Q26 / 57 · JEE Advanced 2021 (Paper 1) · Mathematics
Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations
$$<br>\begin{array}{c}<br>x+2 y+3 z=\alpha \\<br>4 x+5 y+6 z=\beta \\<br>7 x+8 y+9 z=\gamma-1<br>\end{array}<br>$$
is consistent. Let $|M|$ represent the determinant of the matrix
$$<br>M=\left[\begin{array}{ccc}<br>\alpha & 2 & \gamma \\<br>\beta & 1 & 0 \\<br>-1 & 0 & 1<br>\end{array}\right]<br>$$
Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
The value of is
$$<br>\begin{array}{c}<br>x+2 y+3 z=\alpha \\<br>4 x+5 y+6 z=\beta \\<br>7 x+8 y+9 z=\gamma-1<br>\end{array}<br>$$
is consistent. Let $|M|$ represent the determinant of the matrix
$$<br>M=\left[\begin{array}{ccc}<br>\alpha & 2 & \gamma \\<br>\beta & 1 & 0 \\<br>-1 & 0 & 1<br>\end{array}\right]<br>$$
Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
The value of is
Correct answer: 1
Solution
The system of linear equations is consistent it means it has unique solution or infinite solutions
Here,
Hence, there are infinitely many solutions.
If equations have infinitely many solutions, then the equations are linearly connected, i.e.,
Also,
Now,
Hence, the correct answer is 1.