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 HCF of 56, 98, and 42?

A. 14
B. 7
C. 28
D. 21

Solution

The HCF of 56, 98, and 42 is 14, as it is the largest number that divides all three.

Correct Answer: A — 14

Q. What is the HCF of 81 and 27?

A. 27
B. 54
C. 81
D. 9

Solution

The HCF of 81 and 27 is 27, as it is the largest number that divides both.

Correct Answer: A — 27

Q. What is the LCM of 12, 15, and 20?

A. 60
B. 120
C. 180
D. 240

Solution

The LCM of 12, 15, and 20 is 60, as it is the smallest number that is a multiple of all three.

Correct Answer: B — 120

Q. What is the LCM of 4 and 5?

A. 10
B. 20
C. 15
D. 5

Solution

The LCM of 4 and 5 is 20, as it is the smallest number that is a multiple of both.

Correct Answer: A — 10

Q. What is the LCM of 6 and 8?

A. 12
B. 24
C. 18
D. 30

Solution

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

Correct Answer: B — 24

Q. What is the LCM of 6, 8, and 12?

A. 24
B. 48
C. 36
D. 60

Solution

The LCM of 6, 8, and 12 is 24, as it is the smallest number that is a multiple of all three.

Correct Answer: A — 24

Q. What is the LCM of 8, 12, and 15?

A. 120
B. 60
C. 30
D. 90

Solution

The LCM of 8, 12, and 15 is 120, as it is the smallest number that is a multiple of all three.

Correct Answer: A — 120

Q. What is the LCM of 9 and 21?

A. 63
B. 189
C. 27
D. 42

Solution

The LCM of 9 and 21 is 63, as it is the smallest number that is a multiple of both.

Correct Answer: A — 63

Q. What is the product of the roots of the equation x^2 + 3x - 10 = 0?

A. -10
B. 10
C. 3
D. -3

Solution

The product of the roots is c/a = -10/1 = -10.

Correct Answer: A — -10

Q. What is the product of the roots of the equation x^2 + 5x + 6 = 0?

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

Solution

The product of the roots is c/a = 6/1 = 6.

Correct Answer: A — 6

Q. What is the product of the roots of the equation x^2 - 7x + 10 = 0?

A. 10
B. 7
C. 5
D. 3

Solution

The product of the roots is given by c/a = 10/1 = 10.

Correct Answer: A — 10

Q. What is the relationship between HCF and LCM of two numbers a and b?

A. HCF = a + b
B. HCF * LCM = a * b
C. HCF + LCM = a * b
D. HCF - LCM = a - b

Solution

The relationship is HCF * LCM = a * b, which holds for any two integers a and b.

Correct Answer: B — HCF * LCM = a * b

Q. What is the simplified form of (√(2) * √(8))?

A. √(16)
B. 4
C. 2√(2)
D. 8

Solution

√(2) * √(8) = √(16) = 4, but simplified is 2√(2).

Correct Answer: C — 2√(2)

Q. What is the simplified form of (√(a^2) * √(b^2))?

A. ab
B. a^2b^2
C. a + b
D. a + b^2

Solution

√(a^2) * √(b^2) = ab.

Correct Answer: A — ab

Q. What is the sum of the roots of the equation x^2 - 5x + 6 = 0?

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

Solution

The sum of the roots is given by -b/a = 5/1 = 5.

Correct Answer: A — 5

Q. What is the sum of the roots of the equation x^2 - 6x + 8 = 0?

A. 6
B. 8
C. 4
D. 2

Solution

The sum of the roots is given by -b/a = 6/1 = 6.

Correct Answer: A — 6

Q. What is the sum of the roots of the equation x^2 - 7x + 10 = 0?

A. 5
B. 7
C. 10
D. 12

Solution

The sum of the roots is given by -b/a = 7/1 = 7.

Correct Answer: B — 7

Q. What is the value of (3^(2))^(3)?

A. 3^(5)
B. 3^(6)
C. 3^(7)
D. 3^(8)

Solution

Using the property (a^m)^n = a^(m*n), we have (3^(2))^(3) = 3^(2*3) = 3^(6).

Correct Answer: B — 3^(6)

Q. What is the value of (√(2) * √(8))?

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

Solution

√(2) * √(8) = √(16) = 4.

Correct Answer: B — 4

Q. What is the value of (√(2) + √(3))^2?

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

Solution

(√(2) + √(3))^2 = 2 + 3 + 2√(6) = 5 + 2√(6).

Correct Answer: C — 7

Q. What is the value of (√(3) + √(2))^2?

A. 5
B. 7
C. 10
D. 12

Solution

(√(3) + √(2))^2 = 3 + 2 + 2√(6) = 5 + 2√(6).

Correct Answer: C — 10

Q. What is the value of 4^(1/2) * 4^(1/2)?

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

Solution

4^(1/2) = 2, thus 2 * 2 = 4.

Correct Answer: B — 4

Q. What is the value of 4^(1/2)?

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

Solution

4^(1/2) = √(4) = 2.

Correct Answer: A — 2

Q. What is the value of 4^(3/2)?

A. 8
B. 16
C. 32
D. 64

Solution

4^(3/2) = (√4)^3 = 2^3 = 8.

Correct Answer: B — 16

Q. What is the value of 5^(0)?

A. 0
B. 1
C. 5
D. undefined

Solution

Any non-zero number raised to the power of 0 is 1.

Correct Answer: B — 1

Q. What is the value of 9^(1/2) + 16^(1/2)?

A. 5
B. 7
C. 9
D. 10

Solution

9^(1/2) = 3 and 16^(1/2) = 4, so 3 + 4 = 7.

Correct Answer: D — 10

Q. What is the value of log4(64)?

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

Solution

log4(64) = log4(4^3) = 3.

Correct Answer: D — 5

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

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

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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