Sketch the curve \(y = 3x^5-5ax^3\) for positive and negative values of the real number \(a\), and hence determine the number and signs of the real roots of the equation \[ 3x^5 - 5ax^3 + b = 0, \] where \(b\) is real, in the various cases that may arise.
Give (without proof) a rule for multiplying two determinants of \(n\) rows and columns. By multiplying the determinants \[ \begin{vmatrix} x_1^2 + y_1^2 & -2x_1 & -2y_1 & 1 & 0 \\ x_2^2 + y_2^2 & -2x_2 & -2y_2 & 1 & 0 \\ x_3^2 + y_3^2 & -2x_3 & -2y_3 & 1 & 0 \\ x_4^2 + y_4^2 & -2x_4 & -2y_4 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{vmatrix}, \quad \begin{vmatrix} 1 & 1 & 1 & 1 & 0 \\ x_1 & x_2 & x_3 & x_4 & 0 \\ y_1 & y_2 & y_3 & y_4 & 0 \\ x_1^2+y_1^2 & x_2^2+y_2^2 & x_3^2+y_3^2 & x_4^2+y_4^2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{vmatrix} \] or otherwise, find an identical relation connecting the squares of the distances between four points in a plane, and shew that it can be reduced to the form \[ \begin{vmatrix} 2 (14)^2 & (14)^2 + (24)^2 - (12)^2 & (14)^2 + (34)^2 - (13)^2 \\ (14)^2 + (24)^2 - (12)^2 & 2 (24)^2 & (24)^2 + (34)^2 - (23)^2 \\ (14)^2 + (34)^2 - (13)^2 & (24)^2 + (34)^2 - (23)^2 & 2 (34)^2 \end{vmatrix} = 0, \] where (12) denotes the distance between the points 1, 2, etc.
(i) \(f(x)\) is a function of \(x\) (defined for \(x>0\)) whose derivative is \(1/x\). Without using the properties of the logarithmic function, prove that \(f(x) + f(y/x)\) is independent of \(x\). Deduce that, if \(f(x)\) vanishes when \(x=1\), then \(f(x)+f(y) = f(xy)\), for all positive values of \(x\) and \(y\). (ii) \(y\) is defined as an implicit function of \(x\) by the equation \[ x\sqrt{1-y^4} + y\sqrt{1-x^4} = C(1+x^2y^2), \] where \(C\) is a constant. Prove that \(\frac{dy}{dx} = -\frac{\sqrt{1-y^4}}{\sqrt{1-x^4}}\). Hence shew that, if \(f(x) = \int_0^x \frac{dt}{\sqrt{1-t^4}}\), \[ f(x)+f(y) = f\left(\frac{x\sqrt{1-y^4} + y\sqrt{1-x^4}}{1+x^2y^2}\right). \]
If \(y^2 = p(x-\alpha)^2+q(x-\beta)^2\), \(X=r(x-\alpha)^2+s(x-\beta)^2\), where \(\alpha, \beta\) are unequal, prove that the substitution \(\xi = (x-\alpha)/(x-\beta)\) reduces the integral \(\int \frac{dx}{X^{n+1}y}\) to the form \[ k \int \frac{(1-\xi)^{2n+1}}{(r\xi^2+s)^{n+1}\eta} d\xi, \] where \(\eta^2 = p\xi^2+q\), and \(k\) is a constant (to be determined). Prove that this last integral can be expressed as the sum of integrals of the types \[ \text{(i)} \int\frac{d\xi}{(r\xi^2+s)^{m+1}\eta} \quad \text{and} \quad \text{(ii)} \int\frac{d\eta}{(r\eta^2)^{m+1}}, \] and that (i) can be found when \(m=0\) by the substitution \(u = \eta/\xi\).
Prove the theorem of Pappus that, if \(A_1, A_2, A_3\) are three points on a straight line and \(B_1, B_2, B_3\) are three points on a coplanar straight line, then the three points of intersection \((A_2B_3, A_3B_2)\), \((A_3B_1, A_1B_3)\), \((A_1B_2, A_2B_1)\) are collinear. State the (plane) dual of this theorem. \(U, V, L, M, N\) are five given points (no three of which are collinear) in a plane. Prove that the dual of the theorem of Pappus, applied to the sets of lines \[ a_1 = UL, a_2 = UM, a_3 = UN; \quad b_1 = VL, b_2 = VM, b_3 = VN \] defines a point \(O\) such that \(OU, OV\) are the tangents at \(U, V\) to the conic through the points \(U, V, L, M, N\).
The equation of a conic in general homogeneous coordinates is \[ S \equiv ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \] Prove that the line joining the points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\) meets the conic in two points \((x_1+\lambda x_2, y_1+\lambda y_2, z_1+\lambda z_2)\), \((x_1+\mu x_2, y_1+\mu y_2, z_1+\mu z_2)\), where \(\lambda, \mu\) are the roots of an equation in \(\theta\) which can be expressed in the form \[ S_{22}\theta^2 + 2S_{12}\theta+S_{11}=0. \] Deduce the equation of the tangent at the point \((x_1, y_1, z_1)\) of the conic and of the polar of \((x_1, y_1, z_1)\) with respect to the conic, in the form \(S_1=0\), where \(S_1\) is found by replacing \((x_2, y_2, z_2)\) by \((x,y,z)\) in \(S_{12}\).
A thin rectangular window of height \(a\) is smoothly hinged along its upper horizontal edge. The centre of gravity of the window is at its geometrical centre and the weight is \(W\). The window can be held open by a thin light stay smoothly pivoted at a point on the lower horizontal edge of the window. The stay is of length \(a\) and has four equally spaced holes of which the first is distant \(\frac{1}{4}a\) from the end attached to the window and the fourth is distant \(\frac{1}{4}a\) from the opposite end. When the window is shut the stay rests along the bottom edge of the window and the first hole is over a smooth peg on the lower horizontal edge of the fixed window-frame. Find the thrust in the stay when the window is fully open with the peg through the fourth hole of the stay.
A boy stands on level ground in front of a high vertical wall and projects a small smooth ball in such a way that it bounces from the ground and wall in succession and returns to its starting point, distant \(d\) in front of the wall and \(h\) above the ground. The coefficient of restitution at the ground is \(e\) and at the wall is \(e'\). Shew that, if \(V\) is the vertical component of the velocity of projection, the horizontal component must be \[ \frac{(1+e')gd}{e'[(1+e)(V^2+2gh)^{1/2} \pm \{e^2V^2 - (1-e^2)2gh\}^{1/2} \mp V]}. \] Explain fully the significance of the alternative signs, and state whether the result is affected if the boy projects the ball so as to hit the wall first and the ground afterwards, giving reasons.
A smooth thin tube \(ABCDE\) is composed of a pair of horizontal straight sections \(AB, DE\) and a pair of equal curved sections \(BC\) and \(CD\) each in the form of a quadrant of a circle. The tube is mounted in a vertical plane with \(DE\) at a higher level than \(AB\). The tube is used for conveying messages by means of small containers, which are sucked pneumatically through the tube in the direction from \(A\) to \(E\). Each container may be regarded as a heavy particle of mass \(M\), and the suction is equivalent to a constant force \(F\) acting on the particle. Containers may be released from rest at any point on \(AB\) and are required to reach \(E\). Shew that \(F\) must exceed the value \(Mg\sin\theta\), where \(\theta\) is the root of the equation \(\theta = \tan\frac{1}{2}\theta\) lying between \(\frac{1}{2}\pi\) and \(\pi\). Explain why the result is independent of the height of \(DE\) above \(AB\).
A game of shuffle-board is played with a number of equal uniform circular discs of diameter \(d\) which slide, with a flat face downwards, over a horizontal deck, the coefficient of friction being \(\mu\). The deck is marked out with thin lines in a square lattice of edge \(a (>d)\), and four of the squares in line are lettered A, B, C, D. A pair of discs are placed, one at the centre of square A and the other at the centre of square C. A player wishes to project the disc in square A along the line of centres of the two discs in such a way that there is finally a disc wholly within square C and another wholly within square D. Shew that the disc originally within square C is finally within square D provided that the velocity of projection lies between \[ \left[2\mu g(2a-d) + \frac{2}{(1+e)^2}(a+d)\right]^\frac{1}{2} \] and \[ \left[2\mu g(2a-d) + \frac{2}{(1+e)^2}(3a-d)\right]^\frac{1}{2}, \] where \(e\) is the coefficient of restitution. Shew further that, if the above conditions are satisfied, the disc originally within square A automatically ends up within square C, if \(d<\frac{1}{2}a\) and \(e>2-\sqrt{3}\).