Question 2

[1 mark] Simple familiar

Topic guidesPremiumTopic guide 7 · Completing a truth table from a circuit Topic guide 7 · Worked example: 2021 Question 2

Game planPremium6.1 Completing a truth table from a printed circuit

The question

A logic circuit with four inputs, Q, P, R and S, and six gates. Q and P enter an AND gate. Its output branches: one branch feeds the upper input of an OR gate whose lower input is R, and the other branch feeds the upper input of a second AND gate. The OR gate output feeds the lower input of the second AND gate. R also branches down into a NOT gate. The second AND gate output and the NOT gate output feed a second OR gate. That OR output and input S feed an XOR gate, whose output is the output of the circuit.

When the output from the logic circuit is 1, the input to the circuit is

Not quite. With P = Q = 1 the first AND gives 1, that 1 survives the first OR and the second AND, and the second OR outputs 1. With S = 1 the XOR sees two 1s and gives 0. In 2021, 9.5% of Queensland students chose (A).

That is it. Step through the hints below to see why. In 2021, 55.1% of Queensland students chose (B).

Not quite. (C) makes every gate before the XOR output 1, which is the answer you reach if you read the last gate as an AND or an OR. The XOR needs its two inputs to differ, and here both are 1. In 2021, 24.5% of Queensland students chose (C).

Not quite. The top half of the circuit does give 0 here, because P AND Q is 0. But R = 0 makes NOT R = 1, which switches the second OR on, and the XOR then sees two 1s. In 2021, 10.0% of Queensland students chose (D).

Signals are lettered J, K, M and N so you can talk about them. Orange wires carry 1. On a phone, scroll the circuit sideways.

Load an option into the circuit

Or set the inputs yourself (or click the boxes on the circuit)

Check all four options, one column at a time

Click a cell to cycle it through ?, 0 and 1. Click the inputs of a row to send that option to the circuit.

PQRS

This question gives you the output and asks for the input, so work backwards from the XOR. Every option has S = 1.

An XOR gives 1 only when its inputs are different. With S = 1, the other XOR input, N from the second OR, has to be 0.

$$\text{Out}=1,\ S=1\quad\Rightarrow\quad N=0$$

N comes from an OR gate, and an OR gives 0 only when both of its inputs are 0. So M = 0 and NOT R = 0.

$$\text{NOT }R=0\quad\Rightarrow\quad R=1$$

That rules out (C) and (D), which both have R = 0.

M comes from the second AND, which takes J = P AND Q and K = J OR R. With R = 1, K is 1, so M is whatever J is. For M = 0 you need J = 0.

$$J=P\text{ AND }Q=0$$

So P and Q cannot both be 1. That rules out (A) and leaves (B), with P = 0, Q = 1, R = 1 and S = 1. Load it into the circuit and watch the output go to 1.

$$\begin{aligned}\text{Out}&=1,\ S=1\Rightarrow\ N=0\Rightarrow\ M\\&=0\text{ and }R=1\Rightarrow\ P\text{ AND }Q=0\end{aligned}$$

Only (B) has R = 1 with P and Q not both 1. Checked forward, (B) gives J = 0, K = 1, M = 0, NOT R = 0, N = 0 and an output of 0 XOR 1 = 1.

(B)

How Queensland students answered in 2021

(A)

9.5%

(B)

55.1%

(C)

24.5%

(D)

10.0%

The most common wrong answer was (C), 24.5%. It drives every gate before the XOR to 1, which only gives an output of 1 if the last gate is read as an AND or an OR. One more thing worth seeing: M equals J in every one of the 16 rows, so the first OR and the second AND never change the result.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and any figures are from the 2021 QCAA Engineering external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and have been adapted, except where a third-party source is named. Tangent Tuition is not affiliated with the QCAA.

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