JEE Main 2022 PYQ: A and A’ lie on y=2 such that segments AB and A’B (B=(2,3)) subtend angle π/4 at the origin. Find the distance AA’ using the angle-between-two-lines formula.
JEE Main 2022 PYQ: evaluate 16 sin20° sin40° sin80° using the identity sin(60°−θ) sinθ sin(60°+θ) = (sin3θ)/4.
The normal at point P on the parabola y²=6x passes through (5,−8). If the tangent at P meets the directrix at Q, find the ordinate of Q using the parametric form and solving a cubic for the parameter.
For A=[1,−2,α;α,2,−1] and B=[2,α;−1,2;4,−5], find the absolute value of the sum of all complex α for which det(AB)=0, by computing AB directly.
Given two sides of a triangle and its orthocenter, find the distance from the origin to the centroid by locating the altitudes and the third vertex.
Evaluate 2sin(12°)−sin(72°) by splitting 72° as 60°+12°, then recombining into a single sine of a difference angle using sin18°.
Find α³+β³+γ³+δ³ for the roots of x⁴−3x³−2x²+3x+1=0 by factoring the quartic into two quadratics, avoiding solving for the roots directly.
A line through A(4,3) with slope greater than 1 meets x−y−2=0 at B, with AB=√29/3. Find which of four given lines B lies on.
The common tangents to 4(x²+y²)=9 and y²=4x meet at Q. An ellipse has semi-minor axis OQ and semi-major axis 6. Find l/e² for this ellipse.
Circle C passes through A(2,−1) and B(3,4), AB not a diameter, and its centre lies on (x−5)²+(y−1)²=13/2. Find r² using the perpendicular-bisector property.