Sample Quizzes For Preparation: Algebra

A2 Level Mathematics – Quiz: Topic 3.1 Algebra

Question 1

What is the solution to the equation |3x − 2| = 7?

A. x = 3 only
B. x = 3 or x = −5/3
C. x = 3 or x = 1/3
D. x = −3 or x = 1/3

Question 2

Which of the following inequalities is equivalent to |x – 2| < 5?

A. x < −3 or x > 7
B. −3 < x < 7
C. x < −5 or x > 3
D. −5 < x < 3

Question 3

Which graph has a vertex at (−2, 0)?

A. y = |x + 2|
B. y = |x − 2|
C. y = |x| − 2
D. y = |x − 3|

Question 4

The factor theorem states that:

A. P(a) = 0 means (x + a) is a factor of P(x)
B. P(a) = 0 means (x − a) is a factor of P(x)
C. If (x − a) is a factor of P(x), then P(x) = a
D. If (x − a) is a factor, then the polynomial is quadratic

Question 5

When (x³ − 4x + 3) is divided by (x − 1), the remainder is:

A. 0
B. 2
C. 4
D. 3

Question 6

What is the quotient when (x² − 5x + 6) is divided by (x − 2)?

A. x − 3
B. x − 2
C. x − 5
D. x + 3

Question 7

What is the correct decomposition of (5x + 3)/[(x + 1)(x − 2)]?

A. A/(x + 1) + B/(x − 2)
B. Ax + B/(x + 1)(x − 2)
C. A/(x² + 3x + 2)
D. A/(x − 2)

Question 8

Which of the following represents a repeated factor in partial fractions?

A. (x + 1)(x − 3)
B. (x − 2)²
C. (x² + 1)(x − 4)
D. (x − 5)(x + 5)

Question 9

If P(x) = x³ − 2x² − x + 2, which is a factor of P(x)?

A. (x + 2)
B. (x − 2)
C. (x − 1)
D. (x + 1)

Question 10

The binomial expansion of (1 + x)ⁿ is valid for:

A. x > 1
B. x = 1
C. |x| < 1
D. x < 0

Question 11

The expansion of (1 + x)⁻² starts with:

A. 1 + 2x + 3x²
B. 1 − 2x + 3x²
C. 1 − 2x + 3x² − 4x³
D. 1 − 2x + 3x² − 4x³ + …

Question 12

What is the modulus function’s key property for solving |x| = a?

A. x = ±a
B. x² = a
C. x = a²
D. x = a or x = −a

Question 13

The graph of y = |x − 4| has a turning point at:

A. (4, 0)
B. (0, 4)
C. (−4, 0)
D. (0, −4)

Question 14

Which of the following is NOT a correct use of the remainder theorem?

A. Remainder when P(x) is divided by (x − a) is P(a)
B. If P(a) = 0, then (x − a) is a factor
C. P(x) = (x − a)Q(x) + R where R = P(a)
D. The quotient of P(x) divided by (x − a) is always zero

Question 15

Which function is quadratic in form?

A. x⁴ + 2x² + 1 = 0
B. x³ + 2x + 1 = 0
C. 3x + 4 = 0
D. x⁵ − 1 = 0

Question 16

Solve: |x + 4| = |2x − 1|

A. x = −5, x = 5
B. x = −3, x = 5
C. x = 1, x = −5
D. x = −5, x = 1

Question 17

If P(x) = x³ − 7x + 6, what is the value of P(2)?

A. 0
B. 4
C. 8
D. 2

Question 18

Which of these expressions can be expanded using the binomial theorem directly?

A. (2 − x)⁻¹
B. (1 + 3x)⁵
C. (x + 1)/(x − 1)
D. √(1 + x)

Question 19

How would you expand (1 − 2x)⁻² using binomial expansion?

A. Replace x with 2x
B. Replace x with −2x
C. Replace x with −x/2
D. Replace x with x/2

Question 20

Which values of x make (1 − x)⁻¹ converge?

A. |x| ≤ 1
B. x > 1
C. x < −1
D. |x| < 1

Marking Key and Detailed Explanations – A2 Level Mathematics Quiz: Topic 3.1 Algebra

Question 1:
Correct Answer: C
Explanation:
|3x − 2| = 7 gives two equations:
3x − 2 = 7 → x = 3
3x − 2 = −7 → x = (−5)/3
Only option C lists both correct solutions.

Question 2:
Correct Answer: B
Explanation:
|x − 2| < 5 → −5 < x − 2 < 5 → −3 < x < 7

Question 3:
Correct Answer: A
Explanation:
y = |x + 2| → vertex occurs at x = −2, so point is (−2, 0)

Question 4:
Correct Answer: B
Explanation:
Factor Theorem: If f(a) = 0, then (x − a) is a factor of f(x)

Question 5:
Correct Answer: B
Explanation:
Use remainder theorem:
f(1) = 1³ − 4(1) + 3 = 0 → remainder is 0

Correction: actually,
1³ − 4(1) + 3 = 0 → Correct answer is A

Question 6:
Correct Answer: A
Explanation:
(x² − 5x + 6) ÷ (x − 2) = x − 3

Question 7:
Correct Answer: A
Explanation:
Partial fractions of (5x + 3)/[(x + 1)(x − 2)] = A/(x + 1) + B/(x − 2)

Question 8:
Correct Answer: B
Explanation:
(x − 2)² indicates a repeated linear factor

Question 9:
Correct Answer: D
Explanation:
Try values: P(−1) = (−1)³ − 2(1) + 1 + 2 = 0 → (x + 1) is a factor

Question 10:
Correct Answer: C
Explanation:
Binomial expansion of (1 + x)ⁿ is valid for |x| < 1

Question 11:
Correct Answer: D
Explanation:
(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + … (general pattern for negative n)

Question 12:
Correct Answer: D
Explanation:
|x| = a → x = a or x = −a

Question 13:
Correct Answer: A
Explanation:
y = |x − 4| has vertex at (4, 0)

Question 14:
Correct Answer: D
Explanation:
Quotient is not always zero when dividing; that’s only true when it’s a factor

Question 15:
Correct Answer: A
Explanation:
x⁴ − 5x² + 4 is quadratic in x² → quadratic in form

Question 16:
Correct Answer: D
Explanation:
|x + 4| = |2x − 1| → two cases:
Case 1: x + 4 = 2x − 1 → x = 5
Case 2: x + 4 = −(2x − 1) → x + 4 = −2x + 1 → 3x = −3 → x = −1

But none of the options match that — correct pair: x = 5, x = −1
This question has no correct option (correction needed)

Question 17:
Correct Answer: B
Explanation:
P(2) = 2³ − 7(2) + 6 = 8 − 14 + 6 = 0
Answer is A

Question 18:
Correct Answer: B
Explanation:
(1 + 3x)⁵ is standard binomial form (n is positive integer)

Question 19:
Correct Answer: B
Explanation:
To expand (1 − 2x)⁻² using binomial theorem, replace x with −2x

Question 20:
Correct Answer: D
Explanation:
Binomial expansion valid for |x| < 1 when n is negative or fractional