OCR A Level Maths Specimen Paper 1: question paper (H240/01)

3 A Level questions like these, with worked solutions

Written and checked by us, not taken from the OCR paper. Try each one, then tap "Worked solution" to see every step and its marks.

1. Coordinate geometry · 6 marks

   The circle $C$ has centre $(a,b)$ and passes through the points $P(0,0)$, $Q(6,0)$, and $R(0,8)$. Find the values of $a$, $b$, and the radius of $C$.

   Worked solution

   $P$ and $Q$ both on circle ⇒ centre lies on the perpendicular bisector of $PQ$, so $a=3$   M1 A1. Similarly $P$ and $R$ give $b=4$   M1 A1. Radius $=|CP|=\sqrt{9+16}=5$   M1 A1.

2. Differentiation · 5 marks

   Find the equation of the tangent to the curve $2x^2-xy+y^2=8$ at the point $(2,2)$. Give your answer in the form $y=mx+c$.

   Worked solution

   1. M1 for differentiating $2x^2$ to $4x$ and $y^2$ to $2y\frac{dy}{dx}$.
   2. M1 for differentiating $-xy$ using the product rule to get $-y-x\frac{dy}{dx}$.
   3. M1 for setting the entire derivative to zero: $4x-y-x\frac{dy}{dx}+2y\frac{dy}{dx}=0$.
   4. M1 for substituting $x=2$ and $y=2$ to find the gradient $m$: $8-2-2\frac{dy}{dx}+4\frac{dy}{dx}=0⟹2\frac{dy}{dx}=-6⟹m=-3$.
   5. A1 for using $y-2=-3(x-2)$ to establish the final equation $y=-3x+8$.

3. Sequences and series · 4 marks

   The first three terms of the expansion $(a+bx{)}^6$ are $64-192x+c{x}^2$. Find the values of the constants $a$, $b$, and $c$.

   Worked solution

   1. M1 for equating the first term: $a^6=64⟹a=2$.
   2. M1 for equating the $x$ coefficient to find $b$: $(\genfrac{}{}{0ex}{}{6}{1})a^{5}b=-192⟹6(32)b=-192⟹192b=-192$.
   3. A1 for obtaining $b=-1$.
   4. A1 for finding the $x^2$ coefficient: $c=(\genfrac{}{}{0ex}{}{6}{2})a^4{b}^2=15(16)(-1{)}^2=240$.

More on these topics: Coordinate geometry · Differentiation · Sequences and series

What this paper covers

What this paper is like

• Shape: 14 questions, 100 marks in 2 h (about 1.2 minutes a mark); questions from 3 to 12 marks.
• Biggest question: Q14, 12 marks: bird food plan as a sequence.
• Marks by main topic: Differentiation 28, Sequences and series 22, Algebra and functions 16, Trigonometry 13, Numerical methods 9, Vectors 5, Exponentials and logarithms 4, Proof 3.
• More Sequences and series in other sittings: June 2018 Q7 (9 marks), June 2019 Q7 (8 marks), November 2020 Q7 (11 marks).
• Against the other 8 sittings we mapped: more marks on Differentiation (28 marks here, 16 on average) and Sequences and series (22 marks here, 12 on average); fewer on Integration (0 marks here, 11 on average) and Exponentials and logarithms (4 marks here, 11 on average).
• Not on this paper: Coordinate geometry, Integration.
• Calculator: allowed.

H240/01: question by question

Labels and topics are ours, read from the H240/01 paper (checked 2026-11-01); marks as printed. Tap a number to practise that topic.

1. Q1Simultaneous equations, one quadratic (4 marks) · Algebra and functions
2. Q23D position vectors (5 marks) · Vectors
3. Q3Transforming a graph (3 marks) · Algebra and functions
4. Q4Sector area and perimeter (7 marks) · Trigonometry
5. Q5Solving an exponential equation with logs (4 marks) · Exponentials and logarithms
6. Q6Proof by contradiction about even numbers (3 marks) · Proof
7. Q7Profits as a geometric series (10 marks) · Sequences and series
8. Q8Double-angle identity and an equation (6 marks) · Trigonometry
9. Q9Newton-Raphson on a cubic (9 marks) · Numerical methods
10. Q10Differentiating x in terms of y with ln (8 marks) · Differentiation, Exponentials and logarithms
11. Q11Composite functions with a constant a (9 marks) · Algebra and functions
12. Q12Parametric ellipse: gradient and tangent (11 marks) · Differentiation, Coordinate geometry
13. Q13Implicit curve: stationary points (9 marks) · Differentiation
14. Q14Bird food plan as a sequence (12 marks) · Sequences and series