New Curriculum Third Term Mathematics examination for SS 1

Class: SS 1
Term: Third Term
Subject: General Mathematics
Time: 2 hours
Total Marks: 70

SECTION A: OBJECTIVES (40 Marks)
Instruction: Answer all 40 questions. Each question carries 1 mark. Show rough working separately where necessary. Diagrams are not necessarily drawn to scale.

1. ________ are the two roots of x² − 7x + 12 = 0.
   (a) −3 and −4    (b) 2 and 6    (c) 3 and 4    (d) −2 and −6
2. A quadratic has roots 2 + √3 and 2 − √3. Its equation with leading coefficient 1 is
   (a) x² − 2x − 3 = 0    (b) x² − 4x + 1 = 0    (c) x² + 4x + 1 = 0    (d) x² − 4x − 1 = 0
3. Use the plotted parabola to determine the two x-intercepts.
   (a) x = 2 and x = 4    (b) x = 1 and x = 3    (c) x = 0 and x = 4    (d) x = −1 and x = −3
4. Calculate the length of an arc subtending 135° at the centre of a circle of radius 8 cm.
   (a) 9π cm    (b) 3π cm    (c) 12π cm    (d) 6π cm
5. A minor segment is cut from a circle of radius 6 cm by a 60° chord. Its area is
   (a) 6π cm²    (b) (12π − 9√3) cm²    (c) (6π + 9√3) cm²    (d) (6π − 9√3) cm²
6. Find the total surface area of a closed cylinder of radius 3 cm and height 10 cm.
   (a) 78π cm²    (b) 90π cm²    (c) 69π cm²    (d) 60π cm²
7. A frustum of a cone has radii 6 cm and 3 cm and height 4 cm. Determine its volume.
   (a) 108π cm³    (b) 72π cm³    (c) 63π cm³    (d) 84π cm³
8. A class survey records 15 pupils travelling by bicycle, 30 by bus and 45 on foot. What angle represents bus travel in a pie chart?
   (a) 150°    (b) 135°    (c) 90°    (d) 120°
9. Scores 2, 4 and 6 occur with frequencies 3, 2 and 5. What fraction of observations have scores greater than 2?
   (a) 3/10    (b) 7/10    (c) 1/2    (d) 2/5
10. In a histogram, the intervals 0–10 and 10–15 have frequencies 20 and 15. Which class must have the taller bar when frequency density is used?
    (a) 0–10    (b) The intervals cannot be compared    (c) Both have equal heights    (d) 10–15
11. A sector has radius 9 cm and central angle 80°. The total perimeter of the sector is
    (a) 4π cm    (b) (18 + 8π) cm    (c) (9 + 4π) cm    (d) (18 + 4π) cm
12. For a sector of radius 10 cm with a central angle of 90°, the area between the arc and its chord is
    (a) (25π − 50) cm²    (b) (50π − 25) cm²    (c) 25π cm²    (d) (25π + 50) cm²
13. A right cone has base radius 5 cm, perpendicular height 12 cm and slant height 13 cm. Its curved surface area is
    (a) 65π cm²    (b) 25π cm²    (c) 90π cm²    (d) 60π cm²
14. A hemisphere of radius 3 cm sits on a cylinder of the same radius and height 10 cm. The combined volume is
    (a) 90π cm³    (b) 99π cm³    (c) 117π cm³    (d) 108π cm³
15. For continuous classes 10–19 and 20–29 measured to the nearest integer, the second class midpoint is
    (a) 29.5    (b) 25    (c) 24.5    (d) 20
16. A collection of measurements is arranged into non-overlapping classes. The main reason for this is to
    (a) remove every unusually large value    (b) organise observations into a frequency distribution    (c) change the original numerical values    (d) ensure all frequencies are equal
17. A pie-chart sector represents 36 out of 120 responses. Its central angle is
    (a) 72°    (b) 96°    (c) 120°    (d) 108°
18. The polynomial x² + ________x + 12 is equal to (x + 3)(x + 4).
    (a) −7    (b) 12    (c) 7    (d) 1
19. Form a monic quadratic equation with roots −1 and 4.
    (a) x² − 5x + 4 = 0    (b) x² − 3x − 4 = 0    (c) x² − 3x + 4 = 0    (d) x² + 3x − 4 = 0
20. A rectangular plot has width x metres and length (x + 3) metres. Its area is 54 m². Find its width.
    (a) 12 m    (b) 9 m    (c) 6 m    (d) 3 m
21. The volume of a square pyramid with base edge 6 cm and perpendicular height 9 cm is
    (a) 162 cm³    (b) 108 cm³    (c) 54 cm³    (d) 324 cm³
22. The data 2, 2, 4, 4, 4, 5, 5 are tabulated. How many observations belong to the value 4?
    (a) 7    (b) 4    (c) 3    (d) 2
23. A grouped distribution has class intervals 0–10, 10–20 and 20–30 with frequencies 5, 12 and 8. Which point represents the modal class on its frequency polygon?
    (a) (15,8)    (b) (20,12)    (c) (15,12)    (d) (10,12)
24. Monthly enrolment counts are 10, 18, 15 and 22 from January to April. Between which two months is the largest rise?
    (a) January and February    (b) January and March    (c) March and April    (d) February and March
25. All the following equations have two distinct real roots except ________.
    (a) x² − 1 = 0    (b) x² − 6x + 9 = 0    (c) x² + x − 2 = 0    (d) x² − 5x + 6 = 0
26. The minimum point of y = (x − 2)² − 9 occurs at
    (a) (2, −9)    (b) (−2, 9)    (c) (2, 9)    (d) (−2, −9)
27. The roots of a quadratic are both 4. Which of these equations represents it?
    (a) x² − 8x + 16 = 0    (b) x² − 4x + 4 = 0    (c) x² + 8x + 16 = 0    (d) x² − 16 = 0
28. When a cone is opened out along a generator, its curved surface forms a sector. The sector’s radius is the cone’s
    (a) base radius    (b) slant height    (c) base diameter    (d) vertical height
29. The area of a sector of radius 12 cm and angle 150° is
    (a) 90π cm²    (b) 72π cm²    (c) 60π cm²    (d) 30π cm²
30. A prism has a triangular cross-section of area 12 cm² and length 15 cm. Its volume is
    (a) 180 cm³    (b) 90 cm³    (c) 360 cm³    (d) 27 cm³
31. The graph of y = x² − 4x + 3 has axis of symmetry ________.
    (a) x = 2    (b) x = 3    (c) x = −2    (d) x = 4
32. The correct step in obtaining the roots from a plotted quadratic graph is to
    (a) measure the distance from the origin to the vertex    (b) read the y-coordinate of its turning point    (c) read the x-coordinates where the curve meets the x-axis    (d) add the x- and y-intercepts
33. For a 60° circular segment cut from a circle of radius 10 cm, the straight chord has length 10 cm. The segment’s perimeter is ________.
    (a) (10 + 20π/3) cm    (b) (20 + 10π/3) cm    (c) 10π/3 cm    (d) (10 + 10π/3) cm
34. A cone has base radius 5 cm and slant height 13 cm. The area of the sector formed by its curved surface is ________.
    (a) 65π cm²    (b) 25π cm²    (c) 130π cm²    (d) 60π cm²
35. If the edge of a cube is doubled, its volume becomes how many times the original?
    (a) 2 times    (b) 4 times    (c) 16 times    (d) 8 times
36. The curved surface area of a conical frustum with radii 6 cm and 3 cm and slant height 5 cm is ________.
    (a) 60π cm²    (b) 45π cm²    (c) 15π cm²    (d) 30π cm²
37. Which of these is not suitable for displaying grouped continuous data by adjoining rectangles?
    (a) A histogram with contiguous bars for equal widths    (b) A simple separated-bar chart of unrelated categories    (c) A histogram with frequency densities for unequal widths    (d) A histogram with correct class boundaries
38. The frequencies in four successive classes are 8, 14, 11 and 7. The number of observations in the distribution is
    (a) 40    (b) 34    (c) 32    (d) 48
39. A frequency polygon has class midpoints 5, 15 and 25 with frequencies 4, 10 and 6. Its highest plotted point is ________.
    (a) (25,6)    (b) (10,15)    (c) (15,10)    (d) (5,10)
40. Determine the roots of x² − 5x − 14 = 0 by factorisation.
    (a) −7 and 2    (b) 7 and 2    (c) 7 and −2    (d) −7 and −2

SECTION B: THEORY (30 Marks)
Instruction: Answer Question 1 and any other TWO questions. Each question carries 10 marks. Show all necessary workings and label diagrams clearly.

Question 1: Quadratic equations and graph interpretation
(a) Form a monic quadratic equation whose roots are −2 and 5. (3 marks)
(b) For y=x²−4x+3, find the roots and turning point, then sketch a labelled parabola. (4 marks)
(c) A rectangular plot is 3 m longer than it is wide and has area 54 m². Find its width. (3 marks)
Draw the quadratic graph on labelled coordinate axes.

Question 2: Sectors, segments and cones
(a) Find the perimeter of a sector of radius 9 cm and central angle 80°. (3 marks)
(b) Find the minor segment area in a circle of radius 6 cm where the centre angle is 60°. (4 marks)
(c) A cone has radius 5 cm and slant height 13 cm. Find its curved surface area. (3 marks)

Question 3: Frustums and compound solids
(a) Find the volume of a conical frustum of height 4 cm and radii 6 cm and 3 cm. (4 marks)
(b) A hemisphere of radius 3 cm is placed on a cylinder of radius 3 cm and height 10 cm. Find total volume. (3 marks)
(c) The same frustum has slant height 5 cm. Find its curved surface area. (3 marks)

Question 4: Statistical presentation and interpretation
(a) In a survey, 36 of 120 respondents choose a category. Calculate its pie-chart angle. (3 marks)
(b) Draw a frequency table for the marks 2, 4, 2, 6, 4, 6, 2, 6, 6, 6 and state the modal mark. (4 marks)
(c) In a histogram, classes 0–10 and 10–15 have frequencies 20 and 15. Determine which bar is taller and explain. (3 marks)

Objective Marking Scheme
Question 1 – ✅ (c) 3 and 4
Question 2 – ✅ (b) x² − 4x + 1 = 0
Question 3 – ✅ (b) x = 1 and x = 3
Question 4 – ✅ (d) 6π cm
Question 5 – ✅ (d) (6π − 9√3) cm²
Question 6 – ✅ (a) 78π cm²
Question 7 – ✅ (d) 84π cm³
Question 8 – ✅ (d) 120°
Question 9 – ✅ (b) 7/10
Question 10 – ✅ (d) 10–15
Question 11 – ✅ (d) (18 + 4π) cm
Question 12 – ✅ (a) (25π − 50) cm²
Question 13 – ✅ (a) 65π cm²
Question 14 – ✅ (d) 108π cm³
Question 15 – ✅ (c) 24.5
Question 16 – ✅ (b) organise observations into a frequency distribution
Question 17 – ✅ (d) 108°
Question 18 – ✅ (c) 7
Question 19 – ✅ (b) x² − 3x − 4 = 0
Question 20 – ✅ (c) 6 m
Question 21 – ✅ (b) 108 cm³
Question 22 – ✅ (c) 3
Question 23 – ✅ (c) (15,12)
Question 24 – ✅ (a) January and February
Question 25 – ✅ (b) x² − 6x + 9 = 0
Question 26 – ✅ (a) (2, −9)
Question 27 – ✅ (a) x² − 8x + 16 = 0
Question 28 – ✅ (b) slant height
Question 29 – ✅ (c) 60π cm²
Question 30 – ✅ (a) 180 cm³
Question 31 – ✅ (a) x = 2
Question 32 – ✅ (c) read the x-coordinates where the curve meets the x-axis
Question 33 – ✅ (d) (10 + 10π/3) cm
Question 34 – ✅ (a) 65π cm²
Question 35 – ✅ (d) 8 times
Question 36 – ✅ (b) 45π cm²
Question 37 – ✅ (b) A simple separated-bar chart of unrelated categories
Question 38 – ✅ (a) 40
Question 39 – ✅ (c) (15,10)
Question 40 – ✅ (c) 7 and −2

Theory Marking Scheme

Question 1: Quadratic equations and graph interpretation
(a) (x+2)(x−5)=0 (1) giving x²−3x−10=0 (2).
(b) Factor (x−1)(x−3), roots 1 and 3 (2); vertex at x=2, y=−1 (1); upward sketch with intercepts and vertex labelled (1).
(c) x(x+3)=54 (1); x²+3x−54=(x+9)(x−6)=0 (1); reject −9, width 6 m (1).

Question 2: Sectors, segments and cones
(a) Arc = (80/360)×2π×9 =4π cm (2); perimeter =18+4π cm (1).
(b) Sector=(60/360)π(6)²=6π (2); triangle=(1/2)×6²×sin60=9√3 (1); difference 6π−9√3 cm² (1).
(c) πrl = π×5×13 =65π cm²; formula 1, substitution 1, result 1.

Question 3: Frustums and compound solids
(a) V=(πh/3)(R²+Rr+r²) (1);=(4π/3)(36+18+9) (2);=84π cm³ (1).
(b) Cylinder=π×9×10=90π (1); hemisphere=(2/3)π×27=18π (1); sum=108π cm³ (1).
(c) π(R+r)l = π(6+3)(5)=45π cm². Formula 1, substitution 1, result 1.

Question 4: Statistical presentation and interpretation
(a) (36/120)×360°=108°. Fraction 1, calculation 1, result 1.
(b) Frequency table: mark 2→3, mark 4→2, mark 6→5 (3 marks, one per correct frequency); modal mark = 6 (1 mark).
(c) Frequency densities: 20/10=2 (1); 15/5=3 (1); class 10–15 has taller bar because 3>2 (1).