Show that the eliminant of the equations \begin{align*} x+y+z &= 0 \\ \frac{x^2}{a} + \frac{y^2}{b} + \frac{z^2}{c} &= 0 \\ ayz + bzx + cxy &= 0 \end{align*} is \((b+c)(c+a)(a+b) = a^3+b^3+c^3+5abc\).
Show that if \(x^n + a_1 x^{n-1} + \dots + a_n = 0\), where the \(a\)'s are rational numbers, then any polynomial in \(x\) with rational coefficients can be expressed as a polynomial of degree not greater than \((n-1)\) with rational coefficients. Show that if \(x^4 + 4ax^3 + 6bx^2 + 4cx + d = 0\) and \(y=x^2+2ax\), then \(y\) satisfies a quadratic equation provided \(c=3ab-2a^3\) and that the values of \(y\) are real if \(c^2 \geq a^3d\).
\(A_1, A_2, \dots, A_n\) are \(n\) places in succession through which a road passes and \(P, Q\) are places on opposite sides of the road. From each of \(A_1, A_2, \dots, A_n\) a road passes to \(P\) and a road to \(Q\). Find the number of ways of going from \(P\) to \(Q\) by the roads, and also the number of ways of going from \(P\) to \(Q\) without going through \(A_r\), where \(r\) is not \(1\) or \(n\).
Find \(a, b, c, d\) so that the coefficient of \(x^n\) in the expansion of \[ \frac{a+bx+cx^2+dx^3}{(1-x)^4} \] may be \(n^3\). Find the coefficient of \(x^n\) in the expansion of \((1+\lambda x+x^2)^n\), where \(n\) is a positive integer; and if \(\lambda=2\cos\theta\), deduce that \[ c_0^2 + c_1^2 \cos 2\theta + \dots + c_n^2 \cos 2n\theta = n! \cos n\theta \left\{\frac{(2\cos\theta)^n}{n!} + \frac{(2\cos\theta)^{n-2}}{1! (n-2)!} + \frac{(2\cos\theta)^{n-4}}{2! (n-4)!} + \dots \right\}, \] where the indices of \(\cos\theta\) are positive numbers or zero, and \((1+x)^n = c_0+c_1x+\dots+c_nx^n\).
Two equilateral triangles \(ABC, ABD\) have a side \(AB\) common and their planes at right angles. Find (i) the acute angle between \(AC, AD\) and (ii) the distance between the mid-points of \(AC, BD\).
Find the sums of the infinite series
The roots of the quadratic equation \(az^2+2bz+c=0\), where \(a, b, c\) are real and \(ac>b^2\), are represented on an Argand diagram by points \(P, Q\). Prove that \(P\) and \(Q\) are equidistant from the origin, and that \(PQ\) is perpendicular to the axis of real numbers. Hence show that \(P\) and \(Q\) may be found by a geometrical construction which does not require the solution of the equation. Prove also that, if \(a', b', c'\) are real and \(a'c'>b'^2\), the points representing the roots of \(a'z^2+2b'z+c'=0\) lie on the circle through \(P, Q\) and the origin, if \(bc'=b'c\).
A rod \(AB\) moves so that \(A, B\) respectively lie on fixed lines \(OP, OQ\) inclined at an angle \(\alpha\). Prove that, if \(OA=x, OB=y\), and the area of \(OAB=u\), \[ \frac{du}{dx} = \frac{(y^2-x^2)\sin\alpha}{2(y-x\cos\alpha)}. \] Deduce that \(u\) is a maximum when \(x=y\).
Find the equation of the tangent at any point of the curve \(x=f(t), y=F(t)\). If \(Y\) is the foot of the perpendicular from the origin \(O\) on the tangent to the curve \(ay^2=x^3\) at any point \(P(at^2, at^3)\), show that the coordinates of \(Y\) are \(\dfrac{3at^4}{9t^2+4}\) and \(\dfrac{2at^3}{9t^2+1}\). Deduce that, if \(OY=p\) and the inclination of \(OY\) to the axis of \(x\) is \(\psi\), \[ 27p = 4a\cos\psi\cot^2\psi. \]
Explain what is meant by a point of inflexion on a plane curve, and prove that, if \(y=f(x)\) has a point of inflexion whose abscissa is \(x_0\), \(f''(x_0)=0\). The graph of a polynomial of the fourth degree in \(x\) touches the \(x\)-axis at \((a,0)\), and has a point of inflexion at \((-a,0)\). Prove that the graph passes through \((-2a,0)\), and that it has a second point of inflexion whose abscissa is \(a/2\).