1 Find an expansion of (confusion over indexes intended, 2017)
a. cos⁸θ in terms of cos 2nθ
b. cos 8θ in terms of cos²nθ
c. sin 7θ in terms of sinnθ
d. sin⁹θ in terms of sin nθ
2a) Show (prove, effectively) that the centre of mass of a triangular lamina lies on the intersection of its medians.
b) A triangular lamina of sides 8, 6,10 is viewed as having the y-axis along the shortest side and the x-axis along the middle length side. It is folded twice, each fold by a right angle and in opposing directions along parallel lines at x = 3 and x = 7. Find the new position of the centre of mass assuming that the folds make the centre have positive z.
c) Would this folded shape stay upright?
3 A transformation of the Argand diagram is given by z = (λw+1)/(w-1) for a>1 and real. Show that the circle |z|=1 remains a circle and describe its characteristics.
4 Justify your choice of distribution to model these defective components,
a. P(X=x)= 0.02, n = 100
b. P(X=x)= 0.15, n = 15
c. P(X=x)= 0.92, n = 20
d. P(X=x)= 0.01, n = 1000
e. P(X=x)= 0.24, n = 400
5 Find a 3x3 matrix A with eigenvalues -1, 2, 3 and corresponding eigenvectors (0,1,1), (1,0,1) and (1,1,0).
The first row first column entry of An is a. Show that 2a = 2n + 3n and find the least integer so that |a| > 250.
6 Find the eigenvalues and eigenvectors of the matrix A shown below. Hence find a non-singular matrix P and a diagonal matrix D so that A+A²+A³ = PDP-1 =A
7 Show that the centroid (centre of mass) of the region bounded by the x-axis, x=1, x=√3 and the line y = 1/(1+x2) is (6/π) ln 2
8 Two particles P&Q of masses m and 4m respectively lie at rest on a smooth horizontal table. One end of a light elastic spring, natural length l and modulus of elasticity 3mg, is attached to the particle Q. The other end of the spring is attached to the table at O, where OQ = l and PQO is a straight line. P is projected towards Q with speed u and collides with Q. The coefficient of restitution between the particles is 1/5. Show that, after the collision, the speed of P is u/25. Find also the speed with which Q begins to move.[4]
Show that the subsequent motion of Q, provided P and Q do not collide again, is simple harmonic. Find the period and amplitude of this motion.[5]
Find the time it would take from the instant of the collision to the spring being first at maximum extension, in the form k√(l / g), giving k in an exact form. Hence discover whether there will be a second collision between P & Q.[5]
[based on 1995/4/s/Q4].
9 A battleship and a cruiser (types of warship) are initially 16km apart with the battleship on a bearing of 035° from the cruiser. The battleship steams at 14 kph at 151° and the cruiser at 17kph at 050°. The guns on the battleship have a range of 6km (up to 6km). Find (a) the least distance between the ships (how close the two ships get to each other) and (b) the length of time for which the battleship can shell the cruiser. Don’t assume these are modern warships.
10 A golfer wishes to hit a ball from his current tee to a green at the same level; the near and far edges of the green are 160 and 185 metres away. The golfer hits the ball at 49m/s (g is 9.8m/s/s) so find the possible angles of projection, assuming no spin or air resistance.
Reconsider the problem if the green is 3m higher than the tee.
Answers:
2 (2 1/48, 2, 31/48) (really). Easily.
6 lambda = -1,2,8; eigen vectors (4, -9, 8), (5, -6, 4) and (1, 0, 2) (from memory)
8 to add; start is u/25 and 6u/25; end is that obviously they meet again since u/25 is +ve.
9 about 5.5km and 12 minutes. Scale drawing recommended to believe answers.
10 to add; all angles come in pairs around 20-25º and 62-67º
