Study Companion
Topic 04 | Drill packs (timed, mixed)

Drill Pack 5 (Selective pace)

Selective prep: 20 multi-step questions in 25 minutes. Year 8 content plus Year 9 extensions (surds, indices, simple quadratics). Closer to real Selective pace.

25 min timed Printable practice Answer key
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Questions

Reasoning

Drill Pack 5 — 25 minutes

    1. Simplify 75+12\sqrt{75} + \sqrt{12}.
    2. Evaluate 3−2+2−13^{-2} + 2^{-1} as a single fraction.
    3. A shirt marked $80 is discounted by 25%25\%, then GST of 10%10\% is added. Final price?
    4. The mean of five numbers is 1414. If four of them are 10,12,15,1810, 12, 15, 18, find the fifth.
    5. Solve x+32=2x−13\dfrac{x + 3}{2} = \dfrac{2x - 1}{3}.
    6. Two parallel lines are cut by a transversal. One co-interior angle is (3x+10)∘(3x + 10)^\circ and the other is (2x+40)∘(2x + 40)^\circ. Find xx.
    7. A right-angled triangle has legs 66 cm and 88 cm. Find the hypotenuse.
    8. Rewrite 52\dfrac{5}{\sqrt{2}} with a rational denominator.
    9. A bag has 44 red, 55 blue, 33 green marbles. Two are drawn without replacement. Find P(both red)P(\text{both red}).
    10. Expand and simplify (x+3)(x−2)(x + 3)(x - 2).
    11. A train leaves town AA at 9:159{:}15 and arrives at town BB, 240240 km away, at 12:4512{:}45. Average speed in km/h?
    12. If x:y=3:5x : y = 3 : 5 and y:z=2:7y : z = 2 : 7, find x:zx : z in simplest form.
    13. Simplify (2a−2b3)2(2a^{-2} b^3)^2, with positive indices only.
    14. Factorise 12x2−18x12x^2 - 18x.
    15. A rectangle has area 9696 cm2^2 and length 1212 cm. The diagonal has length (in surd form)?
    16. Four people share a bill such that Alex pays twice as much as Ben, Ben pays the same as Chloe, and Dan pays $15 more than Ben. Total is $150. How much does Ben pay?
    17. Evaluate 45÷815\dfrac{4}{5} \div \dfrac{8}{15}.
    18. If P(A)=0.3P(A) = 0.3, P(B)=0.5P(B) = 0.5, and A,BA, B are mutually exclusive, find P(A or B)P(A \text{ or } B) and P(neither)P(\text{neither}).
    19. The sum of three consecutive even integers is 7878. Find the smallest.
    20. A square and a rectangle have the same perimeter. The rectangle is 1414 cm by 66 cm. Find the area of the square.
Answers

Answer key

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Each answer pairs the quick method with a trap to avoid.

    1. 75+12=53+23=73\sqrt{75} + \sqrt{12} = 5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}. Trap: simplify each surd first; don’t just add 75+1275 + 12.
    2. 3−2+2−1=19+12=218+918=11183^{-2} + 2^{-1} = \tfrac{1}{9} + \tfrac{1}{2} = \tfrac{2}{18} + \tfrac{9}{18} = \tfrac{11}{18}. Trap: negative index means reciprocal, not negative number.
    3. $66. Quick: $80 ×0.75=60\times 0.75 = 60; then 60×1.10=6660 \times 1.10 = 66 dollars. Trap: never “subtract 25%25\% then add 10%10\% = subtract 15%15\%” — they compound on different bases.
    4. Fifth =5×14−(10+12+15+18)=70−55=15= 5 \times 14 - (10 + 12 + 15 + 18) = 70 - 55 = 15.
    5. Cross-multiply: 3(x+3)=2(2x−1)3(x + 3) = 2(2x - 1), so 3x+9=4x−23x + 9 = 4x - 2, x=11x = 11.
    6. Co-interior sum =180= 180: (3x+10)+(2x+40)=180(3x + 10) + (2x + 40) = 180, so 5x=1305x = 130, x=26x = 26.
    7. c2=36+64=100c^2 = 36 + 64 = 100, so c=10c = 10 cm. Quick: recognise 6,8,106,8,10 as a Pythagorean triple — 2×(3,4,5)2 \times (3,4,5).
    8. 52×22=522\dfrac{5}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{5\sqrt{2}}{2}.
    9. P(both red)=412×311=12132=111P(\text{both red}) = \dfrac{4}{12} \times \dfrac{3}{11} = \dfrac{12}{132} = \dfrac{1}{11}. Trap: denominator changes from 1212 to 1111 because it’s without replacement.
    10. (x+3)(x−2)=x2+x−6(x + 3)(x - 2) = x^2 + x - 6. Quick: FOIL gives x2−2x+3x−6x^2 - 2x + 3x - 6; middle terms collect.
    11. Time =3= 3 h 3030 min =3.5= 3.5 h. Speed =240÷3.5=4807≈68.6= 240 \div 3.5 = \dfrac{480}{7} \approx 68.6 km/h. Trap: convert the time to a single unit first.
    12. x:y=3:5x : y = 3 : 5 and y:z=2:7y : z = 2 : 7. Make yy match: multiply first by 22 and second by 55: x:y=6:10x : y = 6 : 10; y:z=10:35y : z = 10 : 35. So x:z=6:35x : z = 6 : 35.
    13. (2a−2b3)2=4a−4b6=4b6a4(2a^{-2} b^3)^2 = 4 a^{-4} b^6 = \dfrac{4b^6}{a^4}.
    14. 6x(2x−3)6x(2x - 3). Quick: HCF of 12x212x^2 and 18x18x is 6x6x.
    15. Width =96÷12=8= 96 \div 12 = 8 cm. Diagonal =122+82=144+64=208=413= \sqrt{12^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13} cm.
    16. Let Ben =b= b. Then Chloe =b= b, Alex =2b= 2b, Dan =b+15= b + 15. Sum: 2b+b+b+(b+15)=5b+15=1502b + b + b + (b + 15) = 5b + 15 = 150, so b=27b = 27. Ben pays $27.
    17. 45÷815=45×158=6040=32\dfrac{4}{5} \div \dfrac{8}{15} = \dfrac{4}{5} \times \dfrac{15}{8} = \dfrac{60}{40} = \dfrac{3}{2}.
    18. Mutually exclusive ⇒P(A or B)=0.3+0.5=0.8\Rightarrow P(A \text{ or } B) = 0.3 + 0.5 = 0.8. P(neither)=1−0.8=0.2P(\text{neither}) = 1 - 0.8 = 0.2.
    19. Let the smallest be nn. Then n+(n+2)+(n+4)=3n+6=78n + (n + 2) + (n + 4) = 3n + 6 = 78, so n=24n = 24.
    20. Perimeter of rectangle =2(14+6)=40= 2(14 + 6) = 40. Square side =10= 10. Area =100= 100 cm2^2. Trap: “same perimeter” ≠\neq “same area”.

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