O Level Elementary Mathematics Graphs Coordinate Geometry Quiz

O-Level Elementary Mathematics Quiz - Graphs Coordinate Geometry

ANSWER KEY AND MARKING SCHEME

Total Marks: 50

Section A: Basic Concepts and Techniques (10 marks)

1. Gradient = (17 − 5) / (8 − 2) = 12/6 = 2 [M1] Correct substitution into gradient formula [A1] Correct answer: 2

2. Length = √[(5 − (−3))² + (7 − 1)²] = √(8² + 6²) = √(64 + 36) = √100 = 10 units [M1] Correct substitution into distance formula [A1] Correct answer: 10 units

3. y − (−2) = 3(x − 1) → y + 2 = 3x − 3 → y = 3x − 5 [M1] Correct use of point-gradient form y − y₁ = m(x − x₁) [A1] Correct equation: y = 3x − 5

4. Midpoint = ((−4 + 6)/2, (3 + (−5))/2) = (2/2, −2/2) = (1, −1) [M1] Correct use of midpoint formula [A1] Correct coordinates: (1, −1)

5. x-intercept: set y = 0 → 2x = 12 → x = 6 → (6, 0) y-intercept: set x = 0 → 3y = 12 → y = 4 → (0, 4) [M1] Correct method for finding at least one intercept [A1] Both correct: (6, 0) and (0, 4)

Section B: Straight Line Graphs (14 marks)

6(a). Gradient = (−2 − 4) / (3 − (−1)) = −6/4 = −3/2 or −1.5 [M1] Correct substitution [A1] Correct gradient: −3/2 or −1.5

6(b). Using y − y₁ = m(x − x₁) with point (−1, 4): y − 4 = −3/2(x − (−1)) y − 4 = −3/2(x + 1) y − 4 = −3/2 x − 3/2 y = −3/2 x + 5/2 or y = −1.5x + 2.5 [M1] Correct substitution into point-gradient form [A1] Correct equation in form y = mx + c

7(a). Rearrange: 4x − 5y = 20 → −5y = −4x + 20 → y = (4/5)x − 4 Gradient = 4/5 or 0.8 [M1] Correct rearrangement to y = mx + c [A1] Correct gradient: 4/5 or 0.8

7(b). x-intercept: set y = 0 → 4x = 20 → x = 5 → (5, 0) y-intercept: set x = 0 → −5y = 20 → y = −4 → (0, −4) [M1] Correct method for at least one intercept [A1] Both correct: (5, 0) and (0, −4)

8. Substitute x = 5 into y = 2x − 13: y = 2(5) − 13 = 10 − 13 = −3 Since the calculated y-value (−3) matches the given y-coordinate (−3), the point (5, −3) lies on the line. [M1] Correct substitution [A1] Correct conclusion with reasoning

9. Substitute (2, 1) into y = kx + 7: 1 = k(2) + 7 1 = 2k + 7 2k = −6 k = −3 [M1] Correct substitution [A1] Correct value: k = −3

10. Parallel lines have the same gradient, so m = 4. Passes through (0, 5), so y-intercept c = 5. Equation: y = 4x + 5 [M1] Identifying gradient of parallel line as 4 [A1] Correct equation: y = 4x + 5

Section C: Coordinate Geometry Applications (14 marks)

11(a). AB = √[(5 − 1)² + (8 − 2)²] = √(4² + 6²) = √(16 + 36) = √52 = 2√13 ≈ 7.21 units [M1] Correct substitution into distance formula [A1] Correct length: √52 or 2√13 or 7.21 units

11(b). Gradient of AC = (2 − 2) / (9 − 1) = 0/8 = 0 [A1] Correct gradient: 0

11(c). AC is horizontal (gradient = 0). AB = √52. BC = √[(9 − 5)² + (2 − 8)²] = √(16 + 36) = √52. Since AB = BC = √52, triangle ABC is isosceles. [M1] Finding BC or noting AB = BC [A1] Correct identification: isosceles triangle with justification

12(a). Gradient of L₁ = 2. Perpendicular lines have gradients whose product is −1. Gradient of L₂ = −1/2 [A1] Correct gradient: −1/2

12(b). Using point (4, 3) and m = −1/2: y − 3 = −1/2(x − 4) y − 3 = −1/2 x + 2 y = −1/2 x + 5 [M1] Correct substitution into point-gradient form [A1] Correct equation: y = −1/2 x + 5

13(a). Gradient = (11 − 3) / (8 − 2) = 8/6 = 4/3 Using point (2, 3): y − 3 = 4/3(x − 2) y − 3 = 4/3 x − 8/3 y = 4/3 x + 1/3 [M1] Correct gradient and substitution [A1] Correct equation: y = 4/3 x + 1/3

13(b). Substitute (k, 7) into equation: 7 = 4/3 k + 1/3 Multiply by 3: 21 = 4k + 1 4k = 20 k = 5 [M1] Correct substitution [A1] Correct value: k = 5

14. x-intercept: set y = 0 → 3x = 9 → x = 3 → (3, 0) y-intercept: set x = 0 → y = 9 → (0, 9) Triangle has base 3 and height 9. Area = 1/2 × 3 × 9 = 13.5 square units [M1] Correct intercepts found [A1] Correct area: 13.5 square units

Section D: Graphs and Interpretation (12 marks)

15(a). Gradient = (15 − 3) / (6 − 0) = 12/6 = 2 [M1] Correct substitution [A1] Correct gradient: 2

15(b). y-intercept = 3, gradient = 2 → y = 2x + 3 [A1] Correct equation: y = 2x + 3

15(c). When x = 10: y = 2(10) + 3 = 23 [A1] Correct value: 23

16. y = −2x + 4 y-intercept: (0, 4) x-intercept: set y = 0 → 0 = −2x + 4 → 2x = 4 → x = 2 → (2, 0) Sketch: straight line passing through (0, 4) and (2, 0), extending in both directions. [B1] Correct y-intercept labelled [B1] Correct x-intercept labelled [B1] Correct straight line drawn through both intercepts

17(a). Gradient = (9 − 5) / (2 − 0) = 4/2 = 2 [A1] Correct gradient: 2

17(b). When x = 0, y = 5, so c = 5 [A1] Correct value: c = 5

17(c). y = 2x + 5. When x = 15: y = 2(15) + 5 = 35 [A1] Correct value: 35

18. Any valid misleading feature with explanation, for example:

• "The y-axis does not start at zero, which exaggerates the differences between values and makes small changes appear more significant than they really are."
• "The scale on the axes is not uniform, which distorts the shape of the graph and can lead to incorrect interpretation of trends." [B1] Correct identification of a misleading feature [B1] Clear explanation of why it is misleading

19. Gradient = (10 − 4) / (5 − 1) = 6/4 = 3/2 Using point (1, 4): y − 4 = 3/2(x − 1) y − 4 = 3/2 x − 3/2 y = 3/2 x + 5/2 [M1] Correct gradient and substitution [A1] Correct equation: y = 3/2 x + 5/2

20. y = 2x + 3 y-intercept A: set x = 0 → y = 3 → A(0, 3) x-intercept B: set y = 0 → 0 = 2x + 3 → 2x = −3 → x = −3/2 → B(−3/2, 0) Distance AB = √[(−3/2 − 0)² + (0 − 3)²] = √(9/4 + 9) = √(9/4 + 36/4) = √(45/4) = √45 / 2 = 3√5 / 2 ≈ 3.35 units [M1] Correct intercepts found [A1] Correct distance: 3√5 / 2 or 3.35 units

END OF ANSWER KEY