Q. In a function f(x) = ax^2 + bx + c, what does the coefficient 'a' determine?
A.
The direction of the parabola's opening.
B.
The y-intercept of the graph.
C.
The slope of the graph.
D.
The x-intercepts of the graph.
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Solution
The coefficient 'a' in a quadratic function determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
Correct Answer: A — The direction of the parabola's opening.
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Q. In a function f(x) = ax^2 + bx + c, what does the value of 'a' determine?
A.
The direction in which the parabola opens.
B.
The x-intercepts of the graph.
C.
The y-intercept of the graph.
D.
The maximum value of the function.
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Solution
'a' determines the direction of the parabola; if 'a' is positive, it opens upwards, and if negative, it opens downwards.
Correct Answer: A — The direction in which the parabola opens.
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Q. In a function f(x) = x^3 - 3x, what is the nature of the critical points?
A.
All critical points are local maxima.
B.
All critical points are local minima.
C.
There are both local maxima and minima.
D.
There are no critical points.
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Solution
The function has critical points where the first derivative is zero, which can be analyzed to find both local maxima and minima.
Correct Answer: C — There are both local maxima and minima.
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Q. In a function f(x), if f(a) = f(b) for a ≠ b, what can be inferred about the function?
A.
The function is one-to-one.
B.
The function is constant.
C.
The function is quadratic.
D.
The function is increasing.
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Solution
If f(a) = f(b) for a ≠ b, it indicates that the function is not one-to-one, which means it does not pass the horizontal line test.
Correct Answer: B — The function is constant.
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Q. In a game, the probability of winning is 0.25. If a player plays 4 times, what is the probability of winning at least once?
A.
0.75
B.
0.84
C.
0.93
D.
0.99
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Solution
The probability of losing all 4 games is (0.75)^4 = 0.3164. Therefore, the probability of winning at least once is 1 - 0.3164 = 0.6836, approximately 0.84.
Correct Answer: B — 0.84
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Q. In a geometric progression, if the 1st term is 4 and the 5th term is 64, what is the common ratio?
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Solution
Let the common ratio be r. The 5th term is given by ar^4 = 64. Thus, 4r^4 = 64 => r^4 = 16 => r = 2.
Correct Answer: C — 4
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Q. In a geometric progression, if the 3rd term is 27 and the common ratio is 3, what is the first term?
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Solution
Let the first term be a. The 3rd term is a * r^2 = a * 3^2 = 9a. Setting 9a = 27 gives a = 3.
Correct Answer: B — 9
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Q. In a geometric progression, if the first term is 3 and the common ratio is 2, what is the 5th term?
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Solution
The nth term of a GP is given by a * r^(n-1). Here, a = 3, r = 2, and n = 5. Thus, the 5th term = 3 * 2^(5-1) = 3 * 16 = 48.
Correct Answer: A — 48
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Q. In a geometric progression, if the first term is 4 and the common ratio is 1/2, what is the 6th term?
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Solution
The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 4/32 = 0.125.
Correct Answer: A — 0.25
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Q. In a geometric progression, if the first term is 5 and the common ratio is 0.5, what is the sum of the first 4 terms?
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Solution
The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_4 = 5(1 - 0.5^4) / (1 - 0.5) = 5(1 - 0.0625) / 0.5 = 5 * 0.9375 / 0.5 = 9.375.
Correct Answer: B — 10
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Q. In a geometric progression, if the first term is 5 and the last term is 80, and there are 4 terms in total, what is the common ratio?
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Solution
Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.
Correct Answer: A — 2
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Q. In a geometric progression, if the first term is x and the common ratio is r, what is the expression for the sum of the first n terms?
A.
x(1 - r^n)/(1 - r)
B.
x(1 + r^n)/(1 + r)
C.
xr^n/(1 - r)
D.
xr^n/(1 + r)
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Solution
The sum of the first n terms of a GP is given by S_n = a(1 - r^n)/(1 - r) for r ≠ 1.
Correct Answer: A — x(1 - r^n)/(1 - r)
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Q. In a geometric series where the first term is 4 and the common ratio is 2, what is the sum of the first 5 terms? (2023)
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Solution
The sum of the first n terms of a geometric series is a(1 - r^n) / (1 - r). Here, a = 4, r = 2, n = 5. So, 4(1 - 2^5) / (1 - 2) = 4(1 - 32) / -1 = 124.
Correct Answer: C — 64
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Q. In a geometric series where the first term is 4 and the common ratio is 2, what is the 6th term? (2023)
A.
64
B.
128
C.
256
D.
512
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Solution
The nth term of a geometric series is given by ar^(n-1). Here, a = 4, r = 2, n = 6. So, 4 * 2^(6-1) = 4 * 32 = 128.
Correct Answer: C — 256
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Q. In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?
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Solution
Let the first term be a and the common ratio be r. Then, 3rd term = ar^2 = 27 and 5th term = ar^4 = 243. Dividing gives r^2 = 9, so r = 3. Substituting back gives a = 3.
Correct Answer: B — 9
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Q. In a GP, if the first term is 10 and the common ratio is 0.5, what is the 6th term?
A.
0.625
B.
1.25
C.
2.5
D.
5
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Solution
The 6th term is given by 10 * (0.5)^(6-1) = 10 * (0.5)^5 = 10 * 0.03125 = 0.3125.
Correct Answer: A — 0.625
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Q. In a GP, if the first term is 5 and the common ratio is 1/2, what is the sum of the first four terms?
A.
15
B.
10
C.
12.5
D.
20
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Solution
The first four terms are 5, 2.5, 1.25, and 0.625. Their sum is 5 + 2.5 + 1.25 + 0.625 = 9.375.
Correct Answer: C — 12.5
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Q. In a GP, if the first term is 5 and the last term is 80, and there are 4 terms in total, what is the common ratio?
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Solution
Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.
Correct Answer: A — 2
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Q. In a GP, if the first term is 7 and the common ratio is 1/2, what is the 6th term?
A.
0.4375
B.
0.5
C.
1
D.
1.75
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Solution
The 6th term is given by 7 * (1/2)^(6-1) = 7 * (1/32) = 0.4375.
Correct Answer: A — 0.4375
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Q. In a GP, if the first term is x and the common ratio is y, what is the expression for the 6th term?
A.
xy^5
B.
xy^6
C.
x^6y
D.
x^5y
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Solution
The nth term of a GP is given by a * r^(n-1). Thus, the 6th term = x * y^(6-1) = xy^5.
Correct Answer: A — xy^5
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Q. In a group of 150 people, 90 like basketball, 60 like soccer, and 30 like both. How many people like neither sport?
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Solution
The number of people who like at least one sport is 90 + 60 - 30 = 120, so those who like neither is 150 - 120 = 30.
Correct Answer: A — 30
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Q. In a group of 150 people, 90 like reading fiction, 60 like reading non-fiction, and 30 like both. How many like only non-fiction?
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Solution
The number of people who like only non-fiction is: 60 - 30 = 30.
Correct Answer: A — 30
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Q. In a group of 200 people, 120 like basketball, 80 like football, and 50 like both. How many like only basketball?
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Solution
The number of people who like only basketball is: 120 - 50 = 70.
Correct Answer: A — 70
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Q. In a group of 200 people, 120 like football, 80 like basketball, and 40 like both. How many people like only football?
A.
40
B.
80
C.
120
D.
160
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Solution
The number of people who like only football is 120 - 40 = 80.
Correct Answer: B — 80
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Q. In a group of 200 people, 120 like football, 80 like basketball, and 50 like both. How many people like neither sport?
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Solution
The number of people who like at least one sport is: 120 + 80 - 50 = 150. Therefore, those who like neither is: 200 - 150 = 50.
Correct Answer: C — 70
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Q. In a group of 200 people, 120 like football, 80 like basketball, and 50 like both. How many people like only football?
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Solution
The number of people who like only football is 120 - 50 = 70.
Correct Answer: A — 70
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Q. In a group of 200 people, 120 like sports, 80 like music, and 50 like both. How many people like only music?
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Solution
The number of people who like only music is 80 - 50 = 30.
Correct Answer: A — 30
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Q. In a group of 300 people, 150 like rock music, 100 like pop music, and 50 like both. How many people like only rock music?
A.
50
B.
100
C.
150
D.
200
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Solution
The number of people who like only rock music is 150 - 50 = 100.
Correct Answer: B — 100
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Q. In a harmonic progression, if the first term is 1 and the second term is 1/2, what is the common difference of the corresponding arithmetic progression?
A.
1/2
B.
1/4
C.
1/6
D.
1/8
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Solution
The reciprocals are 1 and 2, which are in arithmetic progression with a common difference of 1/2.
Correct Answer: B — 1/4
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Q. In a harmonic progression, if the first term is 1 and the second term is 1/2, what is the sum of the first three terms?
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Solution
The first term is 1, the second term is 1/2, and the third term can be calculated as 1/(1 + 1/2) = 2/3. The sum is 1 + 1/2 + 2/3 = 2.
Correct Answer: C — 3
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