Level 3 Technical Level

Level 3 Technical Level IT: PROGRAMMING Unit 5 Mathematics for programmers Tuesday 21 May 2019 Afternoon Time allowed: 2 hours Materials For this paper you must have: • a ruler • a scientific calculator (non-programmable) • stencils or other drawing equipment (eg flowchart stencils). Instructions • Use black ink or black ball-point pen. • Fill in the boxes at the top of this page. • Answer all questions. • You must answer the questions in the spaces provided. Do not write outside the box around each page or on blank pages. • Do all rough work in this book. Cross through any work you do not want to be marked. • If you need more space use the additional pages at the back of this booklet. • Include units in all answers, where required, as marks are given for units in some questions. Information • The marks for questions are shown in brackets. • The maximum mark for this paper is 80. • There are 50 marks in Section A and 30 marks in Section B. Both sections should be attempted. Advice • In all calculations, show clearly how you work out your answer. • Use diagrams, where appropriate, to clarify your answers. • You are expected to use a calculator where appropriate. • You are reminded of the need for good English and clear presentation in your answers. Section A Answer all questions in this section. outside the box 0 1 The result of the logical operation AND is Tick () one box. [1 mark] 0 2 De Morgan’s law converts A AND B into: Tick () one box. [1 mark] A OR B A OR B A OR B A OR B 0 3 If f(x) = x2 and g(y) = 1 y then g(f(x)) is equal to Tick () one box. [1 mark] 1 x2 x2 x2 y y x2 Turn over for the next question Turn over ► 0 4 Figure 1 shows a Venn diagram. Figure 1 What is A ∩ B? Tick () one box. [1 mark] { } {7, 11} {1, 5, 6} {1, 2, 4, 5, 6, 9} 0 5 The probability of an event is 1 . 8 Find the probability of the same event happening independently twice in a row. Tick () one box. [1 mark] 1 64 2 64 2 8 7 8 In probability theory, what is the difference between independent events and dependent events? [1 mark] Turn over for the next question Turn over ► . Evaluate 5! [1 mark] box Answer = . Evaluate 5! 3! 4! [2 marks] Answer = 0 8 Express the linear equations x + y = 6 and −3x + y = 2 in matrix form. [3 marks] � � � � = � � Turn over for the next question DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box *07* Turn over ► Figure 2 shows a digital logic circuit. Figure 2 box . Identify the type of logic gate labelled C in Figure 2. [1 mark] . Using the information in Figure 2, complete Table 1. [3 marks] *0* Table 1 X Y L M P 0 0 0 1 1 0 1 1 . Give the logic equation representing the digital logic circuit in Figure 2. [3 marks] box Turn over for the next question Turn over ► . Give an example of an arithmetic sequence with four terms. Explain what makes your example an arithmetic sequence. [2 marks] box . Give an example of a geometric sequence with four terms. Explain what makes your example a geometric sequence. [2 marks] . What is the nth term of the sequence 2, 5, 10, 17, … ? [1 mark] . Solve the equation 3x – 9 = 0 [1 mark] box . Solve the equation (x + 3)  x 6 5 Show your working. [2 marks] Question 11 continues on the next page Turn over ► . Solve the simultaneous equations 2x – 6y = 8 4x + 2y = 2 Show your working. [3 marks] box A straight line on a graph crosses the y-axis at −2 and the x-axis at +2. box . Sketch this line. . The general equation of the straight line is y = mx + c. What is the function f(x) that defines the straight line in Question 12.1? You need to show how you arrived at the function f(x). [1 mark] [3 marks] Turn over for the next question Turn over ► A and B are two matrices defined as 1 2  box A = 1 3 2 and B =  6 1  4 1 5   3 2 . What is the negative of matrix B? [1 mark] . What is the transpose of matrix A? [1 mark] . Calculate the matrix M = 2B. [1 mark] . State why it is not possible to add the matrices A and B. [1 mark] box Turn over for the next question Turn over ► . Convert into hexadecimal. [1 mark] box . Convert the 8-bit signed binary number to a decimal number. [1 mark] . Convert the binary fraction 0.110 to a decimal fraction. [1 mark] Turn over for the next question box *17* Turn over ► An array contains the numbers 25, 15, 10, 8, 3, 1, 17, 32, 20. box . Arrange these numbers into a suitable order for a binary search algorithm. [1 mark] . Describe the steps required in a binary search to find the number 25 in this array. [3 marks] . A binary search algorithm can use a recursive function. Explain what is meant by a recursive function. [1 mark] *1* Turn over for Section B DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Turn over ► Section B Answer all questions in this section. box 1 6 Figure 3 shows the following equations: y = x2 + 2x – 8 y = x + 4 Figure 3 . Using the graph in Figure 3 solve the equation x2 + 2x – 8 = 0 Explain your method. [3 marks] box Question 16 continues on the next page Turn over ► . Using the graph in Figure 3, solve the simultaneous equations y = x2 + 2x – 8 y = x + 4 Give the values for x and y and explain your method. [5 marks] box . One of the graphs in Figure 3 does not represent a function x = f(y). Identify this graph and explain your answer. [3 marks] box Question 16 continues on the next page Turn over ► . Sketch the graph of y = x2 – 1 on Figure 4, clearly showing where the line intersects the x and y-axes. box Give the coordinates where the graph intersects the x and y-axes. [4 marks] Figure 4 . Complete Table 3 for an Exclusive OR logic gate. [1 mark] box Table 3 X Y P 0 0 0 1 1 0 1 1 Question 17 continues on the next page Turn over ► . An Exclusive OR logic gate can be represented by the logic equation P = ( X.Y ) + ( X.Y ) Draw the logic diagram for this equation using only AND, OR and NOT logic gates. • Label the individual gate types in your diagram. • Label the output of each gate with a single letter of the alphabet, eg A, B, C, D etc. [6 marks] box box *27* Question 17 continues on the next page Turn over ► . Complete the truth table, Table 4, with entries for the output of each gate in the logic diagram of Question 17.2. Use the labels you assigned to the output of each gate in Question 17.2 in the first row of Table 4. box [4 marks] Table 4 . Draw the logic diagram for an Exclusive OR logic gate using only NAND logic gates. [4 marks] END OF QUESTIONS *2* If needed, use the following pages to continue your answers. Write the question number beside your answer. box Turn over ► box There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED box Copyright information For confidentiality purposes, from the November 2015 examination series, acknowledgements of third-party copyright material are published in a separate booklet rather than including them on the examination paper or support materials. This booklet is published after each examination series and is available for free download from after the live examination series. Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright-holders may have been unsuccessful and AQA will be happy to rectify any omissions of acknowledgements. If you have any queries please contact the Copyright Team, AQA, Stag Hill House, Guildford, GU2 7XJ. 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