OCR AS Level Mathematics B (MEI) H630/01 – 2026/2027 – 40 Original Pure Mathematics & Mechanics Questions with Mark Schemes, Diagrams & Fully Worked Solutions

OCR AS LEVEL MATHEMATICS B (MEI)
H630/01 PURE MATHEMATICS AND MECHANICS VERIFIED QUESTIONS
WITH MARKING SCHEME & SOLUTIONS 2025/ 2026

Questions • Mark Schemes • Fully Worked Solutions


Coverage
Pure mathematics: proof, algebra, functions, graphs, coordinate geometry, sequences and series, trigonometry,
exponentials and logarithms, calculus, numerical methods and vectors. Mechanics: models and quantities, one-
dimensional kinematics, forces and Newton's laws.




Question 1 — Algebra
Solve 2x² - 7x - 15 = 0.
Working: __________________________________________________________________________
__________________________________________________________________________________

Mark Scheme
M1 Factorise or use quadratic formula.
A1 Obtain the two correct roots.
A1 x = 5 and x = -3/2.

Fully Worked Solution
2x² - 7x - 15 = (2x+3)(x-5).
Therefore (2x+3)(x-5)=0, so x=-3/2 or x=5.
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Question 2 — Algebra
Complete the square for x² - 8x + 11 and hence state its minimum value.
Working: __________________________________________________________________________
__________________________________________________________________________________

Mark Scheme
M1 Complete the square.
A1 (x-4)²-5.
B1 Minimum value -5.

OCR H630/01 Expanded Practice 2026/2027 | 40 Original Questions | Solutions After Each Question

,Fully Worked Solution
x²-8x+11=(x-4)²-16+11=(x-4)²-5.
Since (x-4)²≥0, the minimum value is -5, attained when x=4.
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Question 3 — Algebra
Solve the simultaneous equations y=2x+3 and x²+y=15.
Working: __________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________

Mark Scheme
M1 Substitute the linear equation.
A1 Obtain x²+2x-12=0.
A1 Correct x-values.
A1 Corresponding y-values.

Fully Worked Solution
Substitute y=2x+3 into x²+y=15: x²+2x+3=15, so x²+2x-12=0.
x=-1±√13.
Then y=2x+3, giving y=1±2√13 respectively.
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Question 4 — Functions
Let f(x)=3x-4 and g(x)=x²+1. Find fg(2) and gf(2).
Working: __________________________________________________________________________
__________________________________________________________________________________

Mark Scheme
B1 Correct fg(2).
B1 Correct gf(2).
B1 Demonstrate correct composition order.

Fully Worked Solution
fg(2)=f(g(2)). Since g(2)=5, f(5)=11.
gf(2)=g(f(2)). Since f(2)=2, g(2)=5.
Thus fg(2)=11 and gf(2)=5.
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OCR H630/01 Expanded Practice 2026/2027 | 40 Original Questions | Solutions After Each Question

, Question 5 — Functions
The function f(x)=2x+7. Find f⁻¹(x) and state the domain of f⁻¹ if f has domain x≥1.
Working: __________________________________________________________________________
__________________________________________________________________________________

Mark Scheme
M1 Rearrange y=2x+7.
A1 f⁻¹(x)=(x-7)/2.
B1 Domain x≥9.

Fully Worked Solution
Let y=2x+7. Rearranging gives x=(y-7)/2, so f⁻¹(x)=(x-7)/2.
For x≥1, f(x)≥9. Therefore the inverse has domain x≥9.
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Question 6 — Graphs
Describe fully the transformation taking y=x² to y=3(x-2)²+5.




Diagram for Question 6 (not necessarily to scale)
Working: __________________________________________________________________________
__________________________________________________________________________________

Mark Scheme
B1 Horizontal translation.
B1 Vertical stretch.
B1 Vertical translation.




OCR H630/01 Expanded Practice 2026/2027 | 40 Original Questions | Solutions After Each Question