If $[x]$ denotes the greatest integer $\leq x$, then the system of linear equations $[\sin \theta] x+[-\cos \theta] y=0$ $[\cot \theta] x+y=0$

(A) have infinitely many solutions if $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right)$ and has a unique solution if $\theta \in\left(\pi, \frac{7 \pi}{6}\right)$

(B) have infinitely many solutions if $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$

(C) has a unique solution if $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right)$ and have infinitely many solutions if $\theta \in\left(\pi, \frac{7 \pi}{6}\right)$

(D) has a unique solution if $\theta \in\left(\frac{\pi}{2}, \frac{2 \pi}{3}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Determinants

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[x]

denotes the greatest integer

\leq x

, then the system of linear equations

[sin θ] x + [- cos θ] y = 0

[cot θ] x + y = 0

(A)have infinitely many solutions if $θ \in (\frac{π}{2}, \frac{2 π}{3})$ and has a unique solution if $θ \in (π, \frac{7 π}{6})$ ✓ Correct
(B)have infinitely many solutions if $θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$
(C)has a unique solution if $θ \in (\frac{π}{2}, \frac{2 π}{3})$ and have infinitely many solutions if $θ \in (π, \frac{7 π}{6})$
(D)has a unique solution if $θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$

Answer: (A)

Solution

(\frac{π}{2}, \frac{2 π}{3})

sin θ \in (\frac{3}{2}, 1)

[sin θ] = 0

;

cos θ \in (- \frac{1}{2}, 0)

[- cos θ] = 0

; and

cot θ

is a small negative number so

[cot θ] = - 1

. The system collapses to

0 = 0

together with

- x + y = 0

, a single condition, giving infinitely many solutions. On

(π, \frac{7 π}{6})

[sin θ] = - 1

[- cos θ] = 0

and

[cot θ] \geq 1

, so

- x = 0

and

x + y = 0

, forcing

x = y = 0

, a unique solution. So infinitely many on the first interval and a unique solution on the second.

JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Single correct option

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JEE Main · 2019 · 12 Apr 2019 · Shift 2 · Determinants

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