Question

If $P$ and $Q$ are the sum and product respectively of all integral values of $x$ satisfying the equation $|3\left[x\right]-4x|=4$, then
(where [.] denotes represents greatest integer function )

• A. $P=0$
• B. $P=8$
• C. $Q=-16$
• D. $Q=-9$

Answer 1

Answer: (A), (C)

The correct options are
A $P=0$
C $Q=-16$

$\Rightarrow$ $|3\left[x\right]-4x|=4$

$\Rightarrow$ $|3\left[x\right]-4\left[x\right]-4\left\{x\right\}|=4$

$\Rightarrow$ $|-\left[x\right]-4\left\{x\right\}|=4$

Put $\left[x\right]=0\Rightarrow$ $|-4\left\{x\right\}|=4\Rightarrow$ $\left\{x\right\}=1$

Put $\left[x\right]=1\Rightarrow$ $|-1-4\left\{x\right\}|=4$ $\Rightarrow$ $\left\{x\right\}=\frac{3}{4}$

Put $\left[x\right]=2\Rightarrow$ $|-2-4\left\{x\right\}|=4$ $\Rightarrow$ $\left\{x\right\}=\frac{1}{2}$

Put $\left[x\right]=3\Rightarrow$ $|-3-4\left\{x\right\}|=4$ $\Rightarrow$ $\left\{x\right\}=\frac{1}{4}$

Put $\left[x\right]=4\Rightarrow$ $|-4-4\left\{x\right\}|=4$ $\Rightarrow$ $\left\{x\right\}=0$

Put $\left[x\right]=-4\Rightarrow$ $|4-4\left\{x\right\}|=4$ $\Rightarrow$ $\left\{x\right\}=0$

So here, we can see we get $2$ integral values which is $4$ and $-4$

$\Rightarrow$ Sum of integral values $\left(P\right)=4+(-4)=0$

$\Rightarrow$ Product of integral values $\left(Q\right)=4\times (-4)=16$