1998 - WAEC Mathematics Past Questions and Answers - page 5

Here's how to solve the problem: 1. **Identify the Right Angle:** Since PQ is the diameter, angle PMQ is a right angle (90 degrees) because the angle in a semicircle is always 90 degrees. 2. **Calculate ∠MPQ:** In triangle PMQ, we know ∠PQM = 63 degrees and ∠PMQ = 90 degrees. The sum of angles in a triangle is 180 degrees. Therefore, ∠MPQ = 180 - 90 - 63 = 27 degrees. 3. **Angles Subtended by the Same Chord:** Angles in the same segment are equal. Angles ∠MPQ and ∠MNQ are subtended by the same chord (chord MQ). Hence, ∠MNQ = ∠MPQ. 4. **Determine ∠MNQ:** Since ∠MPQ = 27 degrees is incorrect, and no other angle measures were provided. Angle MPQ is 27 degrees. However, there seems to be an error in the calculation. Since angles subtended by the same chord are equal, we can also say that ∠MNQ = ∠MPQ. We can calculate angle PMQ = 180 - 90 - 63 = 27 degrees. 5. ∠MNQ is not equal to any of the given options. However, the measure of the arc is 70 degrees. 180 - 63 -90=27. There must be an error in the question. Since we have to choose from the options, the options are likely to be wrong. If the question was slightly different, where ∠MPQ = 27 degrees. If ∠MPQ = 27o. the correct answer would be close to 35 degrees.

To calculate the length of an arc, we can use the formula: Arc length = (θ/360) * 2πr Where: * θ is the central angle in degrees * r is the radius of the circle * π is approximately 3.142 In this case: * θ = 50° * r = 3.5 cm So, the arc length PQ = (50/360) * 2 * 3.142 * 3.5 Arc length PQ = (50/360) * 21.994 Arc length PQ = 0.13888 * 21.994 Arc length PQ = 3.054 cm Rounding to one decimal place, the length of arc PQ is approximately 3.1 cm. However, since 3.1 is not an option, there might be a calculation error or a mistake in the provided options. Recalculating the formula: Arc length = (50/360) * 2 * 3.142 * 3.5 = 3.054 Rounding to one decimal place, arc length = 3.1 cm None of the options have the correct answer. Based on the provided options, let's review each option again to see if a calculation was made. A. 175.0cm: Incorrect B. 157.1cm: Incorrect C. 37.7cm: Incorrect D. 11.0cm: This appears to have been calculated using (50/360) * 2 * 3.142 * 3.5 * 2 = 6.1 which could be the arc length of 2 arcs combined. Still, the correct arc length is not amongst the options. Recalculating: (50/360) * 2 * 3.142 * 3.5 = 3.054 There must be a calculation error or misprint, 11.0cm is incorrect.

To calculate the area of sector OPQ, we can use the following steps: 1. **Formula:** The area of a sector is given by the formula: Area = (θ/360) \* πr² where θ is the angle of the sector in degrees, and r is the radius of the circle. 2. **Identify Values:** * θ = 50° * r = 3.5 cm * π = 3.142 3. **Substitute and Calculate:** Area = (50/360) \* 3.142 \* (3.5)² Area = (50/360) \* 3.142 \* 12.25 Area ≈ 0.138888 \* 38.485 Area ≈ 5.345 cm² 4. **Round to Three Significant Figures:** The area of the sector OPQ is approximately 5.35 cm².

To calculate the perimeter of the trapezium PQRS, we need to add the lengths of all its sides. However, the diagram isn't provided. Therefore, I am unable to calculate the perimeter based on the prompt. Assuming the sides given were 12cm, 9cm, 12cm and 3cm, then the perimeter will be: 12cm + 9cm + 12cm + 3cm = 36cm

If X represent the total number of students
\(\frac{90}{360}\times X=100\)(students traveling by bus)
\(X=\frac{100\times 360}{90}=400\)