

A1. Let C be the curve y^{2} = x^{3} (where x takes all nonnegative real values). Let O be the origin, and A be the point where the gradient is 1. Find the length of the curve from O to A.


A2. Let C be the curve y = x^{3} (where x takes all real values). The tangent at A meets the curve again at B. Prove that the gradient at B is 4 times the gradient at A.


A3. The roots of x^{3} + a x^{2} + b x + c = 0 are α, β and γ. Find the cubic whose roots are α^{3}, β^{3}, γ^{3}.


A4. Given 4 lines in Euclidean 3space:
L_{1}: x = 1, y = 0;
L_{2}: y = 1, z = 0;
L_{3}: x = 0, z = 1;
L_{4}: x = y, y = 6z.
Find the equations of the two lines which both meet all of the L_{i}.


A5. Do either (1) or (2)
(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).


A6. Do either (1) or (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 (v^{2}  2gh)^{1/2} of the point directly below the point of impact.


A7. Do either (1) or (2):
(1) Let C_{a} be the curve (y  a^{2})^{2} = x^{2}(a^{2}  x^{2}). 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} x^{i}, prove that (1 + hkx)(1  hkx)^{1}(1  h^{2}x)^{1}(1  k^{2}x)^{1} = ∑_{i≥0} a_{i}^{2} x^{i}.


B1. The points P(a,b) and Q(0,c) are on the curve y/c = cosh (x/c). The line through Q parallel to the normal at P cuts the xaxis at R. Prove that QR = b.


B2. Evaluate ∫_{1}^{3} ( (x  1)(3  x) )^{1/2} dx and ∫_{1}^{∞} (e^{x+1} + e^{3x})^{1} dx.


B3. Given a_{n} = (n^{2} + 1) 3^{n}, find a recurrence relation a_{n} + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0. Hence evaluate ∑_{n≥0} a_{n} x^{n}.


B4. The axis of a parabola is its axis of symmetry and its vertex is its point of intersection with its axis. Find: the equation of the parabola which touches y = 0 at (1,0) and x = 0 at (0,2); the equation of its axis; and its vertex.


B5. Do either (1) or (2):
(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} [x^{2}] 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!]


B6. Do either (1) or (2):
(1) f is continuous on the closed interval [a, b] and twice differentiable on the open interval (a, b). Given x_{0} ∈ (a, b), prove that we can find ξ ∈ (a, b) such that ( (f(x_{0})  f(a))/(x_{0}  a)  (f(b)  f(a))/(b  a) )/(x_{0}  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?


B7. Do either (1) or (2):
(1) Let a_{i} = ∑_{n=0}^{∞} x^{3n+i}/(3n+i)! Prove that a_{0}^{3} + a_{1}^{3} + a_{2}^{3}  3 a_{0}a_{1}a_{2} = 1.
(2) Let O be the origin, λ a positive real number, C be the conic ax^{2} + by^{2} + cx + dy + e = 0, and C_{λ} the conic ax^{2} + by^{2} + λcx + λdy + λ^{2}e = 0. Given a point P and a nonzero 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.

