1. **Find the maximum and minimum value of the function** $$f(x) = 3x^2 - 2x^3 - 12x + 8$$ 2. **Find critical points by differentiating:** $$f'(x) = 6x - 6x^2 - 12$$ 3. **Set derivative equal to zero to find critical points:** $$6x - 6x^2 - 12 = 0$$ Divide by 6: $$x - x^2 - 2 = 0$$ Rearrange: $$-x^2 + x - 2 = 0$$ Multiply by -1: $$x^2 - x + 2 = 0$$ 4. **Calculate discriminant:** $$\Delta = (-1)^2 - 4\times1\times2 = 1 - 8 = -7 < 0$$ Since discriminant is negative, no real critical points; function has no maxima or minima in the real numbers. --- 2. **Number of ways to choose a mixed hockey team** *From 7 men, choose 5:* $$C(7,5) = \frac{7!}{5!2!} = 21$$ *From 9 women, choose 6:* $$C(9,6) = \frac{9!}{6!3!} = 84$$ *Total ways:* $$21 \times 84 = 1764$$ --- 3. **Find the value of $x$ such that 3 consecutive terms of GP are** $$2+x, 2 - x, 5 - x$$ 4. **Condition for GP:** $$\frac{2 - x}{2 + x} = \frac{5 - x}{2 - x}$$ 5. **Cross multiply:** $$(2 - x)^2 = (2 + x)(5 - x)$$ 6. **Expand:** $$ (2 - x)^2 = 4 - 4x + x^2$$ $$ (2 + x)(5 - x) = 10 - 2x + 5x - x^2 = 10 + 3x - x^2$$ 7. **Set equal and simplify:** $$4 - 4x + x^2 = 10 + 3x - x^2$$ $$x^2 + x^2 - 4x - 3x + 4 - 10 = 0$$ $$2x^2 - 7x - 6 = 0$$ 8. **Solve quadratic:** $$\Delta = (-7)^2 - 4 \times 2 \times (-6) = 49 + 48 = 97$$ $$x = \frac{7 \pm \sqrt{97}}{4}$$ --- 4. **Given Universal set $U = \{1,2,3,4,5,6,7,8,9\}$, sets** $$A = \{1,2,4,7\}, B=\{2,4,6,8\}, C=\{3,4,5,6\}$$ (i) Find $$A \cap (B \cup C)$$ 5. Compute $$B \cup C = \{2,3,4,5,6,8\}$$ 6. Then $$A \cap (B \cup C) = \{1,2,4,7\} \cap \{2,3,4,5,6,8\} = \{2,4\}$$ (ii) Find $$B \cup (C - A)$$ 7. Compute $$C - A = \{3,4,5,6\} - \{1,2,4,7\} = \{3,5,6\}$$ 8. Compute $$B \cup (C - A) = \{2,4,6,8\} \cup \{3,5,6\} = \{2,3,4,5,6,8\}$$ --- 5. **Prove if $y = 10^{\log a - \log b x}$, then** $$y = \left(\frac{a}{b}\right)^x$$ 6. Write: $$y = 10^{\log a - \log b x} = 10^{\log a} \times 10^{-\log b x} = a \times 10^{-\log b x}$$ 7. Use property: $$10^{-\log b x} = \frac{1}{10^{\log b x}} = \frac{1}{(10^{\log b})^x} = \frac{1}{b^x}$$ 8. Thus: $$y = a \times \frac{1}{b^x} = a b^{-x} = \left(\frac{a}{b}\right)^x$$ --- 6. **Solve inequality:** $$1 - \frac{3x}{2} \leq x - 4$$ 7. Multiply all terms by 2 to eliminate fraction: $$2 - 3x \leq 2x - 8$$ 8. Bring terms to one side: $$2 + 8 \leq 2x + 3x$$ $$10 \leq 5x$$ 9. Solve for x: $$x \geq 2$$ --- 7. **Find inverse of matrix** $$A = \begin{pmatrix}1 & 5 \\ 2 & 4\end{pmatrix}$$ 8. **Calculate determinant:** $$|A| = 1 \times 4 - 5 \times 2 = 4 - 10 = -6$$ 9. **Inverse:** $$A^{-1} = \frac{1}{-6} \begin{pmatrix}4 & -5 \\ -2 & 1\end{pmatrix} = \begin{pmatrix}-\frac{2}{3} & \frac{5}{6} \\ \frac{1}{3} & -\frac{1}{6}\end{pmatrix}$$ 10. **Solve simultaneous equations:** $$\begin{cases} 1x + 5y = 11 \\ 2x + 4y = 6 \end{cases}$$ 11. Represent as $$AX = B$$ where $$X= \begin{pmatrix}x \\ y\end{pmatrix}$$ and $$B=\begin{pmatrix}11 \\ 6\end{pmatrix}$$ 12. Find $$X = A^{-1}B$$: $$X = \begin{pmatrix}-\frac{2}{3} & \frac{5}{6} \\ \frac{1}{3} & -\frac{1}{6}\end{pmatrix} \begin{pmatrix}11 \\ 6\end{pmatrix}$$ 13. Calculate: $$x = -\frac{2}{3} \times 11 + \frac{5}{6} \times 6 = -\frac{22}{3} + 5 = -\frac{22}{3} + \frac{15}{3} = -\frac{7}{3}$$ $$y = \frac{1}{3} \times 11 - \frac{1}{6} \times 6 = \frac{11}{3} - 1 = \frac{11}{3} - \frac{3}{3} = \frac{8}{3}$$ --- 8. **Brand usage in 800 households:** - Rina (R) = 230 - Avena (A) = 245 - Elianto (E) = 325 - All three = 30 - Rina & Elianto = 70 - Rina only = 110 - Elianto only = 185 (i) **Venn diagram representation:** - Rina only = 110 - Elianto only = 185 - Avena only = \(245 - (\text{Rina & Avena}) - (\text{Avena & Elianto}) - 30\) (unknown) (ii) **Find households who used none:** Let - Sum all single and multiple overlaps and subtract from 800 Sum known households: - Rina only = 110 - Elianto only = 185 - All three = 30 - Rina & Elianto (including all three) = 70; so Rina & Elianto only = 70 - 30 = 40 - Similarly, total Rina = 230, so Rina & Avena only = 230 - 110 - 40 - 30 = 50 - Total Elianto = 325, so Avena & Elianto only = 325 - 185 - 40 - 30 = 70 - Total Avena = 245, so Avena only = 245 - 50 - 70 - 30 = 95 Total using at least one brand: $$110 + 185 + 95 + 50 + 70 + 40 + 30 = 580$$ Households using none: $$800 - 580 = 220$$