Explain clearly and concisely how and why a boy seated on a swing is able to increase the amplitude of successive ``swings'' by his own efforts. Any physical principles assumed in your explanation should be stated precisely both in words and in mathematical terms.
Solve for \(x, y, z\) the three simultaneous equations \[ \begin{cases} ax+3y+2z=b, \\ 5x+4y+3z=1, \\ x+2y+z=b^2, \end{cases} \] explaining in particular the different cases obtained for \(a=3\) with varying values of \(b\).
The cubic equation \(x^3+px+q=0\) has roots \(\alpha, \beta, \gamma\). Find the cubic equation whose roots are \((\beta-\gamma)^2, (\gamma-\alpha)^2, (\alpha-\beta)^2\). Hence or otherwise deduce the condition for the cubic equation \[ a_0x^3+3a_1x^2+3a_2x+a_3=0 \] to have a pair of equal roots in the form \[ (a_0^2a_3+2a_1^3-3a_0a_1a_2)^2+4(a_0a_2-a_1^2)^3=0. \]
Prove that the total number of ways in which a distinct set of three non-zero positive integers can be chosen so that the sum is a given odd positive integer \(p\) is the integer nearest \(p^2/12\).
Two men, \(A\) and \(B\), play a gambling game by tossing together four apparently similar unbiassed coins. If three or four heads are uppermost \(B\) wins, if three or four tails \(A\) wins, and if two of each neither wins. \(A\) and \(B\) start each with \(N\) counters, and after each single game the winner receives a counter from the loser; the first to collect all \(2N\) counters wins the complete game. \(B\), unperceived by \(A\), has arranged that one of the coins is ``double headed'', the other three coins being normal. Show that as a result of this stratagem \(B\)'s chance of winning any single game is four times that of \(A\). Prove further that if when \(A\) has \(r\) counters his chances of winning the complete game is \(U_r\), then \(U_{r+2}-5U_{r+1}+4U_r=0\) for \(0< r< 2N-2\). Hence, or otherwise, deduce that at the start \(A\)'s chance of winning the complete game is \(1/(4^N+1)\). \subsubsection*{SECTION B}
By means of the substitution \[ (1+e\cos\theta)(1-e\cos\phi)=1-e^2 \quad(e<1) \] transform the integral \[ I_n = \int (1+e\cos\theta)^{-n} d\theta \] to an integral involving \(\phi\). Evaluate in terms of \(\phi\) the integrals \(I_0, I_1,\) and \(I_2\).
Prove, under conditions to be stated, that \[ f(b)-f(a)=(b-a)f'(x), \] where \(x\) is some value between \(a\) and \(b\). Illustrate this result geometrically. Extend the formula to obtain a measure of the departure of a curve from its tangent at a given point.
The graph in rectangular coordinates of a polynomial of the fourth degree in \(x\) is found to touch the \(x\)-axis at \((-a,0)\) and have a point of inflection at \((a,0)\). Show that the curve must also pass through the point \((2a,0)\) and possesses another point of inflection on the line \(x=-a/2\).
Solution: Let \(p(x) = c(x+a)^2(x-a)(x-p)\), since it touches \((-a,0)\) and also the point \((a,0)\). Since there is a point of inflection at \((a,0)\) the second derivative is \(0\) there, ie \begin{align*} && p(x) &= c(x^2-a^2)(x^2+(a-p)x-ap) \\ &&&=c(x^4+(a-p)x^3-(ap+a^2)x^2-a^2(a-p)x+a^3p)\\ && p''(x) &= c(12x^2+6(a-p)x-2(ap+a^2))\\ && 0 &= p''(a) \\ &&&= c(12a^2+6(a-p)a-2(ap+a^2)) \\ &&&= c(16a^2-8ap) \\ &&&= 8ac(2a-p) \\ \Rightarrow && p = 2a \\ \Rightarrow && p(2a) &=0 \quad \text{ie curve passes through }(2a,0) \\ && p''(x) &= c(12x^2-6ax-2(2a^2+a^2))\\ &&&= c(12x^2-6ax-6a^2) \\ &&&= 6c(2x^2-ax-a^2)\\ &&&= 6c(2x-a)(x-a) \\ \end{align*} Therefore there is a point of inflection when \(x = -a/2\) since it is clear \(p''\) changes sign at this point.
Find for what ranges of \(x\) the function \(\dfrac{\log x}{x}\) increases as \(x\) increases, and decreases as \(x\) increases. Hence show that if \(n\) is a given positive number and \(x\) is a positive real variable, the equation \(x=n^x\) has two roots, one root, or no root according to the value of \(n\), and state the critical values of \(n\) concerned.
A right circular cone has unit volume. Show that its total surface area, including the base, cannot be less than \(2(9\pi)^{\frac{1}{3}}\). If such a cone has unit total surface area, what would be its maximum volume?