🏆 Grade 9 · Exam Paper 3

Challenge Paper  |  60 marks  |  75 minutes
75:00
Click timer to start / pause
🏆 Challenge Paper — Calculator allowed. Show all working for full marks.
Section A – Number, Surds & Bounds (16 marks)

1. Simplify √48 + √75 − √27. Give your answer in the form k√3.[3]

k =

2. Simplify (5 + √3)² giving your answer in the form a + b√3.[3]

a = , b =

3. (3.6×10⁵) + (4.8×10⁴). Give in standard form.[2]

4. x = 4.2 (to 1 dp), y = 3.5 (to 1 dp). Find the maximum value of x/y to 3 dp.[3]

5. A car depreciated by 12% per year. It cost £18000 new. Find its value after 3 years (to nearest £).[2]

£

6. After a 35% increase followed by a 20% decrease, what is the overall percentage change? (+ for increase)[3]

%
  • Q1: √48=4√3, √75=5√3, √27=3√3; total=(4+5−3)√3=6√3 → k=6
  • Q2: (5+√3)²=25+10√3+3=28+10√3 → a=28, b=10
  • Q3: 3.6×10⁵+0.48×10⁵=4.08×10⁵
  • Q4: max x=4.25, min y=3.45; 4.25/3.45≈1.232
  • Q5: £18000×0.88³=£18000×0.681472≈£12267
  • Q6: 1.35×0.80=1.08 → 8% increase
Section B – Algebra & Sequences (20 marks)

7. Solve x² + 3x − 10 = 0 by factorising. State both roots.[2]

x = or x =

8. Complete the square for x² − 6x + 2. Give in the form (x + p)² + q.[2]

p = , q =

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

x = or x =

10. Solve: 3x + 2y = 16 and 5x − y = 9.[3]

x = , y =

11. f(x) = x² + 1, g(x) = 2x − 3. Find gf(x) and simplify.[2]

gf(x) = 2x² +

12. A quadratic sequence has terms: 4, 11, 22, 37, 56, … Find the nth term.[3]

nth term = n² + n +

13. The line through (2, 5) and (6, 13) meets the y-axis at (0, k). Find k.[2]

k =

14. Solve x² − 4x > 5.[3]

x < or x >
  • Q7: (x+5)(x−2)=0 → x=−5 or x=2
  • Q8: (x−3)²−7 → p=−3, q=−7
  • Q9: disc=4+60=64; x=(−2±8)/6 → x=1 or x=−5/3≈−1.67
  • Q10: from eq2: y=5x−9; sub: 3x+2(5x−9)=16 → 13x=34 → x=34/13≈2.62? Let me redo: 3x+10x−18=16 → 13x=34 → x=34/13. Hmm, not integer. Let me fix: 5x−y=9→y=5x−9; 3x+2y=16→3x+2(5x−9)=16→3x+10x−18=16→13x=34→x=34/13≈2.615, y=5(34/13)−9=170/13−117/13=53/13≈4.08. Accept decimal: x≈2.62, y≈4.08
  • Q11: gf(x)=2(x²+1)−3=2x²−1 → coefficient after 2x² is −1
  • Q12: 2nd differences=4→2n²; subtract: 4,11,22,37,56 minus 2,8,18,32,50 = 2,3,4,5,6 → linear part: n+1; nth term=2n²+n+1
  • Q13: gradient=(13−5)/(6−2)=2; y=2x+c; 5=4+c→c=1; k=1
  • Q14: x²−4x−5>0 → (x−5)(x+1)>0 → x<−1 or x>5
Section C – Geometry, Trig & Vectors (16 marks)

15. A cuboid has length 8 cm, width 5 cm, height 6 cm. Find the length of the space diagonal (to 2 dp).[3]

cm
d=√(l²+w²+h²)

16. In right-angled triangle, angle=55°, hypotenuse=20 cm. Find the adjacent side (to 2 dp).[2]

cm

17. O is the centre. Angle PQR = 64° (inscribed angle). Find angle POR.[2]

°

18. PQRS is a cyclic quadrilateral. Angle P = 3x + 10 and angle R = 2x + 20. Find x.[3]

x = °

19. Vectors OA = (4, 1) and OB = (−2, 7). M is the midpoint of AB. Find the vector OM.[3]

OM = (, )

20. Show that vectors AB = (6, 4) and CD = (9, 6) are parallel. What is the ratio |CD| : |AB|?[3]

Ratio: :
  • Q15: √(64+25+36)=√125≈11.18 cm
  • Q16: cos(55°)×20≈11.47 cm
  • Q17: 128° (angle at centre = 2× circumference)
  • Q18: (3x+10)+(2x+20)=180 → 5x+30=180 → 5x=150 → x=30
  • Q19: OM = OA+½AB = (4,1)+½(−6,6) = (4,1)+(−3,3) = (1,4)
  • Q20: CD=1.5×AB (parallel); |AB|=√52, |CD|=√117=√(9×13)=3√13; ratio=3:2
Section D – Statistics & Probability (8 marks)

21. A histogram has class 40–50 with frequency density 2.4. Find the frequency.[1]

22. From 60 data values, the cumulative frequency at 50 is 45. Estimate the percentage above 50.[1]

%

23. Bag A: 3 red, 2 green. Bag B: 4 red, 1 green. One ball drawn from each bag. Find P(both same colour).[3]

24. P(A)=0.45, P(B|A)=0.6. Find P(A and B).[1]

25. A box plot has: min=12, Q1=20, median=28, Q3=38, max=50. Find the IQR.[1]

26. From a bag of 10 balls numbered 1–10, two are drawn without replacement. Find P(both even).[1]

  • Q21: fd×cw=2.4×10=24
  • Q22: (60−45)/60×100=25%
  • Q23: P(both red)=3/5×4/5=12/25; P(both green)=2/5×1/5=2/25; P(same)=14/25=0.56
  • Q24: P(A∩B)=P(A)×P(B|A)=0.45×0.6=0.27
  • Q25: IQR=38−20=18
  • Q26: 5/10×4/9=20/90=2/9