Algebra & Number Theory for Competitive Exam Success

Algebra & Number Theory

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

Q. What is the value of log7(1)?

A. 0
B. 1
C. 7
D. undefined

Solution

log7(1) = 0 because 7^0 = 1.

Correct Answer: A — 0

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Q. What is the value of x in the equation 2x + 3 = 11?

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

Solution

Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.

Correct Answer: C — 4

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Q. What is the value of x in the equation 3(x - 2) = 9?

A. 1
B. 3
C. 5
D. 7

Solution

Divide both sides by 3: x - 2 = 3. Add 2: x = 5.

Correct Answer: C — 5

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Q. What is the value of x in the equation 7x - 2 = 5x + 10?

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

Solution

Subtract 5x from both sides: 2x - 2 = 10. Add 2: 2x = 12. Divide by 2: x = 6.

Correct Answer: A — 4

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Q. What is the value of x in the equation x^2 - 2x - 8 = 0?

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

Solution

Factoring gives (x - 4)(x + 2) = 0, so x = 4 or x = -2.

Correct Answer: C — 4

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Q. What is the value of √(16) + √(9)?

A. 5
B. 7
C. 9
D. 11

Solution

√(16) = 4 and √(9) = 3, so 4 + 3 = 7.

Correct Answer: B — 7

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Q. What is the value of √(25) - √(16)?

A. 1
B. 3
C. 5
D. 7

Solution

√(25) = 5 and √(16) = 4, so 5 - 4 = 1.

Correct Answer: B — 3

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Q. What is the vertex of the parabola represented by the equation y = x^2 - 6x + 8?

A. (3, -1)
B. (3, -4)
C. (6, 8)
D. (0, 8)

Solution

The vertex is at x = -b/(2a) = 3. Plugging in gives y = -1.

Correct Answer: A — (3, -1)

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Q. Which of the following equations has no real solutions?

A. x^2 + 1 = 0
B. x^2 - 1 = 0
C. x^2 + 4 = 0
D. x^2 - 4 = 0

Solution

x^2 + 1 = 0 leads to x^2 = -1, which has no real solutions.

Correct Answer: A — x^2 + 1 = 0

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Q. Which of the following is a root of the quadratic equation x^2 - 5x + 6 = 0?

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

Solution

Factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.

Correct Answer: B — x = 2

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Q. Which of the following is equal to 5^(2/3)?

A. √(25)
B. 5√(5)
C. 25
D. √(125)

Solution

5^(2/3) = (5^(1/3))^2 = 5√(5).

Correct Answer: B — 5√(5)

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Q. Which of the following is equivalent to (x^(1/2))^4?

A. x^(2)
B. x^(1)
C. x^(1/4)
D. x^(1/8)

Solution

Using the power of a power property, (x^(1/2))^4 = x^(1/2 * 4) = x^(2).

Correct Answer: A — x^(2)

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Q. Which of the following is equivalent to 2^(3) * 2^(2)?

A. 2^(5)
B. 2^(6)
C. 2^(7)
D. 2^(8)

Solution

Using the property a^m * a^n = a^(m+n), we have 2^(3+2) = 2^(5).

Correct Answer: A — 2^(5)

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Q. Which of the following is equivalent to 2^(3/2)?

A. √8
B. 4
C. 2√2
D. 8

Solution

2^(3/2) = √(2^3) = √8.

Correct Answer: A — √8

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Q. Which of the following is equivalent to log10(0.01)?

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

Solution

log10(0.01) = log10(10^-2) = -2.

Correct Answer: B — -2

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Q. Which of the following is equivalent to log5(25)?

A. 1
B. 2
C. 0.5
D. 5

Solution

log5(25) = log5(5^2) = 2.

Correct Answer: B — 2

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Q. Which of the following is true for the expression 3^(2x) = 27?

A. x = 1
B. x = 2
C. x = 3
D. x = 0

Solution

27 = 3^(3), so 2x = 3, thus x = 3/2.

Correct Answer: A — x = 1

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Q. Which of the following is true for the expression 4^(x) = 16?

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

Solution

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

Correct Answer: B — x = 2

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Q. Which of the following is true for the expression 5^(x) = 25?

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

Solution

25 = 5^(2), thus x = 2.

Correct Answer: B — x = 2

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Q. Which of the following is true for √(x^2)?

A. x
B. -x
C.
D. x
.
. x^2

Solution

√(x^2) = |x|.

Correct Answer: C —

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Q. Which of the following pairs of numbers has an HCF of 10?

A. 20 and 30
B. 25 and 35
C. 40 and 50
D. 15 and 25

Solution

The HCF of 40 and 50 is 10, as it is the largest number that divides both.

Correct Answer: C — 40 and 50

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Algebra & Number Theory

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