2001 - JAMB Mathematics Past Questions and Answers - page 6

In Δ PQT,
∠PTQ = 25^o(base ∠s of isosceles Δ)
In Δ QSR,
∠RQS = ∠QPT + ∠QTP
(Extr = sum of interior opposite ∠s)
∠RQS = 25 + 25
= 50^o
Also in Δ QSR,
75 + ∠RQS + ∠QSR = 180^o
(sum of ∠s of Δ)
∴75 + 50 + ∠QSR = 180
125 + ∠QSR = 180
∠QSR = 180 - 125
∠QSR = 55^o
But ∠QSR and ∠RST are the same
∠RST = 55^o

Number of yellow cars = 3
Total number of cars = 3 + 4 + 8 + 2 + 6 + 2 = 25
Fraction of yellow cars = ³/₂₅

< T = \(\frac{x}{1}\) = 25^o (PQ = QT)

< SQR = 2(25^o) = 50^o (sum of interior angle)

< Q + < R + < S = 180^o

50^o + 75^o + < S = 180^o = 125^o + < S = 180^o

< S = 180^o - 125^o = 55^o

Using cosine formula (t\(\sqrt{3}\))² = t² + t² - 2t² cos\(\theta\)

3t² = 2t² - 2t² cos\(\theta\) = 2t²(1 - cos\(\theta\))

1 - cos\(\theta\) = \(\frac{3t^2}{2t^2}\) = \(\frac{3}{2}\)

cos = 1 - \(\frac{3}{2} = -\frac{1}{2}\)

\(\theta\) = cos^-1(-\(\frac{1}{2}\)) = 120^o and 240^o

N.B 0 \(\geq\) \(\theta\) 360

\(\begin{array}{c|c} \text{colour of cars} & \text{Number (frequency)} \ \hline yellow & 3 \white & 4\ red & 8\ green & 2\ blue & 6\ black & 2\ \hline & 25 \ \hline\end{array}\)

Thus, the fraction of the total numbers that are yellow is \(\frac{3}{25}\)

\(\begin{array}{c|c} \text{no. of passengers} & \text{Number of taxis}\ \hline 0.5 - 2.5 & 3\ 2.5 - 4.5 & 4 \ 4.5 - 6.5 & 7\ 6.5 - 8.5 & 5\ 8.5 - 10.5 & 4 \ 10.5 - 12.5 & 1\ \hline \text{Total} & 24 \end{array}\)

Thus, the taxi with more than 4 passengers

= 7 + 5 + 4 + 1 = 17