A Level Mathematics Bridging Unit Preparation for Core and Statistics Mathematic topics to help with the transition from GCSE to AS/A level. It provides support to students through revision of a variety of GCSE math topics essential for Core and Statistics A level Mathematics. NMBEC Sixth Form ___________________________________________________________________ NMBEC Guiding Principles of Sixth Form learning You learn best when you... 1. Use your initiative, have a go and clarify later; it’s your ideas that matter. 2. Take a risk and be creative. 3. Talk about what you are learning and what you know. 4. Work together to explore how you would solve a problem. 5. Reflect on your needs and how you learn best in terms of interests, abilities and styles of learning. Make changes if necessary. 6. Respond positively to feedback from your teachers and your peers. 7. Take responsibility for your own learning. ___________________________________________________________________________ 1 Core 1 Indices What you know (3) Write the laws of indices Now use the laws to find the value of each of the following, giving each answer as an integer or fraction as appropriate. 1 i. 2 ii. iii. 1 4 1 25 (Total 3 marks) 2 Express 125√5 in the form 5 (Total 2 marks) 2 What you know (3) Surds Write the rules for surds Now simplify the following 1) Simplify √48 2√27 (2) 2 Expand and simplify 5 3√2 (3) 3) Express √75 √48 in the form √3 (3) (Total 8 marks) 3 Algebraic Fractions Use your initiative, have a go and clarify later. (1) Simplify the following algebraic fractions. 4 1 5 6 (3) 2 3 3 2 7 6 (3) 3 6 2 30 50 21 3 10 (4) 4 5 4 2 1 (3) 5 1 2 1 (3) 6 3 7 4 1 (4) 5 Quadratics Use your initiative, have a go and clarify later; it’s your ideas that matter (1) Take responsibility for your own learning (7) Write 7 6 in the form (3) State the coordinates of the minimum point on the graph of 7 6 (2) Find the discriminant of 7 6 ; how many roots does this graph have? (2) Find the coordinates of the points where the graph of and sketch the graph showing these points. 7 6 crosses the axes (3) (Total 10 marks) 6 Coordinate Geometry Take a risk and be creative. (2) The sides of a triangle are the lines 0, 3 5 0 and 2 7 0. i. Find the coordinates of the vertices of the triangle. (3) ii. Find the area of the triangle. (2) iii. Is this a right angled triangle? Prove it. (3) (Total 8 marks) 7 Core1 and Core2 Transformation of graphs Take responsibility for your own learning. (7) 1) The point P (5, 4) is on the curve y = f(x). State the coordinates of the image of P when the graph of y = f(x) is transformed to the graph: (i) y = f(x – 5) (2) (ii) y = f(x) + 7 (2) 2) The curve y = f(x) has a minimum point at (3, 5). State the coordinates of the corresponding minimum point on the graph of (i) y = 3f(x) (2) (ii) y = f(2x) (2) 8 3) Fig.4 shows a sketch of the graph of y = f(x). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to A, B and C. (i) y = 2f(x) (2) (ii) y = f(x + 3) (2) 9 4) Fig. 5 shows a sketch of the graph of y = f(x). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to P, Q and R. (i) y = f( x) (2) (ii) y = -f(x) (2) 10 Core 2 Sine rule and cosine rule What you know already. (3) Write the sine and cosine rules. 1) 11 2) 12 Reflect on your needs and how you learn best in terms of abilities and styles of learning (5) Take a risk and be creative (2). 3) O is the centre of the circle, radius 8cm. P and Q are points on the circumference. Angle POQ = 108o Draw a diagram a) Calculate the area of the sector POQ. (2) b) Calculate the difference between the arc PQ and the length of the chord PQ. (4) c) Calculate the area of the segment PQ (area enclosed by the arc PQ and the chord PQ) (4) (Total 10 marks) 13 Statistics 1 Use your initiative, explore how you would solve a problem (1,4) Scatter Graphs 1. The scatter graph shows the average body length and average foot length of different species of rodents. 50 40 Foot length (mm) 30 20 10 0 100 (a) 150 200 Body length (mm) 250 What does the scatter graph tell you about the type of correlation between the body length and foot length for these rodents? (1) (b) Calculate the mean for i) Body length (2) ii) Foot length (2) 14 iii) Plot and circle the mean coordinate. Draw a line of best fit on the scatter diagram going through the mean coordinate. (2) (c) If body length increased by 50 mm, by approximately how many millimetres would you expect foot length to increase? Ring the correct value below. 2 7 15 50 275 (1 ) (d) An animal has a body length of 228 mm, and foot length of 22 mm. Is this animal likely to be one of these species of rodents? Explain your answer. (1 ) (e) Write the equation for the line of best fit (3) (Total 12 marks) 15 Venn Diagram Take responsibility for your own learning. (7) In an office there are 30 people; 12 have ‘A’ levels in Art (A), 8 have ‘A’ levels in Biology (B), 8 have ‘A’ levels in Latin (L), 3 have ‘A’ levels in Art and Biology, 3 have ‘A’ levels in Biology and Latin, 4 have ‘A’ levels in Latin and Art and 2 have ‘A’ levels in Art, Biology and Latin. (a) Draw a Venn diagram to represent these data. (4) (b) One person is chosen at random. Calculate the probability that they have ‘A’ levels in : (i) At least one of the three subjects (2) (ii) Only one of the three subjects (2) (iii) Latin but not Biology (2) (Total 10 marks) 16 Conditional Probability 3. There are What you know already (3) 4 bottles of orange juice, 3 bottles of apple juice, 2 bottles of tomato juice. Viv takes a bottle at random and drinks the juice. Then Caroline takes a bottle at random and drinks the juice. Work out the probability that they both take a bottle of the same type of juice. .................................... (4) Work out the probability they take a different type of juice. .............. (4) (Total 8 marks) 17 Averages from Grouped Data What you know already (3) 4. Charles found out the length of reign of each of 41 kings. He used the information to complete the frequency table. Length of reign (L years) (a) Number of kings 0 < L ≤ 10 14 10 < L ≤ 20 13 20 < L ≤ 30 8 30 < L ≤ 40 4 40 < L ≤ 50 2 Write down the class interval that contains the median. ….……………………. (2) (b) Calculate an estimate for the mean length of reign. …………………. years (4) (Total 6 marks) 18 5. The box plot gives information about the distribution of the weights of bags on a plane. 0 5 10 15 20 25 30 Weight (kg) (a) Jean says the heaviest bag weighs 23 kg. She is wrong. Explain why. ....................................................................................................................... ....................................................................................................................... (1) (b) Write down the median weight. ............................. kg (1) (c) Work out the interquartile range of the weights. ............................. kg (1) There are 240 bags on the plane. (d) Work out the number of bags with a weight of 10 kg or less. ............................. (2) (Total 5 marks) 19 Use your initiative, have a go and clarify later (1) Cumulative Frequency What you know already (3) 6. 200 students took a test. The cumulative frequency graph gives information about their marks. 200 Cumulative frequency 160 120 80 40 0 10 20 30 40 50 60 Mark The lowest mark scored in the test was 10. The highest mark scored in the test was 60. Use this information and the cumulative frequency graph to draw a box plot showing information about the students’ marks. 10 20 30 40 50 60 Mark (Total 3 marks) 20 Explore how you would solve a problem (4) Histograms The table and histogram show information about the length of time it took 165 adults to connect to the internet. 7. Time (t seconds) Frequency 0 < t ≤ 10 20 10 < t ≤ 15 15 < t ≤ 17.5 30 17.5 < t ≤ 20 40 20< t ≤ 25 25< t ≤ 40 None of the adults took more than 40 seconds to connect to the internet. (a) Use the table to complete the histogram. (2) (b) Use the histogram to complete the table. (2) 2003 Frequency density O 5 10 15 20 Time (seconds) 21 25 30 35 40 2004 Frequency density O 5 10 15 20 25 30 35 40 Time (seconds) The histogram shows information about the time it took some children to connect to the internet. None of the children took more than 40 seconds to connect to the internet. 110 children took up to 12.5 seconds to connect to the internet. (c) work out an estimate for the number of children who took 21 seconds or more to connect to the internet. ..................................... (3) (Total 7 marks) 22