These Ten-Year-Series (TYS) worked solutions with video explanations for 2019 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
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A solid cylinder has radius $r$ cm, height $h$ cm and total surface area $900$ cm$^{2}$. Find the exact value of the maximum possible volume of the cylinder. Find also the ratio $r:h$ that gives this maximum volume.
[7]
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A company produces drinking mugs. It is known that, on average, $8\%$ of the mugs are faulty. Each day the quality manager collects $50$ of the mugs at random and checks them; the number of faulty mugs found is the random variable $F$.
(i)
State, in the context of the question, two assumptions needed to model $F$ by a binomial distribution.
[2]
(i) State, in the context of the question, two assumptions needed to model $F$ by a binomial distribution.
[2]
You are now given that $F$ can be modelled by a binomial distribution.
(ii)
Find the probability that, on a randomly chosen day, at least $7$ faulty mugs are found.
[2]
(ii) Find the probability that, on a randomly chosen day, at least $7$ faulty mugs are found.
[2]
(iii)
The number of faulty mugs produced each day is independent of other days. Find the probability that, in a randomly chosen working week of $5$ days, at least $7$ faulty mugs are found on no more than $2$ days.
[2]
(iii) The number of faulty mugs produced each day is independent of other days. Find the probability that, in a randomly chosen working week of $5$ days, at least $7$ faulty mugs are found on no more than $2$ days.
[2]
The company also makes saucers. The number of faulty saucers also follows a binomial distribution. The probability that a saucer is faulty is $p$. Faults on saucers are independent of faults on mugs.
(iv)
Write down an expression in terms of $p$ for the probability that, in a random sample of $10$ saucers, exactly $2$ are faulty.
[1]
(iv) Write down an expression in terms of $p$ for the probability that, in a random sample of $10$ saucers, exactly $2$ are faulty.
[1]
The mugs and saucers are sold in sets of $2$ randomly chosen mugs and $2$ randomly chosen saucers. The probability that a set contains at most $1$ faulty item is $0.97$.
(v)
Write down an equation satisfied by $p$. Hence find the value of $p$.
[4]
(v) Write down an expression in terms of $p$ for the probability that, in a random sample of $10$ saucers, exactly $2$ are faulty.
[1]
Assumptions:
Assumptions:
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