Secondary 2 Mathematics Calculus Quiz

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Secondary 2 Mathematics Quiz - Calculus

Name: ________________________
Class: ________________________
Date: ________________________
Score: ________ / 60

Duration: 60 minutes
Total Marks: 60

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Instructions

• Answer all questions in the spaces provided.
• Show all working clearly. Marks are awarded for correct method as well as final answer.
• Use a calculator where appropriate. Unless otherwise stated, give answers correct to 3 significant figures.
• The number of marks for each question is shown in brackets [ ].
• This quiz covers introductory calculus concepts aligned to the Secondary 2 G3 Mathematics syllabus.

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Section A: Understanding Rate of Change (Questions 1–5)

Answer all questions in this section.

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1. The distance travelled by a car, $d$ metres, after $t$ seconds is given by $d = 5t^2 + 3t$.

(a) Find the distance travelled when $t = 2$.
[2]

(b) Find the change in distance when $t$ increases from 2 to 4.
[2]

 

 

 

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2. A ball is thrown upward. Its height $h$ metres after $t$ seconds is given by $h = 20t - 5t^2$.

(a) Find the height when $t = 1$.
[1]

(b) Find the height when $t = 3$.
[1]

(c) Describe what happens to the height of the ball between $t = 1$ and $t = 3$.
[2]

 

 

 

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3. The volume of water in a tank, $V$ litres, at time $t$ minutes is given by $V = 3t^2 + 10t + 50$.

(a) Find the initial volume of water in the tank (when $t = 0$).
[1]

(b) Find the volume after 5 minutes.
[2]

(c) Calculate the average rate of change of volume between $t = 0$ and $t = 5$.
[2]

 

 

 

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4. A plant grows so that its height $h$ cm after $w$ weeks is given by $h = 2w^2 + 3w + 10$.

(a) Find the height of the plant at week 0.
[1]

(b) Find the height at week 4.
[2]

(c) Find the average growth rate (in cm/week) between week 0 and week 4.
[2]

 

 

 

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5. The cost $C$ dollars of producing $x$ items is given by $C = x^2 + 5x + 100$.

(a) Find the cost of producing 10 items.
[2]

(b) Find the cost of producing 15 items.
[2]

(c) Find the average rate of change of cost when production increases from 10 to 15 items.
[2]

 

 

 

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Section B: Gradient of a Curve (Questions 6–10)

Answer all questions in this section.

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6. The curve $y = x^2 - 4x + 3$ passes through the point $(3, 0)$.

(a) Find the value of $y$ when $x = 1$.
[1]

(b) Find the value of $y$ when $x = 5$.
[1]

(c) Calculate the average gradient of the curve between $x = 1$ and $x = 5$.
[2]

 

 

 

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7. A curve is given by $y = 2x^2 - 3x + 1$.

(a) Complete the table below.

$x$	0	1	2	3	4
$y$

[3]

(b) Find the average gradient between $x = 0$ and $x = 4$.
[2]

 

 

 

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8. The height $h$ metres of a rocket $t$ seconds after launch is given by $h = -2t^2 + 16t$.

(a) Find $h$ when $t = 0$, $t = 2$, $t = 4$, $t = 6$, and $t = 8$.
[3]

(b) Between which two consecutive seconds is the average rate of change of height the greatest?
[2]

 

 

 

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9. The function $f(x) = x^2 - 6x + 8$ is defined for $x \ge 0$.

(a) Find $f(0)$, $f(2)$, $f(4)$, and $f(6)$.
[2]

(b) Find the average rate of change of $f(x)$ between $x = 0$ and $x = 6$.
[2]

(c) Between which two consecutive integer values of $x$ does the function decrease most rapidly?
[2]

 

 

 

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10. A particle moves along a straight line. Its displacement $s$ metres from a fixed point after $t$ seconds is given by $s = t^2 - 8t + 15$.

(a) Find the displacement when $t = 0$.
[1]

(b) Find the displacement when $t = 4$.
[1]

(c) Find the displacement when $t = 8$.
[1]

(d) Find the average velocity between $t = 0$ and $t = 8$.
[2]

(e) At what time does the particle return to the fixed point ($s = 0$)?
[2]

 

 

 

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Section C: Differentiation — Finding the Gradient Function (Questions 11–15)

Answer all questions in this section.

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11. Given $y = x^2$, complete the following:

(a) When $x = 3$, $y = \_\_\_\_$.
[1]

(b) When $x = 3 + h$, $y = (3 + h)^2 = \_\_\_\_$.
[1]

(c) The change in $y$ is $(3+h)^2 - 9 = \_\_\_\_$.
[1]

(d) The average gradient between $x = 3$ and $x = 3+h$ is $\frac{\text{change in } y}{h} = \_\_\_\_$.
[1]

(e) As $h$ approaches 0, the gradient at $x = 3$ approaches ____.
[1]

 

 

 

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12. Use the first principles approach to find the gradient of the curve $y = x^2$ at the point where $x = 2$.

Show all steps clearly.
[4]

 

 

 

 

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13. The gradient function (derivative) of $y = x^2$ is $\frac{dy}{dx} = 2x$.

Use this result to find the gradient of $y = x^2$ at:

(a) $x = 1$
[1]

(b) $x = 5$
[1]

(c) $x = -3$
[1]

(d) At what value of $x$ is the gradient equal to 0?
[1]

 

 

 

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14. Given that the gradient function of $y = x^2 + 3x$ is $\frac{dy}{dx} = 2x + 3$, find:

(a) The gradient at $x = 0$.
[1]

(b) The gradient at $x = 2$.
[1]

(c) The gradient at $x = -1$.
[1]

(d) The value of $x$ where the gradient is 5.
[2]

 

 

 

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15. The gradient function of a curve is $\frac{dy}{dx} = 4x - 6$.

(a) Find the gradient at $x = 1$.
[1]

(b) Find the gradient at $x = 3$.
[1]

(c) Find the value of $x$ where the gradient is zero.
[2]

(d) State whether the gradient is positive or negative when $x = 0$.
[1]

 

 

 

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Section D: Applications of Differentiation (Questions 16–20)

Answer all questions in this section.

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16. The area $A$ cm² of a square of side length $x$ cm is given by $A = x^2$.

(a) Find the area when $x = 5$.
[1]

(b) Find the rate of change of area with respect to side length when $x = 5$.
[2]

(c) Explain what this rate of change means in context.
[2]

 

 

 

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17. A rectangular garden has a fixed perimeter of 40 m. Let the length be $x$ m and the width be $(20 - x)$ m.

(a) Write an expression for the area $A$ in terms of $x$.
[2]

(b) Complete the table:

$x$ (m)	5	8	10	12	15
$A$ (m²)

[3]

(c) What value of $x$ gives the maximum area?
[2]

 

 

 

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18. The profit $P$ dollars from selling $x$ items is given by $P = -x^2 + 20x - 50$.

(a) Find the profit when $x = 5$.
[1]

(b) Find the profit when $x = 10$.
[1]

(c) Find the profit when $x = 15$.
[1]

(d) Use the gradient function $\frac{dP}{dx} = -2x + 20$ to find the value of $x$ that maximises profit.
[2]

(e) Calculate the maximum profit.
[2]

 

 

 

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19. The height $h$ metres of a ball thrown upward after $t$ seconds is given by $h = -5t^2 + 30t + 2$.

(a) Find the initial height of the ball (when $t = 0$).
[1]

(b) Find the gradient function $\frac{dh}{dt}$.
[1]

(c) Find the time at which the ball reaches its maximum height.
[2]

(d) Calculate the maximum height.
[2]

 

 

 

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20. A curve has equation $y = x^2 - 4x + 7$.

(a) Find the gradient function $\frac{dy}{dx}$.
[1]

(b) Find the coordinates of the point on the curve where the gradient is zero.
[3]

(c) Find the equation of the tangent to the curve at the point where $x = 3$.
[3]

 

 

 

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End of Quiz

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Secondary 2 Mathematics Quiz - Calculus

Answer Key

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Section A: Understanding Rate of Change (Questions 1–5)

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1. $d = 5t^2 + 3t$

(a) When $t = 2$:
$d = 5(2)^2 + 3(2) = 5(4) + 6 = 20 + 6 = \boxed{26 \text{ m}}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

(b) When $t = 4$:
$d = 5(4)^2 + 3(4) = 5(16) + 12 = 80 + 12 = 92$ m

Change in distance $= 92 - 26 = \boxed{66 \text{ m}}$
[2 marks] — 1 mark for finding $d$ at $t=4$, 1 mark for correct difference.

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2. $h = 20t - 5t^2$

(a) When $t = 1$:
$h = 20(1) - 5(1)^2 = 20 - 5 = \boxed{15 \text{ m}}$
[1 mark]

(b) When $t = 3$:
$h = 20(3) - 5(3)^2 = 60 - 45 = \boxed{15 \text{ m}}$
[1 mark]

(c) The height is the same at $t = 1$ and $t = 3$ (both 15 m). The ball rises to a maximum height and then falls back to the same height. Between $t = 1$ and $t = 3$, the ball reaches its peak and descends.
[2 marks] — 1 mark for noting the heights are equal, 1 mark for describing the rise and fall.

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3. $V = 3t^2 + 10t + 50$

(a) When $t = 0$:
$V = 3(0)^2 + 10(0) + 50 = \boxed{50 \text{ litres}}$
[1 mark]

(b) When $t = 5$:
$V = 3(25) + 10(5) + 50 = 75 + 50 + 50 = \boxed{175 \text{ litres}}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

(c) Average rate of change $= \frac{V(5) - V(0)}{5 - 0} = \frac{175 - 50}{5} = \frac{125}{5} = \boxed{25 \text{ litres/min}}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

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4. $h = 2w^2 + 3w + 10$

(a) When $w = 0$:
$h = 2(0)^2 + 3(0) + 10 = \boxed{10 \text{ cm}}$
[1 mark]

(b) When $w = 4$:
$h = 2(16) + 3(4) + 10 = 32 + 12 + 10 = \boxed{54 \text{ cm}}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

(c) Average growth rate $= \frac{h(4) - h(0)}{4 - 0} = \frac{54 - 10}{4} = \frac{44}{4} = \boxed{11 \text{ cm/week}}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

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5. $C = x^2 + 5x + 100$

(a) When $x = 10$:
C = (10)^2 + 5(10) + 100 = 100 + 50 + 100 = \boxed{\250}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

(b) When $x = 15$:
C = (15)^2 + 5(15) + 100 = 225 + 75 + 100 = \boxed{\400}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

(c) Average rate of change = \frac{C(15) - C(10)}{15 - 10} = \frac{400 - 250}{5} = \frac{150}{5} = \boxed{\30 \text{ per item}}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

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Section B: Gradient of a Curve (Questions 6–10)

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6. $y = x^2 - 4x + 3$

(a) When $x = 1$:
$y = (1)^2 - 4(1) + 3 = 1 - 4 + 3 = \boxed{0}$
[1 mark]

(b) When $x = 5$:
$y = (5)^2 - 4(5) + 3 = 25 - 20 + 3 = \boxed{8}$
[1 mark]

(c) Average gradient $= \frac{y(5) - y(1)}{5 - 1} = \frac{8 - 0}{4} = \boxed{2}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

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7. $y = 2x^2 - 3x + 1$

(a) Table:

$x$	0	1	2	3	4
$y$	1	0	3	10	21

Working:

• $x = 0$: $y = 0 - 0 + 1 = 1$
• $x = 1$: $y = 2 - 3 + 1 = 0$
• $x = 2$: $y = 8 - 6 + 1 = 3$
• $x = 3$: $y = 18 - 9 + 1 = 10$
• $x = 4$: $y = 32 - 12 + 1 = 21$

[3 marks] — 1 mark for each correct row (3+ correct = 3 marks, 2 correct = 2 marks, 1 correct = 1 mark).

(b) Average gradient $= \frac{y(4) - y(0)}{4 - 0} = \frac{21 - 1}{4} = \frac{20}{4} = \boxed{5}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

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8. $h = -2t^2 + 16t$

(a)

• $t = 0$: $h = -2(0) + 0 = \boxed{0}$
• $t = 2$: $h = -2(4) + 32 = -8 + 32 = \boxed{24}$
• $t = 4$: $h = -2(16) + 64 = -32 + 64 = \boxed{32}$
• $t = 6$: $h = -2(36) + 96 = -72 + 96 = \boxed{24}$
• $t = 8$: $h = -2(64) + 128 = -128 + 128 = \boxed{0}$

[3 marks] — 1 mark for each correct value (3+ correct = 3 marks).

(b) Average rates of change between consecutive seconds:

• $t = 0$ to $t = 2$: $\frac{24 - 0}{2} = 12$
• $t = 2$ to $t = 4$: $\frac{32 - 24}{2} = 4$
• $t = 4$ to $t = 6$: $\frac{24 - 32}{2} = -4$
• $t = 6$ to $t = 8$: $\frac{0 - 24}{2} = -12$

The greatest average rate of change is between $\boxed{t = 0 \text{ and } t = 2}$ (value = 12 m/s).
[2 marks] — 1 mark for calculating rates, 1 mark for identifying the correct interval.

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9. $f(x) = x^2 - 6x + 8$

(a)

• $f(0) = 0 - 0 + 8 = \boxed{8}$
• $f(2) = 4 - 12 + 8 = \boxed{0}$
• $f(4) = 16 - 24 + 8 = \boxed{0}$
• $f(6) = 36 - 36 + 8 = \boxed{8}$

[2 marks] — 1 mark for each pair correct.

(b) Average rate of change $= \frac{f(6) - f(0)}{6 - 0} = \frac{8 - 8}{6} = \boxed{0}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

(c) Rates of change between consecutive integers:

• $x = 0$ to $x = 1$: $f(1) = 1 - 6 + 8 = 3$, rate $= \frac{3-8}{1} = -5$
• $x = 1$ to $x = 2$: rate $= \frac{0-3}{1} = -3$
• $x = 2$ to $x = 3$: $f(3) = 9 - 18 + 8 = -1$, rate $= \frac{-1-0}{1} = -1$
• $x = 3$ to $x = 4$: rate $= \frac{0-(-1)}{1} = 1$
• $x = 4$ to $x = 5$: $f(5) = 25 - 30 + 8 = 3$, rate $= \frac{3-0}{1} = 3$
• $x = 5$ to $x = 6$: rate $= \frac{8-3}{1} = 5$

The function decreases most rapidly between $\boxed{x = 0 \text{ and } x = 1}$ (rate = −5).
[2 marks] — 1 mark for calculating rates, 1 mark for identifying the correct interval.

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10. $s = t^2 - 8t + 15$

(a) When $t = 0$:
$s = 0 - 0 + 15 = \boxed{15 \text{ m}}$
[1 mark]

(b) When $t = 4$:
$s = 16 - 32 + 15 = \boxed{-1 \text{ m}}$
[1 mark]

(c) When $t = 8$:
$s = 64 - 64 + 15 = \boxed{15 \text{ m}}$
[1 mark]

(d) Average velocity $= \frac{s(8) - s(0)}{8 - 0} = \frac{15 - 15}{8} = \boxed{0 \text{ m/s}}$
[2 marks] — 1 mark for correct formula, 1 mark for correct answer.

(e) When $s = 0$:
$t^2 - 8t + 15 = 0$
$(t - 3)(t - 5) = 0$
$t = \boxed{3 \text{ s or } 5 \text{ s}}$
[2 marks] — 1 mark for setting up equation, 1 mark for correct solutions.

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Section C: Differentiation — Finding the Gradient Function (Questions 11–15)

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11. $y = x^2$

(a) When $x = 3$: $y = (3)^2 = \boxed{9}$
[1 mark]

(b) When $x = 3 + h$: $y = (3 + h)^2 = \boxed{9 + 6h + h^2}$
[1 mark]

(c) Change in $y = (9 + 6h + h^2) - 9 = \boxed{6h + h^2}$
[1 mark]

(d) Average gradient $= \frac{6h + h^2}{h} = \boxed{6 + h}$
[1 mark]

(e) As $h \to 0$, the gradient approaches $\boxed{6}$.
[1 mark]

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12. First principles for $y = x^2$ at $x = 2$:

Let $f(x) = x^2$.
$f(2) = 4$
$f(2 + h) = (2 + h)^2 = 4 + 4h + h^2$

Change in $y = f(2+h) - f(2) = (4 + 4h + h^2) - 4 = 4h + h^2$

Average gradient $= \frac{4h + h^2}{h} = 4 + h$

As $h \to 0$, gradient $\to \boxed{4}$

[4 marks] — 1 mark for $f(2)$, 1 mark for $f(2+h)$ expansion, 1 mark for difference quotient, 1 mark for limit.

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13. $\frac{dy}{dx} = 2x$

(a) At $x = 1$: gradient $= 2(1) = \boxed{2}$
[1 mark]

(b) At $x = 5$: gradient $= 2(5) = \boxed{10}$
[1 mark]

(c) At $x = -3$: gradient $= 2(-3) = \boxed{-6}$
[1 mark]

(d) When gradient $= 0$: $2x = 0 \Rightarrow x = \boxed{0}$
[1 mark]

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14. $\frac{dy}{dx} = 2x + 3$

(a) At $x = 0$: gradient $= 2(0) + 3 = \boxed{3}$
[1 mark]

(b) At $x = 2$: gradient $= 2(2) + 3 = \boxed{7}$
[1 mark]

(c) At $x = -1$: gradient $= 2(-1) + 3 = \boxed{1}$
[1 mark]

(d) When gradient $= 5$:
$2x + 3 = 5$
$2x = 2$
$x = \boxed{1}$
[2 marks] — 1 mark for setting up equation, 1 mark for correct solution.

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15. $\frac{dy}{dx} = 4x - 6$

(a) At $x = 1$: gradient $= 4(1) - 6 = \boxed{-2}$
[1 mark]

(b) At $x = 3$: gradient $= 4(3) - 6 = \boxed{6}$
[1 mark]

(c) When gradient $= 0$:
$4x - 6 = 0$
$4x = 6$
$x = \boxed{1.5}$
[2 marks] — 1 mark for setting up equation, 1 mark for correct solution.

(d) At $x = 0$: gradient $= 4(0) - 6 = -6$, which is $\boxed{\text{negative}}$.
[1 mark]

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Section D: Applications of Differentiation (Questions 16–20)

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16. $A = x^2$

(a) When $x = 5$: $A = (5)^2 = \boxed{25 \text{ cm}^2}$
[1 mark]

(b) $\frac{dA}{dx} = 2x$. At $x = 5$: rate of change $= 2(5) = \boxed{10 \text{ cm}^2\text{/cm}}$
[2 marks] — 1 mark for derivative, 1 mark for correct evaluation.

(c) This means that when the side length is 5 cm, the area is increasing at a rate of 10 cm² for every 1 cm increase in side length.
[2 marks] — 1 mark for identifying the meaning (rate of change of area), 1 mark for correct units/context.

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17. Perimeter = 40 m, length = $x$ m, width = $(20 - x)$ m.

(a) $A = x(20 - x) = \boxed{20x - x^2}$
[2 marks] — 1 mark for correct expression, 1 mark for simplification.

(b) Table:

$x$ (m)	5	8	10	12	15
$A$ (m²)	75	96	100	96	75

Working:

• $x = 5$: $A = 20(5) - 25 = 100 - 25 = 75$
• $x = 8$: $A = 20(8) - 64 = 160 - 64 = 96$
• $x = 10$: $A = 20(10) - 100 = 200 - 100 = 100$
• $x = 12$: $A = 20(12) - 144 = 240 - 144 = 96$
• $x = 15$: $A = 20(15) - 225 = 300 - 225 = 75$

[3 marks] — 1 mark for each correct value (3+ correct = 3 marks).

(c) From the table, the maximum area occurs when $x = \boxed{10 \text{ m}}$ (giving a square).
[2 marks] — 1 mark for identifying $x = 10$, 1 mark for stating maximum area = 100 m².

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18. $P = -x^2 + 20x - 50$, $\frac{dP}{dx} = -2x + 20$

(a) When $x = 5$:
$P = -(25) + 100 - 50 = \boxed{25}$
[1 mark]

(b) When $x = 10$:
$P = -(100) + 200 - 50 = \boxed{50}$
[1 mark]

(c) When $x = 15$:
$P = -(225) + 300 - 50 = \boxed{25}$
[1 mark]

(d) Maximum profit when $\frac{dP}{dx} = 0$:
$-2x + 20 = 0$
$2x = 20$
$x = \boxed{10}$
[2 marks] — 1 mark for setting derivative to zero, 1 mark for correct solution.

(e) Maximum profit = P(10) = -(100) + 200 - 50 = \boxed{\50}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

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19. $h = -5t^2 + 30t + 2$

(a) When $t = 0$: $h = -0 + 0 + 2 = \boxed{2 \text{ m}}$
[1 mark]

(b) $\frac{dh}{dt} = \boxed{-10t + 30}$
[1 mark]

(c) At maximum height, $\frac{dh}{dt} = 0$:
$-10t + 30 = 0$
$10t = 30$
$t = \boxed{3 \text{ s}}$
[2 marks] — 1 mark for setting derivative to zero, 1 mark for correct solution.

(d) Maximum height $= h(3) = -5(9) + 30(3) + 2 = -45 + 90 + 2 = \boxed{47 \text{ m}}$
[2 marks] — 1 mark for substitution, 1 mark for correct answer.

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20. $y = x^2 - 4x + 7$

(a) $\frac{dy}{dx} = \boxed{2x - 4}$
[1 mark]

(b) When gradient $= 0$:
$2x - 4 = 0$
$x = 2$

When $x = 2$: $y = (2)^2 - 4(2) + 7 = 4 - 8 + 7 = 3$

Coordinates: $\boxed{(2, 3)}$
[3 marks] — 1 mark for derivative, 1 mark for finding $x$, 1 mark for finding $y$.

(c) At $x = 3$:
$y = (3)^2 - 4(3) + 7 = 9 - 12 + 7 = 4$
Point: $(3, 4)$

Gradient at $x = 3$: $\frac{dy}{dx} = 2(3) - 4 = 2$

Equation of tangent: $y - 4 = 2(x - 3)$
$y - 4 = 2x - 6$
$y = 2x - 2$

Answer: $\boxed{y = 2x - 2}$
[3 marks] — 1 mark for point, 1 mark for gradient, 1 mark for equation.

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End of Answer Key

Total Marks: 60