A Level Further Mathematics Core Pure Practice Test

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• Which of the following expressions represents the volume of a solid formed by revolving y = f(x) about the x-axis, using integration with respect to x?
• If the radius of a cone is doubled while height remains fixed, by what factor does the volume increase?
• For the quadratic x^2 + 2x + 1 = 0, how many real roots does it have?
• In matrix notation, what does the expression |A| denote?
• For the quadratic z^2 - z + 2 = 0, which statement about its roots is true?
• The determinant of a 3x3 matrix is computed by expanding along a row or column using cofactors. Which name describes this method?
• In the Maclaurin series for sin x, what is the coefficient of x^5?
• Which statement is true about a 2x2 matrix A regarding invertibility?
• For z^2 - z + 2 = 0, what is the sum of its roots?
• Consider the quadratic x^2 + 3x + 2 = 0. What is the nature of its roots?
• Given a_0 = 4 and a_n = 3 a_{n-1} − 2. What is a_4?
• Which value equals (α+β)^3?
• Solve y' + 2y = e^{−x}, with y(0)=0. Which is y(x)?
• If z1, z2 are roots of z^2 - z + 2 = 0, what is z1 z2?
• Which condition ensures a matrix is non-singular?
• The volume of a solid formed by rotating the curve y = f(x) about the x-axis through a full rotation is given by which integral?
• The circle equation |z - (a + ib)| = r has its centre at which point?
• Arc length of r(t) = (cos t, sin t, t) for t in [0, 2π] is which of the following?
• In an induction proof, what is the purpose of the base case?
• Which is a correct simplification of 1/α + 1/β?
• For the same quartic, which expression equals αβγδ?
• If the quartic is monic, the sum of triple products αβγ + αβδ + αγδ + βγδ equals which expression?
• The two independent real solutions corresponding to the complex roots m = -1 ± i are:
• In the vector plane equation R = A + λ B + μ C, what are λ and μ?
• Compute cos θ where θ is the angle between a=(1,2,3) and b=(-1,0,1). Which of the following equals cos θ?
• What is the complex conjugate of z = a + bi?
• Compute the value of the telescoping series ∑_{n=1}^∞ (1/n - 1/(n+1)).
• The modulus |z| of a complex number represents what geometric quantity?
• Which expression equals α^3 + β^3 + γ^3 in terms of α+β+γ and αβ+βγ+αγ and αβγ?
• The modulus of the quotient z1/z2 equals which expression?
• The center of the circle |z - (a + ib)| = r has Cartesian coordinates (a, b). Which of the following is the center in the complex plane?
• Which of the following statements characterizes a non-singular matrix?
• The sum of squares S = α^2 + β^2 + γ^2 + δ^2 can be written in terms of a, b, c as which of the following?
• Which statement about the determinant is true?
• The argument of a complex number is defined as what?
• Solve the linear system: x + 2y = 5 and 3x + 4y = 6. Find x and y.
• For A = [[4,1],[0,2]], which statement is true?
• What can be said about z z* for any complex number z?
• What is i^4?
• If a > 0 and all four roots are positive, what can be said about the sign of b?
• Find the stationary points of f(x) = x^4 - 4x^2 + 2 and determine their type using f''.
• Which of the following is a correct description of the roots of z^2 + 4z + 5 = 0?
• The four roots of z^4 = -16 have arguments (angles) of which values?
• Which is the expression for 1/α + 1/β + 1/γ?
• The polynomial p(z) = z^2 - 2z + 5 has complex roots. Find both roots and verify they are conjugates.
• The product of the roots of z^2 - (α+β) z + αβ = 0 is:
• Which expression verifies that a unit vector u = (a,b,c) has magnitude 1?
• True or False: A matrix with two distinct eigenvalues is diagonalizable.
• 1/α + 1/β + 1/γ + 1/δ equals which expression?
• For the quadratic 2x^2 - 3x - 2 = 0 with roots α and β, α + β equals what?
• Which expression equals α^n β^n γ^n δ^n?
• Using Euler's formula e^{i θ} = cos θ + i sin θ, evaluate e^{iπ}.
• The argument of the product z1 z2 equals which expression?
• Let α, β, γ, δ be the roots of the quartic a x^4 + b x^3 + c x^2 + d x + e = 0, with a ≠ 0. Which expression equals α + β + γ + δ?
• If a 2x2 matrix has determinant zero, what can be said about the matrix?
• The modulus of the product of two complex numbers z1 and z2 equals which of the following?
• When evaluating a 3x3 determinant by expanding along the first row, which statement is correct?
• For a quadratic equation ax^2 + bx + c = 0 with real coefficients, if b^2 - 4ac > 0, what is the nature of the roots?
• What is the standard form of a complex number?
• Let b_0 = 0 and b_n = b_{n-1} + n. What is b_5?
• The derivative of which function is 1/(1+x^2)?
• In sigma notation, what does the number below the sigma represent?
• Find stationary points of f(x) = x^3 - 3x and classify using f' and f''. Which statement is correct?
• Which expression equals (α+β+γ+δ)^2?
• If α and β are the roots of ax^2 + bx + c = 0, then α + β equals what?
• α^2 + β^2 equals which of the following?
• Let A = [[1,2],[3,4]]. Which of the following is A^{-1}?
• For A = [[4,1],[0,2]], which vector is an eigenvector corresponding to λ = 4?
• In the cone volume formula V = (1/3) π r^2 h, which variable is squared?
• Express 3 − 4i in polar form z = r(cos θ + i sin θ). What are r and θ (principal value)?
• α^3 + β^3 equals which identity?
• For invertible matrices P and Q, which is the correct formula for the inverse of the product PQ?
• Represent z = -4 in polar form.
• Find the principal argument of z = -1 − i.
• Expanding (z - α)(z - β) = 0 yields which quadratic?
• For the same quartic, which expression equals αβ + αγ + αδ + βγ + βδ + γδ?
• Which statement correctly identifies a singular matrix?
• Which statement best describes a non-singular matrix?
• Let r(t) = (cos t, sin t, t). What is the speed |r'(t)|?
• Are eigenvectors corresponding to distinct eigenvalues always linearly independent?
• For z = 3 − 4i, compute z^5 in a+bi form.
• If |z1| = 3 and |z2| = 4, what is |z1 z2|?
• If z1 = a + bi and z2 = c + di, and z1 = z2, which must be true?
• The principal argument of z = -2 - 2√3 i is which value?
• Volume of a cylinder equals which expression?
• Sum of the first n terms of a geometric sequence with a1 = 2 and r = -1/2 is S_n = (4/3)(1 - (-1/2)^n). Evaluate S_5.
• Compute the indefinite integral ∫ (2x)/(x^2+1) dx.
• For any integer n, the product (α^n)(β^n)(γ^n)(δ^n) equals which expression?
• The volume of a solid formed by rotating the curve x = g(y) about the y-axis is given by which integral?
• Which of the following best defines a singular matrix?
• Which expression gives α^3 + β^3 + γ^3 + δ^3 in terms of sums and products?
• If arg z1 = π/4 and arg z2 = π/6, what is arg(z1 z2) (modulo 2π)?
• If a = 1, the sum α + β + γ + δ equals which of the following?
• Which expression correctly states the conventional order of matrix dimensions?
• The sum of the triple products αβγ + αβδ + αγδ + βγδ equals which expression in terms of a and d?
• Compute (2 cis(π/4))^4.
• Which of the following is not a valid polar form representation of a complex number as given?
• What is the limit lim_{x→0} (sin x)/x?
• On an Argand diagram, the x-axis corresponds to which axis?
• Determine the radius of convergence of ∑ x^n / n!.
• For the matrix A = [ [2, 1], [0, 3] ], which statement about eigenvalues, eigenvectors and diagonalizability is correct?
• Compute the magnitude and argument of z = -3 + 3i.
• For the Euler-Cauchy equation with complex roots m = -1 ± i, which of the following is a correct form of the general real solution?
• Compute the determinant of the matrix [ [1,0,0], [2,1,0], [3,4,1] ].
• In the Maclaurin expansion for ln(1+x) around 0, what is the coefficient of x^2?
• Let A = [ [0, 1], [1, 0] ]. Which of the following is a correct statement about A^2 and A^3?
• Which statement about eigenpairs of A = [[2,1],[0,3]] is true?
• The determinant of a 2x2 matrix [ [a, b], [c, d] ] is:
• Which of the following is equal to (αβ)^n?
• What is i^2?
• For A = [[4,1],[0,2]], which vector is an eigenvector corresponding to λ = 2?
• Solve the first-order linear ODE y' + y = e^{2x} with y(0) = 1. Which is the correct solution?
• Compute the determinant of the matrix A = [ [1,2,3], [4,5,6], [7,8,9] ].
• Which list contains all complex roots of z^4 = -16?
• Decompose (3x+4)/(x^2 - 1) into A/(x-1) + B/(x+1). What are the values of A and B?
• Use De Moivre to compute (−1 + i)^8.
• Write the Maclaurin series for e^x up to and including the x^4 term.
• For a cubic equation with real coefficients, which patterns of roots are possible?
• What is i^3?
• Which description best defines Cartesian form of a complex number?
• Under what condition does a square matrix have an inverse?
• Compute the magnitude of z = -3 − 3i.
• For z^2 − 2z + 5 = 0, the roots are z = 1 ± 2i. What are |z| and the arguments Arg(z) for these roots?
• Which method correctly expresses the determinant of a 3x3 matrix by using 2x2 minors?
• Evaluate the sum of the geometric series ∑_{n=0}^∞ (1/3)^n.
• If the quartic is monic, the product αβγδ equals which expression?
• In sigma notation, what does the number above the sigma represent?
• Distance from P = (1, 2, 3) to the line through the origin in the direction v = (1, −2, 2) equals which value exactly?
• Which statement about non-real roots for polynomials with real coefficients is true?
• Which statement about the general real solution is true?
• For a quartic equation with real coefficients, which is NOT a possible root pattern?
• If z = x + i y, which of the following equals |z|^2?
• The argument of the quotient z1/z2 equals which expression?
• α^2 + β^2 + γ^2 + δ^2 equals which expression?
• Compute a·(b×c) for a = (1,2,3), b = (4,5,6), c = (7,8,9).
• For the polynomial z^2 - 3 z + 4 = 0, what is the sum of its roots?
• For the quadratic x^2 + x + 1 = 0, what is the nature of the roots?
• The principal argument of z = 1 + 2i is which of the following?
• α^3 + β^3 + γ^3 equals which expression?
• The projection of a onto b for a = (4,−2,2), b = (1,0,-1) is which vector?
• How can the determinant of a 3x3 matrix be computed using minors?
• The Maclaurin expansion for ln(1+x) around 0 up to x^3 starts with which term?
• On an Argand diagram, the y-axis corresponds to which axis?
• In the vector equation R = A + λ B + μ C of a plane, which statements are true about A, B, C?
• What is the equivalent principal argument of -2π/3 in the interval (0, 2π)?
• Compute the magnitude of the cross product |a × b| for a = (1,2,3) and b = (−1,0,1).
• The product of a matrix and its inverse equals which matrix?
• The curvature κ(t) of r(t) = (t, t^2, 0) is given by which expression?
• For any integer n, the product (α^n)(β^n)(γ^n) equals which expression?
• Which of the following is the general solution to the Euler-Cauchy equation x^2 y'' + 3 x y' + 2 y = 0?
• For the same quartic, which expression equals αβγ + αβδ + αγδ + βγδ?
• The derivative d/dx arctan x equals which expression?
• Determine the dimension of the null space of A = [[1,2,3],[4,5,6],[7,8,9]].
• If two complex numbers are equal, what must be true?
• If z = x + i y, which expression equals |z|?
• Cube roots of 8 cis(60°) are which of the following?
• Which expression equals α^2 + β^2 + γ^2 in terms of (α+β+γ)^2 and pairwise sums?
• If b^2 - 4ac = 0, what is the nature of the roots?
• Which statement about a non-singular matrix is true?
• If z = x + iy, then |z| equals which expression?
• If the quartic is monic (a = 1), the sum of pairwise products αβ + αγ + αδ + βγ + βδ + γδ equals which expression?
• Compute the determinant of the 2x2 matrix [ [3, 5], [2, 4] ].
• Express z = 2[cos(120°) + i sin(120°)] in rectangular form.
• Which statement describes the form of the identity matrix?
• If b^2 - 4ac < 0, what is the nature of the roots?
• In modulus-argument form, z equals which expression?
• In the quadratic z^2 - (α+β) z + αβ = 0, what is the sum of the roots?
• For z = a + bi, what are Re(z) and Im(z)?
• Write the Maclaurin series for sin x up to and including the x^5 term.
• Evaluate the definite integral ∫_0^{π} cos^2 x dx.
• The determinant of A = [[1,2],[3,4]] is -2. Which statement is true about det(A)?
• What is the formula for the volume of a cone with base radius r and height h?
• For any invertible matrix A with det(A) ≠ 0, which statement is true about det(A^{-1})?
• Solve y'' + y = 0 with y(0) = 2 and y'(0) = 0. Which function satisfies the equation?
• α^2 + β^2 + γ^2 equals which expression?
• For A = [[2,1],[0,3]], which pair of eigenvectors is correct for the eigenvalues 2 and 3 respectively?
• For the circle |z - (a+ib)| = r, what is the radius?
• In a 1x1 matrix [ [k] ], the determinant equals:
• Consider z1 = a + bi and z2 = a - bi. Under what condition are z1 and z2 equal?
• If the quartic is monic (a = 1), the sum of squares α^2 + β^2 + γ^2 + δ^2 equals which of the following?