Study Companion
Selective Entry Mathematics | Practice mode

Practice

88 questions across 4 topics, drawn from every Practice and Challenge block in Selective Entry mathematics. Filter by topic or level, cap the count, shuffle, and start the timer when you want to time a session.

00:00
Topics to include
·
Showing all 88 questions.

Surds & irrational arithmetic

Fluency · Simplify

  1. 1. Simplify 12\sqrt{12}. (show answer)
    Answer
    232\sqrt{3}.
  2. 2. Simplify 27\sqrt{27}. (show answer)
    Answer
    333\sqrt{3}.
  3. 3. Simplify 45\sqrt{45}. (show answer)
    Answer
    353\sqrt{5}.
  4. 4. Simplify 98\sqrt{98}. (show answer)
    Answer
    727\sqrt{2}.
  5. 5. Simplify 2322\sqrt{32}. (show answer)
    Answer
    232=2×42=822\sqrt{32} = 2 \times 4\sqrt{2} = 8\sqrt{2}.
  6. 6. Simplify 200\sqrt{200}. (show answer)
    Answer
    10210\sqrt{2}.

Fluency · Add & subtract

  1. 1. 5+35\sqrt{5} + 3\sqrt{5}. (show answer)
    Answer
    454\sqrt{5}.
  2. 2. 72−427\sqrt{2} - 4\sqrt{2}. (show answer)
    Answer
    323\sqrt{2}.
  3. 3. 8+18\sqrt{8} + \sqrt{18}. (show answer)
    Answer
    8+18=22+32=52\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}.
  4. 4. 50−32\sqrt{50} - \sqrt{32}. (show answer)
    Answer
    52−42=25\sqrt{2} - 4\sqrt{2} = \sqrt{2}.
  5. 5. 75+27\sqrt{75} + \sqrt{27}. (show answer)
    Answer
    53+33=835\sqrt{3} + 3\sqrt{3} = 8\sqrt{3}.

Fluency · Multiply

  1. 1. 3×7\sqrt{3} \times \sqrt{7}. (show answer)
    Answer
    21\sqrt{21}.
  2. 2. 25×322\sqrt{5} \times 3\sqrt{2}. (show answer)
    Answer
    6106\sqrt{10}.
  3. 3. (6)2(\sqrt{6})^2. (show answer)
    Answer
    66.
  4. 4. 8×2\sqrt{8} \times \sqrt{2}. (show answer)
    Answer
    16=4\sqrt{16} = 4.
  5. 5. 3(2+12)\sqrt{3}(2 + \sqrt{12}). (show answer)
    Answer
    23+36=23+62\sqrt{3} + \sqrt{36} = 2\sqrt{3} + 6.

Fluency · Rationalise

  1. 1. Rationalise 13\dfrac{1}{\sqrt{3}}. (show answer)
    Answer
    33\dfrac{\sqrt{3}}{3}.
  2. 2. Rationalise 105\dfrac{10}{\sqrt{5}}. (show answer)
    Answer
    1055=25\dfrac{10\sqrt{5}}{5} = 2\sqrt{5}.
  3. 3. Rationalise 28\dfrac{\sqrt{2}}{\sqrt{8}}. (show answer)
    Answer
    28=28=14=12\dfrac{\sqrt{2}}{\sqrt{8}} = \sqrt{\tfrac{2}{8}} = \sqrt{\tfrac{1}{4}} = \tfrac{1}{2}.

Reasoning · Mixed & apply

  1. 1. A square has area 48 cm248\text{ cm}^2. Find its side length in simplest surd form. (show answer)
    Answer
    side =48=43= \sqrt{48} = 4\sqrt{3} cm.
  2. 2. Find the exact length of the diagonal of a 3×53 \times 5 rectangle in simplest surd form. (show answer)
    Answer
    diagonal =9+25=34= \sqrt{9 + 25} = \sqrt{34}. Already in simplest form.
  3. 3. Simplify 18+8−50\sqrt{18} + \sqrt{8} - \sqrt{50}. (show answer)
    Answer
    18+8−50=32+22−52=0\sqrt{18} + \sqrt{8} - \sqrt{50} = 3\sqrt{2} + 2\sqrt{2} - 5\sqrt{2} = 0.
  4. 4. Show that 2×3×6=6\sqrt{2} \times \sqrt{3} \times \sqrt{6} = 6. (show answer)
    Answer
    2×3×6=2×3×6=36=6\sqrt{2} \times \sqrt{3} \times \sqrt{6} = \sqrt{2 \times 3 \times 6} = \sqrt{36} = 6.

Reasoning · Harder reasoning

  1. 1. Expand and simplify (3+2)(3−2)(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}). What does the pattern resemble? (show answer)
    Answer
    (3)2−(2)2=3−2=1(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1. It's the difference of squares pattern (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2, used for rationalising binomial surds.
  2. 2. Expand (2+5)2(2 + \sqrt{5})^2. (show answer)
    Answer
    (2+5)2=4+45+5=9+45(2 + \sqrt{5})^2 = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5}.
  3. 3. Rationalise 12+3\dfrac{1}{2 + \sqrt{3}}. (Hint: multiply top and bottom by 2−32 - \sqrt{3}.) (show answer)
    Answer
    12+3×2−32−3=2−34−3=2−3\dfrac{1}{2 + \sqrt{3}} \times \dfrac{2 - \sqrt{3}}{2 - \sqrt{3}} = \dfrac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}.
  4. 4. The hypotenuse of a right-angled triangle with legs 11 and 3\sqrt{3} equals the side of a square. Find the area of the square. (show answer)
    Answer
    Hypotenuse =1+3=2= \sqrt{1 + 3} = 2. Area of square =22=4= 2^2 = 4.

Index laws extended

Fluency · Evaluate

  1. 1. 505^0. (show answer)
    Answer
    11.
  2. 2. (−3)0(-3)^0. (show answer)
    Answer
    11.
  3. 3. 70−207^0 - 2^0. (show answer)
    Answer
    1−1=01 - 1 = 0.
  4. 4. 2−22^{-2}. (show answer)
    Answer
    14\dfrac{1}{4}.
  5. 5. 3−33^{-3}. (show answer)
    Answer
    127\dfrac{1}{27}.
  6. 6. 10−1+10−210^{-1} + 10^{-2}. (show answer)
    Answer
    110+1100=0.11\dfrac{1}{10} + \dfrac{1}{100} = 0.11.

Fluency · Simplify

  1. 1. x5×x−3x^5 \times x^{-3}. (show answer)
    Answer
    x5+(−3)=x2x^{5 + (-3)} = x^2.
  2. 2. y2y−4\dfrac{y^2}{y^{-4}}. (show answer)
    Answer
    y2−(−4)=y6y^{2 - (-4)} = y^6.
  3. 3. (a−2)3(a^{-2})^3. (show answer)
    Answer
    a−2×3=a−6=1a6a^{-2 \times 3} = a^{-6} = \dfrac{1}{a^6}.
  4. 4. (2m)−2(2m)^{-2}. (show answer)
    Answer
    (2m)−2=1(2m)2=14m2(2m)^{-2} = \dfrac{1}{(2m)^2} = \dfrac{1}{4m^2}.
  5. 5. 6a32a7\dfrac{6a^3}{2a^7}. (show answer)
    Answer
    3a3−7=3a−4=3a43a^{3-7} = 3a^{-4} = \dfrac{3}{a^4}.
  6. 6. 4x0+(5x)04x^0 + (5x)^0. (show answer)
    Answer
    4(1)+1=54(1) + 1 = 5.

Reasoning · Mixed algebraic

  1. 1. Simplify 8p5q−34p2q−1\dfrac{8p^5 q^{-3}}{4p^2 q^{-1}} with positive indices. (show answer)
    Answer
    84⋅p5−2q−3−(−1)=2p3q−2=2p3q2\dfrac{8}{4} \cdot p^{5-2} q^{-3-(-1)} = 2p^3 q^{-2} = \dfrac{2p^3}{q^2}.
  2. 2. Simplify (3a−1b2)2(3a^{-1} b^2)^2. (show answer)
    Answer
    9a−2b4=9b4a29a^{-2}b^4 = \dfrac{9b^4}{a^2}.
  3. 3. Simplify (2x3)24x−1\dfrac{(2x^3)^2}{4x^{-1}}. (show answer)
    Answer
    4x64x−1=x6−(−1)=x7\dfrac{4x^6}{4x^{-1}} = x^{6 - (-1)} = x^7.
  4. 4. Write 1a−5×a−3\dfrac{1}{a^{-5}} \times a^{-3} as a single power of aa. (show answer)
    Answer
    1a−5×a−3=a5×a−3=a2\dfrac{1}{a^{-5}} \times a^{-3} = a^5 \times a^{-3} = a^2.
  5. 5. Show that x−2y3x−4y−1=x2y4\dfrac{x^{-2} y^3}{x^{-4} y^{-1}} = x^2 y^4. (show answer)
    Answer
    x−2x−4⋅y3y−1=x−2−(−4)y3−(−1)=x2y4\dfrac{x^{-2}}{x^{-4}} \cdot \dfrac{y^3}{y^{-1}} = x^{-2 - (-4)} y^{3 - (-1)} = x^2 y^4. ✓\checkmark

Reasoning · Harder reasoning

  1. 1. If ax=8a^x = 8 and ay=2a^y = 2, find ax−ya^{x - y}. (show answer)
    Answer
    ax−y=axay=82=4a^{x - y} = \dfrac{a^x}{a^y} = \dfrac{8}{2} = 4.
  2. 2. Solve 2n=1322^n = \dfrac{1}{32} for nn. (show answer)
    Answer
    132=2−5\dfrac{1}{32} = 2^{-5}, so n=−5n = -5.
  3. 3. Simplify (2a−3b2)−2\left(\dfrac{2a^{-3}}{b^2}\right)^{-2} with positive indices. (show answer)
    Answer
    (2a−3b2)−2=(b22a−3)2=b44a−6=a6b44\left(\dfrac{2a^{-3}}{b^2}\right)^{-2} = \left(\dfrac{b^2}{2a^{-3}}\right)^2 = \dfrac{b^4}{4a^{-6}} = \dfrac{a^6 b^4}{4}.
  4. 4. Given x>0x > 0, order from smallest to largest: x3,x−1,x0,x1/2x^3, x^{-1}, x^0, x^{1/2} when x=4x = 4. (show answer)
    Answer
    With x=4x = 4: x−1=0.25x^{-1} = 0.25, x0=1x^0 = 1, x1/2=2x^{1/2} = 2, x3=64x^3 = 64. Order: x−1<x0<x1/2<x3x^{-1} < x^0 < x^{1/2} < x^3.

Drill Pack 1 (easy mix)

Fluency · Drill Pack 1 — 25 minutes

  1. 1. Evaluate −7+12−5-7 + 12 - 5. (show answer)
    Answer
    00. Quick: group signs, (−7−5)+12=−12+12=0(-7 - 5) + 12 = -12 + 12 = 0. Trap: don't compute left-to-right if a cancellation is obvious.
  2. 2. 34+12\dfrac{3}{4} + \dfrac{1}{2} as a single fraction. (show answer)
    Answer
    54\dfrac{5}{4} (or 1141\tfrac{1}{4}). Quick: common denominator 44: 34+24\tfrac{3}{4} + \tfrac{2}{4}. Trap: don't add denominators.
  3. 3. 25%25\% of 160160. (show answer)
    Answer
    4040. Quick: 25%25\% is 14\tfrac{1}{4}, so divide by 44. Trap: 25%25\% of 160160 is not 25×1.6=4025 \times 1.6 = 40 by coincidence — always convert the percent to a fraction first.
  4. 4. Simplify the ratio 18:2418 : 24. (show answer)
    Answer
    3:43 : 4. Quick: HCF of 1818 and 2424 is 66, divide both by 66.
  5. 5. Solve 3x+4=193x + 4 = 19. (show answer)
    Answer
    x=5x = 5. Quick: 3x=153x = 15, x=5x = 5. Trap: do "undo + 4" before "undo ×3\times 3".
  6. 6. Round 3.7483.748 to 11 decimal place. (show answer)
    Answer
    3.73.7. Quick: the digit after the 77 is a 44, so round down.
  7. 7. A rectangle is 77 cm by 44 cm. Perimeter? (show answer)
    Answer
    2222 cm. Quick: 2(7+4)=222(7 + 4) = 22.
  8. 8. Area of a triangle with base 66 cm and height 55 cm? (show answer)
    Answer
    1515 cm2^2. Quick: 12×6×5\tfrac{1}{2} \times 6 \times 5. Trap: don't forget the 12\tfrac{1}{2}.
  9. 9. 0.60.6 written as a fraction in simplest form. (show answer)
    Answer
    35\dfrac{3}{5}. Quick: 0.6=610=350.6 = \tfrac{6}{10} = \tfrac{3}{5}.
  10. 10. Next term of 2,5,8,11,…2, 5, 8, 11, \ldots (show answer)
    Answer
    1414. Quick: common difference is +3+3.
  11. 11. The mean of 4,6,8,10,124, 6, 8, 10, 12. (show answer)
    Answer
    88. Quick: the middle number of a symmetric arithmetic set equals the mean.
  12. 12. P(even)P(\text{even}) when rolling a fair six-sided die. (show answer)
    Answer
    12\dfrac{1}{2}. Quick: 2,4,62, 4, 6 out of 66. Trap: don't rewrite as 36\tfrac{3}{6} if you can spot 12\tfrac{1}{2} immediately.
  13. 13. Expand 4(x−3)4(x - 3). (show answer)
    Answer
    4x−124x - 12. Quick: distributive law.
  14. 14. If x=5x = 5, evaluate 2x2−32x^2 - 3. (show answer)
    Answer
    4747. Quick: 2(25)−3=50−3=472(25) - 3 = 50 - 3 = 47. Trap: "2x22x^2" means 2⋅x22 \cdot x^2, not (2x)2(2x)^2.
  15. 15. A car travels 180180 km in 33 hours. Average speed in km/h? (show answer)
    Answer
    6060 km/h. Quick: 180÷3=60180 \div 3 = 60.
  16. 16. Write 35\dfrac{3}{5} as a percentage. (show answer)
    Answer
    60%60\%. Quick: 35=60100\tfrac{3}{5} = \tfrac{60}{100}.
  17. 17. A triangle has angles 50∘50^\circ, 60∘60^\circ, and xx. Find xx. (show answer)
    Answer
    70∘70^\circ. Quick: 180−50−60=70180 - 50 - 60 = 70.
  18. 18. Simplify 8a−3a+2a8a - 3a + 2a. (show answer)
    Answer
    7a7a. Quick: 8−3+2=78 - 3 + 2 = 7.
  19. 19. 12212^2. (show answer)
    Answer
    144144. Memorise squares up to 152=22515^2 = 225 — comes up constantly.
  20. 20. Which is larger: 23\dfrac{2}{3} or 0.650.65? (show answer)
    Answer
    0.650.65. Quick: 23≈0.667\tfrac{2}{3} \approx 0.667, just above 0.650.65. So 23\tfrac{2}{3} is larger. Trap: reread "which is larger" — test-writers put the tempting wrong answer first.

Drill Pack 5 (Selective pace)

Reasoning · Drill Pack 5 — 25 minutes

  1. 1. Simplify 75+12\sqrt{75} + \sqrt{12}. (show answer)
    Answer
    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. 2. Evaluate 3−2+2−13^{-2} + 2^{-1} as a single fraction. (show answer)
    Answer
    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. 3. A shirt marked $80 is discounted by 25%25\%, then GST of 10%10\% is added. Final price? (show answer)
    Answer
    $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. 4. The mean of five numbers is 1414. If four of them are 10,12,15,1810, 12, 15, 18, find the fifth. (show answer)
    Answer
    Fifth =5×14−(10+12+15+18)=70−55=15= 5 \times 14 - (10 + 12 + 15 + 18) = 70 - 55 = 15.
  5. 5. Solve x+32=2x−13\dfrac{x + 3}{2} = \dfrac{2x - 1}{3}. (show answer)
    Answer
    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. 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. (show answer)
    Answer
    Co-interior sum =180= 180: (3x+10)+(2x+40)=180(3x + 10) + (2x + 40) = 180, so 5x=1305x = 130, x=26x = 26.
  7. 7. A right-angled triangle has legs 66 cm and 88 cm. Find the hypotenuse. (show answer)
    Answer
    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. 8. Rewrite 52\dfrac{5}{\sqrt{2}} with a rational denominator. (show answer)
    Answer
    52×22=522\dfrac{5}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{5\sqrt{2}}{2}.
  9. 9. A bag has 44 red, 55 blue, 33 green marbles. Two are drawn without replacement. Find P(both red)P(\text{both red}). (show answer)
    Answer
    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. 10. Expand and simplify (x+3)(x−2)(x + 3)(x - 2). (show answer)
    Answer
    (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. 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? (show answer)
    Answer
    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. 12. If x:y=3:5x : y = 3 : 5 and y:z=2:7y : z = 2 : 7, find x:zx : z in simplest form. (show answer)
    Answer
    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. 13. Simplify (2a−2b3)2(2a^{-2} b^3)^2, with positive indices only. (show answer)
    Answer
    (2a−2b3)2=4a−4b6=4b6a4(2a^{-2} b^3)^2 = 4 a^{-4} b^6 = \dfrac{4b^6}{a^4}.
  14. 14. Factorise 12x2−18x12x^2 - 18x. (show answer)
    Answer
    6x(2x−3)6x(2x - 3). Quick: HCF of 12x212x^2 and 18x18x is 6x6x.
  15. 15. A rectangle has area 9696 cm2^2 and length 1212 cm. The diagonal has length (in surd form)? (show answer)
    Answer
    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. 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? (show answer)
    Answer
    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. 17. Evaluate 45÷815\dfrac{4}{5} \div \dfrac{8}{15}. (show answer)
    Answer
    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. 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}). (show answer)
    Answer
    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. 19. The sum of three consecutive even integers is 7878. Find the smallest. (show answer)
    Answer
    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. 20. A square and a rectangle have the same perimeter. The rectangle is 1414 cm by 66 cm. Find the area of the square. (show answer)
    Answer
    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".