AA HL Q/A

Sample question

Express (3x + 7)/((x + 1)(x − 2)) as a sum of partial fractions.

Sample question

Solve 2^(x+1) = 3 · 5^x, giving your answer in terms of ln 2 and ln 5.

Sample question

The sum of the first n terms of a GP is S_n. Show that S_∞ exists iff |r| < 1 and find it for r = 1/3, a = 9.

Sample question

Prove that the sum of any three consecutive integers is always divisible by 3.

Sample question

Prove by induction that n³ − n is divisible by 6 for all positive integers n.

Sample question

Find the coefficient of x⁵ in the expansion of (2x − 1/x)⁹.

Sample question

How many 4-digit numbers can be formed from 0–9 with no digit repeating and the number being even?

Sample question

Express z = 1 + i√3 in modulus-argument form and hence find z⁶.

Sample question

Find all 5th roots of z⁵ = 32, giving each answer in exponential form.

Sample question

Find the values of k for which the system has a unique solution, no solution, or infinitely many solutions.

Sample question

Find the equation of the line passing through (2, −1) perpendicular to 3x − 4y = 7.

Sample question

For what values of k does kx² + (k + 2)x + 1 = 0 have two distinct real roots?

Sample question

Given f(x) = 2x + 3 and g(x) = x² − 1, find (f ∘ g)(x) and state the domain of f⁻¹.

Sample question

Sketch y = e^x − 2 and y = ln(x + 2), labelling all intercepts and asymptotes.

Sample question

Sketch f(x) = (x² − 1)/(x − 2). State all asymptotes and intercepts.

Sample question

Sketch y = −2 f(x − 3) + 1 given the graph of y = f(x). Identify each transformation in order.

Sample question

Given P(x) = x³ − 4x² + ax + b has a factor (x − 2), and P(1) = −6, find a and b.

Sample question

Solve (x − 1)/(x + 2) ≥ 3, giving your answer using interval notation.

Sample question

Sketch y = |2x − 5| and solve |2x − 5| < 3.

Sample question

A sector has radius 6 cm and angle 5π/12 radians. Find arc length and sector area in exact form.

Sample question

A right cone has slant height 13 cm and base radius 5 cm. Find total surface area and volume.

Sample question

In triangle ABC, a = 8, b = 10, C = 70°. Find side c and the area of the triangle.

Sample question

Without a calculator, find the exact value of sin(7π/6) + cos(5π/4) − tan(π/3).

Sample question

h(t) = 3 sin(πt/6) + 8 models tidal height. Find amplitude, period and the time of first maximum.

Sample question

Solve 2 cos²x − cos x − 1 = 0 for 0 ≤ x ≤ 2π. Give answers in exact form.

Sample question

Find the exact value of arctan(1) + arccos(−1/2) and state the range of arcsin(x).

Sample question

Prove that sin(2θ)/(1 + cos(2θ)) ≡ tan θ, stating any restrictions on θ.

Sample question

Find the angle between vectors a = (2, −1, 3) and b = (1, 4, −2). Are they perpendicular?

Sample question

Find the shortest distance between the lines r₁ = (1,0,2) + t(2,1,−1) and r₂ = (3,−1,4) + s(1,2,1).

Sample question

Find the Cartesian equation of the plane through A(1,2,3), B(2,0,−1), and C(0,1,2).

Sample question

From the frequency table given, find the mean, standard deviation, and identify any outliers using 1.5×IQR.

Sample question

For the data given, find r, the regression line y on x, and predict y when x = 12. Comment on reliability.

Sample question

Two cards are drawn without replacement from a standard deck. Find P(both kings | at least one face card).

Sample question

A discrete random variable X has P(X = x) = kx for x = 1, 2, 3, 4. Find k, E(X) and Var(X).

Sample question

X ~ B(12, 0.4). Find P(X ≥ 5), E(X) and Var(X). State the conditions for using a binomial model.

Sample question

Heights ~ N(170, 8²). Find P(X > 180), the 90th percentile, and use inverse normal where appropriate.

Sample question

For pdf f(x) = kx(2 − x) on [0, 2], find k, E(X) and P(X > 1).

Sample question

Differentiate y = x² sin(3x) using the product rule, then simplify your answer fully.

Sample question

Find the equation of the tangent and normal to y = x³ − 2x at the point (1, −1).

Sample question

Find ∫ (3x² − 4x + 2)/√x dx, simplifying your answer fully.

Sample question

Find the area enclosed between y = x² and y = x + 2, giving your answer as an exact fraction.

Sample question

A box with square base has total surface area 100 cm². Find dimensions that maximise volume.

Sample question

A particle’s velocity is v(t) = 3t² − 12. Find total distance travelled for 0 ≤ t ≤ 4.

Sample question

Evaluate lim x→2 of (x² − 4)/(x − 2) and discuss continuity of f(x) = x²/(x − 2) at x = 2.

Sample question

For x² + xy + y² = 7, find dy/dx by implicit differentiation. Hence find the gradient at (1, 2).

Sample question

Find ∫ x · e^(2x) dx using integration by parts.

Sample question

Solve dy/dx = xy with y(0) = 2 using separation of variables.

Sample question

Find the Maclaurin series for f(x) = ln(1 + x) up to and including the term in x⁴.

Sample question

Evaluate lim x→0 of (sin x − x)/x³ using l’Hôpital’s rule and verify using the Maclaurin series of sin x.

Sample question

Long-form questions covering SL-level Section B style content with multi-part synthesis.

Sample question

HL-only Section B and Paper 3-style problems demanding deep synthesis across topics.