All 5 questions from the Junior Secondary Certificate Examination (JSCE) Mathematics 2022 Theory paper, with the correct answer and a full explanation for each. Free, no signup needed.
1(a) In a triangle PQR, angle Q is three-quarter of angle P and angle R is half of angle P. If the measure of angle P is (x + 5)°, form an equation in x and hence calculate the angles of triangle PQR.
Model answer
Angle P = (x + 5)°
Angle Q = 3/4 (x + 5)°
Angle R = 1/2 (x + 5)°
Angles of a triangle add up to 180°:
(x + 5) + 3/4 (x + 5) + 1/2 (x + 5) = 180
Multiply through by 4: 4(x + 5) + 3(x + 5) + 2(x + 5) = 720
9(x + 5) = 720
x + 5 = 80, so x = 75
Angle P = 80°
Angle Q = 3/4 × 80 = 60°
Angle R = 1/2 × 80 = 40°
Check: 80 + 60 + 40 = 180°
1(b) Find the smallest number by which 240 must be multiplied to get a perfect square.
Model answer
Prime factors: 240 = 2 × 2 × 2 × 2 × 3 × 5 = 2⁴ × 3 × 5
The powers of 3 and 5 are odd, so multiply by 3 × 5 = 15.
240 × 15 = 3600 = 60²
The smallest number is 15.
2(a) A car travels at a distance of 60km. Construct a table of value for speed and time. Given that the time for the journey vary as: 1 1/2 hrs, 2 hrs, 2 1/2 hrs and 3 hrs.
Model answer
Speed = distance ÷ time = 60 ÷ time
Time (hrs) | 1 1/2 | 2 | 2 1/2 | 3
Speed (km/h) | 60 ÷ 1.5 = 40 | 60 ÷ 2 = 30 | 60 ÷ 2.5 = 24 | 60 ÷ 3 = 20
Speed varies inversely as time (speed × time = 60).
2(b) A crate containing 24 bottles of drinks is made up of 10 bottles of Coca cola, 8 bottles of Fanta and the rest are Sprite. If one bottle is picked at random, what is the probability of picking: (i) a coca cola drink (ii) a sprite drink
2(c) Calculate the unknown angles a and b in the diagram above. (XY and PQ are parallel lines, with 45° at Y and 55° at Q.)
Model answer
Draw a line through the vertex parallel to XY and PQ.
Alternate angles split angle a into 45° and 55°.
a = 45° + 55° = 100°
b is the reflex angle at the same point: b = 360° - 100° = 260°
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