A Level Maths Year 1 starter questions: weeks 1–13
39 short questions to open a lesson for A Level Maths Year 1 (AS), weeks 1–13 of the teaching year, three or four a week. Each question is shown as an image with its text underneath; answers and mark schemes stay with teachers.
Week 1
Question 1: Laws of indices
Simplify (6x⁵ × 2x⁻²)/(4x²).
Question 2: Expanding & factorising polynomials
Factorise completely 2x³ - 8x.
Question 3: Surds: simplifying & rationalising
Rationalise the denominator of 4/(√5 - 1).
Week 2
Question 1: Solving quadratics & the discriminant
Solve x² - 4x - 1 = 0, giving your answers in surd form.
Question 2: Completing the square
Write 2x² - 12x + 5 in the form a(x + b)² + c.
Question 3: Quadratic graphs & roots
For y = x² - 2x - 8, find the x-intercepts, the y-intercept and the vertex.
Week 3
Question 1: Simultaneous equations (linear & quadratic)
Solve the simultaneous equations 2x + 3y = 13 and 4x - y = 5.
Question 2: Simultaneous equations (linear & quadratic)
Solve the simultaneous equations x + y = 5 and xy = 6.
Question 3: Linear & quadratic inequalities
Solve x² + 2x - 15 ≤ 0, giving your answer in set notation.
Week 4
Question 1: Linear & quadratic inequalities
Solve 6/x > 2, x ≠ 0.
Question 2: Sketching cubics, quartics & reciprocals
For y = (x - 1)²(x + 2), find where the curve meets the axes. Where does it touch the x-axis?
Question 3: Sketching cubics, quartics & reciprocals
Write down the equations of the asymptotes of y = 1/(x - 2).
Week 5
Question 1: Graph transformations (translations, stretches, reflections)
The graph of y = f(x) has a maximum point at (2, 5). Write down the maximum point of y = f(x + 3) and of y = 2f(x).
Question 2: Graph transformations (translations, stretches, reflections)
Describe the single transformation that maps y = x³ onto y = (2x)^3.
Question 3: Straight lines: gradient, equation & intersection
Find the gradient and the y-intercept of the line 3x - 2y + 8 = 0.
Week 6
Question 1: Straight lines: gradient, equation & intersection
Find the point of intersection of y = 3x - 2 and x + 2y = 10.
Question 2: Parallel & perpendicular lines
Find the equation of the line through (3, 4) parallel to y = -2x + 7.
Question 3: Parallel & perpendicular lines
Show that the lines 2x + 3y = 6 and 3x - 2y = 1 are perpendicular.
Week 7
Question 1: Equation & properties of a circle
Write down the equation of the circle with centre (-1, 4) and radius 6, and expand it.
Question 2: Equation & properties of a circle
Find the centre and radius of the circle x² + y² + 4x - 10y + 20 = 0.
Question 3: Tangents & chords of a circle
Find the equation of the tangent to the circle (x - 1)² + (y - 2)² = 25 at the point (4, 6).
Week 8
Question 1: Tangents & chords of a circle
The line y = x + k is a tangent to the circle x² + y² = 8. Find the possible values of k.
Question 2: Algebraic division & the factor theorem
Show that (x + 3) is a factor of x³ + x² - 9x - 9, and factorise it fully.
Question 3: Algebraic division & the factor theorem
Simplify (x² + 5x + 6)/(x² - 4).
Week 9
Question 1: Proof by deduction
Prove that the square of any odd number is odd.
Question 2: Proof by exhaustion
Prove by exhaustion that no square number ends in the digit 7.
Question 3: Disproof by counter-example
Disprove by counter-example: 'if x > y then x² > y²'.
Week 10
Question 1: Binomial expansion (positive integer n)
Find the first three terms, in ascending powers of x, of (1 - 2x)^7.
Question 2: Binomial expansion (positive integer n)
Find the coefficient of x³ in the expansion of (3 + 2x)^5.
Question 3: Sine & cosine rules
A triangle has sides a = 7 cm, b = 8 cm and c = 9 cm. Find angle A to 1 decimal place.
Week 11
Question 1: Sine & cosine rules
In triangle PQR, PQ = 12 cm, angle P = 40^∘ and angle Q = 75^∘. Find QR to 3 significant figures.
Question 2: Area of a triangle (½ab sinC)
A triangle has sides of 6 cm and 9 cm with an included angle of 120^∘. Find its exact area.
Question 3: Graphs of sin, cos & tan
State the period of y = tan x and of y = cos x, in degrees, and the range of y = 3sin x.
Week 12
Question 1: Trig ratios & exact values
Find the exact values of cos 210^∘ and tan 135^∘.
Question 2: Fundamental trig identities
Given cosθ = -5/13 and 180^∘ < θ < 270^∘, find sinθ and tanθ.
Question 3: Solving basic trig equations
Solve cos x = -0.5 for 0^∘ ≤ x ≤ 360^∘.
Week 13
Question 1: Solving quadratic-form trig equations
Solve 2sin² x + sin x - 1 = 0 for 0^∘ ≤ x ≤ 360^∘.
Question 2: Vectors in 2D: notation, magnitude & direction
Find the magnitude of -2i + 2√3j and the angle it makes with the positive i direction.
Question 3: Vector arithmetic & scalar multiples
Given a = 2i - 3j and b = -i + 4j, find 3a - 2b and |a + b|.
More starter questions
Other weeks: Weeks 1–13 · Weeks 14–26 · Weeks 27–40. Answers and mark schemes are in the starter tasks for teachers.