The locus of the midpoint of the segment joining the focus of y²=4ax to a moving point on the parabola is another parabola. Find its directrix using the parametric form and midpoint formula.
Circle C1: x²+y²−2(x+y)+1=0 and C2 have centre (−1,0), radius 2. Their common chord meets the y-axis at P. Find the square of the distance from P to the centre of C1, using the radical axis.
Find the sum of the squares of all roots of x²+|2x−3|−4=0, by splitting into two cases at the modulus’s sign-change point and checking each candidate root against its case’s domain.
m arithmetic means are inserted between 1 and 31 so the ratio of the 7th and (m−1)th means is 5:9. Find m by expressing both means via the common difference D=30/(m+1).
Find the maximum area of a rectangle inscribed in the ellipse x²/a²+y²/b²=1, using the parametric point (a cosθ, b sinθ) and maximising over θ.
A girl throws a die: on 5 or 6 she tosses a coin 3 times, otherwise once. Given she got exactly one head, find the probability she threw 1, 2, 3, or 4, using Bayes’ theorem.
Find how many words can be formed from the letters of DAUGHTER such that all the vowels are not together, by subtracting the all-vowels-together count from the total.
Find the equation of a common tangent to the circle x²+y²−6x=0 and the parabola y²=4x, by writing the parabola’s tangent in slope form and applying the circle’s tangent-distance condition.
In an arithmetic progression, if m times the mth term equals n times the nth term (m≠n), prove that the (m+n)th term must equal zero.
Given x, y, z are unit vectors in 3D space, find the minimum value of |x+y|²+|y+z|²+|z+x|² by relating the sum to |x+y+z|² and using that a squared magnitude can’t be negative.