Q. If the first term of an arithmetic sequence is 5 and the common difference is 3, what is the 10th term? (2023)
Solution
The nth term of an arithmetic sequence is given by a + (n-1)d. Here, a = 5, d = 3, n = 10. So, 5 + (10-1)3 = 32.
Correct Answer:
A
— 32
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Q. If the first three terms of a harmonic progression are 1, 1/2, and 1/3, what is the common difference of the corresponding arithmetic progression?
-
A.
1/6
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
The reciprocals are 1, 2, and 3. The common difference is 2 - 1 = 1.
Correct Answer:
A
— 1/6
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Q. If the first three terms of a harmonic progression are 1, 1/2, and 1/3, what is the fourth term?
-
A.
1/4
-
B.
1/5
-
C.
1/6
-
D.
1/7
Solution
The reciprocals are 1, 2, and 3, which are in arithmetic progression. The next term in the sequence of reciprocals is 4, so the fourth term is 1/4.
Correct Answer:
C
— 1/6
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Q. If the first three terms of a harmonic progression are 1/2, 1/3, and 1/x, what is the value of x?
Solution
The reciprocals of the terms form an arithmetic progression: 2, 3, and x. The common difference is 1. Therefore, x = 6.
Correct Answer:
B
— 6
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Q. If the first three terms of a harmonic progression are a, b, and c, which of the following equations holds true?
-
A.
1/a + 1/b = 1/c
-
B.
1/a + 1/c = 1/b
-
C.
1/b + 1/c = 1/a
-
D.
1/a + 1/b + 1/c = 0
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression, hence 1/a + 1/b = 1/c.
Correct Answer:
A
— 1/a + 1/b = 1/c
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Q. If the first three terms of a harmonic progression are a, b, and c, which of the following is true?
-
A.
1/a + 1/c = 2/b
-
B.
a + b + c = 0
-
C.
a*b*c = 1
-
D.
a + b = c
Solution
In a harmonic progression, the reciprocals of the terms form an arithmetic progression, which leads to the relationship 1/a + 1/c = 2/b.
Correct Answer:
A
— 1/a + 1/c = 2/b
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Q. If the first three terms of a harmonic progression are a, b, c, which of the following is true?
-
A.
1/a, 1/b, 1/c are in AP
-
B.
a, b, c are in AP
-
C.
1/a, 1/b, 1/c are in GP
-
D.
b = (a+c)/2
Solution
In a harmonic progression, the reciprocals of the terms are in arithmetic progression, hence 1/a, 1/b, 1/c are in AP.
Correct Answer:
A
— 1/a, 1/b, 1/c are in AP
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Q. If the function f(x) is defined as f(x) = 2x + 1, what is the value of f(3)?
Solution
Substituting x = 3 into the function gives f(3) = 2(3) + 1 = 6 + 1 = 7.
Correct Answer:
C
— 7
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Q. If the function g(x) = 2x + 3 is transformed to g(x) = 2(x - 1) + 3, what type of transformation has occurred?
-
A.
Vertical shift up.
-
B.
Vertical shift down.
-
C.
Horizontal shift left.
-
D.
Horizontal shift right.
Solution
The transformation g(x) = 2(x - 1) + 3 indicates a horizontal shift to the right by 1 unit.
Correct Answer:
D
— Horizontal shift right.
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Q. If the GCD of two numbers is 1, what can be inferred about the two numbers? (2023)
-
A.
They are both even.
-
B.
They are both odd.
-
C.
They are coprime.
-
D.
They are multiples of each other.
Solution
If the GCD of two numbers is 1, it means they have no common factors other than 1, thus they are coprime.
Correct Answer:
C
— They are coprime.
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Q. If the GCD of two numbers is 1, which of the following statements is true?
-
A.
The numbers are multiples of each other
-
B.
The numbers are co-prime
-
C.
The numbers are both even
-
D.
The numbers are both odd
Solution
If the GCD of two numbers is 1, it means they have no common factors other than 1, hence they are co-prime.
Correct Answer:
B
— The numbers are co-prime
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Q. If the graph of a function f(x) intersects the x-axis at x = 1 and x = 3, which of the following can be inferred?
-
A.
f(1) = 0 and f(3) = 0.
-
B.
The function is linear.
-
C.
The function has no real roots.
-
D.
The function is increasing.
Solution
If the graph intersects the x-axis at x = 1 and x = 3, it means that f(1) = 0 and f(3) = 0, indicating the roots of the function.
Correct Answer:
A
— f(1) = 0 and f(3) = 0.
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Q. If the graph of a function f(x) intersects the x-axis at x = 3, what can be concluded?
-
A.
f(3) = 0.
-
B.
f(3) > 0.
-
C.
f(3) < 0.
-
D.
f(3) is undefined.
Solution
The intersection of the graph with the x-axis indicates that the function value at that point is zero.
Correct Answer:
A
— f(3) = 0.
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Q. If the graph of a function f(x) is symmetric about the y-axis, which of the following must be true?
-
A.
f(x) = f(-x) for all x.
-
B.
f(x) = -f(-x) for all x.
-
C.
f(x) is always positive.
-
D.
f(x) has a maximum value.
Solution
A function that is symmetric about the y-axis satisfies the property f(x) = f(-x) for all x.
Correct Answer:
A
— f(x) = f(-x) for all x.
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Q. If the graph of a function is a parabola opening upwards, which of the following can be inferred about the function?
-
A.
The function has a maximum value.
-
B.
The function has a minimum value.
-
C.
The function is linear.
-
D.
The function is constant.
Solution
A parabola that opens upwards indicates that the function has a minimum value at its vertex.
Correct Answer:
B
— The function has a minimum value.
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Q. If the graph of a function is symmetric about the y-axis, which of the following types of functions could it represent?
-
A.
Linear function
-
B.
Odd function
-
C.
Even function
-
D.
Exponential function
Solution
A function is symmetric about the y-axis if it is an even function, which satisfies the condition f(x) = f(-x).
Correct Answer:
C
— Even function
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Q. If the graph of a function is symmetric about the y-axis, which of the following types of functions could it be?
-
A.
Linear function
-
B.
Odd function
-
C.
Even function
-
D.
Exponential function
Solution
A function is symmetric about the y-axis if it is an even function, which satisfies the condition f(x) = f(-x).
Correct Answer:
C
— Even function
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Q. If the graph of a function is symmetric about the y-axis, which of the following must be true?
-
A.
The function is linear.
-
B.
The function is even.
-
C.
The function is odd.
-
D.
The function has no intercepts.
Solution
A function is even if it is symmetric about the y-axis, meaning f(x) = f(-x) for all x.
Correct Answer:
B
— The function is even.
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Q. If the HCF of 18 and 24 is 6, what is the LCM of these two numbers? (2023)
Solution
Using the relation HCF * LCM = Product of the numbers, we have 6 * LCM = 18 * 24. Thus, LCM = (18 * 24) / 6 = 72.
Correct Answer:
A
— 72
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Q. If the HCF of 24 and 36 is 12, what is the LCM of these two numbers? (2023)
Solution
Using the relation HCF * LCM = Product of the numbers, we have LCM = (24 * 36) / 12 = 72.
Correct Answer:
A
— 72
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Q. If the HCF of three numbers is 1, which of the following can be true?
-
A.
All three numbers are prime
-
B.
All three numbers are even
-
C.
All three numbers are odd
-
D.
Two numbers are even and one is odd
Solution
If the HCF is 1, it is possible that all three numbers are prime, as they would not share any common factors.
Correct Answer:
A
— All three numbers are prime
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Q. If the HCF of three numbers is 5 and their LCM is 300, what can be said about the numbers?
-
A.
They are all multiples of 5
-
B.
They are all prime
-
C.
They are all even
-
D.
They are all odd
Solution
Since the HCF is 5, all three numbers must be multiples of 5.
Correct Answer:
A
— They are all multiples of 5
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Q. If the HCF of three numbers is 5 and their LCM is 300, what is the product of the three numbers? (2023)
-
A.
1500
-
B.
3000
-
C.
6000
-
D.
7500
Solution
The product of the three numbers is equal to HCF * LCM = 5 * 300 = 1500.
Correct Answer:
B
— 3000
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Q. If the HCF of three numbers is 7 and their LCM is 420, what is the maximum possible value of the product of these three numbers? (2023)
-
A.
2940
-
B.
840
-
C.
1260
-
D.
2100
Solution
The product of the three numbers is equal to HCF * LCM. Thus, 7 * 420 = 2940.
Correct Answer:
C
— 1260
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Q. If the HCF of three numbers is 7 and their LCM is 420, what is the product of the three numbers? (2023)
-
A.
2940
-
B.
840
-
C.
2100
-
D.
1470
Solution
The product of the three numbers = HCF * LCM = 7 * 420 = 2940.
Correct Answer:
A
— 2940
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Q. If the HCF of two numbers is 1, what can be inferred about the two numbers? (2023)
-
A.
They are both prime.
-
B.
They are coprime.
-
C.
They are multiples of each other.
-
D.
They are both even.
Solution
If the HCF is 1, it means the two numbers are coprime, meaning they have no common factors other than 1.
Correct Answer:
B
— They are coprime.
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Q. If the HCF of two numbers is 1, what can be said about the two numbers? (2023)
-
A.
They are prime
-
B.
They are coprime
-
C.
They are even
-
D.
They are odd
Solution
If the HCF is 1, the two numbers are coprime, meaning they have no common factors other than 1.
Correct Answer:
B
— They are coprime
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Q. If the HCF of two numbers is 12 and their LCM is 120, what is the product of the two numbers?
-
A.
144
-
B.
1200
-
C.
240
-
D.
100
Solution
The product of two numbers is equal to the product of their HCF and LCM. Therefore, the product = 12 * 120 = 1440.
Correct Answer:
B
— 1200
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Q. If the HCF of two numbers is 12 and their LCM is 144, what is the product of the two numbers? (2023)
-
A.
1728
-
B.
864
-
C.
144
-
D.
288
Solution
The product of two numbers is equal to the product of their HCF and LCM. Therefore, 12 * 144 = 1728.
Correct Answer:
A
— 1728
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Q. If the HCF of two numbers is 5 and their LCM is 100, what is the sum of the two numbers if one of them is 25?
-
A.
50
-
B.
75
-
C.
100
-
D.
125
Solution
Let the other number be x. Then, HCF(25, x) = 5 and LCM(25, x) = 100. Therefore, 25 * x = 5 * 100, which gives x = 20. The sum is 25 + 20 = 45.
Correct Answer:
A
— 50
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