Secondary 4 Additional Mathematics Algebra Functions Quiz

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

Total Marks: 40
Topic: Algebra Functions

Section A: Quadratic Functions and Equations

1. [2 marks]
Complete the square:
$x^2 + 6x + 4 = (x^2 + 6x + 9) - 9 + 4 = (x + 3)^2 - 5$
Answer: $(x + 3)^2 - 5$
Teaching note: Halve the coefficient of $x$ (6 → 3), square it (9), add and subtract.
Marks: 1 for correct bracket, 1 for correct constant.

2. [2 marks]
$f(x) = 2(x^2 - 2x) + 7 = 2[(x - 1)^2 - 1] + 7 = 2(x - 1)^2 + 5$
Vertex at $(1, 5)$.
Answer: $(1, 5)$
Teaching note: Minimum since $a = 2 > 0$. Vertex from completed square.

3. [2 marks]
$a = -1$, $b = 2$, $c = -5$. $\Delta = b^2 - 4ac = 4 - 4(-1)(-5) = 4 - 20 = -16 < 0$.
Since $a < 0$ and $\Delta < 0$, curve is always below x-axis → always negative.
Answer: Yes, always negative.
Marks: 1 for discriminant, 1 for conclusion.

4. [2 marks]
$x^2 - 3x - 4 = (x - 4)(x + 1) > 0$ → $x < -1$ or $x > 4$.
Answer: $x \in (-\infty, -1) \cup (4, \infty)$
Teaching note: Sketch parabola opening upward; positive outside roots.

5. [2 marks]
Equal roots → $\Delta = 0$: $k^2 - 4(1)(9) = 0$ → $k^2 = 36$ → $k = \pm 6$.
Answer: $k = 6$ or $k = -6$
Marks: 1 for equation, 1 for both values.

Section B: Surds and Polynomials

6. [2 marks]
$\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$.
Answer: $\frac{5\sqrt{3}}{3}$
Marks: 1 for multiplying numerator/denominator by $\sqrt{3}$, 1 for simplified form.

7. [2 marks]
Square both sides: $x + 3 = 25$ → $x = 22$. Check: $\sqrt{25} = 5$ ✓
Answer: $x = 22$

8. [2 marks]
$P(1) = 1 - 4 - 7 + 10 = 0$ → by factor theorem, $(x - 1)$ is a factor.
Answer: Shown.

9. [2 marks]
Remainder theorem: $P(2) = 2(8) + a(4) - 5(2) + 6 = 16 + 4a - 10 + 6 = 12 + 4a = 12$ → $4a = 0$ → $a = 0$.
Answer: $a = 0$

10. [2 marks]
$\frac{3x + 1}{(x + 2)(x - 1)} = \frac{A}{x + 2} + \frac{B}{x - 1}$
$3x + 1 = A(x - 1) + B(x + 2)$
Let $x = 1$: $4 = 3B$ → $B = 4/3$
Let $x = -2$: $-5 = -3A$ → $A = 5/3$
Answer: $\frac{5/3}{x + 2} + \frac{4/3}{x - 1}$

Section C: Functions, Exponential and Logarithmic

11. [2 marks]
$g(2) = 4$, $f(4) = 2(4) + 3 = 11$.
Answer: 11

12. [2 marks]
$y = \frac{x - 1}{2}$ → $2y = x - 1$ → $x = 2y + 1$ → $f^{-1}(x) = 2x + 1$.
Answer: $f^{-1}(x) = 2x + 1$

13. [2 marks]
$2^x = 2^4$ → $x = 4$.
Answer: $x = 4$

14. [2 marks]
$\log_2(x(x - 2)) = 3$ → $x(x - 2) = 8$ → $x^2 - 2x - 8 = 0$ → $(x - 4)(x + 2) = 0$
$x = 4$ (since $x > 2$ for log domain).
Answer: $x = 4$

15. [2 marks]
$y = e^{3x + 1}$.
Answer: $y = e^{3x + 1}$

Section D: Structured and Applied Problems

16. [3 marks]
(a) $2x^2 - 8x + 5 = 2(x^2 - 4x) + 5 = 2[(x - 2)^2 - 4] + 5 = 2(x - 2)^2 - 3$
(b) Min value = $-3$ at $x = 2$.
Marks: 2 for (a), 1 for (b).

17. [3 marks]
Set equal: $x^2 + kx + 4 = 2x - 1$ → $x^2 + (k - 2)x + 5 = 0$.
No intersection → $\Delta < 0$: $(k - 2)^2 - 20 < 0$ → $(k - 2)^2 < 20$ → $-\sqrt{20} < k - 2 < \sqrt{20}$
$2 - 2\sqrt{5} < k < 2 + 2\sqrt{5}$.
Answer: $2 - 2\sqrt{5} < k < 2 + 2\sqrt{5}$

18. [4 marks]
(a) Form stated.
(b) $5x + 4 = A(x - 2)^2 + B(x + 1)(x - 2) + C(x + 1)$
$x = -1$: $-1 = 9A$ → $A = -1/9$
$x = 2$: $14 = 3C$ → $C = 14/3$
Coeff $x^2$: $0 = A + B$ → $B = 1/9$
Answer: $A = -1/9, B = 1/9, C = 14/3$
Marks: 1 for form, 3 for values.

19. [4 marks]
(a) $P = 200 e^{0.2 \times 5} = 200 e^1 \approx 200 \times 2.718 = 543.6$
(b) $400 = 200 e^{0.2t}$ → $2 = e^{0.2t}$ → $\ln 2 = 0.2t$ → $t = \frac{\ln 2}{0.2} \approx 3.47$ h
Marks: 2 each part.

20. [4 marks]
(a) $f(2) = 8 - 12 - 8 + 12 = 0$ → $(x - 2)$ factor. Divide: $x^3 - 3x^2 - 4x + 12 = (x - 2)(x^2 - x - 6) = (x - 2)(x - 3)(x + 2)$
(b) $x = 2, 3, -2$
(c) $(-2, 0), (2, 0), (3, 0)$
Marks: 2 for factorise, 1 for solve, 1 for intercepts.