Core Practice
Worksheet

💡 Explanation Numerator: \(x^5 \cdot x^{-2} = x^{5+(-2)} = x^3\). Then \(\dfrac{x^3}{x^3} = x^{3-3} = x^0 = \mathbf{1}\).
Key rule: Any non-zero base to the power 0 equals 1.

💡 Explanation Move the decimal 4 places right to get \(3.47\). Since the original number is < 1, the exponent is negative: \(3.47 \times 10^{-4}\).
Options A & D have wrong sign or wrong coefficient range (must satisfy \(1 \le a < 10\)).

💡 Explanation \((a-b)^2 = a^2 - 2ab + b^2\). Here \(a=2x,\ b=3\):
\((2x)^2 - 2(2x)(3) + 3^2 = 4x^2 - 12x + 9\).
B is the classic trap — squaring each term and ignoring \(2ab\).

💡 Explanation Expand: \(6x - 12 = 2x + 10\) → \(4x = 22\) → \(x = \dfrac{22}{4} = \dfrac{11}{2}\).
Always expand first, then collect like terms on each side.

💡 Explanation \(m = \dfrac{-2-4}{3-0} = \dfrac{-6}{3} = -2\). The \(y\)-intercept \(c = 4\) (given by point \((0,4)\)).
Equation: \(y = -2x + 4\).

💡 Explanation \(x^2 - 4x + 3 = 0\) → \((x-1)(x-3) = 0\) → \(x = 1\) or \(x = 3\).
The vertex is at \(x = 2\) (axis of symmetry between the roots), not an x-intercept!

💡 Explanation \((x+3)\) shifts the graph left by 3 (counter-intuitive — adding inside moves left!).
\(-5\) outside shifts the graph down by 5. Vertex moves from \((0,0)\) to \((-3,-5)\).

💡 Explanation \(c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\) cm.
The triple 5-12-13 is one of the most tested. Also memorise: 3-4-5, 8-15-17.

💡 Explanation Area scale factor = (linear scale factor)² = \((3)^2 : (5)^2 = 9 : 25\).
Option D uses cubing — that's for volume, not area.

💡 Explanation Base: \(\pi r^2 = 3.14 \times 36 = 113.04\) cm².
Lateral: \(\pi r l = 3.14 \times 6 \times 10 = 188.4\) cm².
Total: \(113.04 + 188.4 = \mathbf{301.44}\) cm². Option B gives only lateral; D gives only base.

💡 Explanation \(\sin 30° = \dfrac{\text{opp}}{\text{hyp}}\) → \(\text{opp} = 20 \times \sin 30° = 20 \times 0.5 = 10\) cm.
Option A uses \(\cos 30°\) — that gives the adjacent side, not opposite.

💡 Explanation \(c^2 = a^2 + b^2 - 2ab\cos C = 64 + 25 - 2(8)(5)\cos 60°\)
\(= 89 - 80 \times 0.5 = 89 - 40 = 49\) → \(c = 7\).

💡 Explanation Area \(= \dfrac{1}{2} \times 10 \times 7 \times \sin 45° = 35 \times \dfrac{\sqrt{2}}{2} = \dfrac{35\sqrt{2}}{2} \approx 24.75\) cm².
Option A ignores the sin factor; B doubles the answer.

💡 Explanation 7 values → median = 4th value = 8. Mode = value appearing most = 7 (appears twice).
C gives the mean, not the median. Never confuse these three measures!

💡 Explanation P(1st red) = \(\dfrac{4}{10}\). After one red removed: P(2nd red) = \(\dfrac{3}{9}\).
P(both red) = \(\dfrac{4}{10} \times \dfrac{3}{9} = \dfrac{12}{90} = \dfrac{2}{15}\).
Option D uses \(\dfrac{4}{10} \times \dfrac{4}{10}\) — the with-replacement answer (wrong!).

💡 Explanation \(|F \cup T| = 18 + 12 - 5 = 25\). Neither = \(30 - 25 = \mathbf{5}\).
Inclusion-exclusion: always subtract the overlap to avoid counting twice!

💡 Explanation From eq. 2: \(x = y + 1\). Sub into eq. 1: \(2(y+1) + 3y = 12\) → \(5y = 10\) → \(y = 2\), \(x = 3\).
Check: \(2(3)+3(2)=12\) ✓ and \(3-2=1\) ✓

💡 Explanation \(-2 < 0\) → opens down → max at vertex = 8.
Set \(y = 0\): \(-2(x-1)^2 + 8 = 0\) → \((x-1)^2 = 4\) → \(x-1 = \pm 2\) → \(x = 3\) or \(x = -1\).

💡 Explanation Height: \(\sqrt{10^2 - 6^2} = \sqrt{64} = 8\) m.
Angle with ground: \(\cos\theta = \dfrac{6}{10} = 0.6\) → \(\theta = \cos^{-1}(0.6) \approx 53.1°\).
Option B uses the wrong angle: \(36.9°\) is the angle at the top of the wall.

💡 Explanation Favourable pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 pairs.
P = \(\dfrac{6}{36} = \dfrac{1}{6}\). Note: A and C are the same — watch for reformulated correct answers as distractors!