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Mathematics questions for ss2 third term with answers

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Subject: Mathematics                                                                                               Duration: 1 hr 30 mins

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  • Week 1:

    • Review of Last Term’s Work

    • Resumption Test

    • Copying of Scheme of Work

  • Week 2:

    • Trigonometry: Derivation of Sine Rule

    • Application of Sine Rule

    • Derivation and Application of Cosine Rule

  • Week 3:

    • Bearings:

      • Revision of Trigonometric Ratios

      • Angles of Elevation and Depression

      • Drawing of Cardinal Points (4, 8, 16)

      • Cardinal and 3-Digit Notation for Bearings

      • Practical Problems on Bearings

  • Week 4:

    • Statistics: Measures of Central Tendency

      • Meaning and Computation of Mean, Median, Mode (Ungrouped Data)

  • Week 5:

    • Statistics: Measures of Dispersion

      • Range, Variance, Standard Deviation

      • Practical Application in Real Life

  • Week 6:

    • Histograms (Revision):

      • Grouping, Class Boundaries, Class Intervals

      • Drawing of Histograms

      • Estimating Mode from Histograms

    • Cumulative Frequency Graph (Ogive):

      • Drawing and Interpretation (Median, Quartiles, Percentiles)

  • Week 7:

    • Mid-Term Test and Break

  • Week 8:

    • Grouped Data:

      • Determining Mean, Median, and Mode of Grouped Frequency

  • Week 9:

    • Probability:

      • Experimental Outcomes, Sample Space, Events

      • Chance Instruments (Die, Coin, Cards)

      • Theoretical Probability and Equi-probable Events

  • Week 10:

    • Probability (continued):

      • Addition and Multiplication Rules

      • Mutually Exclusive, Complementary, and Independent Events

      • Practical Applications of Probability

  • Week 11:

    • Revision

  • Week 12:

    • Examination, Marking, Result Compilation

 

Section A: Objective Question

Each question carries equal marks. Choose the option that best answers each question.

  1. What is the sine of 30°?
    A. 1
    B. 0.5
    C. √3/2
    D. 0
  2. In a triangle, if a = 5, b = 7, and angle C = 60°, use the Cosine Rule to find c².
    A. 109
    B. 65
    C. 49
    D. 25
  3. The formula for Sine Rule is:
    A. a² = b² + c² – 2bc cosA
    B. sin A / a = sin B / b = sin C / c
    C. sin A × a = sin B × b
    D. cos A / a = cos B / b
  4. Bearings are measured in what unit?
    A. Metres
    B. Kilometres
    C. Degrees
    D. Radians
  5. How many cardinal points are there in a 3-digit bearing system?
    A. 4
    B. 8
    C. 12
    D. 16
  6. The bearing of East is:
    A. 090°
    B. 045°
    C. 180°
    D. 135°
  7. What is the mode of the following ungrouped data: 3, 4, 5, 5, 6, 6, 6, 7?
    A. 4
    B. 5
    C. 6
    D. 7
  8. Find the mean of these values: 10, 20, 30, 40, 50
    A. 30
    B. 25
    C. 35
    D. 40
  9. The variance of 2, 4, 4, 4, 5, 5, 7, 9 is:
    A. 4
    B. 5
    C. 3
    D. 6
  10. The range of the data set 15, 21, 28, 31 is:
    A. 10
    B. 16
    C. 15
    D. 17
  11. Standard deviation is the square root of:
    A. Mean
    B. Range
    C. Variance
    D. Median
  12. A histogram is best used to represent:
    A. Ungrouped data
    B. Discrete data
    C. Grouped data
    D. Frequency polygon
  13. What is the class interval of a class with lower boundary 20 and upper boundary 29?
    A. 10
    B. 8
    C. 9
    D. 11
  14. Cumulative frequency is best represented using:
    A. Pie chart
    B. Bar chart
    C. Ogive
    D. Line graph
  15. In an ogive, the median is located at:
    A. Highest point
    B. First quartile
    C. 50th percentile
    D. X-axis
  16. What is the mid-point of the class 10–20?
    A. 10
    B. 15
    C. 20
    D. 30
  17. What is the cumulative frequency of a class with frequencies 2, 5, 7, 3?
    A. 17
    B. 15
    C. 16
    D. 12
  18. If the median of grouped data lies in the class 30–39, it means:
    A. It is the smallest class
    B. It has the most frequency
    C. It lies in the middle position
    D. It is the last class
  19. Probability of an impossible event is:
    A. 0
    B. 1
    C. 0.5
    D. 2
  20. If a fair die is tossed, the probability of getting 4 is:
    A. 1/4
    B. 1/5
    C. 1/6
    D. 1/3
  21. Sample space of tossing two coins is:
    A. 2
    B. 4
    C. 6
    D. 8
  22. An event that cannot happen is called:
    A. Certain
    B. Independent
    C. Impossible
    D. Mutually exclusive
  23. If P(A) = 1/2 and P(B) = 1/3, what is P(A ∪ B) assuming A and B are mutually exclusive?
    A. 1/6
    B. 5/6
    C. 2/3
    D. 1
  24. Which of the following is not a chance instrument?
    A. Dice
    B. Coin
    C. Playing card
    D. Ruler
  25. A class boundary is calculated by:
    A. Subtracting 0.5 from the lower limit
    B. Adding 0.5 to the upper limit
    C. Both A and B
    D. None of the above
  26. The mode is the:
    A. Middle value
    B. Most frequent value
    C. Average of all values
    D. Difference between highest and lowest
  27. What is the probability of picking a red ball from a bag containing 3 red, 2 blue, and 5 green balls?
    A. 1/3
    B. 3/10
    C. 2/5
    D. 1/2
  28. In a class interval 20–24, what is the class mark?
    A. 22
    B. 21
    C. 23
    D. 24
  29. The sum of probabilities of all possible outcomes is:
    A. 2
    B. 1
    C. 0
    D. 0.5
  30. Which is a true statement about variance?
    A. It’s always negative
    B. It’s the square of standard deviation
    C. It equals mean
    D. It’s less than zero
  31. A probability of 1 means:
    A. Impossible event
    B. Unlikely event
    C. Certain event
    D. Random guess
  32. If a spinner has 5 equal sections, what is the probability of landing on one section?
    A. 1/4
    B. 1/5
    C. 2/5
    D. 5/1
  33. A die is thrown once. What is the probability of getting an even number?
    A. 1/6
    B. 1/2
    C. 1/3
    D. 5/6
  34. An event and its complement always add up to:
    A. 0
    B. 1
    C. 2
    D. 100
  35. If P(E) = 0.75, what is P(E′)?
    A. 0.25
    B. 0.75
    C. 1.75
    D. 1
  36. Mean of 5, 8, 10, 2, 5 is:
    A. 5
    B. 6
    C. 10
    D. 7
  37. In trigonometry, the Cosine Rule applies when:
    A. Two sides and the included angle are known
    B. One angle is right
    C. All angles are 90°
    D. Triangle is equilateral
  38. Bearings are measured from:
    A. South
    B. North
    C. East
    D. West
  39. If angle of elevation is 30°, the opposite side is 4 cm, find hypotenuse using sine rule.
    A. 6 cm
    B. 8 cm
    C. 12 cm
    D. 2 cm
  40. What is the standard deviation of values 5, 5, 5, 5?
    A. 0
    B. 1
    C. 5
    D. 10

Section B: Theory

 

1. Trigonometry – Sine Rule
In triangle PQR, ∠P = 40°, ∠Q = 60°, and side p = 8 cm (opposite ∠P).
Use the sine rule to find the length of side q (opposite ∠Q). Give your answer to 2 decimal places.

2. Trigonometry – Cosine Rule
Triangle XYZ has sides x = 6 cm, y = 9 cm, and included angle ∠Z = 60°.
Use the cosine rule to calculate the third side z. Round off to two decimal places.

3. Bearings
A ship sails 10 km on a bearing of 045°, then turns and travels 8 km on a bearing of 135°.
(a) Represent the journey with a diagram.
(b) Use trigonometry to find the distance between the starting point and the final point.

4. Measures of Central Tendency (Ungrouped Data)
Given the following scores of students in a test: 6, 8, 7, 9, 10, 6, 8, 7, 6, 9.
(a) Find the mean
(b) Find the median
(c) Find the mode

5. Measures of Dispersion
Given this data set: 4, 5, 7, 8, 9
(a) Find the range
(b) Find the variance
(c) Find the standard deviation

6. Cumulative Frequency Graph (Ogive)
Using the grouped frequency table below,

Class Interval Frequency
0 – 9 5
10 – 19 8
20 – 29 12
30 – 39 7
40 – 49 3

(a) Construct the cumulative frequency table
(b) Draw the Ogive
(c) Estimate the median from the graph

7. Measures of Central Tendency (Grouped Data)
Using the data below, calculate the mean, mode, and median:

Class Interval Frequency
1 – 5 2
6 – 10 5
11 – 15 8
16 – 20 4
21 – 25 1

8. Probability – Basic Concepts
A box contains 3 red balls, 4 green balls, and 5 blue balls.
(a) What is the probability of picking:
i. a red ball
ii. a green ball
iii. not a blue ball
(b) If a ball is picked at random and replaced, what is the probability of picking a green ball twice?

9. Probability – Addition and Multiplication Rules
Events A and B are such that:
P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2
(a) Find P(A ∪ B)
(b) Are A and B independent? Show working.

10. Statistics – Histogram and Frequency Polygon
Given the grouped data:

Class Interval Frequency
10 – 14 4
15 – 19 6
20 – 24 9
25 – 29 5
30 – 34 2

(a) Construct a histogram
(b) Draw the frequency polygon
(c) Estimate the modal class and explain why

Section C: Objective Answers

  1. B — The sine of 30° is 0.5. It’s a standard trigonometric value students should memorize.
  2. A — Using the Cosine Rule: c² = 5² + 7² – 2(5)(7)cos60° = 25 + 49 – 70(0.5) = 74 – 35 = 39
  3. B — The Sine Rule is: sin A / a = sin B / b = sin C / c. This helps solve triangles that aren’t right-angled.
  4. C — Bearings are always measured in degrees (0°–360°) from the North.
  5. D — There are 16 cardinal points in a full directional system (N, NNE, NE, etc.).
  6. A — East is always at 90° bearing from North.
  7. C — Mode is the most frequent number. Here, 6 occurs 3 times — more than any other number.
  8. A — Add the numbers: (10+20+30+40+50) ÷ 5 = 150 ÷ 5 = 30
  9. C — The variance is 4. The data has a mean of 5. The formula gives 3 for variance.
  10. B — Range = highest – lowest = 31 – 15 = 16
  11. C — Standard deviation is the square root of variance.
  12. C — Histograms best represent grouped frequency data.
  13. C — Class interval = Upper boundary – Lower boundary = 29 – 20 = 9
  14. C — Cumulative frequency is represented using an ogive (a curve).
  15. C — The median corresponds to the 50th percentile in cumulative frequency.
  16. B — Mid-point = (10 + 20) ÷ 2 = 15
  17. A — Add all: 2+5+7+3 = 17 is the total cumulative frequency.
  18. C — The median class is the one that contains the middle frequency when data is arranged.
  19. A — An impossible event means it cannot happen. Its probability = 0
  20. C — A die has 6 sides. The chance of getting 4 is 1 out of 6 = 1/6
  21. B — Two coins give 4 outcomes: HH, HT, TH, TT.
  22. C — An event that cannot happen is called impossible.
  23. B — Mutually exclusive events: P(A ∪ B) = P(A) + P(B) = 1/2 + 1/3 = 5/6
  24. D — A ruler is not a chance instrument. Dice, coins, and cards are.
  25. C — Class boundary includes: Lower – 0.5 and Upper + 0.5. So both are used.
  26. B — The mode is the most frequently occurring value.
  27. B — Total balls = 10, Red = 3. So 3/10 is the probability.
  28. A — Class mark = (20 + 24) ÷ 2 = 44 ÷ 2 = 22
  29. B — All probabilities must add up to 1.
  30. B — Variance is the square of the standard deviation. It is always positive or zero.
  31. C — A probability of 1 means the event is certain to happen.
  32. B — Spinner has 5 equal sections: 1 out of 5 chance = 1/5
  33. B — Even numbers on die: 2, 4, 6. So 3/6 = 1/2
  34. B — An event and its complement always sum to 1 (e.g., P(A) + P(not A) = 1).
  35. A — Complement: 1 – 0.75 = 0.25
  36. B — Sum = 5+8+10+2+5 = 30. Mean = 30 ÷ 5 = 6
  37. A — Cosine rule applies when 2 sides and the angle between them are known.
  38. B — Bearings are measured clockwise from North (0°).
  39. A — sin(30°) = opposite/hypotenuse → sin 30° = 4/hypotenuse → 0.5 = 4/x → x = 8 cm
  40. A — If all values are the same, deviation from the mean is 0, so standard deviation = 0

 

Section D: Theory Answers

1. Trigonometry – Sine Rule

Question: In triangle PQR, ∠P = 40°, ∠Q = 60°, and side p = 8 cm (opposite ∠P). Use the sine rule to find the length of side q (opposite ∠Q).

Answer:
From triangle angle sum: ∠R = 180° – 40° – 60° = 80°

Use the Sine Rule

2. Trigonometry – Cosine Rule

Question: Triangle XYZ has sides x = 6 cm, y = 9 cm, and included angle ∠Z = 60°. Use the cosine rule to calculate the third side z.

Answer:
Cosine Rule:
z^2=x^2+y^-2xycosZ
z^2=6^2+9^2-2(6)(9) cos60
z^2=36+81-108.05=117-54=63
z=(square-root) of 63 = 7.94cm

3. Bearings

Question: A ship sails 10 km on a bearing of 045°, then turns and travels 8 km on a bearing of 135°.

(a) Diagram: Draw a North-facing line, first vector at 45°, second vector at 135°.

(b) Use Cosine Rule: Angle between paths = 135° – 45° = 90°
d2=102+82−2(10)(8)cos⁡90°=100+64=164d^2 = 10^2 + 8^2 – 2(10)(8)\cos90° = 100 + 64 = 164
d=(square-root) of 164 = 12.81km

4. Measures of Central Tendency (Ungrouped Data)

Scores: 6, 8, 7, 9, 10, 6, 8, 7, 6, 9

(a) Mean:
Mean=6+8+7+9+10+6+8+7+6+910=7610=7.6\text{Mean} = \frac{6+8+7+9+10+6+8+7+6+9}{10} = \frac{76}{10} = 7.6

(b) Median: Arrange = 6,6,6,7,7,8,8,9,9,10
Median = (5th + 6th)/2 = (7 + 8)/2 = 7.5

(c) Mode: Most frequent = 6 (3 times)

5. Measures of Dispersion

Data: 4, 5, 7, 8, 9

(a) Range: 9 – 4 = 5

(b) Mean: (4+5+7+8+9)/5 = 33/5 = 6.6

(c) Variance:
Deviations: (-2.6)^2,(-1.6)^2,0.4^2,1.4^2,2.4^2
Squares:6.76 + 2.56 + 0.16 + 1.96 + 5.76 = 17.2
Variance = 17.2/5 = 3.44
Standard deviation = (square-root) of 3.44 = 1.85

6. Ogive

Cumulative Frequency Table:

Class Frequency Cumulative Freq
0 – 9 5 5
10 – 19 8 13
20 – 29 12 25
30 – 39 7 32
40 – 49 3 35

(b) Draw Ogive: X-axis = Upper class boundaries, Y-axis = Cumulative frequency

(c) Estimate Median: Total freq = 35, Half = 17.5. Find class containing this on graph: around 20–29

7. Measures of Central Tendency (Grouped)

Class Midpoints: 3, 8, 13, 18, 23

(a) Mean:
Sum = (3.2 + 8.5 + 13.8 + 18.4 + 23.1) = 6 + 40 + 104 + 72 + 23 = 245
Freq Total = 20
Mean = 245/20 = 12.25

(b) Median Class: CF: 2, 7, 15, 19, 20
Median = Class 11–15 (8th to 10th item)

(c) Mode: Class with highest frequency = 11–15 (modal class)

8. Probability: Basic

Red: 3, Green: 4, Blue: 5. Total = 12

(a)
(i) P(Red) = 3/12 = 1/4
(ii) P(Green) = 4/12 = 1/3
(iii) P(Not Blue) = (3+4)/12 = 7/12

(b) P(Green twice) = (4/12) × (4/12) = 1/3 × 1/3 = 1/9

9. Probability: Addition and Multiplication

P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2

(a) P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = 0.4 + 0.5 – 0.2 = 0.7

(b) Check Independence:
P(A) × P(B) = 0.4 × 0.5 = 0.2 = P(A ∩ B) ✓
Yes, A and B are independent.

10. Histogram and Frequency Polygon

Class Midpoints: 12, 17, 22, 27, 32

(a) Histogram: Draw bars for each class with height = frequency

(b) Frequency Polygon: Plot midpoints against frequencies; join with straight lines

(c) Modal Class: 20–24 has highest frequency = 9, hence it is the modal class.

Student-Focused Conclusion

As we wrap up this comprehensive review of SS2 Mathematics for the third term, it’s important to remember that consistent practice and a clear understanding of each topic are the keys to success in both internal and external examinations like WAEC and NECO. From trigonometry and bearings to statistics and probability, every concept covered from Week 1 to Week 12 is essential for building a strong mathematical foundation.

Students are advised to regularly revisit these questions, master the solving techniques, and familiarize themselves with real-life applications of mathematics. Avoid rushing through revision or depending on examination malpractice, true confidence comes from preparation, discipline, and dedication. Stay committed, stay honest, and success will follow.

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