Pre-College Workbook

Transcription

1 Pre-College Workbook Bexhill College Maths Department The questions in this workbook represent a non-exhaustive selection of skills required to succeed on the A Level mathematics course - it is your responsibility to ensure you can correctly answer each of these questions before starting in September. Additional help videos for each of the topics, along with the answers for each exercise, can be found at. You should use the videos to help you revise topic areas that you are not completely comfortable with before, or during, attempting the associated questions. You must also use the answers to check your work and ensure you have a correct worked solution to each question. You must bring completed, self-marked, worked solutions for every question to your first maths lesson in September. In your first week there will be an induction test based on these topics, amongst others from Higher GCSE. NO CALCULATORS should be used, except where specifically allowed. If you find a mistake in this workbook, on the website, or in the solutions, please rhysmills@bexhillcollege.ac.uk.

2 Simplifying Surds Write each of the following as a single surd, in the form a b More Complex Surds Simplify the following fully e) Rationalising Surds Simplify the following fully e) Linear Equations Solve the following equations. w + 5 = ( + v) (5 v) = 5 k = 6 4 (x ) (4 x) = x + e) x + = 7 x + 9 = x + x

3 5 Linear Inequalities Solve the following linear inequalities. x + 5 ( x) > 9 x < 4 ( x) 5( x) 6 Rearranging the Subject of a Formula Rearrange these equations to make the letter in the bracket the subject of the formula. y = x + a b + c (x) a = 5 d ( j a = (j + (j) f = t + r t (t) e) p y = g + r g b (g) w = x 4y (y) 7 Indices Simplify the following, without using a calculator. 6 ( ) ( ) 7 (.5).5 5 Write the following in the form ax n. x e) x 5 ( ) g) 9 h) 6x(x) x 8 Factorising Factorise the following expressions. a + 8b 4c 49 x x + 8x + 7 4x 9 e) x + 6x 4pqr p q g) x 8x + h) x 8 i) x + x 6 j) x 7x k) x 4x l) x 75

4 9 Solving Quadratic Equations Solve the following equations. x + 7x + = 0 x 0x + = 0 0 = 4 x x x 8x = 9 e) x = x (x + 5) = 5 x g) x + x + = x h) 4 x 7 5x = 0 i) 6 x + 6 = x Quadratic Simultaneous Equations Solve the following simultaneous equations. x + y = x + y = x + y = 5 x + y = y = x 4 x + y = 8 x xy = 8 x + xy = Algebraic Fractions Express each of the following as a single fraction and simplify where possible. e) x 5 x x x 4 x + x 8 x 6 x + 5 6x x x + 4 x x + 4 (x + ) + x +

5 Statistics You may use a calculator in this section.. A large calculator manufacturer decided it needed to know how many calculators the average person owned. It surveyed 50 maths teachers and most of the results are shown below. Additionally, there was one calculator enthusiast that owned calculators and is not included in the table below. Calculators Frequency ( Explain an error by the manufacturer with their survey technique. suggest a way they could improve their data in future surveys. Hence, ( Calculate the mean and median of the 50 data points. Is the mean or median more useful to represent the average number of calculators owned? ( A second survey asked the same question to a group of science teachers, and found that there was a mean number of calculators of.5 and a standard deviation of 0.7. Calculate the standard deviation for the first sample, and then compare the results.. The time taken for a group of students to complete their Bexhill College Pre-College Mathematics Workbook was recorded in the table below. Time Taken (mins) Frequency 0 < t < t < t < t < t 40 6 ( How many students completed the workbook? ( Estimate the mean time taken, and explain why this is an estimate. ( Explain why the range cannot be calculated for this data. ( A histogram needs to be drawn. group. Calculate the frequency density for each (e) Given that the actual height of the 0 < t 40 class is 6cm, what is the actual height of the 90 < t 50 class? ( Estimate the number of students that took longer than hours to complete the workbook.

6 Venn Diagrams You may use a calculator in this section. A B C The Venn diagram above shows the number of students in an A Level mathematics class and whether they use popular mathematics revision websites. One of these students is chosen at random. Find the probability that the student uses more than one of these websites. Find the probability that the student uses website A or website B (or both). Find the probability that the student uses both website A and website C. Given that the student uses website A, find the probability they also use website B.. There are 80 students on a postgraduate mathematics course. Students on this course can choose to take up to three extra options: take Axiomatic Set Theory, 70 take Quantum Information Theory, 8 take Stochastic Optimisation, 5 take Quantum Information Theory and Axiomatic Set Theory, 8 take Stochastic Optimisation and Quantum Information Theory, 40 take Axiomatic Set Theory and Stochastic Optimisation, 4 take all three options. Draw a Venn Diagram to represent this information. A student is chosen at random. What is the probability that they only take one option? What is the probability that a random student studying Quantum Information Theory is actually taking all three options?

7 4 Tree Diagrams You may use a calculator in this section.. The maths teacher gets the bus to school with a probability of, otherwise he walks. If he gets the bus, the probability that he buys a cup of tea at school is. 4 If he walks, the probability is. Draw a tree diagram to represent this information and find the probability that the maths teacher buys a cup of tea at work.. Two events, C and D, are represented on the tree diagram below: 7 C C 4 4 D D D D ( Write down: ( Find: i. P (D C) ii. P (D C ) i. P (C D) ii. P (D )