A|=5 and |B|=7?
- A. 34
- B. 38
- C. 35
- D. 36
Show correct answer and explanation
Correct answer: C. 35
The Cartesian product has 5×7=35 pairs.
Q144. How many ordered pairs are in A×B if |A|=6 and |B|=8?
- A. 51
- B. 48
- C. 49
- D. 47
Show correct answer and explanation
Correct answer: B. 48
The Cartesian product has 6×8=48 pairs.
Q145. How many ordered pairs are in A×B if |A|=7 and |B|=9?
- A. 63
- B. 64
- C. 62
- D. 66
Show correct answer and explanation
Correct answer: A. 63
The Cartesian product has 7×9=63 pairs.
Q146. How many ordered pairs are in A×B if |A|=8 and |B|=10?
- A. 81
- B. 79
- C. 83
- D. 80
Show correct answer and explanation
Correct answer: D. 80
The Cartesian product has 8×10=80 pairs.
Q147. How many ordered pairs are in A×B if |A|=9 and |B|=11?
- A. 98
- B. 102
- C. 99
- D. 100
Show correct answer and explanation
Correct answer: C. 99
The Cartesian product has 9×11=99 pairs.
Q148. How many ordered pairs are in A×B if |A|=10 and |B|=12?
- A. 123
- B. 120
- C. 121
- D. 119
Show correct answer and explanation
Correct answer: B. 120
The Cartesian product has 10×12=120 pairs.
Q149. How many ordered pairs are in A×B if |A|=11 and |B|=13?
- A. 143
- B. 144
- C. 142
- D. 146
Show correct answer and explanation
Correct answer: A. 143
The Cartesian product has 11×13=143 pairs.
Q150. How many ordered pairs are in A×B if |A|=12 and |B|=14?
- A. 169
- B. 167
- C. 171
- D. 168
Show correct answer and explanation
Correct answer: D. 168
The Cartesian product has 12×14=168 pairs.
Interactive MCQ Practice
Prefer to practise one question at a time? Open the interactive question sets, reveal each answer with its explanation, and continue to the next question.
Start Pre-Calculus I Interactive Practice — 150 MCQs