Pearson Edexcel International GCSE (9–1) · MATHEMATICS A · Unit
Unit 10
108 practice questions
Topics covered
Practice questions
11–18 of 18 · Hard
Question 11
4 marks · Hard
Answer and see the solution →Explain why the formula can be used to calculate the area of any triangle, not just right-angled triangles.
Question 12
2 marks · Hard
Answer and see the solution →A bag contains 5 red counters and 3 blue counters. Two counters are taken from the bag one after the other, without replacement. What is the probability that both counters are red? Give your answer as a fraction in its simplest form.
Question 13
4 marks · Hard
Answer and see the solution →A company has two machines, A and B, producing light bulbs. - Machine A produces 60% of the bulbs, and machine B produces 40%. - 3% of the bulbs produced by machine A are defective. - 5% of the bulbs produced by machine B are defective. A bulb is selected at random. Draw a tree diagram showing all possible outcomes and use it to calculate the probability that a randomly chosen bulb is defective.
Question 14
4 marks · Hard
Answer and see the solution →A rare disease affects 2 out of every 1000 people. A test for the disease gives a positive result in 99% of cases when the person actually has the disease, but also gives a false positive result in 2% of cases when the person does not have the disease. If a person is selected at random and tests positive, how likely is it that they actually have the disease? Explain your reasoning.
Question 15
10 marks · Hard
Answer and see the solution →A fruit seller is combining baskets of apples and pears. The amount of apples and pears after a trade can be represented as algebraic fractions, but the trade involves uncertain conditions shown by a probability tree.
(a)Express the following as a single fraction in its simplest form:
(b)[4 marks]A customer selects a basket at random. There are 3 baskets labelled A, B and C. The probability that she selects basket A is 0.5. If she does not select A, the probability she selects B is 0.7.
i) Draw a tree diagram to show all possible outcomes.
(c)[6 marks]ii) Find the probability that she selects basket C.
Question 16
10 marks · Hard
Answer and see the solution →A manufacturer fits security systems in homes. The company records the number of alarms installed each month. The number of alarms, n, is related to the cost per alarm, c, by the equation:
On one occasion:
(a)[3 marks]Given that 8 alarms were installed that month, solve the equation above to find the cost per alarm, c.
(b)[2 marks]A follow-up promotion means that the number of alarms in a month is now a random variable with the following probabilities:
Number of alarms, n Probability 8 0.2 12 0.3 16 0.4 20 0.1 A supervisor picks a month at random from the coming year and uses the value of n from that month in the same cost equation as above, with c = 12. (i) For each possible value of n, state if the equation has a solution for n > 0. Briefly explain your answer.
(c)[3 marks]The supervisor now wants to know, "What is the probability that the equation has a valid solution for n > 0, based on the probability distribution in part b?"
Complete a probability tree diagram for randomly selecting a value of n and checking if the equation can be solved for n > 0. Clearly label the branches with numbers and probabilities.
(d)[2 marks]Use your tree diagram to find the probability that a randomly selected month leads to a valid value of n from part (c).
Question 17
10 marks · Hard
Answer and see the solution →Lia rolls a fair six-sided die. The probability of rolling each number is .
Part of her scoring game is described by the equation:
where is the score Lia rolls.
(a)[3 marks]Solve the equation for x, giving all integer solutions.
(b)[2 marks]Given that x must be a possible die score, what is the probability that Lia's roll will give a solution to the equation?
(c)[3 marks]In another round, Lia rolls the die twice. She finds the solution to the equation for each value of x, and records whether the equation is solvable (using only possible die scores).
Draw a tree diagram to show all possible outcomes and the probabilities for both rolls, where S means the equation is solvable for the roll, and N means it is not solvable.
(d)[2 marks]Use your diagram to calculate the probability that exactly one of the two rolls gave a score making the equation solvable.
Question 18
11 marks · Hard
Answer and see the solution →This question is about algebraic fractions and probability tree diagrams.
(a)[2 marks]Simplify the following expression, giving your answer in its simplest form:
(b)A bag contains only red and blue counters. The probability that a counter selected at random is red is given by
where .
Two counters are selected from the bag, one after the other, with replacement.
i) Draw a tree diagram to show all possible outcomes for the colours of the two counters. Label each branch clearly with its probability in terms of .
ii) Express, in terms of , the probability that exactly one of the two counters selected is red.
(i)[2 marks]Draw and label the tree diagram. Show probabilities for red and blue at each stage in terms of .
(ii)[3 marks]Calculate the probability that exactly one counter is red using your tree diagram.
(c)[4 marks]If the probability that exactly one of the counters chosen is red is , find the value of .