GATE 2018 IN Q.28: solution and answer

GATE 2018 · INQ.28 · 2 marks · MCQ

Consider the following system of linear equations:

3 x + 2 k y k x + 6 y = - 2 = 2

Here $x$ and $y$ are the unknowns and $k$ is a real constant. The value of $k$ for which there are infinite number of solutions is

(A)
$3$
(B)
$1$
(C)
$- 3$
(D)
$- 6$

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Answer and solution

ANSWER

$k = - 3$ , option (C). At this value the two equations are the same line, so the system has infinitely many solutions.

Official answer: CSource: Official answer key, GATE 2018 IN (IIT Guwahati)

Write the system as $A X = B$ with $A = [3 k 2 k 6]$ and $B = [- 2 2]$ . For infinitely many solutions we need $ρ (A) = ρ ([A ∣ B]) < 2$ , where 2 is the number of unknowns.
For the rank of $A$ to be below 2 we need $det A = 0$ : $3 \cdot 6 - (2 k) (k) = 18 - 2 k^{2} = 0 \Rightarrow k^{2} = 9 \Rightarrow k = \pm 3.$
Check $k = 3$ : the equations become $3 x + 6 y = - 2$ and $3 x + 6 y = 2$ . The left sides are the same but the right sides differ, so $ρ (A) = 1$ while $ρ ([A ∣ B]) = 2$ . The system is inconsistent and has no solution.
Check $k = - 3$ : the equations become $3 x - 6 y = - 2$ and $- 3 x + 6 y = 2$ . Multiplying the second by $- 1$ gives $3 x - 6 y = - 2$ , which is the first equation again. So $ρ (A) = ρ ([A ∣ B]) = 1 < 2$ .
Rank 1 with two unknowns leaves one free variable, so the solutions are $x = \frac{6 y - 2}{3}$ for every real $y$ . That is infinitely many solutions.
Only $k = - 3$ gives infinitely many solutions, which is option (C).

KEY IDEA

A $2 \times 2$ system has infinitely many solutions only when $ρ (A) = ρ ([A ∣ B]) < 2$ . Put another way, the coefficient ratios must all be equal: $\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$ . Here that reads $\frac{3}{k} = \frac{2 k}{6} = \frac{- 2}{2} = - 1$ , and both equalities give $k = - 3$ .

WATCH OUT FOR

Stopping at $det A = 0$ and choosing $k = 3$ (option A). A zero determinant only rules out a unique solution. At $k = 3$ the two lines are parallel and distinct ( $3 x + 6 y = - 2$ against $3 x + 6 y = 2$ ), so the system has no solution. You have to check that the augmented matrix has the same rank before calling the solutions infinite.

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GATE 2018 IN Q.28: solution and answer

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