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JSCE: MATHEMATICS

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Question 1:

(a) In a triangle PQR, angle Q is three-quarters of angle P and angle R is half of angle P. If the measure of angle P is (x + 5)º, form an equation in x and hence, calculate the angles of triangle PQR.

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(b) Find the smallest number by which 240 must be multiplied to get a perfect square.

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Question 2:

(a) A car travels at a distance of 60 km. Construct a table of values for speed and time. Given that the time for the journey varies as: ⇒ \( \scriptsize 1\frac{1}{2}\: hrs, \:2 \:hrs, \:2\frac{1}{2}\: hrs, \: and \: 3 \:hrs \)

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(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

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Question 3:

Screenshot 2023 05 01 at 12.53.22
Calculate the unknown angles a and b in the diagram above.

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Question 1a

In a triangle PQR, angle Q is three-quarters of angle P and angle R is half of angle P. If the measure of angle P is (x + 5)º, form an equation in x and hence, calculate the angles of triangle PQR.

Solution:

P ⇒ \(\scriptsize \left(x\: + \: 5 \right)^{\circ}\)

Q is three-quarters of angle P ⇒ \( \frac{3}{4} \scriptsize \left(x\: + \: 5 \right)^{\circ} \)

R is half of angle P ⇒ \( \frac{1}{2} \scriptsize \left(x\: + \: 5 \right)^{\circ} \)

P + Q + R = 180º (sum of angles in a Δ)

∴ \( \scriptsize (x\:+\:5)^{\circ}  \: + \: \normalsize \frac{3}{4}\scriptsize (x\:+\:5)^{\circ}\: + \: \normalsize \frac{1}{2}\scriptsize (x\:+\:5)^{\circ} \)

Use the L.C.M of the denominators to solve the expression.

⇒ \( \frac{4(x\:+\:5)^{\circ}  \: + \: 3(x\:+\:5)^{\circ} \: + \: 2(x\:+\:5)^{\circ} }{4}  \scriptsize = 180^{\circ}\)

⇒ \( \frac{9(x\:+\:5)^{\circ} }{4} = \frac{180^{\circ}}{1}\)

cross multiply

⇒ \( \scriptsize 9(x \: +\:5)^{\circ}  = 4(180)^{\circ}  \)

⇒ \( \scriptsize 9x \: +\: 45 = 720 \)

⇒ \( \scriptsize 9x = 720\:  – \: 45 \)

⇒ \( \scriptsize 9x = 675 \)

⇒ \( \scriptsize x = \normalsize \frac{675}{9} \)

⇒ \( \scriptsize x = 75 \)

Use the value of x to find angles P, Q and R

∴ P ⇒ \(\scriptsize \left(x\: + \: 5 \right)^{\circ} \\ \scriptsize = \left(75\: + \: 5 \right)^{\circ} = 80^{\circ} \)

∴ Q ⇒ \( \frac{3}{4} \scriptsize \left(x\: + \: 5 \right)^{\circ} \\ \frac{3}{4} \scriptsize \: \times \:  \left(75\: + \: 5 \right)^{\circ} \\ \frac{3}{4} \scriptsize \: \times \:  \left(80\right)^{\circ} \\ \scriptsize = 3 \: \times \: 20\\ \scriptsize = 60^{\circ}\)

∴ R ⇒ \( \frac{1}{2} \scriptsize \left(x\: + \: 5 \right)^{\circ} \\ \frac{1}{2} \scriptsize \: \times \:  \left(75\: + \: 5 \right)^{\circ} \\ \scriptsize\frac{1}{2} \: \times \:  \left(80 \right)^{\circ} \\ \scriptsize = 40^{\circ}\)

∴ P, Q, R = 80º, 60º, 40º respectively.

Question 1b

Find the smallest number by which 240 must be multiplied to get a perfect square.

Solution:

Step 1: Express 240 as a product of its prime factors.

Step 2: Pick the product of prime factors in pairs of the same digits. Then multiply digits without pairs (stand-alone digits) together. This gives the required number. i.e

‘ 3 and 5’ stand alone

The required number = 3 × 5 = 15

Question 2a

A car travels at a distance of 60 km. Construct a table of values for speed and time. Given that the time for the journey varies as: ⇒ \( \scriptsize 1\frac{1}{2}\: hrs, \:2 \:hrs, \:2\frac{1}{2}\: hrs, \: and \: 3 \:hrs \)

Solution:

Distance 60 60 60 60
Time \( \scriptsize 1 \frac{1}{2} \) \( \scriptsize 2\) \( \scriptsize 2 \frac{1}{2} \) \( \scriptsize 3\)
Speed 40 30 24 20

 

Workings:

Speed = \( \frac{Distance}{Time} \)

Speed 1 = \( \frac{60}{1.5} \scriptsize = 40 \: km/hr \)

Speed 2 = \( \frac{60}{2} \scriptsize = 30 \: km/hr \)

Speed 3 = \( \frac{60}{2.5} \scriptsize = 24 \: km/hr \)

Speed 4 = \( \frac{60}{3} \scriptsize = 20 \: km/hr \)

Question 2b

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

Solution:

Total bottles of drinks = 24

Coca-cola bottles = 10

Fanta bottles = 8

Sprite bottles = 24 – (10 + 8) = 24 – 18 = 6

(i) Pr (coca-cola) = \( \frac{No \: of \: bottles \: of \: coca-cola}{Total\: no \: of \: bottles} \\ = \frac{10}{24} \\ = \frac{5}{12}\)

(ii) Pr (sprite) = \( \frac{No \: of \: bottles \: of \: sprite}{Total\: no \: of \: bottles} \\ = \frac{6}{24} \\ = \frac{1}{4}\)

Question 3

Calculate the unknown angles a and b in the diagram above.

Solution:

Draw a line from the end of Q to meet at X.

 

∠X = ∠Q =  55º (alternate angles are equal)

Find aº

aº∠X +  Y (exterior angles are equal to sum of two interior angles)

aº = 55º + 45º

aº = 100º

Find bº

aº + bº = 360º (sum of angles at a point)

100º + bº = 360º

bº = 360º – 100º

bº = 260º

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