Pioneer Junior College H2 Math papers, with video solutions

1. In the triangle \(ABC\) as shown below, \(BC=3\), angle \(BAC=\frac{\pi }{3}+\theta \) radians and angle \(ACB=\frac{\pi }{2}\) radians.
   
   Show that \(AC=\frac{3\left( 1-\sqrt{3}\tan \theta \right)}{\sqrt{3}+\tan \theta }\).[3]
   Given that \(\theta \) is a sufficiently small angle, deduce that \(AC\approx \sqrt{3}+a\theta \), where \(a\) is a constant to be determined.[3]

The weight, $Y$, of a Yummy cereal bar is normally distributed with mean $\left( 120-k \right)$g and standard deviation 10g. The weight, F, of a Fullness cereal bar is normally distributed with mean 180 g and standard deviation 20 g.

1. Given that $\mathrm{P}\left( Y<80 \right)=\mathrm{P}\left( Y>150 \right)$, show that the value of $k$ is 5.

   [1]

2. 5 Yummy cereal bars are randomly chosen. Find the probability that exactly one bar weighs lesser than the lower quartile weight and exactly one bar weighs more than the median weight.

   [2]

3. Find the probability that the weight of 2 randomly chosen Yummy cereal bars differs from one and a half times the weight of a randomly chosen Fullness cereal bar by at most 5g. State the assumption needed for your working.

   [4]

The Fullness cereal bars are sold at $2 per 100g.

4. Find the probability that a randomly chosen Fullness cereal bar costs more than $3.50.

   [2]

5. A random sample of 100 Fullness cereal bars is taken. Find the probability that there are at least 55 bars but at most 65 bars with each costing more than $3.50.

   [3]