Ordinary level 2026 technical North West regional mock mathematics guide

I have extracted the text from the newest image, which appears to contain a mark scheme and detailed solutions for various mathematics problems, including Venn diagrams, set theory, and trigonometry.

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SECTION A: Solutions & Mark Scheme

1. Prime Factors, HCF, and LCM

• a) $30 = 2 \times 3 \times 5$; $\quad 36 = 2 \times 2 \times 3 \times 3$

• b) HCF: $2 \times 3 = 6$

• c) LCM: $2 \times 2 \times 3 \times 3 \times 5 = 180$

2. Fractions & Order of Operations

• $2\frac{1}{2} + 3\frac{1}{4} \div 2\frac{1}{6} – 1\frac{1}{6}$

• $= \frac{5}{2} + \left( \frac{13}{4} \div \frac{13}{6} \right) – \frac{7}{6}$

• $= \frac{5}{2} + \left( \frac{13}{4} \times \frac{6}{13} \right) – \frac{7}{6} = \frac{17}{6} = 2\frac{5}{6}$

3. Rationalizing Denominators

• $A = \frac{(3 – \sqrt{2})(1 – \sqrt{2})}{(1 + \sqrt{2})(1 – \sqrt{2})} = -5 + 4\sqrt{2}$

4. Composite Functions

• Let $g(x) = y \leftrightarrow f(y) = 4y – 5$

• $\therefore 4y – 5 = 6x – 3 \rightarrow y = \frac{3x+1}{2} = g(x)$

5. Standard Form

• $\frac{0.32 \times 0.042}{0.32} = \frac{32 \times 10^{-3} \times 42 \times 10^{-3}}{32 \times 10^{-2}} = 4.2 \times 10^{-3}$

6. Percentages & Circles

• a) Weight decrease: $(80 – 71)\text{kg} = 9\text{kg}$

  • $\%$ increase $= \frac{9}{80} \times 100\% = 11.25\%$

• b) Radius: $\frac{10}{2} = 5\text{ m}$

  • Area $= \pi r^2 = \pi(5 \times 5)\text{ m}^2 = 25\pi\text{ m}^2$

7. Map Scale & Shaded Area

• a) Actual distance: $\frac{14 \times 20000}{100000}\text{ km} = 2.8\text{ km}$

• b) Area ABCD: $19 \times 9 = 171\text{ cm}^2$

  • Area PQRS: $7 \times 3 = 21\text{ cm}^2$

  • Shaded part: $171 – 21 = 150\text{ cm}^2$

8. Trigonometry & Geometry

• a) Right-angled triangle with sides 2, 3, and hypotenuse $\sqrt{13}$.

  • $\sin x = \frac{3}{\sqrt{13}}; \quad \cos x = \frac{2}{\sqrt{13}}$

• b) $x = 55^\circ$; $\quad 4y = 110^\circ \rightarrow y = 27.5^\circ$

  • $x + y = 82.5^\circ$

9. Inequalities

• $2x – 1 \le 3x + 9 \rightarrow -x \le 10 \rightarrow x \ge -10$

• (Number line shown starting at -10 with a solid circle and arrow pointing right)

10. Discriminant for Equal Roots

• $4x^2 + kx + 1 = 0 \rightarrow \Delta = k^2 – 16$

• For equal roots, $\Delta = 0 \rightarrow k = \pm 4$

• But $k < 0 \rightarrow k = -4$

11. Probability

• a) $\frac{1}{4} \times 28 = 7$ Students for lawn A

• b) Left: $28 – 7 = 21; \quad \frac{1}{3} \times 21 = 7$ Students for B

• c) $28 – 7 – 7 = 14$ Students for C

• d) P(Lawn C): $\frac{14}{28} = \frac{1}{2}$

12. Perpendicular Lines

• $(L_1): y = \frac{1}{7}x + \frac{1}{14} \rightarrow m_1 = \frac{1}{7}$

• For $(L_2): m_2 = -\frac{1}{m_1} = -7$

• $\therefore (L_2): y – \frac{5}{2} = -7(x + 1) \rightarrow y = -7x – \frac{9}{2}$

13. Mean and Estimated Height

• $\bar{x} = \frac{3(2)+4(3)+5(7)+6(6)+7(2)}{20} = \frac{103}{20} = 5.15$

• $\approx 5.2\text{ m}$

14. Set Theory & Venn Diagrams

• a) $A = \{0,1,2,3,4\}; B = \{-2,-1,0,1,2,3\}$; $\varepsilon = \{-3,-2,-1,0,1,2,3,4,5\}$

• b) Venn Diagram:

  * Intersection $A \cap B = \{0, 1, 2, 3\}$

  • Only A: $\{4\}$

  • Only B: $\{-2, -1\}$

  • Outside A and B: $\{-3, 5\}$

15. Matrices

• a) $U_5 = S_5 – S_4 = 21 – 13 = 8$

• b) $P – Q = \begin{pmatrix} -1 & 3 \\ 2 & 5 \end{pmatrix} – \begin{pmatrix} 4 & 1 \\ 0 & -3 \end{pmatrix} = \begin{pmatrix} -5 & 2 \\ 2 & 8 \end{pmatrix}$

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Would you like me to explain the logic behind the “equal roots” discriminant in question 10 or the method for finding perpendicular gradients in question 12?