Question

Consider the following linear equations
$$
\begin{aligned}
& a x+b y+c z=0 \
& b x+c y+a z=0 \
& c x+a y+b z=0
\end{aligned}
$$
Match the conditions/expressions in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

| Column I | Column II |
| --- | --- |
| (A) $a+b+c \neq 0$ and $a^{2}+b^{2}+c^{2}=ab+bc+ca$ | (p) the equations represent planes meeting only at a single point. |
| (B) $a+b+c=0$ and $a^{2}+b^{2}+c^{2} \neq ab+bc+ca$ | (q) the equations represent the line $x=y=z$. |
| (C) $a+b+c \neq 0$ and $a^{2}+b^{2}+c^{2} \neq ab+bc+ca$ | (r) the equations represent identical planes. |
| (D) $a+b+c=0$ and $a^{2}+b^{2}+c^{2}=ab+bc+ca$ | (s) the equations represent the whole of the three dimensional space. |

Accepted Answer

The determinant is . **(A)** and implies , so and the equations represent identical planes → **(r)**. **(B)** and implies with infinitely many solutions. Solving gives → **(q)**. **(C)** and implies , so unique solution → **(p)**. **(D)** and implies , representing all of 3D space → **(s)**.

JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Determinants

Solution

More JEE Advanced questions on Determinants

Column I	Column II
(A) $a + b + c \neq = 0$ and $a^{2} + b^{2} + c^{2} = ab + b c + c a$	(p) the equations represent planes meeting only at a single point.
(B) $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} \neq = ab + b c + c a$	(q) the equations represent the line $x = y = z$ .
(C) $a + b + c \neq = 0$ and $a^{2} + b^{2} + c^{2} \neq = ab + b c + c a$	(r) the equations represent identical planes.
(D) $a + b + c = 0$ and $a^{2} + b^{2} + c^{2} = ab + b c + c a$	(s) the equations represent the whole of the three dimensional space.