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GCSE · OCR

P2.4 Boolean Logic

OCR J277 · Calculators not allowed · about 100 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

In computer science, a Boolean is a data type that can only take one of two values.

(a)State the two values that a variable of Boolean data type can hold.(1)

(b)Give an example of a real-world question, other than a question about an alarm system, that has a Boolean (yes/no) answer.(1)

(Total for Question 1 is 2 marks)

The NOT operator inverts (reverses) the value of a single Boolean input A. Complete the truth table below for NOT A.

A	NOT A
0
1

(Total for Question 2 is 2 marks)

The AND and OR operators combine two Boolean inputs, A and B, to produce a single output.

(a)Complete the truth table for the AND operator.

A	B	A AND B
0	0
0	1
1	0
1	1

(4)

(b)Complete the truth table for the OR operator.

A	B	A OR B
0	0
0	1
1	0
1	1

(4)

(Total for Question 3 is 8 marks)

A student sets A = 1 and B = 0.

(a)What is the value of A AND B?(1)

A) 0
B) 1
C) Cannot be determined
D) Both 0 and 1

(b)What is the value of A OR B?(1)

A) 0
B) 1
C) Cannot be determined
D) Both 0 and 1

(Total for Question 4 is 2 marks)

A house alarm system has two sensors: a door sensor D and a window sensor W. Each sensor outputs 1 if triggered and 0 if not triggered. Let A represent whether the alarm sounds (1 = sounds, 0 = silent).

(a)The alarm should sound if either sensor is triggered. Write a Boolean expression for A in terms of D and W.(1)

(b)The owner finds the alarm sounds too often because of a cat setting off one sensor. Write a new Boolean expression for A so that both sensors must be triggered at the same time before the alarm sounds.(1)

(Total for Question 5 is 2 marks)

Consider the Boolean expression X = NOT A AND B, where A and B are single-bit inputs.

(a)Complete the truth table for X.

A	B	NOT A	X
0	0
0	1
1	0
1	1

(4)

(b)State how many of the four input combinations of A and B give an output of X = 1.(1)

(Total for Question 6 is 5 marks)

A smart heating system has a temperature sensor that outputs tempLow = True when the room temperature is below 18C, a window sensor windowClosed = True when the window is shut, and a manual override switch override, which the user can set to True to force the heating on regardless of the other sensors.

(a)Write a Boolean expression for heatingOn that turns the heating on when the temperature is low AND the window is closed, or when the override switch is on.(2)

(b)State the value of heatingOn when tempLow = True, windowClosed = False and override = False.(1)

(Total for Question 7 is 3 marks)

A logic circuit has two inputs, P and Q. Input P is connected to the input of a NOT gate. The output of the NOT gate, together with input Q, are the two inputs to an AND gate. The output of the AND gate is X.

(a)Write the Boolean expression for X in terms of P and Q.(1)

(b)Complete the truth table for the circuit.

P	Q	X
0	0
0	1
1	0
1	1

(4)

(c)State the input values of P and Q that produce X = 1.(1)

(Total for Question 8 is 6 marks)

A programmer wants to build a logic circuit that produces the output of the Boolean expression X = A OR NOT B, using two inputs A and B.

(a)State the two logic gates that are needed to build this circuit.(2)

(b)Describe how the two gates would be connected to produce the correct output X.(2)

(Total for Question 9 is 4 marks)

Complete the truth table below for the expression Output = A AND (B OR C).

A	B	C
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

(Total for Question 10 is 4 marks)

A program is written using OCR Exam Reference Language:

age <- 16
hasID <- True
IF age >= 18 OR hasID == True THEN
    OUTPUT "Entry allowed"
ELSE
    OUTPUT "Entry denied"
ENDIF

(a)State the output produced when this code is run with the values shown above.(1)

(b)The line hasID <- True is changed to hasID <- False, with age still equal to 16. State the new output.(1)

(c)Complete a trace table showing the values of age, hasID and the OUTPUT produced for both runs described in parts (a) and (b).

Run	age	hasID	OUTPUT
1
2

(3)

(Total for Question 11 is 5 marks)

A login validation routine uses three variables: usernameLength (an integer), hasDigit (a Boolean, True if the password contains at least one digit) and hasSpecialChar (a Boolean, True if the password contains at least one special character such as ! or #). Using OCR Exam Reference Language, write an IF statement that outputs "Valid" if usernameLength is at least 8 AND both hasDigit and hasSpecialChar are True, and otherwise outputs "Invalid".

(Total for Question 12 is 4 marks)

A student claims that NOT (A AND B) always produces the same output as (NOT A) OR (NOT B) for every combination of A and B.

(a)Complete the two truth tables below to test this claim.

A	B	A AND B	NOT (A AND B)
0	0
0	1
1	0
1	1

A	B	NOT A	NOT B	(NOT A) OR (NOT B)
0	0
0	1
1	0
1	1

(4)

(b)State whether the student's claim is correct, using evidence from your completed tables.(1)

(Total for Question 13 is 5 marks)

Complete the truth table below for the expression Output = (A OR B) AND NOT C.

A	B	C
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

(Total for Question 14 is 4 marks)

A programmer is designing a form-validation routine for a website that collects a user's age (which must be an integer between 5 and 100 inclusive) and a consent Boolean (True if the user has ticked a consent box).

Explain how Boolean logic operators (AND, OR, NOT) could be used to validate this data, including at least one example of a condition that combines more than one operator.

(Total for Question 15 is 4 marks)

A program is written using OCR Exam Reference Language:

count <- 0
total <- 0
FOR i <- 1 TO 5
    INPUT mark
    IF mark >= 40 AND mark <= 100 THEN
        total <- total + mark
        count <- count + 1
    ENDIF
NEXT i
IF count > 0 THEN
    average <- total / count
    OUTPUT average
ELSE
    OUTPUT "No valid marks"
ENDIF

The program is run with the five inputs, in order: 35, 72, 100, 45, 105.

(a)Complete a trace table for this program using the five inputs given.

i	mark	condition true?	total	count
1	35
2	72
3	100
4	45
5	105

(5)

(b)State the value output by the program.(1)

(Total for Question 16 is 6 marks)

A vending machine dispenses an item if a coin has been inserted (coinInserted) and the selected item is in stock (stockAvailable), or if an engineer's override switch (overrideSwitch) is on, which bypasses the coin and stock requirements.

(a)Write a Boolean expression for dispense in terms of coinInserted, stockAvailable and overrideSwitch.(2)

(b)The engineer adds a safety interlock: a doorOpen sensor must be False before the machine is allowed to dispense anything, even when the override switch is on. Rewrite the expression for dispense to include this new rule.(2)

(Total for Question 17 is 4 marks)

The truth table below shows the output of a Boolean expression using three inputs A, B and C.

A	B	C	Output
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

(a)Which expression produces this truth table?(1)

A) (A AND B) OR (B AND C) OR (A AND C)
B) A AND B AND C
C) A OR B OR C
D) NOT A AND B AND C

(b)Justify your answer using one row from the table.(1)

(Total for Question 18 is 2 marks)

A programmer wants an algorithm that repeatedly asks the user to input a PIN (a string) until the PIN entered is exactly 4 characters long AND every character in it is a digit. You may assume a function IsDigit(character) that returns True if the character passed to it is a digit and False otherwise. Characters in a string can be accessed using position numbers starting at 0, e.g. pin[0] is the first character.

Using OCR Exam Reference Language, write an algorithm that repeats the input process until a valid PIN is entered, then outputs "PIN accepted".

(Total for Question 19 is 6 marks)

Complete the truth table below for the expression Output = NOT A OR (B AND NOT C).

A	B	C
0	0	0
0	0	1
0	1	0
0	1	1
1	0	0
1	0	1
1	1	0
1	1	1

(a)Complete the Output column of the table.(4)

(b)State the number of the eight input combinations for which the expression is False.(1)

(Total for Question 20 is 5 marks)

Mark scheme · P2.4 Boolean Logic

Question 1

(a) B1 Both values given, oe (True and False, or 1 and 0)
(a) Answer: True and False (also accept 1 and 0)
(b) B1 Any valid question with a genuine yes/no answer, oe
(b) Answer: e.g. 'Is the front door locked?' (accept any sensible yes/no question)

Question 2

B1 NOT A = 1 when A = 0
B1 NOT A = 0 when A = 1
Answer: A=0 -> NOT A=1; A=1 -> NOT A=0

Question 3

(a) B1 Row 1 (0,0) -> 0
(a) B1 Row 2 (0,1) -> 0
(a) B1 Row 3 (1,0) -> 0
(a) B1 Row 4 (1,1) -> 1
(a) Answer: 0, 0, 0, 1
(b) B1 Row 1 (0,0) -> 0
(b) B1 Row 2 (0,1) -> 1
(b) B1 Row 3 (1,0) -> 1
(b) B1 Row 4 (1,1) -> 1
(b) Answer: 0, 1, 1, 1

Question 4

(a) B1 A) 0 selected
(a) Answer: A) 0
(b) B1 B) 1 selected
(b) Answer: B) 1

Question 5

(a) B1 A = D OR W, oe
(a) Answer: A = D OR W
(b) B1 A = D AND W, oe
(b) Answer: A = D AND W

Question 6

(a) B1 Row (0,0): NOT A=1, X=0
(a) B1 Row (0,1): NOT A=1, X=1
(a) B1 Row (1,0): NOT A=0, X=0
(a) B1 Row (1,1): NOT A=0, X=0
(a) Answer: X values in row order: 0, 1, 0, 0
(b) B1 1, cao
(b) Answer: 1

Question 7

(a) M1 tempLow AND windowClosed combined correctly, oe
(a) A1 Full expression: heatingOn = (tempLow AND windowClosed) OR override, oe, brackets not required
(a) Answer: heatingOn = (tempLow AND windowClosed) OR override
(b) B1 False (heating off), cao, ft from part (a) expression
(b) Answer: False (heating stays off)

Question 8

(a) B1 X = NOT P AND Q, oe
(a) Answer: X = NOT P AND Q
(b) B1 Row (0,0) -> 0
(b) B1 Row (0,1) -> 1
(b) B1 Row (1,0) -> 0
(b) B1 Row (1,1) -> 0
(b) Answer: X values in row order: 0, 1, 0, 0
(c) B1 P=0 and Q=1, cao, ft from part (b)
(c) Answer: P = 0, Q = 1

Question 9

(a) B1 NOT gate identified
(a) B1 OR gate identified
(a) Answer: A NOT gate and an OR gate
(b) B1 Input B is connected to the input of the NOT gate
(b) B1 The output of the NOT gate and input A are connected to the two inputs of the OR gate, producing output X
(b) Answer: B feeds into the NOT gate; the NOT gate's output and A both feed into the OR gate, which outputs X.

Question 10

B1 Correct Output for rows 1-2 (A=0): 0, 0
B1 Correct Output for rows 3-4 (A=0): 0, 0
B1 Correct Output for rows 5-6 (A=1): 0, 1
B1 Correct Output for rows 7-8 (A=1): 1, 1
Answer: Output column in row order: 0, 0, 0, 0, 0, 1, 1, 1

Question 11

(a) B1 "Entry allowed", cao
(a) Answer: Entry allowed
(b) B1 "Entry denied", cao
(b) Answer: Entry denied
(c) B1 Run 1: age=16, hasID=True, OUTPUT='Entry allowed'
(c) B1 Run 2: age=16, hasID=False, OUTPUT='Entry denied'
(c) B1 Both rows fully consistent with the code and with the answers to (a) and (b)
(c) Answer: Run 1: 16, True, Entry allowed. Run 2: 16, False, Entry denied.

Question 12

M1 Correct relational operator used for length, e.g. usernameLength ≥ 8
M1 All three conditions correctly combined using AND
M1 THEN branch outputs "Valid"
M1 ELSE branch outputs "Invalid", with ENDIF used to close the statement; any correct logic, ignore minor syntax errors

Answer:

IF usernameLength >= 8 AND hasDigit == True AND hasSpecialChar == True THEN
    OUTPUT "Valid"
ELSE
    OUTPUT "Invalid"
ENDIF

Question 13

(a) B1 First table: NOT(A AND B) column correct = 1, 1, 1, 0
(a) B1 Second table: NOT A and NOT B columns correct in all 4 rows
(a) B1 Second table: (NOT A) OR (NOT B) column correct = 1, 1, 1, 0
(a) B1 Both final output columns fully correct and matching row for row
(a) Answer: Both final columns read: 1, 1, 1, 0
(b) B1 Yes, the claim is correct because the output columns for NOT(A AND B) and (NOT A) OR (NOT B) are identical for all four rows, ft from part (a)
(b) Answer: Yes, correct; both expressions give the output pattern 1, 1, 1, 0 for all four input combinations.

Question 14

B1 Correct Output for rows 1-2: 0, 0
B1 Correct Output for rows 3-4: 1, 0
B1 Correct Output for rows 5-6: 1, 0
B1 Correct Output for rows 7-8: 1, 0
Answer: Output column in row order: 0, 0, 1, 0, 1, 0, 1, 0

Question 15

Level 1 (1-2): Basic or partial explanation of how a single Boolean operator could be used in validation. Little or no reference to combining conditions, or an example given without explanation.
Level 2 (3-4): Clear explanation of how AND, OR and/or NOT can be combined to validate both fields, supported by at least one correctly constructed combined condition (e.g. using AND to require the age to be in range and consent to be given).
Indicative content:
- AND can be used to require the age to be within range AND consent to be True at the same time, e.g. age >= 5 AND age <= 100 AND consent == True.
- The two boundary checks (age >= 5 and age <= 100) must both be True, which is exactly what AND tests for.
- OR could be used if there were multiple acceptable ways to satisfy a rule, e.g. accepting the record if consent == True OR an adult override flag is set.
- NOT could be used to invert a check, e.g. NOT consent to flag records where consent has not been given.
- A single combined condition reduces the need for multiple nested IF statements and makes the validation clearer and more efficient.

Question 16

(a) B1 i=1, mark=35: condition False, total=0, count=0
(a) B1 i=2, mark=72: condition True, total=72, count=1
(a) B1 i=3, mark=100: condition True, total=172, count=2
(a) B1 i=4, mark=45: condition True, total=217, count=3
(a) B1 i=5, mark=105: condition False, total=217, count=3
(a) Answer: Final values after the loop: total=217, count=3
(b) B1 average = 217/3, awrt 72.3, ft from part (a) totals
(b) Answer: awrt 72.3 (217/3 = 72.333...)

Question 17

(a) M1 coinInserted AND stockAvailable combined correctly, oe
(a) A1 Full expression: dispense = (coinInserted AND stockAvailable) OR overrideSwitch, oe
(a) Answer: dispense = (coinInserted AND stockAvailable) OR overrideSwitch
(b) M1 NOT doorOpen used as an additional condition
(b) A1 Full expression: dispense = NOT doorOpen AND ((coinInserted AND stockAvailable) OR overrideSwitch), oe, ft from part (a)
(b) Answer: dispense = NOT doorOpen AND ((coinInserted AND stockAvailable) OR overrideSwitch)

Question 18

(a) B1 A) selected, cao
(a) Answer: A
(b) B1 A valid row used correctly to support the choice, e.g. row A=1,B=1,C=0 gives Output=1 because A AND B = 1, matching option A but not options B, C or D
(b) Answer: e.g. row A=1, B=1, C=0 gives Output=1; in option A, A AND B=1 so the expression is 1, matching the table. Option B needs all three inputs to be 1 so would give 0 here, so B is wrong.

Question 19

1 mark A loop structure is used that repeats the input process (e.g. DO...UNTIL or WHILE...ENDWHILE)
1 mark The length of the entered PIN is correctly checked, e.g. LEN(pin) == 4 (or ≠ 4 to flag invalid)
2 marks Iteration (e.g. FOR i <- 0 TO 3) is used with IsDigit correctly applied to check that every character is a digit
1 mark The length check and the digit check are correctly combined so that the loop only ends once both conditions are satisfied
1 mark OUTPUT "PIN accepted" is produced once, after a valid PIN has been entered