Question

The point, which does not lie in the half plane, \[2x + 3y - 12 \leqslant 0\] is
A.(1, 2)
B.(2, 1)
C.(2, 3)
D.(-3, 2)

Answer 1

Hint: Here in this question we are asked to find the point that does not lie in the half plane. We know that the given inequality is in linear form with two variables, also this can be represented graphically. To find the point we first substitute the given points in the inequality and then examine whether the points satisfy the given inequality.

Complete step-by-step answer:
Considering the inequality
\[ \Rightarrow \] \[2x + 3y - 12 \leqslant 0\] --------- (1)
On substituting the first point (1, 2) in the inequality (1)
\[ \Rightarrow 2\left( 1 \right) + 3\left( 2 \right) - 12 \leqslant 0\]
On simplifying the above inequality we get
\[ \Rightarrow 2 + 6 - 12 \leqslant 0\]
\[ \Rightarrow - 4 \leqslant 0\] --------- (2)
The given point (1, 2) satisfies the inequality and thus lie in the given half plane.
On substituting the second point (2, 1) in the inequality (1)
\[ \Rightarrow 2\left( 2 \right) + 3\left( 1 \right) - 12 \leqslant 0\]
On simplifying the above inequality we get
\[ \Rightarrow 4 + 3 - 12 \leqslant 0\]
\[ \Rightarrow - 5 \leqslant 0\] ---------- (3)
The given point (2, 1) satisfies the inequality and thus lie in the given half plane.
On substituting the third point (2, 3) in the inequality (1)
\[ \Rightarrow 2\left( { - 3} \right) + 3\left( 2 \right) - 12 \leqslant 0\]
On simplifying the above inequality we get
\[ \Rightarrow 4 + 9 - 12 \leqslant 0\]
\[ \Rightarrow 1 \leqslant 0\] --------- (4)
The given point (2, 3) satisfies the inequality and it does not lie in the given half plane.
On substituting the fourth point (-3, 2) in the inequality (1)
\[ \Rightarrow 2\left( { - 3} \right) + 3\left( 2 \right) - 12 \leqslant 0\]
On simplifying the above inequality we get
\[ \Rightarrow - 6 + 6 - 12 \leqslant 0\]
\[ \Rightarrow - 12 \leqslant 0\] --------- (5)
The given point (1, 2) satisfies the inequality and thus lie in the given half plane.
From equations (2), (3), (4), (5) we can clearly note that inequality not satisfying the inequality condition thus the point (2, 3) does not lie in the half plane. We can also use the graphical representation for verification.

Therefore, plotting the graph makes it clearer that the point that does not lie in the given plane is (2, 3) the correct option is ‘c’.
So, the correct answer is “Option C”.

Note: Remember the given inequality is a closed half-plane as the inequality is ‘\[ \leqslant \]’, This plane will be having a solid boundary line with the shaded region. The inequality with two variables, will have a line dividing the plane into two half planes. A non-vertical line will divide the plane into lower and upper half planes, whereas a vertical line will divide the plane in the left and right half plane.