Question

pls answer this question with explanation​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\: - 1 \leqslant x < 0$

So, by definition of Greatest Integer function,

$\red{\rm :\longmapsto\:[x] = - 1}$

Also,

$\rm :\longmapsto\:0 \leqslant y < 1$

So,

$\red{\rm :\longmapsto\:[y] = 0}$

Also,

$\rm :\longmapsto\:1 \leqslant z < 2$

So,

$\red{\rm :\longmapsto\:[z] = 1}$

Now, Consider

$\rm :\longmapsto\:\rm \: =\: \begin{gathered}\sf \left | \begin{array}{ccc}1 + [x]&[y]&[z]\\ \: [x]&1 + [y]& [z]\\ \: [x]&[y]&[z] + 1\end{array}\right | \end{gathered}$

On substituting the values of [x], [y] and [z], we get

$\rm \: = \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}0&0&1\\ - 1&1& 1\\ - 1& 0& 2\end{array}\right | \end{gathered}$

On expanding along Row 1, we get

$\rm \:  =  \:  \: 1 \times (0 - ( - 1))$

$\rm \:  =  \:  \: 1 \times 1$

$\rm \:  =  \:  \: 1$

$\rm \:  =  \:  \: [z]$

• Hence, Option (c) is correct.

Additional Information :-

1. The determinant value remains unaltered if rows or columns are interchanged.

2. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

3. The determinant value is 0, of its two rows or columns are identical.

4. The determinant value is 0, if every element of any row or column is 0.

5. The determinant value remains unaltered if rows or columns are added or subtracted.