Drill Pack 5 (Selective pace)

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

$\sqrt{75} + \sqrt{12} = 5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$ . Trap: simplify each surd first; don’t just add $75 + 12$ .
$3^{-2} + 2^{-1} = \tfrac{1}{9} + \tfrac{1}{2} = \tfrac{2}{18} + \tfrac{9}{18} = \tfrac{11}{18}$ . Trap: negative index means reciprocal, not negative number.
$66. Quick: $80 $\times 0.75 = 60$ ; then $60 \times 1.10 = 66$ dollars. Trap: never “subtract $25\%$ then add $10\%$ = subtract $15\%$ ” — they compound on different bases.
Fifth $= 5 \times 14 - (10 + 12 + 15 + 18) = 70 - 55 = 15$ .
Cross-multiply: $3(x + 3) = 2(2x - 1)$ , so $3x + 9 = 4x - 2$ , $x = 11$ .
Co-interior sum $= 180$ : $(3x + 10) + (2x + 40) = 180$ , so $5x = 130$ , $x = 26$ .
$c^2 = 36 + 64 = 100$ , so $c = 10$ cm. Quick: recognise $6,8,10$ as a Pythagorean triple — $2 \times (3,4,5)$ .
$\dfrac{5}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{5\sqrt{2}}{2}$ .
$P(\text{both red}) = \dfrac{4}{12} \times \dfrac{3}{11} = \dfrac{12}{132} = \dfrac{1}{11}$ . Trap: denominator changes from $12$ to $11$ because it’s without replacement.
$(x + 3)(x - 2) = x^2 + x - 6$ . Quick: FOIL gives $x^2 - 2x + 3x - 6$ ; middle terms collect.
Time $= 3$ h $30$ min $= 3.5$ h. Speed $= 240 \div 3.5 = \dfrac{480}{7} \approx 68.6$ km/h. Trap: convert the time to a single unit first.
$x : y = 3 : 5$ and $y : z = 2 : 7$ . Make $y$ match: multiply first by $2$ and second by $5$ : $x : y = 6 : 10$ ; $y : z = 10 : 35$ . So $x : z = 6 : 35$ .
$(2a^{-2} b^3)^2 = 4 a^{-4} b^6 = \dfrac{4b^6}{a^4}$ .
$6x(2x - 3)$ . Quick: HCF of $12x^2$ and $18x$ is $6x$ .
Width $= 96 \div 12 = 8$ cm. Diagonal $= \sqrt{12^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13}$ cm.
Let Ben $= b$ . Then Chloe $= b$ , Alex $= 2b$ , Dan $= b + 15$ . Sum: $2b + b + b + (b + 15) = 5b + 15 = 150$ , so $b = 27$ . Ben pays $27.
$\dfrac{4}{5} \div \dfrac{8}{15} = \dfrac{4}{5} \times \dfrac{15}{8} = \dfrac{60}{40} = \dfrac{3}{2}$ .
Mutually exclusive $\Rightarrow P(A \text{ or } B) = 0.3 + 0.5 = 0.8$ . $P(\text{neither}) = 1 - 0.8 = 0.2$ .
Let the smallest be $n$ . Then $n + (n + 2) + (n + 4) = 3n + 6 = 78$ , so $n = 24$ .
Perimeter of rectangle $= 2(14 + 6) = 40$ . Square side $= 10$ . Area $= 100$ cm $^2$ . Trap: “same perimeter” $\neq$ “same area”.

Self-check

Score out of 20: _______ / 20
Time taken: _______ minutes
Most time lost on: _______
Topics to revisit: _______