Secondary 4 Elementary Mathematics Graphs Coordinate Geometry Quiz

Secondary 4 Elementary Mathematics Quiz - Graphs Coordinate Geometry (Answer Key)

Total Marks: 40

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Section A: Gradient, Distance, and Midpoint (Questions 1–5) [10 marks]

1. [2 marks]

Given: $A(-3, 7)$, $B(5, -1)$

(a) Gradient of $AB$: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 7}{5 - (-3)} = \frac{-8}{8} = -1$ Answer: $-1$ [1]

(b) Midpoint of $AB$: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-3 + 5}{2}, \frac{7 + (-1)}{2} \right) = \left( \frac{2}{2}, \frac{6}{2} \right) = (1, 3)$ Answer: $(1, 3)$ [1]

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2. [2 marks]

Given: $P(2, k)$, $Q(8, 3)$, $PQ = 10$

Distance formula: $PQ = \sqrt{(8 - 2)^2 + (3 - k)^2} = 10$ $\sqrt{6^2 + (3 - k)^2} = 10$ $\sqrt{36 + (3 - k)^2} = 10$

Square both sides: $36 + (3 - k)^2 = 100$ $(3 - k)^2 = 64$ $3 - k = \pm 8$

Case 1: $3 - k = 8 \Rightarrow k = -5$
Case 2: $3 - k = -8 \Rightarrow k = 11$

Answer: $k = -5$ or $k = 11$ [2]

Marking note: 1 mark for correct equation setup, 1 mark for both correct values.

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3. [2 marks]

Given: Points $(4, -2)$ and $(-2, 10)$

(a) Equation of line: Gradient: $m = \frac{10 - (-2)}{-2 - 4} = \frac{12}{-6} = -2$

Using point-slope form with $(4, -2)$: $y - (-2) = -2(x - 4)$ $y + 2 = -2x + 8$ $y = -2x + 6$

Answer: $y = -2x + 6$ [1]

(b) $y$-intercept: From $y = -2x + 6$, the $y$-intercept is $c = 6$. Answer: $6$ (or $(0, 6)$) [1]

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4. [2 marks]

Given: $3x - 4y + 12 = 0$

(a) Gradient: Rearrange to $y = mx + c$: $-4y = -3x - 12$ $y = \frac{3}{4}x + 3$ Gradient $m = \frac{3}{4}$.

Answer: $\frac{3}{4}$ [1]

(b) $x$-intercept (where $y = 0$): $3x - 4(0) + 12 = 0$ $3x + 12 = 0$ $3x = -12$ $x = -4$ Coordinates: $(-4, 0)$

Answer: $(-4, 0)$ [1]

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5. [2 marks]

Given: $R(-1, 4)$, $S(3, -2)$, $T(7, 2)$ are three vertices of parallelogram $RSTU$.

In a parallelogram, diagonals bisect each other. Midpoint of $RT$ = Midpoint of $SU$.

Midpoint of $RT$: $\left( \frac{-1 + 7}{2}, \frac{4 + 2}{2} \right) = (3, 3)$

Let $U = (x, y)$. Midpoint of $SU$: $\left( \frac{3 + x}{2}, \frac{-2 + y}{2} \right) = (3, 3)$

$\frac{3 + x}{2} = 3 \Rightarrow 3 + x = 6 \Rightarrow x = 3$ $\frac{-2 + y}{2} = 3 \Rightarrow -2 + y = 6 \Rightarrow y = 8$

Answer: $(3, 8)$ [2]

Alternative method: Vector approach $\overrightarrow{RS} = \overrightarrow{TU}$ or $\overrightarrow{RU} = \overrightarrow{ST}$.

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Section B: Parallel and Perpendicular Lines (Questions 6–10) [10 marks]

6. [2 marks]

Given: $L_1$ through $(0, 5)$ and $(4, 1)$; $L_2 \parallel L_1$ through $(-2, 3)$

Gradient of $L_1$: $m = \frac{1 - 5}{4 - 0} = \frac{-4}{4} = -1$

Since $L_2 \parallel L_1$, gradient of $L_2$ is also $-1$.

Equation of $L_2$ using point-slope form: $y - 3 = -1(x - (-2))$ $y - 3 = -x - 2$ $y = -x + 1$

Answer: $y = -x + 1$ [2]

Marking note: 1 mark for correct gradient, 1 mark for correct equation.

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7. [2 marks]

Given: $L_1: y = -\frac{2}{3}x + 4$; $L_2 \perp L_1$ through $(6, -1)$

Gradient of $L_1$: $m_1 = -\frac{2}{3}$

Since $L_2 \perp L_1$, $m_1 \times m_2 = -1$: $-\frac{2}{3} \times m_2 = -1 \Rightarrow m_2 = \frac{3}{2}$

Equation of $L_2$: $y - (-1) = \frac{3}{2}(x - 6)$ $y + 1 = \frac{3}{2}x - 9$ $y = \frac{3}{2}x - 10$

Convert to $ax + by = c$ with integer coefficients: Multiply by 2: $2y = 3x - 20$ Rearrange: $3x - 2y = 20$

Answer: $3x - 2y = 20$ [2]

Marking note: 1 mark for correct perpendicular gradient, 1 mark for correct equation in required form.

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8. [2 marks]

Given: $L: 2x + 5y = 20$; $M \perp L$ through $y$-intercept of $L$

$y$-intercept of $L$ (set $x = 0$): $2(0) + 5y = 20 \Rightarrow 5y = 20 \Rightarrow y = 4$ Point: $(0, 4)$

Gradient of $L$: Rearrange $2x + 5y = 20 \Rightarrow 5y = -2x + 20 \Rightarrow y = -\frac{2}{5}x + 4$ $m_L = -\frac{2}{5}$

Since $M \perp L$, $m_M = \frac{5}{2}$ (negative reciprocal)

Equation of $M$ through $(0, 4)$: $y = \frac{5}{2}x + 4$

Answer: $y = \frac{5}{2}x + 4$ [2]

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9. [2 marks]

Given: $A(1, 2)$, $B(5, 6)$, $C(7, 2)$

(a) Show $AB \perp BC$:

Gradient of $AB$: $m_{AB} = \frac{6 - 2}{5 - 1} = \frac{4}{4} = 1$

Gradient of $BC$: $m_{BC} = \frac{2 - 6}{7 - 5} = \frac{-4}{2} = -2$

Product: $m_{AB} \times m_{BC} = 1 \times (-2) = -2 \neq -1$

Wait — this is not perpendicular! Let me recalculate.

Actually: $m_{AB} = 1$, $m_{BC} = -2$. Product = $-2$. Not perpendicular.

Let me check the question again. The question says "Show that AB is perpendicular to BC" — but with these coordinates, they are not perpendicular. This is an error in the question design. Let me adjust the coordinates to make it work.

Correction for answer key: With the given coordinates, $AB$ is not perpendicular to $BC$. The product of gradients is $-2$, not $-1$.

However, for the answer key, I'll show the correct working with the given numbers and note the discrepancy.

Working: $m_{AB} = \frac{6-2}{5-1} = 1$ $m_{BC} = \frac{2-6}{7-5} = -2$ $m_{AB} \times m_{BC} = -2 \neq -1$

Therefore, $AB$ is not perpendicular to $BC$ with these coordinates.

(b) Type of triangle: Since $AB \not\perp BC$, we check other properties. $AB = \sqrt{(5-1)^2 + (6-2)^2} = \sqrt{16+16} = \sqrt{32} = 4\sqrt{2}$ $BC = \sqrt{(7-5)^2 + (2-6)^2} = \sqrt{4+16} = \sqrt{20} = 2\sqrt{5}$ $AC = \sqrt{(7-1)^2 + (2-2)^2} = \sqrt{36} = 6$

No two sides equal, no right angle. Scalene triangle.

Answer: Scalene triangle [1]

Marking note: This question has a flaw — the coordinates don't produce perpendicular lines. In a real exam, coordinates would be chosen to make $m_{AB} \times m_{BC} = -1$. For example, if $C$ were $(9, 2)$, then $m_{BC} = \frac{2-6}{9-5} = -1$, product = $-1$.

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10. [2 marks]

Given: $L_1: y = 3x - 7$; $L_2$ through $(2, 1)$ and $(k, 13)$; $L_1 \perp L_2$

Gradient of $L_1$: $m_1 = 3$

Since $L_1 \perp L_2$, $m_2 = -\frac{1}{3}$

Gradient of $L_2$ using two points: $m_2 = \frac{13 - 1}{k - 2} = \frac{12}{k - 2}$

Set equal to $-\frac{1}{3}$: $\frac{12}{k - 2} = -\frac{1}{3}$ Cross-multiply: $36 = -(k - 2)$ $36 = -k + 2$ $k = 2 - 36 = -34$

Answer: $k = -34$ [2]

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Section C: Coordinate Geometry Problems (Questions 11–15) [10 marks]

11. [2 marks]

Given: $y = 2x + 3$ and $x + y = 9$

Substitute $y = 2x + 3$ into $x + y = 9$: $x + (2x + 3) = 9$ $3x + 3 = 9$ $3x = 6$ $x = 2$

Then $y = 2(2) + 3 = 7$

Answer: $(2, 7)$ [2]

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12. [2 marks]

Given: Line through $(4, -1)$ with $x$-intercept $6$.

$x$-intercept $6$ means point $(6, 0)$.

Gradient: $m = \frac{0 - (-1)}{6 - 4} = \frac{1}{2}$

Equation using point $(6, 0)$: $y - 0 = \frac{1}{2}(x - 6)$ $y = \frac{1}{2}x - 3$

Answer: $y = \frac{1}{2}x - 3$ [2]

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13. [2 marks]

Given: $A(2, 5)$, $B(8, 1)$, $C(4, -3)$

(a) Line through $A$ parallel to $BC$:

Gradient of $BC$: $m_{BC} = \frac{-3 - 1}{4 - 8} = \frac{-4}{-4} = 1$

Line through $A(2, 5)$ with gradient $1$: $y - 5 = 1(x - 2)$ $y = x + 3$

Answer: $y = x + 3$ [1]

(b) Altitude from $A$ to $BC$:

Altitude is perpendicular to $BC$. Since $m_{BC} = 1$, gradient of altitude $= -1$.

Line through $A(2, 5)$ with gradient $-1$: $y - 5 = -1(x - 2)$ $y - 5 = -x + 2$ $y = -x + 7$

Answer: $y = -x + 7$ [1]

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14. [2 marks]

Given: $L$ through $(-2, 4)$ and $(4, -2)$; $M: y = x + 1$

Gradient of $L$: $m_L = \frac{-2 - 4}{4 - (-2)} = \frac{-6}{6} = -1$

Equation of $L$ using $(-2, 4)$: $y - 4 = -1(x + 2)$ $y - 4 = -x - 2$ $y = -x + 2$

Intersection with $M: y = x + 1$: $-x + 2 = x + 1$ $2x = 1$ $x = 0.5$

Then $y = 0.5 + 1 = 1.5$

Answer: $(0.5, 1.5)$ [2]

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15. [2 marks]

Given: $P(-2, 1)$, $Q(4, 3)$, $R(6, -1)$, $S(0, -3)$

(a) Show $PQ \parallel RS$:

$m_{PQ} = \frac{3 - 1}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}$

$m_{RS} = \frac{-3 - (-1)}{0 - 6} = \frac{-2}{-6} = \frac{1}{3}$

Since $m_{PQ} = m_{RS} = \frac{1}{3}$, $PQ \parallel RS$. [1]

(b) Is $PQRS$ a parallelogram?

Check $QR$ and $PS$: $m_{QR} = \frac{-1 - 3}{6 - 4} = \frac{-4}{2} = -2$ $m_{PS} = \frac{-3 - 1}{0 - (-2)} = \frac{-4}{2} = -2$

Since $m_{QR} = m_{PS} = -2$, $QR \parallel PS$.

Both pairs of opposite sides are parallel, so $PQRS$ is a parallelogram.

Answer: Yes, $PQRS$ is a parallelogram because both pairs of opposite sides are parallel ($PQ \parallel RS$ and $QR \parallel PS$). [1]

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Section D: Graphs of Functions and Applications (Questions 16–20) [10 marks]

16. [2 marks]

Graph: $y = x^2 - 4x + 3 = (x - 2)^2 - 1$

(a) Vertex: From completed square form $(x - 2)^2 - 1$, vertex is at $(2, -1)$.

Answer: $(2, -1)$ [1]

(b) Line of symmetry: Vertical line through vertex: $x = 2$.

Answer: $x = 2$ [1]

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17. [2 marks]

Graph: $y = \frac{6}{x}$

(a) Estimate $y$ when $x = 1.5$: Exact value: $y = \frac{6}{1.5} = 4$. From graph, reading at $x = 1.5$ should give approximately $4$.

Answer: $4$ (accept $3.9$–$4.1$) [1]

(b) Gradient at $x = 2$ by tangent: At $x = 2$, $y = \frac{6}{2} = 3$. Point: $(2, 3)$.

Derivative: $\frac{dy}{dx} = -\frac{6}{x^2}$. At $x = 2$, gradient $= -\frac{6}{4} = -1.5$.

By drawing tangent on graph, estimated gradient should be approximately $-1.5$.

Answer: $-1.5$ (accept $-1.4$ to $-1.6$) [1]

Marking note: For graphical estimation, accept reasonable range. Exact calculus value is $-1.5$.

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18. [2 marks]

Graph: $y = 2^x$

(a) $y$-intercept: When $x = 0$, $y = 2^0 = 1$. Coordinates: $(0, 1)$.

Answer: $(0, 1)$ [1]

(b) Solve $2^x = 5$: From graph, find $x$ when $y = 5$. $2^2 = 4$, $2^3 = 8$, so $x$ is between 2 and 3. $2^{2.3} \approx 4.92$, $2^{2.32} \approx 5.00$. To 1 decimal place: $x \approx 2.3$.

Answer: $x = 2.3$ [1]

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19. [2 marks]

Given: $d = 80t$

(a) Complete table:

$t$ (hours)	0	1	2	3	4
$d$ (km)	0	80	160	240	320

Answer: $0, 160, 240, 320$ [1]

(b) Graph: Straight line through origin $(0, 0)$ and points $(1, 80)$, $(2, 160)$, $(3, 240)$, $(4, 320)$. Line passes through all points, labelled axes.

Answer: Graph drawn [1]

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20. [2 marks]

Speed-time graph: Three segments

• $0 \le t \le 4$: line from $(0, 0)$ to $(4, 8)$
• $4 \le t \le 7$: horizontal at $v = 8$
• $7 \le t \le 10$: line from $(7, 8)$ to $(10, 0)$

(a) Acceleration in first 4 seconds: Acceleration = gradient of speed-time graph $a = \frac{8 - 0}{4 - 0} = \frac{8}{4} = 2 \text{ m/s}^2$

Answer: $2 \text{ m/s}^2$ [1]

(b) Total distance in 10 seconds: Distance = area under speed-time graph

Area 1 (triangle, $0$–$4$ s): $\frac{1}{2} \times 4 \times 8 = 16 \text{ m}$ Area 2 (rectangle, $4$–$7$ s): $3 \times 8 = 24 \text{ m}$ Area 3 (triangle, $7$–$10$ s): $\frac{1}{2} \times 3 \times 8 = 12 \text{ m}$

Total distance = $16 + 24 + 12 = 52 \text{ m}$

Answer: $52 \text{ m}$ [1]

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End of Answer Key