Find all the solutions of the equations \begin{align*} x+2y+4z &= 12, \\ xy+2xz+4yz &= 22, \\ xyz &= 6. \end{align*}
If \(f(x)\) is a polynomial and \(f'(x)\) its derivative, state, without proof, what you can deduce about the roots of the equation \(f(x)=0\) from a knowledge of the roots of the equation \(f'(x)=0\). \newline Prove that the equation \[ 1-x+\frac{x^2}{2} - \frac{x^3}{3} + \dots + (-1)^n \frac{x^n}{n} = 0 \] has one real root if \(n\) is odd, and no real root if \(n\) is even. \newline Hence or otherwise find the number of real roots of \[ 1 - \frac{x}{1 \cdot 2} + \frac{x^2}{2 \cdot 3} - \dots + (-1)^n \frac{x^n}{n(n+1)} = 0. \]
Establish necessary and sufficient conditions that \(ax^2+2bx+c\) shall be positive for all real values of \(x\). \newline If the \(a\)'s, \(b\)'s and \(c\)'s are real numbers such that \begin{gather*} a_1 > 0, \quad a_2 > 0, \quad a_3 > 0, \\ a_1c_1-b_1^2 > 0, \quad a_2c_2-b_2^2 > 0, \quad a_3c_3-b_3^2 > 0, \end{gather*} prove that \[ (a_1+a_2+a_3)(c_1+c_2+c_3) - (b_1+b_2+b_3)^2 > 0. \]
Prove that, if \(a, b, c\) are the sides of a triangle of area \(\Delta\), \[ \begin{vmatrix} (b-c)^2 & b^2 & c^2 & 1 \\ a^2 & (c-a)^2 & c^2 & 1 \\ a^2 & b^2 & (a-b)^2 & 1 \\ 1 & 1 & 1 & 0 \end{vmatrix} = -16\Delta^2. \]
Sum to \(N\) terms, and where possible to infinity, the series whose \(n\)th terms are \[ \text{(i)} \ (n+2)n, \quad \text{(ii)} \ (n+2)x^n, \quad \text{(iii)} \ (n+2)\cos n\theta. \]
Lines drawn from the vertices \(A, B, C\) of a triangle through a variable point \(O\) within the triangle meet the opposite sides at \(D, E, F\) respectively.
A variable circle through two fixed points \(A\) and \(B\) cuts a fixed circle at \(P\) and \(Q\). Prove that the ratio of the areas of the triangles \(APQ, BPQ\) is constant.
A fixed point \(A\) and a variable point \(P\) are taken on a given sphere. \(AP\) is produced to \(Q\) so that \(PQ\) is of constant length. Prove that the plane through \(Q\) perpendicular to \(PQ\) touches a fixed sphere.
Prove that the locus \[ x=a_1 t^2 + 2b_1 t, \quad y=a_2 t^2 + 2b_2 t, \] where \(t\) is a parameter, is, in general, a parabola. \newline Find the condition that the line \[ y-y_0 = m(x-x_0) \] may touch the parabola, and prove that the directrix is \[ a_1x + a_2y + b_1^2 + b_2^2 = 0. \]
The lines joining a variable point \(P\) on the ellipse \(x^2/a^2+y^2/b^2=1\) to the fixed points \((ka,0)\) and \((k'a,0)\) cut the ellipse again at \(Q\) and \(R\). Prove that, in general, the line \(QR\) envelops the conic \[ \frac{x^2}{a^2} + \frac{(1-kk')^2}{(1-k^2)(1-k'^2)} \frac{y^2}{b^2} = 1. \] Examine the case in which \(kk'=1\).