Question

Write true (T) or false (F) for the following statements:
\[\left( i \right)\] $\sqrt{0.9}=0.3$
\[\left( ii \right)\] If \[a\]is a natural number, then $\sqrt{a}$ is a rational number.
\[\left( iii \right)\] If \[a\] is negative, then ${{a}^{2}}$ is also negative.
\[\left( iv \right)\] If $p$and $q$are perfect squares, then $\sqrt{\dfrac{p}{q}}$ is a rational number.
\[\left( v \right)\] The square root if a prime number may be obtained approximately, but never exactly.

Answer 1

Hint: For solving the given questions we have to read each and every question twice and then we will solve it accordingly in true and false many times we do mistake for solving this question we must know about the rational number A rational number can be made by dividing two integers A integer is a number with no fractional part.

Complete step-by-step answer:
So, let’s start with our first question
(i) $\sqrt{0.9}=0.3$
Squaring both sides, we get,
 $0.9={{\left( {{0.3}^{{}}} \right)}^{2}}$
But we know that \[0.3\times 0.3=0.09\] .
Since the square of a number with one decimal will have two decimal places that is $0.9\ne 0.09={{\left( 0.3 \right)}^{2}}$. Thus, the given statement is false.
(ii) Let’s consider $\alpha \to $ Natural number
If we take the square root $\sqrt{\alpha }\to $ Rational number
Since the square root of a natural number is not a rational number. Thus we get that the statement given is false.
(iii) Let’s take $\alpha \to $ Negative number
Say,
$\alpha =\left( -1 \right)$
Then
$\begin{align}
  & {{\alpha }^{2}}=\left( -1 \right)\times \left( -1 \right) \\
 & \therefore {{\alpha }^{2}}=1 \\
\end{align}$
Since the square of any negative number will be a positive number. Thus we get that the given statement is also false.
(iv) If p and q are the perfect squares then $\sqrt{\dfrac{p}{q}}$ it will be a rational number. Thus, we can say that the given statement is true.
(v) The square root of a prime number may be obtained approximately, but never exactly. Thus, we can say that the given statement is true.

Note: For solving such types of questions we have to be very careful about the judgement. Sometimes we think that the answer is true but actually it is false. This happens many times so be careful about that, read it twice before solving the question. And also, remember this will help you solve many algebraic Problems . Here $a$ is a positive number.
$\begin{align}
  & -a\times -a={{a}^{2}} \\
 & -a\times a=-{{a}^{2}} \\
 & a\times -a=-{{a}^{2}} \\
 & a\times a={{a}^{2}} \\
\end{align}$