The Value of 2sin(12°) − sin(72°)
Evaluate 2sin(12°)−sin(72°) by splitting 72° as 60°+12°, then recombining into a single sine of a difference angle using sin18°.
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Sum of Cubes of All Roots of x⁴−3x³−2x²+3x+1=0
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.
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Line Through A(4,3), Slope>1, Meets x−y−2=0 at B — Find B’s Line
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.
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Common Tangents to 4(x²+y²)=9 and y²=4x Meet at Q — Find l/e² of an Ellipse
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.
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Circle Through A(2,−1), B(3,4), Center on (x−5)²+(y−1)²=13/2, Find r²
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.
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A∩B for |(z+1)/(z−1)|<1 and arg((z−1)/(z+1))=2π/3
Let A: |(z+1)/(z−1)|<1 and B: arg((z−1)/(z+1))=2π/3 in the complex plane. Find A∩B by converting both conditions into geometric loci.
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The Remainder When 2021^2023 Is Divided by 7
Find the remainder when (2021)^2023 is divided by 7 — a modular arithmetic problem solved by reducing the base first, then finding a repeating power cycle.
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Let A=[0 -2,2 0], M=ΣA²ᵏ, N=ΣA²ᵏ⁻¹ (k=1 to 10): Find MN²
Find MN² for a 2×2 matrix A where M and N are sums of its even and odd powers. Solved by spotting that A² is a scalar multiple of the identity.
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Let A, B=A−I, ω=(√3i−1)/2: Count n With Aⁿ+(ωB)ⁿ=A+B
Given A^n + (ωB)^n = A + B for ω a complex cube root of unity, how many n in 1..100 satisfy it? JEE Main 2022 matrix and complex numbers problem.
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If sin²10°sin20°sin40°sin50°sin70°=α−1/16sin10°, Find 16+1/α
A five-term trigonometric product from JEE Main 2022, reduced step by step using complementary angles and the sin a cos a identity.