Further Algebra Practice Questions

A-Levels · A-Level Further Mathematics · 141 free MCQs with instant results and detailed explanations.

141

Total

Easy

Medium

Hard

Start Practicing Further Algebra

Take a timed quiz or customize your practice session

Quick Quiz (10 Qs) → Mock Test (25 Qs) ⚙ Customize

Sample Questions from Further Algebra

Here are 10 sample questions. Start a quiz to get randomized questions with scoring.

Easy

What is the maximum degree of the polynomial obtained by multiplying (x + 1)(x^2 - 3x + 2)?

A. 1

B. 2

C. 3

D. 4

Show Answer & Explanation

Correct Answer: C

The maximum degree of a product of polynomials is the sum of their degrees. (x + 1) has degree 1 and (x^2 - 3x + 2) has degree 2, so the maximum degree is 1 + 2 = 3.

Easy

If the quadratic equation x^2 - 5x + 6 = 0 is solved, what are the roots?

A. 2 and 3

B. 1 and 6

C. 3 and 4

D. 2 and 4

Show Answer & Explanation

Correct Answer: A

The roots of the quadratic equation x^2 - 5x + 6 = 0 can be calculated using factoring. The equation factors to (x - 2)(x - 3) = 0, giving roots x = 2 and x = 3.

Easy

If f(x) = 3x + 2, what is f(4)?

A. 12

B. 14

C. 10

D. 16

Show Answer & Explanation

Correct Answer: B

To find f(4), substitute 4 into the function: f(4) = 3(4) + 2 = 12 + 2 = 14.

Medium

If the quadratic equation ax^2 + bx + c = 0 has roots α and β, which of the following expressions represents the sum of the roots?

A. -b/a

B. b/a

C. c/a

D. -c/b

Show Answer & Explanation

Correct Answer: A

The sum of the roots of a quadratic equation is given by the formula -b/a according to Vieta's formulas.

Medium

What is the value of k if the polynomial f(x) = x^3 - 3x^2 + kx + 4 has a local maximum at x=1?

A. 2

B. 3

C. 4

D. 5

Show Answer & Explanation

Correct Answer: B

To find k, we set the first derivative f'(x) = 0 at x=1. f'(x) = 3x^2 - 6x + k. Setting f'(1) = 0 gives 3(1)^2 - 6(1) + k = 0, leading to k = 3.

Medium

Given the expression (x^2 - 1)/(x - 1), what is its simplified form when x is not equal to 1?

A. x + 1

B. x - 1

C. x^2 + 1

D. 1

Show Answer & Explanation

Correct Answer: A

The expression can be factored as (x - 1)(x + 1)/(x - 1). When x ≠ 1, the (x - 1) terms cancel out, leaving us with x + 1.

Medium

If a polynomial P(x) has a factor (x - 2) and leaves a remainder of 3 when divided by (x + 1), what is P(2)?

A. 3

B. 5

C. 7

D. 0

Show Answer & Explanation

Correct Answer: A

Since (x - 2) is a factor, P(2) = 0. The remainder when divided by (x + 1) does not affect the value of P(2), hence P(2) = 0 + 3 = 3.

Hard

If the polynomial P(x) = 3x^4 - 8x^3 + 2x^2 - k has a repeated root at x = 2, what is the value of k?

A. 10

B. 14

C. 18

D. 22

Show Answer & Explanation

Correct Answer: B

To find k, we first substitute x = 2 into P(x) and also find P'(x) to ensure it is zero at the same point. Setting P(2) = 0 and P'(2) = 0 gives k = 14, confirming its validity.

Hard

Consider the polynomial function f(x) = 2x^3 - 6x^2 + 4x - 12. What is the value of x for which f(x) = 0?

A. 2

B. 3

C. 4

D. 1

Show Answer & Explanation

Correct Answer: A

The polynomial can be factored by finding its roots. By applying synthetic division or the Rational Root Theorem, we can find that f(2) = 0. Thus, x = 2 is a root of the polynomial f(x).

Q10

Hard

If the sequence defined by a_n = 3n^2 + 2n + 1 has its nth term multiplied by a factor of k = 2, what will be the sum of the first 5 terms of the new sequence?

A. 130

B. 132

C. 134

D. 136

Show Answer & Explanation

Correct Answer: B

The original sequence's first five terms are calculated as a_1 = 6, a_2 = 17, a_3 = 34, a_4 = 57, and a_5 = 86. After multiplying by k = 2, the new sequence's sum is 2*(6 + 17 + 34 + 57 + 86) = 132.

Showing 10 of 141 questions. Start a quiz to practice all questions with scoring and timer.

Practice All 141 Questions →

More A-Level Further Mathematics Chapters

119 Qs →

149 Qs →

142 Qs →

151 Qs →

150 Qs →

6 Hyperbolic Functions

149 Qs →

7 Differential Equations

149 Qs →

Other A-Levels Subjects

🧬 A-Level Biology → 🧪 A-Level Chemistry → 💻 A-Level Computer Science → 📐 A-Level Mathematics → ⚛️ A-Level Physics →

Further Algebra — A-Levels A-Level Further Mathematics Practice Questions Online

This page contains 141 practice MCQs for the chapter Further Algebra in A-Levels A-Level Further Mathematics. The questions are organized by difficulty — 36 easy, 81 medium, 24 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.