Secondary 4 Additional Mathematics Algebra Functions Quiz

Secondary 4 Additional Mathematics Quiz - Algebra Functions (Answer Key)

Total Marks: 50
Topic: Algebra & Functions (AI-generated, syllabus-aligned, not past-year derived)

Section A Answers

1. [2 marks]

• $g(2) = 2^2 + 1 = 5$
• $f(g(2)) = f(5) = 2(5) - 3 = 7$
  Answer: 7
  Teaching note: Composite function $f(g(x))$ means substitute $g(x)$ into $f$. Always evaluate inner function first.

2. [2 marks]

• Let $y = \frac{x+4}{3}$
• $3y = x + 4 \Rightarrow x = 3y - 4$
• So $f^{-1}(x) = 3x - 4$
  Answer: $f^{-1}(x) = 3x - 4$
  Teaching note: Inverse swaps $x$ and $y$, then solve for $y$.

3. [2 marks]

• $x^2 + 6x + 5 = (x^2 + 6x + 9) - 9 + 5 = (x+3)^2 - 4$
  Answer: $(x+3)^2 - 4$
  Teaching note: Halve the coefficient of $x$ (6→3), square it (9), add and subtract.

4. [2 marks]

• Discriminant $\Delta = k^2 - 4(1)(4) = k^2 - 16$
• No real roots when $\Delta < 0 \Rightarrow k^2 - 16 < 0 \Rightarrow -4 < k < 4$
  Answer: $-4 < k < 4$
  Teaching note: No real roots means parabola does not touch x-axis.

5. [2 marks]

• $x^2 - 5x + 6 = (x-2)(x-3) > 0$
• Critical values: $x=2, 3$. Positive outside: $x < 2$ or $x > 3$
  Answer: $x < 2$ or $x > 3$
  Teaching note: Sketch parabola opening upward; inequality >0 is above axis.

6. [2 marks]

• $3^x = 27 = 3^3 \Rightarrow x = 3$
  Answer: 3
  Teaching note: Express both sides with same base.

7. [2 marks]

• $\log_2 8 = 3$, $\log_2 4 = 2$ (since $2^3=8, 2^2=4$)
• Sum = $3 + 2 = 5$
  Answer: 5
  Teaching note: $\log_a a^n = n$.

Section B Answers

8. [3 marks]

• $h(x) = x^2 - 4x + 7 = (x-2)^2 - 4 + 7 = (x-2)^2 + 3$
• Min at $x=2$, min height = 3 m
  Answer: Min height 3 m at $x=2$ s
  Marking: 1 for completing square, 1 for vertex x, 1 for height.

9. [3 marks]

• $P(1) = 1 + a - 3 + b = a + b - 2 = 5 \Rightarrow a + b = 7$
• $P(-2) = -8 + 4a + 6 + b = 4a + b - 2 = -4 \Rightarrow 4a + b = -2$
• Subtract: $3a = -9 \Rightarrow a = -3$, then $b = 10$
  Answer: $a=-3, b=10$

10. [3 marks]

• $\frac{5x+1}{(x+1)(x-2)} = \frac{A}{x+1} + \frac{B}{x-2}$
• $5x+1 = A(x-2) + B(x+1)$
• $x=2: 11 = 3B \Rightarrow B = 11/3$
• $x=-1: -4 = -3A \Rightarrow A = 4/3$
  Answer: $\frac{4/3}{x+1} + \frac{11/3}{x-2}$

11. [3 marks]

• Square: $2x+3 = x^2 \Rightarrow x^2 - 2x - 3 = 0 \Rightarrow (x-3)(x+1)=0$
• $x=3$ or $x=-1$
• Check: $x=3$: $\sqrt{9}=3$ ok. $x=-1$: $\sqrt{1} = 1 \neq -1$ extraneous.
  Answer: $x=3$

12. [3 marks]

• $g(3) = \ln 3$, $f(g(3)) = e^{2\ln 3} = e^{\ln 9} = 9$
• Domain of $f(g(x)) = e^{2\ln x}$ requires $x > 0$
  Answer: 9, domain $x>0$

13. [3 marks]

• $(1+2x)^5 = \binom{5}{0}1^5 + \binom{5}{1}1^4(2x) + \binom{5}{2}1^3(2x)^2 + \dots$
• = $1 + 5(2x) + 10(4x^2) = 1 + 10x + 40x^2$
  Answer: $1 + 10x + 40x^2$

14. [3 marks]

• $\log_3((x+1)(x-1)) = 2 \Rightarrow (x+1)(x-1) = 3^2 = 9$
• $x^2 - 1 = 9 \Rightarrow x^2 = 10 \Rightarrow x = \pm\sqrt{10}$
• Domain: $x>1$, so $x = \sqrt{10}$
  Answer: $x = \sqrt{10}$

Section C Answers

15. [3 marks]

• $x^2 - 4x + 3 = x - 1 \Rightarrow x^2 - 5x + 4 = 0 \Rightarrow (x-1)(x-4)=0$
• $x=1 \Rightarrow y=0$; $x=4 \Rightarrow y=3$
  Answer: A(1,0), B(4,3)

16. [3 marks]

• Centre $(h,h)$: $(h-1)^2+(h-3)^2 = (h-5)^2+(h-1)^2$
• $(h-3)^2 = (h-5)^2 \Rightarrow h=4$
• Radius$^2 = (4-1)^2+(4-3)^2 = 10$
  Answer: $(x-4)^2+(y-4)^2 = 10$

17. [2 marks]

• $f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x)$
• Odd function
  Answer: Odd

18. [2 marks]

• $40 = 5 \cdot 2^x \Rightarrow 2^x = 8 = 2^3 \Rightarrow x=3$
  Answer: 3

19. [3 marks]

• $P(x) = -2(x^2 - 30x) - 400 = -2((x-15)^2 - 225) - 400 = -2(x-15)^2 + 50$
• Max profit $50 at $x=15$ items
  Answer: $50, 15 items

20. [2 marks]

• $\frac{3}{1+\sqrt{2}} \cdot \frac{1-\sqrt{2}}{1-\sqrt{2}} = \frac{3(1-\sqrt{2})}{1-2} = -3(1-\sqrt{2}) = 3\sqrt{2}-3$
  Answer: $3\sqrt{2} - 3$