Grade 10 Practice Paper 1 – Non-Calculator

Section A — Algebra & Number (22 marks)

Questions 1–9 | Show all working in your exercise book.

Q1 2 marks

Simplify fully: (x² − 4) / (x² + x − 2)
Substitute x = 5 into your simplified expression. Enter the decimal answer.

Q1 answer:

Q2 3 marks

Write as a single fraction: 2/(x + 3) + 3/(x − 1)
Your answer should be in the form (ax + b) / [(x+3)(x−1)].
Enter the value of a + b.

Q2: a + b =

Q3 3 marks

Write x² − 6x + 2 in the form (x − p)² + q.

(a) Enter the value of p.

(b) Enter the value of q.

Q3a: p =

Q3b: q =

Q4 3 marks

Solve x² − 6x + 2 = 0 by completing the square. Enter the larger root correct to 2 decimal places.

Q4: larger root ≈

Q5 3 marks

Solve the inequality x² − 7x + 10 < 0.
The solution is a < x < b. Enter a + b.

Q5: a + b =

Q6 4 marks

Given f(x) = 4x − 1 and g(x) = x² + 2:

(a) Find fg(3).

(b) Find the coefficient of x² in the expression gf(x).

Q6a: fg(3) =

Q6b: coeff of x² =

Q7 2 marks

Find f⁻¹(x) where f(x) = 4x − 1. Calculate f⁻¹(11).

Q7: f⁻¹(11) =

Q8 2 marks

The function h(x) = √(2x − 6). State the smallest value of x in the domain of h.

Q8: smallest x =

Q9 2 marks

Prove that n² + (n+2)² is always even for all integers n.
Write the expression in its simplest factored form 2(an² + bn + c). Enter a + b + c.

Q9: a + b + c =

Section B — Sequences & Graphs (18 marks)

Questions 10–15

Q10 4 marks

A geometric sequence has first term a = 5 and common ratio r = 2.

(a) Find the 7th term.

(b) Find the sum of the first 6 terms.

Q10a: 7th term =

Q10b: S₆ =

Q11 3 marks

An infinite geometric series has first term 12 and sum to infinity 30. Find the common ratio r.

Q11: r =

Q12 3 marks

A sequence has nth term u_n = 2n² − n + 1.

(a) Find u₅.

(b) Find the value of n for which u_n = 55.

Q12a: u₅ =

Q12b: n =

Q13 3 marks

The graph of y = f(x) is transformed.

(a) The transformation y = f(x + 3) is applied. State the horizontal shift (positive = right, negative = left).

(b) The point (5, 7) is on y = f(x). Write down the image of this point under y = 2f(x) − 1. Enter the y-coordinate of the image.

Q13a: shift =

Q13b: y-coord =

Q14 2 marks

A curve has equation y = x² − 4x + 1. Using the derivative dy/dx = 2x − 4, find the x-coordinate of the turning point.

Q14: x =

Q15 3 marks

Estimate the area under y = x² between x = 0 and x = 4, using 4 strips of width 1. Use the trapezium rule: A ≈ h/2 × (y₀ + 2y₁ + 2y₂ + 2y₃ + y₄).
Enter your estimate.

Q15: area estimate =

Q16 3 marks

Solve the simultaneous equations: y = 2x + 1 and y = x² − 3.
The solutions are (x₁, y₁) and (x₂, y₂) where x₁ < x₂.

(a) Enter x₁ (the smaller x-value).

(b) Enter x₂ (the larger x-value).

Q16a: x₁ =

Q16b: x₂ =

Section C — Geometry & Vectors (12 marks)

Questions 17–21

Q17 2 marks

In triangle OAB, OA = a, OB = b. M is the midpoint of AB.
Express OM in the form pa + qb. Enter the value of p + q.

Q17: p + q =

Q18 3 marks

OABC is a parallelogram. OA = p, OC = q. D is the midpoint of BC.

(a) Find vector OD in the form αp + βq. Enter α.

(b) Show that O, D, and the midpoint M of diagonal AC are collinear by finding the ratio OM : MD. Enter this ratio as a single number k where OM = k × OD. (Hint: find OM and compare.)

Q18a: α =

Q18b: ratio OM:MD = 1:

Q19 3 marks

O is the centre of a circle. A, B, C are points on the circumference.

(a) Angle AOB = 108°. Find angle ACB. 1 mark

(b) ABCD is a cyclic quadrilateral with angle ABC = 79°. Find angle ADC. 1 mark

(c) TP is a tangent to the circle at P. O is the centre. Angle OTP = 36°. Find angle TOP. 1 mark

Q19a: angle ACB =°

Q19b: angle ADC =°

Q19c: angle TOP =°

Q20 2 marks

A cuboid has dimensions 6 cm × 2 cm × 3 cm. Find the length of the space diagonal, giving your answer in the form √k. Enter the value of k.

Q20: k =

Q21 2 marks

Prove that the sum of opposite angles in a cyclic quadrilateral is 180°.
The key step uses the fact that the angle at the centre is k times the angle at the circumference. Enter the value of k, and then enter the sum ∠DAB + ∠BCD that you would prove equals 180.

Q21: k =

Q21: ∠DAB + ∠BCD =°

Section D — Statistics & Probability (8 marks)

Questions 22–24

Q22 3 marks

In a group of 40 students: 22 study French (F), 16 study Spanish (S), 8 study both.

(a) How many study neither French nor Spanish?

(b) A student is chosen at random. Find P(studies French only). Give as a fraction.

(c) A student who studies Spanish is chosen at random. Find P(also studies French). Give as a fraction in lowest terms — enter as decimal.

Q22a: neither =

Q22b: P =

Q22c: P(F|S) =

Q23 3 marks

P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.12.

(a) Find P(A ∪ B).

(b) Find P(A | B).

(c) Are A and B independent? Enter 1 for Yes, 0 for No.

Q23a: P(A∪B) =

Q23b: P(A|B) =

Q23c: independent? (1/0) =

Q24 2 marks

A bag contains 4 red and 3 blue counters. Two counters are drawn without replacement.
Find P(both the same colour). Enter as a decimal (3 d.p.).

Q24: P =

Grade 10 — Practice Paper 1

Final Score