Surds 1. Good form. ab = a b. b = a. t is an integer such that

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Dominick Matthews
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1 Surds 1 You can give exact answers to calculations by leaving some numbers as square roots. This square has a side length of 10 cm. You can t write 10 exactly as a decimal number. It is called a surd. Remember these rules: 1 ab = a b 2 a NUMBER b = a b 7? Area 5 10 cm 2 Good form If you are asked to write a number or answer in a certain form, you need to fi nd integers for all the letters in an expression is in the form p + q 2 The integers can be positive or negative. t is an integer such that = t 3 Find the value of t. A t = 8 B t = 3 C t = 15 D t = Turn over for the answer.

2 And the answer is A t = = = = W r i t e = 8 3 A d d t h e t w o t e r m s i n 3. Use the rule to split each square root. 9 a n d 25 as whole numbers. Look for factors of 27 and 75 which are square numbers. Rationalising the denominator Rationalising the denominator of a fraction means making the denominator a whole number. You can do this by multiplying the top and bottom of the fraction by the surd Remember that part in the So 3 2 x denominator x 2 = 2 2 = 3 x 2 = 6 The surd part of the denominator is 2 Watch out You can check your working with a calculator, but your answers must always be written in terms of surds. If you only remember two things... ab = a b a b = a b

3 ALGEBRA 11 Arithmetic sequences An arithmetic or linear sequence is a sequence of numbers where the difference between consecutive terms is constant. In your exam, you might need to work out the nth term of a sequence. Simple steps There are simple steps to work out the nth term in a sequence. Write the difference between each pair of terms. 1, 5, 9, 13, 17,... The difference is 4. Work backwards to fi nd the zero term of the sequence. Subtract 4 from the fi rst term. 1 4 = 3 Write down the formula for the nth term and substitute the values for the difference and the zero term. nth term = difference n + zero term nth term = 4n 3 Here are the fi rst fi ve terms of an arithmetic sequence. 2, 6, 10, 14, 18 Write down an expression for the nth term. A 4n + 2 B 2n 4 C 2n 2 D 4n 2 22 Turn over for the answer.

And the answer is D 4n 2 6 2 = 4 2 4 = 2 n th term is 4n 2 Find the difference between pairs of terms. Find the zero term. n th term is difference n + zero term Check it!

4 And the answer is D 4n = = 2 n th term is 4n 2 Find the difference between pairs of terms. Find the zero term. n th term is difference n + zero term Check it! Check your answer by substituting values of n into your nth term. 1st term: when n = 1, 4n 2 = = 2 2nd term: when n = 2, 4n 2 = = 6 You can also generate any term of the sequence. For the 20th term: when n = 20, 4n 2 = = 78 So the 20th term is 78. Watch out If an exam question asks if a particular number is a term in a given sequence, remember that n must be a positive whole number. If you only remember one thing... nth term = difference n + zero term

5 RATIO Ratio and proportion Ratios are used to compare quantities. Find equivalent ratios by multiplying or dividing by the same number. Two quantities are in direct proportion when both quantities increase at the same rate : 9 10 : : 1.8 Number of theatre Total cost tickets bought Two quantities are in inverse proportion when one quantity increases at the same rate as the other quantity decreases. 9 Average speed 40 km/h 2 80 km/h 405 Time taken 2 hours 1 hour 2 Karen, Liam and Amir share some money in the ratio 5 : 2 : 4. In total, Liam and Amir get 150. How much does Karen get? A 125 B 25 C 100 D Turn over for the answer.

6 And the answer is A 125 The money is divided in the ratio 5 : 2 : 4, so there are = 11 parts in total. Liam and Amir get = 6 parts. 6 parts = 150, so 1 part = = 25. Karen gets 5 parts: 5 25 = 125 Simplest form To write a ratio in its simplest form, fi nd an equivalent ratio with the smallest possible whole-number values. Simplest form: 5 : 1 2 : 3 : 4 Not simplest form: 10 : 2 1 : 1.5 : 2 Watch out Make sure your answer makes sense! If you only remember two things... Direct Inverse proportion means two quantities increase together. proportion means that as one quantity increases, the other quantity deceases at the same rate.

7 GEOMETRY 55 Pythagoras theorem Use Pythagoras theorem to fi nd the length of a missing side in right-angled triangle. a 2 + b 2 = c 2 The long side c is the hypotenuse. c b a Calculator skills Use these keys to fi nd squares and square roots. You might need to use the answer as a decimal number. key to get your his right-angled triangle has sides x cm, 7 cm and 10 cm. Calculate the value of x. Give your answer correct to 2 d.p. A cm B 7.14 cm C 12 cm D 149 cm 7 cm cm x 10 cm 76 Turn over for the answer.

8 And the answer is A cm a 7 cm c x cm Label the sides a, b and c. 10 cm b a 2 + b 2 = c 2 c 2 = = 149 c = = cm (2 d.p.) Write the formula for Pythagoras theorem. Substitute the vales for a and b in the formula and solve. Write the units in your answer. Watch out Label the sides of your triangle and be careful if the missing side is one of the shorter sides. If you only remember two things... a2 The + b 2 = c 2 hypotenuse is the longest side.

9 Histograms STATISTICS 91 Histograms can be used to represent grouped data with different class widths. No gaps between the bars. Area of each bar is proportional to frequency. Vertical axis is labelled Frequency density. Bars can be different widths. Frequency density = frequency class width The histogram shows patients waiting times in a doctors surgery during one week. Estimate the numbers of patients who waited less than 25 minutes. Waiting times Frequency density Time (minutes) A 18 B 330 C 33 D Turn over for the answer.

10 And the answer is D 240 Waiting times Frequency density Time (minutes) = = 240 Calculate the total area of the bars to the left of time = 25 minutes. Watch out Use a histogram when the class intervals are different widths. If you are given a frequency table, add a third column to show the frequency density. If you only remember one thing... Use histograms for grouped data with different class widths.

11 Tree diagrams Tree diagrams are used to show conditional probability. This tree diagram shows Holly s journey to school. At each branch the probabilities add up to = 1 Write the probability for each event on the branch. 2 Catch bus Miss bus PROBABILITY Late On time Late On time The outcome of the fi rst event can affect the probability of the second event. Multiply along the branches to fi nd the probability of each outcome. The probability that Holly misses the bus and is late for school is = 4 15 The probability that a student at Holly s school cycles to school is If a student cycles to school, the probability that they are on time is 0.8. If a student does not cycle to school, the probability that they are on time is A student is chosen at random. Work out the probability that they are late. A 0.05 B C D Turn over for the answer.

12 And the answer is D Draw a tree diagram and complete each branch Probabilities in each branch add up to 1: = = 1 Cycles = 1 Does not cycle Probability they are late if they cycle = = 0.05 Probability they are late if they do not cycle = = Probability they are late = = Multiply along the branches. Late On time Late On time Add the separate probabilities. If you only remember two things... At Multiply each branch, probabilities add up to 1. along the branches.

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram.

What you may need to do: 1. Formulate a quadratic expression or equation. Generate a quadratic expression from a description or diagram. Dealing with a quadratic What it is: A quadratic expression is an algebraic expression containing an x 2 term, as well as possibly an x term and/or a number, but nothing else - eg, no x 3 term. The general