Pearson Edexcel International GCSE (9–1) · MATHEMATICS A · Unit
Unit 10
108 practice questions
Topics covered
Practice questions
41–49 of 49 · Normal
Question 41
4 marks · Normal
Answer and see the solution →The probability that a machine produces a defective part is 0.1. The probability that a randomly chosen part was produced by machine A is 0.6, and by machine B is 0.4. The events are independent.
A part is selected at random. The possible outcomes are that the part was produced by machine A or B, and that the part is defective or not defective.
Construct a tree diagram to show all possible outcomes and their probabilities. Complete the probability table below.
Complete the missing probabilities in the table. Write all probabilities as decimals. Produced by Defective? Probability A Yes A No B Yes B No Question 42
3 marks · Normal
Answer and see the solution →A spinner can land on red, blue, or green. The probability of red is 0.3, the probability of blue is 0.4, and the probability of green is 0.3. Two independent spins are made.
Explain, with reference to a tree diagram, why the sum of the probabilities of all the possible outcomes is equal to 1.
Question 43
9 marks · Normal
Answer and see the solution →This question is about rational and irrational numbers and simplifying algebraic fractions.
(a)[4 marks]State, with a reason, whether each of the following is rational or irrational:
(i) (ii) (iii)
(b)[3 marks]Simplify the following algebraic fraction as much as possible:
(c)[2 marks]Given that , write the answer to part (b) in its simplest form.
Question 44
9 marks · Normal
Answer and see the solution →This question is about simplifying surds and combining algebraic fractions.
(a)[3 marks]Simplify the following expression, giving your answer as a surd in its simplest form:
(b)[3 marks]Express as a single algebraic fraction and simplify as far as possible:
(c)[3 marks]Let . Substitute this value into your answer for part (b) and write your answer as a surd in its simplest form.
Question 45
9 marks · Normal
Answer and see the solution →This question connects knowledge of number types and algebraic fractions.
(a)[2 marks]Write the number in its simplest surd form and state whether it is rational or irrational.
(b)[4 marks]Simplify the following algebraic fraction as much as possible:
(c)[3 marks]If , write your answer to part (b) in its simplest form.
Question 46
9 marks · Normal
Answer and see the solution →This question is about surds in probability calculations.
(a)[1 mark]Simplify . Give your answer as a surd in its simplest form.
(b)[1 mark]A bag contains 6 red balls and blue balls. A ball is chosen at random.
i) Write the total number of balls in the bag as a rational number.
(c)[2 marks]ii) Write down, as a fraction in its simplest form, the probability of choosing a blue ball.
(d)[3 marks]A second ball is chosen from the bag without replacement. Calculate the probability that both balls selected are red. Give your answer as a fraction in its simplest form.
(e)[2 marks]Is it possible to get an irrational probability from drawing balls from this bag? Explain your answer clearly.
Question 47
10 marks · Normal
Answer and see the solution →This question involves Venn diagrams and tree diagrams.
(a)[2 marks]Sets , and are defined as follows:
List all the elements of sets , , , and .
(b)[2 marks]Represent the sets , and in a Venn diagram. Clearly indicate and label the region .
(c)[2 marks]A number is selected at random from set .
Draw a tree diagram to show the possible outcomes and probabilities for 'multiple of 2' and 'multiple of 3' in set .
(d)[2 marks]Calculate the probability that a number chosen at random from set is both a multiple of 2 and a multiple of 3.
(e)[2 marks]Using your tree diagram or otherwise, calculate the probability that the number chosen is a multiple of 2 or a multiple of 3 (or both). Give your answer as a fraction in its simplest form.
Question 48
9 marks · Normal
Answer and see the solution →This question connects algebraic surds with conditional probability.
(a)[2 marks]Leo is working with the expression . Decide whether is rational or irrational. Explain your answer.
(b)[2 marks]A numbered spinner has 5 sections: 2 labelled 'A' and 3 labelled 'B'. Each time it is spun, the outcome is equally likely.
If the spinner is spun twice, find, with reasons, the probability of getting 'A' on both spins. Write your answer as a fraction in its lowest terms.
(c)[2 marks]Expand the expression and comment on whether the result is rational or irrational.
(d)[3 marks]The spinner is spun twice. Given that 'A' appeared at least once, what is the probability that the first spin was 'A'? Give your reasoning and write your answer as a fraction in its simplest form.
Question 49
9 marks · Normal
Answer and see the solution →This problem links rational and irrational numbers with probability.
(a)[2 marks]State, with a reason, whether the decimal is rational or irrational.
(b)[2 marks]A die has faces numbered . Find the probability that two dice, both thrown once, will show numbers that sum to an irrational number. Explain your answer.
(c)[2 marks]Now suppose a red die is numbered 1, 2, 3, 4, 5, 6, and a blue die is numbered 1, 4, 9, 16, 25, 36. A red and blue die are rolled together and the product of the two numbers is recorded.
i) Find the probability that the product is a rational number.
(d)[3 marks]ii) The dice are rolled again. Find the probability the square root of their product is irrational.