A Level Further Mathematics Core Pure Practice Test
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Prepare with the A Level Further Mathematics Core Pure Practice Test practice quiz. This question bank includes 10 questions covering matrix, determinant, roots, level, and further. Use it to review important concepts, identify knowledge gaps, and build confidence for the related exam, course, or assessment.
Sample Questions
Question
1
Which method correctly expresses the determinant of a 3x3 matrix by using 2x2 minors?
Explanation:
The determinant of a 3×3 matrix can be computed by expanding along a row into determinants of 2×2 minors. This means you take each entry in a chosen row, multiply it by the determinant of the 2×2 submatrix you get when you remove the row and the column of that entry, and sum with alternating signs. For a matrix with rows [a b c], [d e f], [g h i], the expansion along the first row is det = a*(ei − fh) − b*(di − fg) + c*(dh − eg). Each bracket is a determinant of a 2×2 minor. This is the method described: using 2×2 minors to form the determinant via expansion along a row (or a column). The other options don’t express the determinant in terms of 2×2 minors: the product of diagonals is not generally correct for arbitrary 3×3 matrices, and inversion or a simple sum of diagonal products does not capture the full determinant in general.
Question
2
Express z = 2[cos(120°) + i sin(120°)] in rectangular form.
Explanation:
Converting from polar (trigonometric) form to rectangular form uses the relationships x = r cos θ and y = r sin θ, where z = r[cos θ + i sin θ] = x + iy. Here r = 2 and θ = 120°. Knowing cos 120° = -1/2 and sin 120° = √3/2, the real part is 2(-1/2) = -1 and the imaginary part is 2(√3/2) = √3. So z = -1 + i√3. This places z in the second quadrant, where cosine is negative and sine is positive, matching the given angle.
Question
3
For A = [[4,1],[0,2]], which vector is an eigenvector corresponding to λ = 2?
Explanation:
To find an eigenvector for a given eigenvalue, solve Av = λv, which is the same as (A − λI)v = 0. Here A − 2I = [[4−2, 1], [0, 2−2]] = [[2, 1], [0, 0]]. The equation (A − 2I)v = 0 gives 2x + y = 0, so y = −2x. Thus any nonzero vector of the form (x, −2x) is an eigenvector for λ = 2, i.e., multiples of (1, −2). The vector (1, −2) satisfies A(1, −2) = (2, −4) = 2(1, −2), confirming it is an eigenvector with eigenvalue 2. Other listed vectors don’t satisfy Av = 2v.
Question
4
The Maclaurin expansion for ln(1+x) around 0 up to x^3 starts with which term?
Explanation:
Start with the pattern of the Maclaurin expansion for ln(1+x). It comes from integrating the geometric series for 1/(1+x), which is 1 - x + x^2 - x^3 + ... for |x| < 1. Integrating term by term from 0 to x gives ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ..., with the constant chosen so ln(1+0) = 0. So up to x^3, the expansion is x - x^2/2 + x^3/3. The x^2 term must be negative and the x^3 term positive, which is consistent with this pattern and rules out the other sign choices.
Question
5
Which expression equals α^n β^n γ^n δ^n?
Explanation:
Raising a product to a power distributes the power to each factor: (αβγδ)^n = α^n β^n γ^n δ^n. So the expression with α^n β^n γ^n δ^n is exactly the nth power of the product αβγδ. The other forms don’t match in general: a sum α^n β^n + γ^n δ^n is not a single product and would not simplify to α^n β^n γ^n δ^n; α^n β^n γ^n δ has the δ exponent only 1, not n; and α^n β^n γ^n / δ^n is α^n β^n γ^n δ^{-n}, which puts δ with exponent -n.
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A Level Further Mathematics Core Pure Practice Test
This practice set contains 10 questions from the matching question bank and focuses on matrix, determinant, roots, level, and further. Work through each question carefully, review the provided solutions, and revisit topics that need more study before your next attempt.
This is an independent study resource intended for practice and review; it is not an official examination or an endorsement by any organization named in the title.
Frequently Asked Questions
This quiz contains a total of 10 practice questions carefully selected to test your knowledge on this subject.
Yes, you will have exactly 0 minutes to complete the exam. A countdown timer will be visible once you start.
Yes, you can retake this practice test as many times as you need. The questions and options may be randomized on subsequent attempts to ensure comprehensive learning.
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