A line L through P(2,−1,3) is perpendicular to two given lines. Find where L meets the yz-plane and the distance PQ, using the cross product for L’s direction.
Evaluate the sum 1²·15C1+2²·15C2+…+15²·15C15 using the standard identity for Σr²·nCr, then find m+n+k from its prime factorisation.
Roots α, β of 2x²+cosθ·x−1=0 depend on θ∈(0,2π). Find 16(M+m) where m, M are the minimum and maximum values of α⁴+β⁴ over all θ.
Find the equation of the chord of the ellipse x²/25+y²/16=1 whose midpoint is (3,1), using the T=S₁ chord-of-midpoint formula.
If α, β are roots of 2z²−3z−2i=0, find 16·Re(E)·Im(E) where E=(α^19+β^19+α^11+β^11)/(α^15+β^15), using a Newton’s-identity shortcut.
Solve the differential equation y=(x−y dx/dy)sin(x/y) with x(1)=π/2, using the substitution v=x/y to reduce it to a separable equation.
A circle x²+y²−αx+βy+γ=0 touches the x-axis at (a,0) and cuts a y-axis intercept of length b, lying below the x-axis. Find the ordered pair (2a, b²).
An AP has first term 3, and the sum of its first 4 terms equals 1/5 of the sum of the next 4 terms. Find the sum of the first 20 terms.
A 2×2 matrix has entries from {0,1}. Let X be its determinant. Find the variance of X by listing all 16 matrices and their determinant probabilities.
A 4×4 board has 16 squares. Two are chosen at random. Find the probability they share no side, by counting adjacent pairs directly.