Algebra & Number Theory for Competitive Exam Success

Algebra & Number Theory

HCF and LCM Linear and Quadratic Equations Logarithms and Applications Surds and Indices

Q. Find the HCF of 48, 60, and 72.

A. 12
B. 24
C. 6
D. 18

Solution

The HCF of 48, 60, and 72 is 12, as it is the largest number that divides all three.

Correct Answer: B — 24

Q. Find the HCF of 48, 64, and 80.

A. 8
B. 16
C. 24
D. 32

Solution

The HCF of 48, 64, and 80 is 16, as it is the largest number that divides all three.

Correct Answer: B — 16

Q. Find the LCM of 6 and 8.

A. 24
B. 12
C. 48
D. 36

Solution

The LCM of 6 and 8 is 24, as it is the smallest number that is a multiple of both.

Correct Answer: A — 24

Q. Find the LCM of 9 and 15.

A. 45
B. 30
C. 60
D. 75

Solution

The LCM of 9 and 15 is 45, as it is the smallest number that is a multiple of both.

Correct Answer: A — 45

Q. Find the value of x in the equation 3x^2 - 12 = 0.

A. -2
B. 2
C. 4
D. 0

Solution

Add 12 to both sides: 3x^2 = 12. Divide by 3: x^2 = 4. Thus, x = ±2.

Correct Answer: B — 2

Q. Find the value of x in the equation 4x^2 + 8x + 3 = 0.

A. x = -1
B. x = -3/2
C. x = -1/2
D. x = -3

Solution

Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a gives x = [-8 ± √(64 - 48)] / 8 = [-8 ± 4] / 8.

Correct Answer: B — x = -3/2

Q. Find the value of x in the equation 4x^2 - 16x + 15 = 0.

A. 1
B. 3
C. 5
D. 4

Solution

Factoring gives (4x - 3)(x - 5) = 0, so x = 3 or x = 5.

Correct Answer: B — 3

Q. Find the value of x in the equation 5x - 2 = 3x + 6.

A. x = 2
B. x = 3
C. x = 4
D. x = 5

Solution

Subtract 3x from both sides: 2x - 2 = 6. Add 2: 2x = 8. Divide by 2: x = 4.

Correct Answer: A — x = 2

Q. Find the value of x in the equation 5x - 2(3 - x) = 4.

A. x = 1
B. x = 2
C. x = 3
D. x = 4

Solution

Distributing gives 5x - 6 + 2x = 4. Combining like terms: 7x - 6 = 4. Thus, 7x = 10, x = 10/7.

Correct Answer: B — x = 2

Q. Find the value of x in the equation x^2 + 6x + 9 = 0.

A. x = -3
B. x = 3
C. x = 0
D. x = -9

Solution

This is a perfect square: (x + 3)^2 = 0, so x = -3.

Correct Answer: A — x = -3

Q. If 2x^2 + 3x - 5 = 0, what is the value of x using the quadratic formula?

A. x = 1
B. x = -1
C. x = 2
D. x = -2

Solution

Using the quadratic formula x = [-b ± √(b² - 4ac)] / 2a, we find x = [-3 ± √(9 + 40)] / 4.

Correct Answer: B — x = -1

Q. If 2x^2 + 8x + 6 = 0, what is the value of x?

A. x = -1
B. x = -3
C. x = -2
D. x = -4

Solution

Dividing the equation by 2 gives x^2 + 4x + 3 = 0, which factors to (x + 1)(x + 3) = 0.

Correct Answer: B — x = -3

Q. If 2^(x) = 16, what is the value of x?

A. 2
B. 3
C. 4
D. 5

Solution

16 = 2^(4), so x = 4.

Correct Answer: C — 4

Q. If 2^(x) = 32, what is the value of x?

A. 4
B. 5
C. 6
D. 7

Solution

32 = 2^(5), so x = 5.

Correct Answer: B — 5

Q. If 3^(x) = 81, what is the value of x?

A. 2
B. 3
C. 4
D. 5

Solution

81 = 3^(4), so x = 4.

Correct Answer: C — 4

Q. If 4x + 2 = 18, what is the value of x?

A. 3
B. 4
C. 5
D. 6

Solution

Subtract 2 from both sides: 4x = 16. Divide by 4: x = 4.

Correct Answer: A — 3

Q. If 4x^2 - 16 = 0, what is the value of x?

A. x = 2
B. x = -2
C. x = 4
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. If 5x - 2 = 3x + 6, what is the value of x?

A. x = 2
B. x = 3
C. x = 4
D. x = 5

Solution

Subtract 3x from both sides: 2x - 2 = 6. Add 2: 2x = 8. Divide by 2: x = 4.

Correct Answer: B — x = 3

Q. If 5x - 7 = 3x + 5, what is the value of x?

A. 1
B. 6
C. 8
D. 4

Solution

Subtract 3x from both sides: 2x - 7 = 5. Add 7 to both sides: 2x = 12. Divide by 2: x = 6.

Correct Answer: A — 1

Q. If 5x - 7 = 3x + 9, what is the value of x?

A. 8
B. 7
C. 6
D. 5

Solution

Subtract 3x from both sides: 2x - 7 = 9. Add 7: 2x = 16. Divide by 2: x = 8.

Correct Answer: A — 8

Q. If 5x^2 - 20 = 0, what is the value of x?

A. -2
B. 2
C. 4
D. 0

Solution

Add 20 to both sides: 5x^2 = 20. Divide by 5: x^2 = 4. Thus, x = ±2.

Correct Answer: B — 2

Q. If a = 2^(3) and b = 2^(2), what is the value of a/b?

A. 2^(1)
B. 2^(2)
C. 2^(3)
D. 2^(5)

Solution

a/b = 2^(3)/2^(2) = 2^(3-2) = 2^(1).

Correct Answer: A — 2^(1)

Q. If a = 3 and b = 2, what is the value of a^(b+1)?

A. 6
B. 9
C. 27
D. 81

Solution

a^(b+1) = 3^(2+1) = 3^3 = 27.

Correct Answer: C — 27

Q. If a = 3^(1/2), what is a^4?

A. 9
B. 27
C. 81
D. 3

Solution

a^4 = (3^(1/2))^4 = 3^2 = 9.

Correct Answer: C — 81

Q. If b = 2^(1/3), what is b^6?

A. 4
B. 8
C. 16
D. 32

Solution

b^6 = (2^(1/3))^6 = 2^2 = 4.

Correct Answer: B — 8

Q. If log2(8) = x, what is the value of x?

A. 2
B. 3
C. 4
D. 5

Solution

log2(8) = log2(2^3) = 3.

Correct Answer: B — 3

Q. If log3(27) = x, what is the value of x?

A. 1
B. 2
C. 3
D. 4

Solution

27 = 3^3, so x = 3.

Correct Answer: C — 3

Q. If log3(9) + log3(3) = x, what is the value of x?

A. 1
B. 2
C. 3
D. 4

Solution

log3(9) = 2 and log3(3) = 1, so x = 2 + 1 = 3.

Correct Answer: B — 2

Q. If logx(16) = 4, what is the value of x?

A. 2
B. 4
C. 8
D. 16

Solution

x = 16^(1/4) = 2.

Correct Answer: B — 4

Q. If the HCF of two numbers is 1, what can be said about the numbers?

A. They are even
B. They are odd
C. They are coprime
D. They are multiples of each other

Solution

If the HCF is 1, the numbers are coprime, meaning they have no common factors other than 1.

Correct Answer: C — They are coprime

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