1
A population of 2000 grows by 10% each year. What will the population be after 2 years?
Medium
ExplanationPopulation after 2 years = 2000 × (1 + 10/100)^2 = 2420.
2
If f(x) = 12x − 16, for what value of x does f(x) = 128?
Medium
Explanation12x − 16 = 128, so x = 12.
3
If f(x) = x − 2, what is f(-1)?
Medium
Explanationf(-1) = (1)(-1) − 2 = -3.
4
An arithmetic sequence has first term 13 and common difference 7. What is the 31st term?
Medium
Explanationaₙ = a₁ + (n−1)d = 13 + (31−1)(7) = 223.
5
If f(x) = -6x + 13 and g(x) = 4x − 8, what is g(f(10))?
Medium
Explanationf(10) = -47, so g(f(10)) = g(-47) = 4(-47) − 8 = -196.
6
A geometric sequence has first term 9 and common ratio -2. What is the 5th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = 9 × (-2)^4 = 144.
7
A geometric sequence has first term -7 and common ratio 2. What is the 8th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = -7 × (2)^7 = -896.
8
In an arithmetic sequence, the 12th term is -85 and the 18th term is -145. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-145 − (-85))/(18−12) = -10.
9
In an arithmetic sequence, the 10th term is -29 and the 28th term is -173. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-173 − (-29))/(28−10) = -8.
10
If f(x) = 6x − 9 and g(x) = f(x − 3) + 5, what is g(7)?
Medium
Explanationg(7) = f(4) + 5 = 15 + 5 = 20.
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11
If f(x) = -6x + 16, what is f(13) − f(-8)?
Medium
Explanationf(13) = -62 and f(-8) = 64, so f(13) − f(-8) = -126.
12
An arithmetic sequence has first term 7 and common difference -4. What is the 33rd term?
Medium
Explanationaₙ = a₁ + (n−1)d = 7 + (33−1)(-4) = -121.
13
For the function f(x) = 1/(x − (-18)), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − (-18) cannot equal 0, so x = -18 must be excluded from the domain.
14
An arithmetic sequence has first term -47 and common difference 13. What is the 19th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -47 + (19−1)(13) = 187.
15
An employee's salary starts at Rs. 29816 and increases by Rs. 873 each year. What will the salary be after 11 years?
Medium
ExplanationSalary after 11 years = 29816 + 11 × 873 = Rs. 39419.
16
If f(x) = 5x + 19, what is f⁻¹(-26)?
Medium
Explanationf⁻¹(-26) is the x-value such that f(x) = -26: 5x + 19 = -26, so x = -9.
17
What is the sum of the first 23 terms of an arithmetic sequence with first term -26 and common difference 7?
Medium
ExplanationThe 23rd term is 128. Sum = n(a₁ + aₙ)/2 = 23(-26 + 128)/2 = 1173.
18
A sequence is defined by a₁ = 10 and aₙ = aₙ₋₁ − 8 for n > 1. What is a11?
Medium
ExplanationEach term adds -8 to the previous one, so a11 = a₁ + (11−1)(-8) = -70.
19
What is the sum of the first 13 terms of an arithmetic sequence with first term -8 and common difference -8?
Medium
ExplanationThe 13th term is -104. Sum = n(a₁ + aₙ)/2 = 13(-8 + (-104))/2 = -728.
20
If f(x) = -x − 11 and g(x) = f(x − 1) + 3, what is g(2)?
Medium
Explanationg(2) = f(1) + 3 = -12 + 3 = -9.
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21
A geometric sequence has first term 6 and common ratio 4. What is the 3rd term?
Medium
Explanationaₙ = a₁ × r^(n−1) = 6 × (4)^2 = 96.
22
For the function f(x) = √(x − 51), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − 51 ≥ 0, so x ≥ 51. The smallest such value is 51.
23
f(x) = 8x − 1 if x < -3, and f(x) = 6x − 6 if x ≥ -3. What is f(0)?
Medium
ExplanationSince 0 ≥ -3, use f(x) = 6x − 6: f(0) = -6.
24
If f(x) = 8x + 4, what is f(-10) + f(-14)?
Medium
Explanationf(-10) = -76 and f(-14) = -108, so f(-10) + f(-14) = -184.
25
For the function f(x) = 1/(x − (-20)), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − (-20) cannot equal 0, so x = -20 must be excluded from the domain.
26
A sequence is defined by a₁ = -8 and aₙ = aₙ₋₁ − 9 for n > 1. What is a5?
Medium
ExplanationEach term adds -9 to the previous one, so a5 = a₁ + (5−1)(-9) = -44.
27
If f(x) = 6x − 8, for what value of x does f(x) = -92?
Medium
Explanation6x − 8 = -92, so x = -14.
28
What is the sum of the first 13 terms of an arithmetic sequence with first term -19 and common difference 2?
Medium
ExplanationThe 13th term is 5. Sum = n(a₁ + aₙ)/2 = 13(-19 + 5)/2 = -91.
29
f(x) = -5x + 6 if x < 3, and f(x) = 5x − 15 if x ≥ 3. What is f(5)?
Medium
ExplanationSince 5 ≥ 3, use f(x) = 5x − 15: f(5) = 10.
30
What is the sum to infinity of a geometric sequence with first term 17 and common ratio -4/7?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 17/(1 − (-4)/7) = 119/11.
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31
In an arithmetic sequence, the 9th term is -31 and the 13th term is -51. What is the 23rd term?
Medium
Explanationd = (-51 − (-31))/(13−9) = -5. Then the 23rd term = -31 + (23−9)(-5) = -101.
32
In an arithmetic sequence, the 5th term is 33 and the 12th term is 61. What is the 11th term?
Medium
Explanationd = (61 − 33)/(12−5) = 4. Then the 11th term = 33 + (11−5)(4) = 57.
33
A population of 35000 grows by 10% each year. What will the population be after 3 years?
Medium
ExplanationPopulation after 3 years = 35000 × (1 + 10/100)^3 = 46585.
34
A geometric sequence has first term -3 and common ratio -2. What is the 7th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = -3 × (-2)^6 = -192.
35
What is the sum to infinity of a geometric sequence with first term -14 and common ratio -4/5?
Medium
ExplanationSum to infinity = a₁/(1 − r) = -14/(1 − (-4)/5) = -70/9.
36
If f(x) = -x + 20, what is f(13) + f(-12)?
Medium
Explanationf(13) = 7 and f(-12) = 32, so f(13) + f(-12) = 39.
37
If f(x) = -6x − 4 and g(x) = f(x − 11) − 4, what is g(9)?
Medium
Explanationg(9) = f(-2) − 4 = 8 − 4 = 4.
38
For the function f(x) = √(x − (-13)), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − (-13) ≥ 0, so x ≥ -13. The smallest such value is -13.
39
For the function f(x) = 1/(x − 23), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − 23 cannot equal 0, so x = 23 must be excluded from the domain.
40
A sequence is defined by a₁ = -13 and aₙ = aₙ₋₁ + 3 for n > 1. What is a13?
Medium
ExplanationEach term adds 3 to the previous one, so a13 = a₁ + (13−1)(3) = 23.
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41
If f(x) = -12x + 18, what is f(2) + f(3)?
Medium
Explanationf(2) = -6 and f(3) = -18, so f(2) + f(3) = -24.
42
For the function f(x) = √(x − (-72)), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − (-72) ≥ 0, so x ≥ -72. The smallest such value is -72.
43
f(x) = -8x + 8 if x < 4, and f(x) = 4x − 5 if x ≥ 4. What is f(3)?
Medium
ExplanationSince 3 < 4, use f(x) = -8x + 8: f(3) = -16.
44
A geometric sequence has first term -9 and common ratio -2. What is the 4th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = -9 × (-2)^3 = 72.
45
A sequence is defined by a₁ = -18 and aₙ = aₙ₋₁ − 4 for n > 1. What is a4?
Medium
ExplanationEach term adds -4 to the previous one, so a4 = a₁ + (4−1)(-4) = -30.
46
A sequence is defined by a₁ = -5 and aₙ = aₙ₋₁ − 3 for n > 1. What is a10?
Medium
ExplanationEach term adds -3 to the previous one, so a10 = a₁ + (10−1)(-3) = -32.
47
If f(x) = 2x − 10 and g(x) = f(x − 10) + 2, what is g(11)?
Medium
Explanationg(11) = f(1) + 2 = -8 + 2 = -6.
48
A sequence is defined by a₁ = -14 and aₙ = aₙ₋₁ + 6 for n > 1. What is a10?
Medium
ExplanationEach term adds 6 to the previous one, so a10 = a₁ + (10−1)(6) = 40.
49
If f(x) = 6x + 9, what is f⁻¹(87)?
Medium
Explanationf⁻¹(87) is the x-value such that f(x) = 87: 6x + 9 = 87, so x = 13.
50
A population of 2000 grows by 10% each year. What will the population be after 3 years?
Medium
ExplanationPopulation after 3 years = 2000 × (1 + 10/100)^3 = 2662.
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51
If f(x) = 4x − 10, what is f(-7)?
Medium
Explanationf(-7) = (4)(-7) − 10 = -38.
52
If f(x) = 5x − 19, for what value of x does f(x) = 36?
Medium
Explanation5x − 19 = 36, so x = 11.
53
In an arithmetic sequence, the 7th term is -29 and the 15th term is -77. What is the 30th term?
Medium
Explanationd = (-77 − (-29))/(15−7) = -6. Then the 30th term = -29 + (30−7)(-6) = -167.
54
In an arithmetic sequence, the 7th term is -67 and the 24th term is -271. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-271 − (-67))/(24−7) = -12.
55
For the function f(x) = √(x − 5), what is the smallest value of x in the domain?
Medium
ExplanationThe expression under the square root must be ≥ 0: x − 5 ≥ 0, so x ≥ 5. The smallest such value is 5.
56
If f(x) = 7x, what is f⁻¹(77)?
Medium
Explanationf⁻¹(77) is the x-value such that f(x) = 77: 7x = 77, so x = 11.
57
In an arithmetic sequence, the 10th term is -34 and the 19th term is -70. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-70 − (-34))/(19−10) = -4.
58
For the function f(x) = 1/(x − 63), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − 63 cannot equal 0, so x = 63 must be excluded from the domain.
59
What is the sum of the first 26 terms of an arithmetic sequence with first term -7 and common difference 5?
Medium
ExplanationThe 26th term is 118. Sum = n(a₁ + aₙ)/2 = 26(-7 + 118)/2 = 1443.
60
If f(x) = -6x + 7 and g(x) = 7x + 11, what is f(g(9))?
Medium
Explanationg(9) = 74, so f(g(9)) = f(74) = -6(74) + 7 = -437.
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61
If f(x) = -9x + 23, for what value of x does f(x) = 32?
Medium
Explanation-9x + 23 = 32, so x = -1.
62
An employee's salary starts at Rs. 42142 and increases by Rs. 3691 each year. What will the salary be after 3 years?
Medium
ExplanationSalary after 3 years = 42142 + 3 × 3691 = Rs. 53215.
63
An arithmetic sequence has first term -1 and common difference 7. What is the 22nd term?
Medium
Explanationaₙ = a₁ + (n−1)d = -1 + (22−1)(7) = 146.
64
An arithmetic sequence has first term -43 and common difference 4. What is the 32nd term?
Medium
Explanationaₙ = a₁ + (n−1)d = -43 + (32−1)(4) = 81.
65
A geometric sequence has first term 10 and common ratio 4. What is the 6th term?
Medium
Explanationaₙ = a₁ × r^(n−1) = 10 × (4)^5 = 10240.
66
If f(x) = -x − 6, for what value of x does f(x) = 1?
Medium
Explanation-x − 6 = 1, so x = -7.
67
A population of 4608 grows by 25% each year. What will the population be after 4 years?
Medium
ExplanationPopulation after 4 years = 4608 × (1 + 25/100)^4 = 11250.
68
If f(x) = -2x − 2 and g(x) = f(x − 8) + 5, what is g(2)?
Medium
Explanationg(2) = f(-6) + 5 = 10 + 5 = 15.
69
A geometric sequence has first term -14 and second term 56. What is the common ratio?
Medium
ExplanationCommon ratio = a₂/a₁ = 56/-14 = -4.
70
If f(x) = 8x − 13, what is f⁻¹(3)?
Medium
Explanationf⁻¹(3) is the x-value such that f(x) = 3: 8x − 13 = 3, so x = 2.
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71
What is the sum to infinity of a geometric sequence with first term -11 and common ratio -3/5?
Medium
ExplanationSum to infinity = a₁/(1 − r) = -11/(1 − (-3)/5) = -55/8.
72
If f(x) = 3x − 16, what is f⁻¹(17)?
Medium
Explanationf⁻¹(17) is the x-value such that f(x) = 17: 3x − 16 = 17, so x = 11.
73
If f(x) = x + 3, for what value of x does f(x) = -3?
Medium
Explanationx + 3 = -3, so x = -6.
74
If f(x) = 8x + 18, what is f⁻¹(-38)?
Medium
Explanationf⁻¹(-38) is the x-value such that f(x) = -38: 8x + 18 = -38, so x = -7.
75
In an arithmetic sequence, the 4th term is -15 and the 5th term is -4. What is the common difference?
Medium
Explanationd = (aⱼ − aᵢ)/(j−i) = (-4 − (-15))/(5−4) = 11.
76
For the function f(x) = 1/(x − (-5)), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − (-5) cannot equal 0, so x = -5 must be excluded from the domain.
77
For the function f(x) = 1/(x − (-46)), which value of x must be excluded from the domain?
Medium
ExplanationThe denominator x − (-46) cannot equal 0, so x = -46 must be excluded from the domain.
78
In an arithmetic sequence, the 3rd term is 54 and the 16th term is 145. What is the 17th term?
Medium
Explanationd = (145 − 54)/(16−3) = 7. Then the 17th term = 54 + (17−3)(7) = 152.
79
In an arithmetic sequence, the 5th term is 37 and the 9th term is 61. What is the 18th term?
Medium
Explanationd = (61 − 37)/(9−5) = 6. Then the 18th term = 37 + (18−5)(6) = 115.
80
If f(x) = 2x + 13, what is f⁻¹(11)?
Medium
Explanationf⁻¹(11) is the x-value such that f(x) = 11: 2x + 13 = 11, so x = -1.
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81
If f(x) = -11x − 24, for what value of x does f(x) = 196?
Medium
Explanation-11x − 24 = 196, so x = -20.
82
If f(x) = -6x + 7, for what value of x does f(x) = -5?
Medium
Explanation-6x + 7 = -5, so x = 2.
83
f(x) = -8x − 13 if x < 10, and f(x) = -3x + 9 if x ≥ 10. What is f(6)?
Medium
ExplanationSince 6 < 10, use f(x) = -8x − 13: f(6) = -61.
84
If f(x) = x^2 − 5x − 25, what is f(-6)?
Medium
Explanationf(-6) = 1(-6)^2 + (-5)(-6) + (-25) = 41.
85
If f(x) = -5x + 11, what is f(9) + f(4)?
Medium
Explanationf(9) = -34 and f(4) = -9, so f(9) + f(4) = -43.
86
An arithmetic sequence has first term -38 and common difference 7. What is the 14th term?
Medium
Explanationaₙ = a₁ + (n−1)d = -38 + (14−1)(7) = 53.
87
If f(x) = -2x − 22, for what value of x does f(x) = 12?
Medium
Explanation-2x − 22 = 12, so x = -17.
88
If f(x) = 12x + 22, for what value of x does f(x) = 46?
Medium
Explanation12x + 22 = 46, so x = 2.
89
If f(x) = -x − 21, for what value of x does f(x) = -41?
Medium
Explanation-x − 21 = -41, so x = 20.
90
What is the sum to infinity of a geometric sequence with first term 9 and common ratio 1/6?
Medium
ExplanationSum to infinity = a₁/(1 − r) = 9/(1 − 1/6) = 54/5.
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91
A sequence is defined by a₁ = 0 and aₙ = aₙ₋₁ + 2 for n > 1. What is a5?
Medium
ExplanationEach term adds 2 to the previous one, so a5 = a₁ + (5−1)(2) = 8.
92
If f(x) = -7x − 20, what is f(-10)?
Medium
Explanationf(-10) = (-7)(-10) − 20 = 50.
93
In an arithmetic sequence, the 6th term is -21 and the 21st term is -126. What is the 1st term?
Medium
Explanationd = (-126 − (-21))/(21−6) = -7. Then the 1st term = -21 + (1−6)(-7) = 14.
94
What is the sum of the first 17 terms of an arithmetic sequence with first term 14 and common difference -6?
Medium
ExplanationThe 17th term is -82. Sum = n(a₁ + aₙ)/2 = 17(14 + (-82))/2 = -578.
95
If f(x) = 11x + 7, what is f⁻¹(18)?
Medium
Explanationf⁻¹(18) is the x-value such that f(x) = 18: 11x + 7 = 18, so x = 1.
96
f(x) = 5x + 12 if x < 8, and f(x) = 2x − 9 if x ≥ 8. What is f(-4)?
Medium
ExplanationSince -4 < 8, use f(x) = 5x + 12: f(-4) = -8.
97
If f(x) = 7x + 19, what is f⁻¹(54)?
Medium
Explanationf⁻¹(54) is the x-value such that f(x) = 54: 7x + 19 = 54, so x = 5.
98
What is the sum of the first 20 terms of an arithmetic sequence with first term 5 and common difference 2?
Medium
ExplanationThe 20th term is 43. Sum = n(a₁ + aₙ)/2 = 20(5 + 43)/2 = 480.
99
If f(x) = -x + 2 and g(x) = f(x − 8) − 14, what is g(-11)?
Medium
Explanationg(-11) = f(-19) − 14 = 21 − 14 = 7.
100
What is the sum of the first 16 terms of an arithmetic sequence with first term -27 and common difference 3?
Medium
ExplanationThe 16th term is 18. Sum = n(a₁ + aₙ)/2 = 16(-27 + 18)/2 = -72.