A construction job has 0.65 probability of a strike, with different completion probabilities depending on whether there’s a strike. Find the overall probability of on-time completion using total probability.
The locus |(z−2)/(z−3)|=2 in the complex plane is a circle with center (α,β) and radius γ. Find 3(α+β+γ) by converting to Cartesian form.
Circle C passes through (4,0) and touches x²+y²+4x−6y=12 externally at (1,−1). Find C’s radius using the tangent-point-lies-on-the-line-of-centres property.
Two circles in the first quadrant with radii r₁, r₂ touch both coordinate axes, each cutting a 2-unit intercept on x+y=2. Find r₁²+r₂²−r₁r₂.
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.
If α, β are roots of x²−6x−2=0 and an=α^n−β^n, find (a10−2a8)/(2a9) using Newton’s recurrence formula for power sums.
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.
Prove that the sum of the interior angles of any quadrilateral is 360 degrees, by drawing a diagonal to split it into two triangles.
The perpendicular distance from (β,0,β) to the line x/1=(y−1)/0=(z+1)/(−1) is √(3/2). Find β using the cross-product distance formula.
Two positive numbers p, q satisfy p+q=2 and p^4+q^4=272. Find pq using power-sum identities, then write the quadratic having p, q as roots.