Consider three points P( α, β), Q( α, β) and R(0,0), where 0

Correct Answer is : P,Q,RP, Q, RP,Q,R are non-collinear.

P lies on the line segment R Q.

Q lies on the line segment P R .

R lies on the line segment P Q .

Correct Answer is : P,Q,RP, Q, RP,Q,R are non-collinear.

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WBJEE 2025 Math Paper

Question : 27 of 75

Marks: +1, -0

Consider three points

P(\cos α, \sin β), Q(\sin α, \cos β)

R(0,0)

0 < α, β < \frac{π}{4}

. Then

Solution:

We check collinearity by comparing the slopes of the three pairs:

Slope of PR

m_{PR} = \frac{y_P - y_R}{x_P - x_R} = \frac{\sin β - 0}{\cos α - 0} = \frac{\sin β}{\cos α} > 0 \;\; (.

since

0 < β < \frac{π}{4})

Slope of RQ

m_{RQ} = \frac{y_Q - y_R}{x_Q - x_R} = \frac{\cos β - 0}{\sin α - 0} = \frac{\cos β}{\sin α} > 0 \;\; \left(\text{ since } 0 < α < \frac{π}{4}\right) .

Slope of PQ

m_{PQ} = \frac{\cos β - \sin β}{\sin α - \cos α} = \frac{\cos β - \sin β}{-(\cos α - \sin α)} = -\frac{\cos β - \sin β}{\cos α - \sin α} < 0

because in

\left(0, \frac{π}{4}\right)

we have

\cos t > \sin t

, so both

\cos β - \sin β

and

\cos α - \sin α

are positive.

Since two of the slopes are positive and one is negative, no two of the three points lie on the same straight line. Hence

Option D:

P, Q, R

are non-collinear.

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