L₁:   x = 1, y = 0;
L₂:   y = 1, z = 0;
L₃:   x = 0, z = 1;
L₄:   x = y, y = -6z.

Find the equations of the two lines which both meet all of the L_i.

(1)   x and y are functions of t. Solve x' = x + y - 3, y' = -2x + 3y + 1, given that x(0) = y(0) = 0.

(2)   A weightless rod is hinged at O so that it can rotate without friction in a vertical plane. A mass m is attached to the end of the rod A, which is balanced vertically above O. At time t = 0, the rod moves away from the vertical with negligible initial angular velocity. Prove that the mass first reaches the position under O at t = √(OA/g) ln (1 + √2).

(1)   A circle radius r rolls around the inside of a circle radius 3r, so that a point on its circumference traces out a curvilinear triangle. Find the area inside this figure.

(2)   A frictionless shell is fired from the ground with speed v at an unknown angle to the vertical. It hits a plane at a height h. Show that the gun must be sited within a radius v/g (v² - 2gh)^1/2 of the point directly below the point of impact.

(1)   Let C_a be the curve (y - a²)² = x²(a² - x²). Find the curve which touches all C_a for a > 0. Sketch the solution and at least two of the C_a.

(2)   Given that (1 - hx)^-1(1 - kx)^-1 = ∑_i≥0  a_i xⁱ, prove that (1 + hkx)(1 - hkx)^-1(1 - h²x)^-1(1 - k²x)^-1 = ∑_i≥0  a_i² xⁱ.

(1)   Prove that ∫₁^k [x] f '(x) dx = [k] f(k) - ∑₁^[k] f(n), where k > 1, and [z] denotes the greatest integer ≤ z. Find a similar expression for:   ∫₁^k [x²] f '(x) dx.

(2)   A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from 1,000 ft/s to 900 ft/s over 1,200 ft. Find the time taken to the nearest 0.01 s. [No calculators or log tables allowed!]

(1)   f is continuous on the closed interval [a, b] and twice differentiable on the open interval (a, b). Given x₀ ∈ (a, b), prove that we can find ξ ∈ (a, b) such that ( (f(x₀) - f(a))/(x₀ - a) - (f(b) - f(a))/(b - a) )/(x₀ - b) = f ''(ξ)/2.

(2)   AB and CD are identical uniform rods, each with mass m and length 2a. They are placed a distance b apart, so that ABCD is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?

(1)   Let a_i = ∑_n=0^∞ x³ⁿ⁺ⁱ/(3n+i)!   Prove that a₀³ + a₁³ + a₂³ - 3 a₀a₁a₂ = 1.

(2)   Let O be the origin, λ a positive real number, C be the conic   ax² + by² + cx + dy + e = 0, and C_λ the conic   ax² + by² + λcx + λdy + λ²e = 0. Given a point P and a non-zero real number k, define the transformation D(P,k) as follows. Take coordinates (x',y') with P as the origin. Then D(P,k) takes (x',y') to (kx',ky'). Show that D(O,λ) and D(A,-λ) both take C into C_λ, where A is the point (-cλ/(a(1 + λ)), -dλ/(b(1 + λ)) ). Comment on the case λ = 1.