Three equal smooth spheres \(A\), \(B\), \(C\) are at rest on a table with their centres at three successive vertices of a square of side \(a\). The first sphere \(A\) is projected along the table so as to collide first with \(B\) and then with \(C\). If the impacts are perfectly elastic, show that the direction of projection must lie within an angle \[\tan^{-1} \frac{2b^2}{a^2 - b\sqrt{(a^2 - 4b^2)}},\] where \(b\) is the diameter of a sphere.
\(\Delta_n (-\infty < n < \infty)\) is a sequence of triangles, the vertices of \(\Delta_{n+1}\) being the feet of the altitudes of \(\Delta_n\). If \(\Delta_0\) is not equilateral show that
For each real value of \(y\) the number of real values of \(x\) which satisfy the equation $$x^4 - 8x^3 + 22x^2 - 24x + 7 + 9y^2 - 6y^3 = 0$$ is denoted by \(n(y)\). Illustrate graphically the relation between \(y\) and \(n(y)\). [Precise numerical solutions are not required.]
Solve the equations \begin{align} x + y^3 + z^3 &= 0,\\ x^3 + y + z^3 &= 0,\\ x^3 + y^3 + z &= 0, \end{align} given that no two of \(x, y, z\) are equal.
If \(n\) is a positive integer and \(p\) a prime number, \(\alpha_p(n)\) denotes the greatest integer \(k\) such that \(p^k\) divides \(n\). If \(n\) is written in the form \(n = \sum_{r=0}^N a_r p^r \quad (0 \leq a_r \leq p-1),\) show that \(\alpha_p(n!) = \frac{n - \sum_{r=0}^N a_r}{p-1}.\)
Five points \(A\), \(B\), \(C\), \(D\), \(E\) are given in a plane; \(BD\) meets \(CE\) in \(P\). A variable triangle \(XYZ\) is drawn such that \(X\), \(Y\) lie on \(AB\), \(AC\) respectively, and \(YZ\), \(ZX\), \(XY\) pass through \(D\), \(E\), \(P\) respectively. Show that the locus of \(Z\) is a conic. Determine which of the points \(A\), \(B\), \(C\), \(D\), \(E\), \(P\) lie on this conic.
Two circles of radius \(a\) intersect in \(A\), \(B\), the length of the common chord \(AB\) being equal to \(a\). The figure formed by the interiors of the two circles is rotated about the line through \(B\) perpendicular to \(AB\). Determine the volume of the solid of revolution so formed.
A pile of \(n\) bricks is in equilibrium, each brick resting horizontally on the one and their long sides lying in the same vertical north-south planes. The bricks are uniform rectangular blocks of the same material, of length \(a\) and height \(b\). The sun is due south at an elevation \(\alpha\). Find the minimum length of the shadow of the pile (in the north-south direction) in the following two cases:
The two ends of a cricket pitch are denoted by \(A\), \(B\) and are at a distance \(l\) apart. The bowler bowls from \(A\), the ball leaving his hand at a height \(a\) from the ground at an angle \(\theta\) above the horizontal. The ball bounces at a point which divides \(AB\) in the ratio \(1 : \alpha\), and then hits the stumps at \(B\) at a height \(b\). The ground is assumed to be smooth and the coefficient of restitution between the ball and the ground is \(e\), where \(e > \sqrt{b/a}\). Show that, if \(\alpha > 0\), one value of \(\alpha\) lies between \(0\) and \(e\) while the other lies between \(e\) and \(2e\).
A heavy particle is attached at one end of a long string. The string is wound round a rough circular cylinder of radius \(a\) whose axis is horizontal, and the weight hangs freely at a height \(c\) below the axis of the cylinder. The particle is given a horizontal velocity \(u\), in the direction away from and perpendicular to the vertical plane through the axis of the cylinder. If \(a/c\) is small, and if \(n\) is an integer such that \(0 \leq 2n \leq \frac{u^2 - 5gc}{3\pi ga} < 2n + 1,\) show that the string first slackens after rotating through an angle of approximately \((2n + 1)\pi\).