Let $A = \{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\}$ where $P = \begin{bmatrix} 1 & 2 & 1…

Let $A = \{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\}$ where $P = \begin{bmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{bmatrix}$, then the set $A$.
  1. Is a singleton.
  2. Is an empty set.
  3. Contains more than two elements
  4. Contains exactly two elements

Solution

∵  P=0

∴ System of equations 

x+2 y+z=0 

2x-3y+4z=0 

x+9y-z=0 has infinitely many solutions.

Let z=k∈R then x=-11k7, y=2k7, z=k 

Also x2+y2+z2=1 

∴k=±7174 

∴ Two solutions are only possible.

Hence, set A contains exactly two elements.

Asked in: JEE Main 2020 (02 Sep Shift 2)

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