# Write true (T) or false (F) for the following statements:(i) 392 is a perfect cube.(ii) 8640 is not a perfect cube.(iii) No cube can end with exactly two zeros.(iv) There is no perfect cube which ends in 4.(v) For an integer $a, a^3$ is always greater than $a^2$.(vi)If $a$ and $b$ are integers such that $a^{2}>b^{2}$, then $a^{3}>b^{3} .$.(vii) If $a$ divides $b$, then $a^{3}$ divides $b^{3}$.(viii) If $a^{2}$ ends in 9, then $a^{3}$ ends in $7$.(ix) If $a^{2}$ ends in 5 , then $a^{3}$ ends in 25 .(x) If $a^{2}$ ends in an even number of zeros, then $a^{3}$ ends in an odd number of zeros.

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To find:

We need to check whether the given statements are true or false.

Solution:

(i) Prime factorisation of 392 is,

$392=2\times2\times2\times7\times7$

$=2^3\times7^2$

Grouping the factors in triplets of equal factors, we see that two factors $7 \times 7$ are left.

Therefore, 392 is not a perfect cube.

The given statement is false.

(ii) Prime factorisation of 8640 is,

$8640=2\times2\times2\times2\times2\times2\times3\times3\times3\times5$

$=2^3\times2^3\times3^3\times5$

Grouping the factors in triplets of equal factors, we see that 5 is left.

Therefore, 8640 is not a perfect cube.

(iii) For every zero at the end in a number which when cubed gives three zeros at the end.

The given statement is true.

(iv) $4^3=4\times4\times4$ $=64$

Therefore,

The given statement is false.

(v) For $\frac{1}{2}$

$(\frac{1}{2})^3=\frac{1}{8}$

$(\frac{1}{2})^2=\frac{1}{4}$

$\frac{1}{8}<\frac{1}{4}$

If $n$ is a proper fraction, then the given statement is not possible.

Therefore,

The given statement is false.

(vi) For example,

$(-4)^2 = 16$ and $(-3)^2=9$

$16 > 9$

$(-4)^3 =-64$ and $(-3)^3 = -27$ $-64 < -27$

Therefore,

The given statement is false.

(vii) Let $a$ divides $b$, this implies,

$\frac{b}{a}= k$

$b=ak$

$\frac{b^3}{a^3} = \frac{(ak)^3}{a^3}$

$= \frac{a^3k^3}{a^3}$

$= k^3$

This is true for each value of $b$ and $a$.

Therefore,

The given statement is true.

(viii) For $a = 7$,

$7^2 = 49$

$7^3 = 343$

Therefore,

The given statement is false.

(ix) For $a = 15$,

$(15)^2 = 225$

$(15)^3 = 3375$

Therefore,

The given statement is false.

(x) For $a = 100$,

$(100)^2 = 10000$

$(100)^3 = 1000000$

Therefore,

The given statement is false.

Updated on 10-Oct-2022 12:46:26