📝 Grade 9 · Exam Paper 2

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Section A – Number & Percentages (12 marks)

1. (2.4×10⁵) × (3.0×10³). Give your answer in standard form.[2]

2. A price increases from £85 to £102. Find the percentage increase.[2]

%

3. £5000 is invested at 3.5% compound interest per annum. Find the value after 4 years (to the nearest penny).[3]

£

4. After a 15% decrease, a salary is £25,500. Find the original salary.[2]

£

5. A length is measured as 6.3 cm to 1 dp. Find the upper bound.[1]

cm

6. (9.6×10⁷) ÷ (3.2×10³). Give your answer in standard form.[2]

  • Q1: 7.2×10⁸
  • Q2: 20%
  • Q3: £5000×1.035⁴ = £5744.04
  • Q4: £25500÷0.85 = £30000
  • Q5: 6.35 cm
  • Q6: 3.0×10⁴
Section B – Algebra & Graphs (20 marks)

7. Solve 2x² − 5x − 3 = 0 using the quadratic formula. Give roots to 2 dp.[3]

x = or x =

8. Solve simultaneously: y = x + 2 and y = x² − 4. Give both solutions for x.[4]

x = or x =

9. Write the nth term of the quadratic sequence: 3, 8, 15, 24, 35, …[3]

nth term = n² + n +

10. f(x) = 2x + 1 and g(x) = x² − 3. Find fg(3).[2]

11. The line L has equation 2y = 6x − 4. Find the gradient of a line perpendicular to L.[2]

12. Solve −3 ≤ 2x + 1 < 9. List all integers satisfying this inequality.[3]

13. Write the coordinates of the vertex of y = (x − 4)² − 7.[2]

(, )

14. The 5th term of an arithmetic sequence is 23 and the 9th term is 43. Find the common difference.[1]

d =
  • Q7: x=3 or x=−0.5 (2x²−5x−3=0; a=2,b=−5,c=−3; discriminant=25+24=49; x=(5±7)/4)
  • Q8: x²−4=x+2 → x²−x−6=0 → (x−3)(x+2)=0 → x=3 or x=−2
  • Q9: 3,8,15,24,35… second diff=2→n²; n²: 1,4,9,16,25; diff: 2,4,6,8,10 → 2n−1? Check: n²+2n−? at n=1: 1+2+c=3→c=0. nth term = n²+2n. Check: n=1→3✓, n=2→8✓. So coeffs: 2, 0
  • Q10: g(3)=9−3=6; f(6)=13
  • Q11: 2y=6x−4 → y=3x−2, gradient=3; perpendicular gradient=−1/3≈−0.33
  • Q12: −3≤2x+1<9 → −4≤2x<8 → −2≤x<4; integers: −2,−1,0,1,2,3
  • Q13: vertex=(4,−7)
  • Q14: terms differ by 5 over 4 steps → d=5
Section C – Geometry & Trigonometry (12 marks)

15. A ladder 10 m long leans against a wall. The foot is 4 m from the wall. How high does it reach? (to 2 dp)[2]

m

16. In a right-angled triangle, adjacent = 8 cm, hypotenuse = 17 cm. Find the angle at the base (to 1 dp).[2]

°

17. A right-angled triangle has angle 35° and adjacent side 12 cm. Find the opposite side (to 2 dp).[2]

cm

18. Vectors p = (2, 5) and q = (−1, 3). Find p + q.[1]

(, )

19. Vectors p = (2, 5) and q = (−1, 3). Find |2p − q|. Give to 2 dp.[3]

20. An angle in a semicircle is inscribed from AB (diameter). What is the angle at the circumference?[1]

°

21. Angle at centre = 2 × angle at circumference. Angle ACB = 38°. Find angle AOB.[1]

°
  • Q15: √(100−16)=√84≈9.17 m
  • Q16: cos⁻¹(8/17)≈61.9°
  • Q17: tan(35°)×12≈8.40 cm
  • Q18: (1, 8)
  • Q19: 2p=(4,10), 2p−q=(5,7); |(5,7)|=√(25+49)=√74≈8.60
  • Q20: 90°
  • Q21: 76°
Section D – Statistics & Probability (6 marks)

The grouped frequency table shows heights of 50 plants:

Height (cm)FreqClass width
10–20810
20–301510
30–501820
50–70920

22. Find the frequency density for the 30–50 class.[1]

23. Estimate the mean height.[2]

cm

24. A bag has 4 red and 6 blue balls. Two drawn without replacement. Find P(red then blue).[2]

25. A and B are independent. P(A)=0.35, P(B)=0.6. Find P(A or B).[1]

  • Q22: 18÷20=0.9
  • Q23: midpoints 15,25,40,60; Σfx=8×15+15×25+18×40+9×60=120+375+720+540=1755; mean=1755/50=35.1 cm
  • Q24: 4/10×6/9=24/90=4/15≈0.267
  • Q25: P(A∪B)=0.35+0.6−0.35×0.6=0.95−0.21=0.74