Video: AQA GCSE Mathematics Higher Tier Pack 2 β€’ Paper 2 β€’ Question 12

Tick true or false for each of the following statements. Give reasons for each of your answers. (a) When 𝑛 is a positive integer, the value of 72𝑛 will always be divisible by 8. (b) When 𝑦 > π‘₯, the value of (π‘₯ βˆ’ 𝑦)Β² will always be positive. (c) When 𝑧 is positive, the value of βˆšπ‘§ will always be less than 𝑧.

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Video Transcript

Tick true or false for each of the following statements. Give reasons for each of your answers. Part a) When 𝑛 is a positive integer, the value of 72𝑛 will always be divisible by eight.

The positive integers are one, two, three, and so on. The number 72 can be written as eight multiplied by nine. This means that 72𝑛 can be written as eight multiplied by nine 𝑛.

Our aim in this question is to prove whether 72𝑛 will always be divisible by eight. 72𝑛 divided by eight is equal to eight multiplied by nine 𝑛 divided by eight. The eights here cancel. Therefore, 72𝑛 divided by eight is equal to nine 𝑛.

As 𝑛 is an integer, we can also say that nine 𝑛 will be an integer. Multiplying any integer or whole number by nine will also give us a whole number or integer answer. We can, therefore, say that 72𝑛 is a multiple of eight and is, therefore, always divisible by eight.

The statement β€œwhen 𝑛 is a positive integer, the value of 72𝑛 will always be divisible by eight” is true.

Part b) When 𝑦 is greater than π‘₯, the value of π‘₯ minus 𝑦 all squared will always be positive.

π‘₯ minus 𝑦 all squared is a square number. And square numbers are always positive. For example, four squared is equal to 16 and negative four squared is also equal to 16 as a negative multiplied by a negative is a positive number.

In this question, as 𝑦 is greater than π‘₯, the number inside the bracket will be a negative number. Despite this, when we square the negative number, we’ll get a positive answer.

This means that the statement β€œwhen 𝑦 is greater than π‘₯, the value of π‘₯ minus 𝑦 or squared will always be positive” is true.

Part c) When 𝑧 is positive, the value of the square root of 𝑧 will always be less than 𝑧.

Initially, this statement appears to be true as the square root of four is equal to two; the square root of nine is equal to three. So the value of the square root of 𝑧 is less than 𝑧.

However, when we consider 𝑧 equals one, then the square root of 𝑧 is also equal to one as the square root of one equals one. In this case, 𝑧 is equal to the square root of 𝑧.

In a specific case, when 𝑧 is one, then the square root of 𝑧 is not less than 𝑧. This means that the statement β€œwhen 𝑧 is positive, the value of the square root of 𝑧 will always be less than 𝑧” is false.

When trying to disprove a statement, we only need to give one example where the statement is false.

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