Assertion :Points $P (- sin (β - α), - cos β), Q (cos (β - α), sin β)$ and $R (cos (β - α + θ), sin (β - θ))$ , where $β = \frac{π}{4} + \frac{α}{2}$ are non-collinear. Reason: Three given points are non-collinear if they form a triangle of non-zero area.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but the Reason is correct

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Solution

The correct option is D Assertion is incorrect but the Reason is correct
Reason is wrong as three points are collinear when area of triangle formed is zero.
Area of triangle $P Q R$
$= \frac{1}{2} ∣ ∣ ∣ ∣ ∣ \begin{matrix} - sin (β - α) & - cos β & 1 cos (β - α) & sin β & 1 cos (β - α + θ) & sin (β - θ) & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} - sin (\frac{π}{4} + \frac{α}{2} - α) & - cos (\frac{π}{4} + \frac{α}{2}) & 1 cos (\frac{π}{4} + \frac{α}{2} - α) & sin (\frac{π}{4} + \frac{α}{2}) & 1 cos (\frac{π}{4} + \frac{α}{2} - α + θ) & sin (\frac{π}{4} + \frac{α}{2} - θ) & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} - sin (\frac{π}{4} - \frac{α}{2}) & - cos (\frac{π}{4} + \frac{α}{2}) & 1 cos (\frac{π}{4} - \frac{α}{2}) & sin (\frac{π}{4} + \frac{α}{2}) & 1 cos (\frac{π}{4} - \frac{α}{2} + θ) & sin (\frac{π}{4} + \frac{α}{2} - θ) & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
Applying $R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - R_{1}$
$= \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} - sin (\frac{π}{4} - \frac{α}{2}) & - cos (\frac{π}{4} + \frac{α}{2}) & 1 cos (\frac{π}{4} - \frac{α}{2}) + sin (\frac{π}{4} - \frac{α}{2}) & sin (\frac{π}{4} + \frac{α}{2}) + cos (\frac{π}{4} + \frac{α}{2}) & 0 cos (\frac{π}{4} - \frac{α}{2} + θ) + sin (\frac{π}{4} - \frac{α}{2}) & sin (\frac{π}{4} + \frac{α}{2} - θ) + cos (\frac{π}{4} + \frac{α}{2}) & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{1}{2} (cos (\frac{π}{4} - \frac{α}{2}) + sin (\frac{π}{4} - \frac{α}{2})) (cos (\frac{π}{4} - \frac{α}{2} + θ) + sin (\frac{π}{4} - \frac{α}{2})) \times$
$∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} - sin (\frac{π}{4} - \frac{α}{2}) & - cos (\frac{π}{4} + \frac{α}{2}) & 1 1 & 1 & 0 1 & 1 & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ = 0$
Hence point $P, Q$ and $R$ are collinear
Therefore assertion is wrong.

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Assertion :Points P(−sin(β−α),−cosβ),Q(cos(β−α),sinβ) and R(cos(β−α+θ),sin(β−θ)), where β=π4+α2 are non-collinear. Reason: Three given points are non-collinear if they form a triangle of non-zero area.

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