1
Is the sequence -12, -8, -4, 0, ... arithmetic, geometric, or neither?
Medium
A
Arithmetic
B
Neither
C
Geometric
D
Cannot be determined
Explanation Each term differs from the previous by a constant amount, so this sequence is arithmetic.
2
What is the sum of the first 3 terms of a geometric sequence with first term -6 and common ratio 4?
Medium
A
-121
B
-126
C
-122
D
-135
Explanation Sum = a₁(rⁿ − 1)/(r − 1) = -6((4)^3 − 1)/(4 − 1) = -126.
3
Complete the square: x^2 + 6x − 8 = (x + h)^2 + k. What are the values of h and k?
Medium
A
h = -3, k = -17
B
h = 4, k = -17
C
h = 3, k = -17
D
h = 3, k = -16
Explanation x^2 + 6x − 8 = (x + 3)^2 − 17, since 6 = 2×3 and -8 − 9 = -17.
4
For the function f(x) = √(x − (-34)), what is the smallest value of x in the domain?
Medium
A
-34
B
-37
C
-35
D
-36
Explanation The expression under the square root must be ≥ 0: x − (-34) ≥ 0, so x ≥ -34. The smallest such value is -34.
5
Solve for x: 9x + 5 = -4
Medium
A
-4
B
1
C
-1
Explanation 9x = -4 − (5) = -9, so x = -9/9 = -1.
6
What is the value of -14x + 15 when x = -9?
Medium
A
141
B
145
C
139
D
134
Explanation -14(-9) + 15 = 126 + 15 = 141.
7
In an arithmetic sequence, the 4th term is -47 and the 11th term is -75. What is the common difference?
Medium
A
-3
B
-4
C
-6
D
-1
Explanation d = (aⱼ − aᵢ)/(j−i) = (-75 − (-47))/(11−4) = -4.
8
If f(x) = -5x − 13, what is f(-4)?
Medium
A
11
B
13
C
9
D
7
Explanation f(-4) = (-5)(-4) − 13 = 7.
9
Solve for x: x − 36 = -33
Medium
A
5
B
7
C
3
D
6
Explanation x = -33 − (-36) = 3, so x = 3/1 = 3.
10
Solve for x: -3(x − 18) = 6
Medium
A
12
B
16
C
13
D
14
Explanation Dividing both sides by -3: x − 18 = -2, so x = 16.
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11
If f(x) = -2x + 23, what is f(-8) − f(-9)?
Medium
A
3
B
-2
C
-3
D
-8
Explanation f(-8) = 39 and f(-9) = 41, so f(-8) − f(-9) = -2.
12
Solve the system for y: 6x − 7y = -122, 2x + 4y = 48
Medium
A
15
B
14
C
16
D
18
Explanation Solving the system by elimination or substitution gives y = 14 (and x = -4).
13
Solve for x: 3x + 9 ≥ -63
Medium
A
x ≥ -24
B
x ≥ -25
C
x < -24
D
x ≥ -23
Explanation 3x ≥ -63 − (9) = -72, so x ≥ -24.
14
How many solutions does the equation 11x + 0 = 7x + 13 have?
Medium
A
Exactly two solutions
B
No solution
C
Exactly one solution
D
Infinitely many solutions
Explanation Since the coefficients of x differ, the equation has exactly one solution.
15
What is the y-intercept of the line 13x + 7y = -84?
Medium
A
-12
B
-9
C
-10
D
-16
Explanation Set x = 0: 7y = -84, so y = -84/7 = -12.
16
If f(x) = -7x − 20, what is f(-10)?
Medium
A
54
B
48
C
50
D
49
Explanation f(-10) = (-7)(-10) − 20 = 50.
17
What is the value of -8x + 4 when x = 14?
Medium
A
-109
B
-99
C
-121
D
-108
Explanation -8(14) + 4 = -112 + 4 = -108.
18
If f(x) = 6x + 4, for what value of x does f(x) = 76?
Medium
A
12
B
8
C
13
D
16
Explanation 6x + 4 = 76, so x = 12.
19
If f(x) = 15x + 5, what is f(-14)?
Medium
A
-203
B
-223
C
-217
D
-205
Explanation f(-14) = (15)(-14) + 5 = -205.
20
Solve the inequality: x^2 − 6x − 91 < 0
Medium
A
x < -7 or x > 13
B
-7 < x < 13
C
x < -8 or x > 14
D
-6 < x < 12
Explanation x^2 − 6x − 91 = (x + 7)(x - 13), with roots -7 and 13. Since the parabola opens upward, the expression is negative between the roots, giving -7 < x < 13.
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21
If f(x) = -7x − 11, what is f(5) − f(-2)?
Medium
A
-44
B
-45
C
-53
D
-49
Explanation f(5) = -46 and f(-2) = 3, so f(5) − f(-2) = -49.
22
A geometric sequence has first term 3 and second term 12. What is the common ratio?
Medium
A
5
B
-4
C
4
D
1/4
Explanation Common ratio = a₂/a₁ = 12/3 = 4.
23
What is the sum of the first 6 terms of a geometric sequence with first term -2 and common ratio 4?
Medium
A
-2730
B
-2805
C
-2640
D
-2764
Explanation Sum = a₁(rⁿ − 1)/(r − 1) = -2((4)^6 − 1)/(4 − 1) = -2730.
24
What is the sum of the first 6 terms of a geometric sequence with first term 7 and common ratio 3?
Medium
A
2548
B
2633
C
2618
D
2474
Explanation Sum = a₁(rⁿ − 1)/(r − 1) = 7((3)^6 − 1)/(3 − 1) = 2548.
25
What are the solutions to x^2 − 9x − 90 = 0?
Medium
A
x = -7 or x = 16
B
x = -6 or x = 14
C
x = 6 or x = -15
D
x = -6 or x = 15
Explanation x^2 − 9x − 90 = (x - 15)(x + 6), so x = -6 or x = 15.
26
A cinema sells adult tickets for Rs. 7 and child tickets for Rs. 3. If 17 tickets were sold for a total of Rs. 87, how many adult tickets were sold?
Medium
A
10
B
9
C
6
D
8
Explanation Let a = adult tickets, c = child tickets. a + c = 17 and 7a + 3c = 87. Solving gives a = 9, c = 8.
27
What is the y-intercept of y = -6x^2 + 26x − 30?
Medium
A
-32
B
-31
C
-36
D
-30
Explanation The y-intercept occurs at x = 0: y = -6(0)^2 + (26)(0) − 30 = -30.
28
Solve for x: -14x + 16 > 198
Medium
A
x > -13
B
x < -12
C
x < -13
D
x < -14
Explanation Dividing both sides by the negative coefficient -14 reverses the inequality sign, giving x < -13.
29
What is the nature of the roots of 6x^2 − 17x − 3 = 0?
Medium
A
One repeated real root
B
Two distinct real roots
C
Cannot be determined
D
No real roots
Explanation The discriminant is b^2 − 4ac = (-17)^2 − 4(6)(-3) = 361, which is positive, so the equation has two distinct real roots.
30
What is the y-intercept of y = -8x^2 + 18x − 30?
Medium
A
-26
B
-30
C
-29
D
-27
Explanation The y-intercept occurs at x = 0: y = -8(0)^2 + (18)(0) − 30 = -30.
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31
A sequence is defined by a₁ = -11 and aₙ = aₙ₋₁ − 4 for n > 1. What is a13?
Medium
A
-66
B
-53
C
-54
D
-59
Explanation Each term adds -4 to the previous one, so a13 = a₁ + (13−1)(-4) = -59.
32
A sequence is defined by a₁ = 3 and aₙ = aₙ₋₁ − 8 for n > 1. What is a5?
Medium
A
-24
B
-29
C
-27
D
-23
Explanation Each term adds -8 to the previous one, so a5 = a₁ + (5−1)(-8) = -29.
33
A sequence is defined by a₁ = 15 and aₙ = aₙ₋₁ + 4 for n > 1. What is a4?
Medium
A
25
B
27
C
33
D
31
Explanation Each term adds 4 to the previous one, so a4 = a₁ + (4−1)(4) = 27.
34
If f(x) = -7x − 8 and g(x) = f(x + 2) + 4, what is g(4)?
Medium
A
-40
B
-46
C
-47
D
-48
Explanation g(4) = f(6) + 4 = -50 + 4 = -46.
35
What is the minimum value of y = 3x^2 + 12x − 11?
Medium
A
-26
B
-25
C
-23
D
-27
Explanation Since a = 3 is positive, the parabola opens upward and has a minimum at its vertex, where y = -23.
36
What is the sum to infinity of a geometric sequence with first term 13 and common ratio -2/3?
Medium
A
65/3
B
34/5
C
44/5
D
39/5
Explanation Sum to infinity = a₁/(1 − r) = 13/(1 − (-2)/3) = 39/5.
37
Does the parabola y = 6x^2 + 10x − 29 open upward or downward?
Medium
A
Downward
B
Cannot be determined
C
Upward
D
It is a straight line
Explanation Since a = 6 is positive, the parabola opens upward.
38
What is the value of -4x^2 − 20x + 20 when x = -4?
Medium
A
35
B
33
C
32
D
36
Explanation -4(-4)^2 + (-20)(-4) + (20) = (-64) + (80) + (20) = 36.
39
Solve for x: x/9 + x/11 = -40
Medium
A
-198
B
-169
C
-229
D
-226
Explanation Combining terms: x(1/9 + 1/11) = -40, so x × (20)/99 = -40, giving x = -198.
40
Is the sequence -6, -12, -24, -48, ... arithmetic, geometric, or neither?
Medium
A
Cannot be determined
B
Arithmetic
C
Neither
D
Geometric
Explanation Each term is a constant multiple of the previous one, so this sequence is geometric.
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41
If -2x + 5y = 56 and 9x − 6y = -186, what is x + y?
Medium
A
-18
B
-12
C
-14
D
-10
Explanation Solving the system gives x = -18 and y = 4, so x + y = -14.
42
A delivery service charges a fixed fee of Rs. 23 plus Rs. 12 per item. If a customer has a budget of Rs. 125, what is the maximum number of items they can order?
Medium
A
5
B
11
C
10
D
8
Explanation 23 + 12n ≤ 125, so n ≤ 102/12 ≈ 8.50, meaning at most 8 items.
43
Is the sequence -15, -21, -27, -33, ... arithmetic, geometric, or neither?
Medium
A
Geometric
B
Arithmetic
C
Neither
D
Cannot be determined
Explanation Each term differs from the previous by a constant amount, so this sequence is arithmetic.
44
Factor: x^2 − 9x − 90
Medium
A
(x + 15)(x + 6)
B
(x - 15)(x - 6)
C
(x + 15)(x - 6)
D
(x - 15)(x + 6)
Explanation x^2 − 9x − 90 factors as the product of (x − r) for each root: (x - 15)(x + 6).
45
If f(x) = 5x, what is f⁻¹(-60)?
Medium
A
-15
B
-12
C
-16
D
-8
Explanation f⁻¹(-60) is the x-value such that f(x) = -60: 5x = -60, so x = -12.
46
If -2x − 6y = -150 and 6x + 5y = 190, what is x + y?
Medium
A
35
B
31
C
37
D
38
Explanation Solving the system gives x = 15 and y = 20, so x + y = 35.
47
Factor: 50x^2 − 35x
Medium
A
5x(10x − 7)
B
5(10x − 7)
C
5x(10x + 7)
D
x(50x − 35)
Explanation The GCF of 50 and -35 (besides x) is 5, and factoring it out (along with one x) gives 5x(10x − 7).
48
What is the minimum value of y = x^2 + 2x − 1?
Medium
A
-5
B
-2
C
-3
Explanation Since a = 1 is positive, the parabola opens upward and has a minimum at its vertex, where y = -2.
49
What is the vertex of the parabola y = -x^2 + 20x + 24?
Medium
A
(11, 124)
B
(10, 125)
C
(10, 124)
D
(-10, 124)
Explanation The vertex x-coordinate is −b/(2a) = −(20)/(2×-1) = 10. Substituting gives y = 124, so the vertex is (10, 124).
50
Solve for x: 26 < 5x + 1 < 101
Medium
A
6 < x < 20
B
5 < x < 19
C
4 < x < 21
D
5 < x < 20
Explanation Subtracting 1 from all parts: 25 < 5x < 100. Dividing by 5: 5 < x < 20.
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51
What is the sum of the roots of 3x^2 − 21x + 9 = 0?
Medium
A
7
B
4
C
11
D
6
Explanation By Vieta's formulas, the sum of the roots equals −b/a = −(-21)/3 = 7.
52
If f(x) = -x + 6, for what value of x does f(x) = 9?
Medium
A
-3
B
-4
C
-5
D
-7
Explanation -x + 6 = 9, so x = -3.
53
What is the sum of the first 3 terms of a geometric sequence with first term 2 and common ratio -2?
Medium
A
6
B
4
C
-4
D
13
Explanation Sum = a₁(rⁿ − 1)/(r − 1) = 2((-2)^3 − 1)/(-2 − 1) = 6.
54
What is the value of 7x^2 + 5x + 4 when x = -12?
Medium
A
952
B
972
C
909
D
920
Explanation 7(-12)^2 + (5)(-12) + (4) = (1008) + (-60) + (4) = 952.
55
If 8x − 2y = -68 and 4x − 4y = -88, what is x + y?
Medium
A
16
B
18
C
10
D
14
Explanation Solving the system gives x = -4 and y = 18, so x + y = 14.
56
Which equation has roots -6 and 4?
Medium
A
x^2 + 2x − 24 = 0
B
x^2 + 2x + 24 = 0
C
x^2 − 2x − 24 = 0
D
x^2 + 4x − 24 = 0
Explanation Sum of roots = -2, product of roots = -24, giving x^2 − (-2)x + (-24) = 0, i.e. x^2 + 2x − 24 = 0.
57
What is the maximum value of y = -x^2 + 26x − 27?
Medium
A
135
B
156
C
141
D
142
Explanation Since a = -1 is negative, the parabola opens downward and has a maximum at its vertex, where y = 142.
58
If f(x) = 9x + 10, what is f(12) + f(-5)?
Medium
A
83
B
75
C
76
D
93
Explanation f(12) = 118 and f(-5) = -35, so f(12) + f(-5) = 83.
59
A sequence is defined by a₁ = -16 and aₙ = aₙ₋₁ + 7 for n > 1. What is a9?
Medium
A
40
B
35
C
37
D
34
Explanation Each term adds 7 to the previous one, so a9 = a₁ + (9−1)(7) = 40.
60
If f(x) = 6x + 12, what is f⁻¹(-12)?
Medium
A
-8
B
-7
C
-4
D
-5
Explanation f⁻¹(-12) is the x-value such that f(x) = -12: 6x + 12 = -12, so x = -4.
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61
Solve for x: x + 30 = 31
Medium
A
1
B
-2
C
4
Explanation x = 31 − (30) = 1, so x = 1/1 = 1.
62
In an arithmetic sequence, the 3rd term is -31 and the 6th term is -22. What is the 18th term?
Medium
A
15
B
14
C
10
D
19
Explanation d = (-22 − (-31))/(6−3) = 3. Then the 18th term = -31 + (18−3)(3) = 14.
63
If f(x) = -5x + 29, what is f(5)?
Medium
A
-1
B
4
C
-2
Explanation f(5) = (-5)(5) + 29 = 4.
64
A delivery service charges a fixed fee of Rs. 21 plus Rs. 4 per item. If a customer has a budget of Rs. 49, what is the maximum number of items they can order?
Medium
A
9
B
8
C
6
D
7
Explanation 21 + 4n ≤ 49, so n ≤ 28/4 ≈ 7.00, meaning at most 7 items.
65
Solve for x: -22 < 7x − 8 < 13
Medium
A
-2 < x < 2
B
-3 < x < 4
C
-1 < x < 3
D
-2 < x < 3
Explanation Subtracting -8 from all parts: -14 < 7x < 21. Dividing by 7: -2 < x < 3.
66
What is the value of -9x^2 + x + 14 when x = 6?
Medium
A
-299
B
-304
C
-301
D
-286
Explanation -9(6)^2 + (1)(6) + (14) = (-324) + (6) + (14) = -304.
67
A rectangle has length (x + 7) and width (x + 1). If the area of the rectangle is 187 square units, what is the value of x (where x is a positive number)?
Medium
A
10
B
8
C
12
D
7
Explanation (x + 7)(x + 1) = 187. Testing x = 10: (17)(11) = 187, which checks out (the other algebraic root is negative and not valid here).
68
Solve for x: 4x + 30 = -14
Medium
A
-15
B
-13
C
-12
D
-11
Explanation 4x = -14 − (30) = -44, so x = -44/4 = -11.
69
Solve for x: 8(x − 7) = 40
Medium
A
15
B
10
C
12
D
9
Explanation Dividing both sides by 8: x − 7 = 5, so x = 12.
70
f(x) = -x + 11 if x < -1, and f(x) = -6x − 7 if x ≥ -1. What is f(-6)?
Medium
A
12
B
15
C
23
D
17
Explanation Since -6 < -1, use f(x) = -x + 11: f(-6) = 17.
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71
If x − 5y = -81 and 4x + 9y = 256, what is x + y?
Medium
A
40
B
35
C
39
D
47
Explanation Solving the system gives x = 19 and y = 20, so x + y = 39.
72
What is the nature of the roots of x^2 − 16x − 14 = 0?
Medium
A
Cannot be determined
B
Two distinct real roots
C
No real roots
D
One repeated real root
Explanation The discriminant is b^2 − 4ac = (-16)^2 − 4(1)(-14) = 312, which is positive, so the equation has two distinct real roots.
73
What is the vertex of the parabola y = 4x^2 − 112x + 23?
Medium
A
(-14, -761)
B
(15, -761)
C
(14, -760)
D
(14, -761)
Explanation The vertex x-coordinate is −b/(2a) = −(-112)/(2×4) = 14. Substituting gives y = -761, so the vertex is (14, -761).
74
Solve for x: -7x + 35 = 0
Medium
A
2
B
5
C
4
D
9
Explanation -7x = 0 − (35) = -35, so x = -35/-7 = 5.
75
Solve the system for y: 7x + 6y = -73, -6x + 2y = -66
Medium
A
-22
B
-17
C
-19
D
-18
Explanation Solving the system by elimination or substitution gives y = -18 (and x = 5).
76
For the function f(x) = 1/(x − (-52)), which value of x must be excluded from the domain?
Medium
A
-52
B
-51
C
-50
D
-54
Explanation The denominator x − (-52) cannot equal 0, so x = -52 must be excluded from the domain.
77
Solve for x: -2(x + 8) = -10
Medium
A
-6
B
-2
C
-3
Explanation Dividing both sides by -2: x + 8 = 5, so x = -3.
78
Factor: x^2 + 25x + 150
Medium
A
(x - 15)(x + 10)
B
(x + 15)(x + 10)
C
(x - 15)(x - 10)
D
(x + 15)(x - 10)
Explanation x^2 + 25x + 150 factors as the product of (x − r) for each root: (x + 15)(x + 10).
79
What is the sum of the first 7 terms of a geometric sequence with first term -1 and common ratio -3?
Medium
A
-547
B
-512
C
-580
D
-575
Explanation Sum = a₁(rⁿ − 1)/(r − 1) = -1((-3)^7 − 1)/(-3 − 1) = -547.
80
What is the discriminant of 7x^2 + 22x − 9 = 0?
Medium
A
736
B
771
C
798
D
728
Explanation Discriminant = b^2 − 4ac = (22)^2 − 4(7)(-9) = 484 − -252 = 736.
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81
What is the discriminant of 9x^2 − 5x + 3 = 0?
Medium
A
-82
B
-86
C
-85
D
-83
Explanation Discriminant = b^2 − 4ac = (-5)^2 − 4(9)(3) = 25 − 108 = -83.
82
Is x^2 + 20x + 100 a perfect square trinomial?
Medium
A
Yes
B
Cannot be determined
C
No
D
Only if x is positive
Explanation A trinomial x^2 + bx + c is a perfect square exactly when c = (b/2)^2. Here (b/2)^2 = (10)^2 = 100, which matches c, so it is a perfect square trinomial.
83
Solve the system for y: 3x − 5y = 7, -4x + 2y = -56
Medium
A
13
B
10
C
7
D
11
Explanation Solving the system by elimination or substitution gives y = 10 (and x = 19).
84
Solve for x: x/7 + x/12 = 114
Medium
A
465
B
504
C
420
D
531
Explanation Combining terms: x(1/7 + 1/12) = 114, so x × (19)/84 = 114, giving x = 504.
85
What is the y-intercept of y = 7x^2 + 13x + 19?
Medium
A
19
B
20
C
23
D
18
Explanation The y-intercept occurs at x = 0: y = 7(0)^2 + (13)(0) + 19 = 19.
86
A population of 1792 grows by 25% each year. What will the population be after 3 years?
Medium
A
3433
B
3493
C
3500
D
3518
Explanation Population after 3 years = 1792 × (1 + 25/100)^3 = 3500.
87
What is the sum to infinity of a geometric sequence with first term -5 and common ratio 1/5?
Medium
A
-21/4
B
-4
C
-25/4
D
-29/4
Explanation Sum to infinity = a₁/(1 − r) = -5/(1 − 1/5) = -25/4.
88
Complete the square: x^2 + 20x + 87 = (x + h)^2 + k. What are the values of h and k?
Medium
A
h = -10, k = -13
B
h = 10, k = -12
C
h = 11, k = -13
D
h = 10, k = -13
Explanation x^2 + 20x + 87 = (x + 10)^2 − 13, since 20 = 2×10 and 87 − 100 = -13.
89
Solve for x: -5x − 21 > 24
Medium
A
x < -9
B
x < -10
C
x < -8
D
x > -9
Explanation Dividing both sides by the negative coefficient -5 reverses the inequality sign, giving x < -9.
90
What is the sum of the first 25 terms of an arithmetic sequence with first term 27 and common difference 9?
Medium
A
3375
B
3313
C
3453
D
3369
Explanation The 25th term is 243. Sum = n(a₁ + aₙ)/2 = 25(27 + 243)/2 = 3375.
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91
If f(x) = -7x − 9, what is f(5) + f(-15)?
Medium
A
46
B
52
C
58
D
55
Explanation f(5) = -44 and f(-15) = 96, so f(5) + f(-15) = 52.
92
If f(x) = -8x + 4 and g(x) = f(x − 10) − 6, what is g(3)?
Medium
A
49
B
59
C
54
D
58
Explanation g(3) = f(-7) − 6 = 60 − 6 = 54.
93
An employee's salary starts at Rs. 76191 and increases by Rs. 1558 each year. What will the salary be after 9 years?
Medium
A
90120
B
90209
C
90182
D
90213
Explanation Salary after 9 years = 76191 + 9 × 1558 = Rs. 90213.
94
Is the sequence -10, 30, -90, 270, ... arithmetic, geometric, or neither?
Medium
A
Arithmetic
B
Neither
C
Geometric
D
Cannot be determined
Explanation Each term is a constant multiple of the previous one, so this sequence is geometric.
95
A line has slope -5/9 and another line has slope -4/3. Are the lines parallel, perpendicular, or neither?
Medium
A
Neither
B
Cannot be determined
C
Perpendicular
D
Parallel
Explanation Two lines are parallel if their slopes are equal, and perpendicular if their slopes multiply to −1. Here, -5/9 and -4/3 are neither.
96
How many solutions does the equation 8x + 7 = 8x + 11 have?
Medium
A
Exactly two solutions
B
Exactly one solution
C
No solution
D
Infinitely many solutions
Explanation The x-terms cancel leaving a false statement (since 7 ≠ 11), so there is no solution.
97
In an arithmetic sequence, the 11th term is 53 and the 30th term is 167. What is the common difference?
Medium
A
9
B
6
C
3
D
5
Explanation d = (aⱼ − aᵢ)/(j−i) = (167 − 53)/(30−11) = 6.
98
What is the slope of the line through (-14, -19) and (-15, -24)?
Medium
A
5
B
-5
C
1/5
D
6
Explanation Slope = (y2 − y1)/(x2 − x1) = (-24 − -19)/(-15 − -14) = -5/-1 = 5.
99
An arithmetic sequence has first term 40 and common difference 12. What is the 12th term?
Medium
A
172
B
151
C
159
D
175
Explanation aₙ = a₁ + (n−1)d = 40 + (12−1)(12) = 172.
100
A line has slope 7/4 and passes through (-32, 30). What is its y-intercept?
Medium
A
86
B
73
C
89
D
105
Explanation Using y = mx + b: 30 = (7/4)(-32) + b, so b = 30 − -56 = 86.