Algebra & Number Theory
Q. If the HCF of two numbers is 15 and their LCM is 150, what is the product of the two numbers?
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A.
225
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B.
1500
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C.
300
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D.
75
Solution
The product of the two numbers is equal to the product of their HCF and LCM, which is 15 * 150 = 2250.
Correct Answer: B — 1500
Q. If the HCF of two numbers is equal to one of the numbers, what can be inferred?
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A.
The numbers are equal
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B.
One number is a multiple of the other
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C.
The numbers are coprime
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D.
The numbers are both prime
Solution
If the HCF is equal to one of the numbers, it means that one number is a multiple of the other.
Correct Answer: B — One number is a multiple of the other
Q. If the LCM of two numbers is 120 and one of the numbers is 30, what is the other number?
Solution
The other number can be found using the formula: LCM = (a * b) / HCF. Here, 120 = (30 * b) / HCF. Assuming HCF is 30, b = 120 / 30 = 40.
Correct Answer: A — 40
Q. If the LCM of two numbers is 36 and their HCF is 6, what is the product of the two numbers?
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A.
72
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B.
108
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C.
216
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D.
144
Solution
The product of the two numbers is LCM * HCF = 36 * 6 = 216.
Correct Answer: C — 216
Q. If the LCM of two numbers is 72 and one of the numbers is 8, what is the other number?
Solution
The other number can be found using the formula LCM(a, b) = (a * b) / HCF(a, b). Here, 72 = (8 * x) / HCF(8, x). The other number is 18.
Correct Answer: B — 18
Q. If the LCM of two numbers is 84 and one of the numbers is 12, what is the other number?
Solution
The other number can be found using the formula: LCM = (a * b) / HCF. Here, 84 = (12 * b) / HCF. The other number is 28.
Correct Answer: C — 28
Q. If the roots of the equation x^2 + px + q = 0 are 3 and -2, what is the value of p?
Solution
Using the sum of roots: p = -(3 + (-2)) = -1.
Correct Answer: B — 5
Q. If the roots of the equation x^2 - px + q = 0 are 3 and 4, what is the value of p?
Solution
The sum of the roots is p = 3 + 4 = 7.
Correct Answer: A — 7
Q. If x = 2^(3) and y = 2^(4), what is the value of x/y?
Solution
x/y = 2^(3)/2^(4) = 2^(3-4) = 2^(-1) = 1/2.
Correct Answer: C — 2
Q. If x = 2^(3/4), what is x^4?
Solution
x^4 = (2^(3/4))^4 = 2^3 = 8.
Correct Answer: C — 8
Q. If x = √(25), what is the value of x^2?
Solution
x = √(25) = 5, thus x^2 = 5^2 = 25.
Correct Answer: C — 25
Q. If x^2 + 4x + 4 = 0, what is the value of x?
Solution
This is a perfect square: (x + 2)^2 = 0, so x = -2.
Correct Answer: A — -2
Q. If x^2 - 5x + 6 = 0, what are the roots of the equation?
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A.
x = 1, 6
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B.
x = 2, 3
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C.
x = -2, -3
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D.
x = 0, 6
Solution
Factoring gives (x - 2)(x - 3) = 0, so x = 2 and x = 3.
Correct Answer: B — x = 2, 3
Q. If x^2 - 9 = 0, what are the values of x?
Solution
Factoring gives (x - 3)(x + 3) = 0, so x = -3 or x = 3.
Correct Answer: A — -3, 3
Q. Solve for x: 2x + 3 = 11
Solution
Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.
Correct Answer: B — 3
Q. Solve for x: 3x^2 - 12 = 0.
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A.
x = 2
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B.
x = -2
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C.
x = 4
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D.
x = -4
Solution
Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.
Correct Answer: A — x = 2
Q. Solve for x: 4x^2 - 12x + 9 = 0.
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A.
x = 1
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B.
x = 3
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C.
x = 2
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D.
x = 4
Solution
This factors to (2x - 3)(2x - 3) = 0, giving the double root x = 3.
Correct Answer: B — x = 3
Q. Solve for x: 4x^2 - 16 = 0.
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A.
x = 2
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B.
x = -2
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C.
x = 4
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D.
x = -4
Solution
Add 16 to both sides: 4x^2 = 16. Divide by 4: x^2 = 4. Thus, x = ±2.
Correct Answer: A — x = 2
Q. Solve for x: 5x^2 + 10x = 0.
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A.
x = 0, -2
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B.
x = 2, 0
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C.
x = -2, 2
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D.
x = 5, 0
Solution
Factoring gives 5x(x + 2) = 0, so x = 0 or x = -2.
Correct Answer: A — x = 0, -2
Q. Solve for x: x^2 + 6x + 9 = 0.
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A.
x = -3
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B.
x = 3
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C.
x = 0
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D.
x = -9
Solution
This factors to (x + 3)(x + 3) = 0, so x = -3.
Correct Answer: A — x = -3
Q. Solve for y: 3y + 4 = 19.
Solution
Subtract 4 from both sides: 3y = 15. Divide by 3: y = 5.
Correct Answer: C — 5
Q. What are the roots of the equation x^2 - 5x + 6 = 0?
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A.
1 and 6
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B.
2 and 3
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C.
3 and 4
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D.
0 and 6
Solution
Factoring gives (x - 2)(x - 3) = 0, so x = 2 and x = 3.
Correct Answer: B — 2 and 3
Q. What are the solutions to the equation x^2 + 2x - 8 = 0?
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A.
x = 2, -4
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B.
x = -2, 4
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C.
x = 4, -2
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D.
x = -4, 2
Solution
Factoring gives (x + 4)(x - 2) = 0, so x = -4 and x = 2.
Correct Answer: C — x = 4, -2
Q. What are the solutions to the equation x^2 + 4x + 4 = 0?
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A.
x = -2
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B.
x = 2
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C.
x = 0
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D.
x = -4
Solution
This factors to (x + 2)(x + 2) = 0, so the double root is x = -2.
Correct Answer: A — x = -2
Q. What is the discriminant of the equation 2x^2 + 3x + 1 = 0?
Solution
The discriminant is b^2 - 4ac = 3^2 - 4(2)(1) = 9 - 8 = 1.
Correct Answer: A — 1
Q. What is the discriminant of the equation 2x^2 - 4x + 2 = 0?
Solution
Discriminant D = b² - 4ac = (-4)² - 4(2)(2) = 16 - 16 = 0.
Correct Answer: A — 0
Q. What is the discriminant of the equation 3x^2 + 6x + 2 = 0?
Solution
The discriminant is b^2 - 4ac = 6^2 - 4*3*2 = 36 - 24 = 12.
Correct Answer: A — 0
Q. What is the HCF of 100 and 250?
Solution
The HCF of 100 and 250 is 50, as it is the largest number that divides both.
Correct Answer: A — 50
Q. What is the HCF of 14, 28, and 42?
Solution
The HCF of 14, 28, and 42 is 14, as it is the largest number that divides all three.
Correct Answer: A — 14
Q. What is the HCF of 24 and 36?
Solution
The HCF of 24 and 36 is 6, as it is the largest number that divides both.
Correct Answer: A — 6
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