Further Algebra Practice Questions

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

141
Total
36
Easy
81
Medium
24
Hard

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Sample Questions from Further Algebra

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Q1
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.
Q2
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.
Q3
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.
Q4
Medium
If the quadratic equation ax^2 + bx + c = 0 has roots ฮฑ and ฮฒ, 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.
Q5
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.
Q6
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 โ‰  1, the (x - 1) terms cancel out, leaving us with x + 1.
Q7
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.
Q8
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.
Q9
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.

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Further Algebra โ€” 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 โ€” 36 easy, 81 medium, 24 hard โ€” 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.