Roots of polynomials

Showing 1-15 of 15 problems
2025 Paper 3 Q6
D: 1500.0 B: 1500.0

  1. Let \(a\), \(b\) and \(c\) be three non-zero complex numbers with the properties \(a + b + c = 0\) and \(a^2 + b^2 + c^2 = 0\). Show that \(a\), \(b\) and \(c\) cannot all be real. Show further that \(a\), \(b\) and \(c\) all have the same modulus.
  2. Show that it is not possible to find three non-zero complex numbers \(a\), \(b\) and \(c\) with the properties \(a + b + c = 0\) and \(a^3 + b^3 + c^3 = 0\).
  3. Show that if any four non-zero complex numbers \(a\), \(b\), \(c\) and \(d\) have the properties \(a + b + c + d = 0\) and \(a^3 + b^3 + c^3 + d^3 = 0\), then at least two of them must have the same modulus.
  4. Show, by taking \(c = 1\), \(d = -2\) and \(e = 3\) that it is possible to find five real numbers \(a\), \(b\), \(c\), \(d\) and \(e\) with distinct magnitudes and with the properties \(a + b + c + d + e = 0\) and \(a^3 + b^3 + c^3 + d^3 + e^3 = 0\).

Show Solution
  1. If \(a,b,c\) were all real then \(a^2+b^2+c^2 = 0 \Rightarrow a,b,c = 0\) but they are non-zero. Therefore they cannot all be real. Since \((a+b+c)^2 = 0\) we must have \(ab+bc+ca = 0\). Therefore \(a,b,c\) must satisfy \(x^3 -abc = 0 \Rightarrow\) they all have the same modulus, since they are all cube roots of the same number.
  2. Notice that \(a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab-bc-ca) \Rightarrow abc = 0\) but therefore they cannot all be non-zero.
  3. Suppose \(a+b+c+d = 0\) then note that \(\displaystyle a^2+b^2+c^2+d^2 = (a+b+c+d)^2 - 2\sum_{sym} ab\) and \(\displaystyle a^3+b^3+c^3+d^3 = (a+b+c+d)^3 - 3(a+b+c+d)(ab+ac+ad+bc+bd+cd) + 3(abc+abd+acd+bcd) \Rightarrow abc+abd+acd+bcd = 0\). Therefore \(a,b,c,d\) are roots of a polynomial of the form \(x^4 -kx^2 + l = 0\), but this means they must come in pairs with the same modulus.
  4. Suppose \(c = 1, d = -2, e = 3\) so \(c+d+e = 2\) and \(c^3 + d^3 + e^3 = 1 - 8 + 27 = 20\), so we need to find \(a,b\) satisfying \(a+b = -2, a^2+b^2 = -20\), ie \(4 = (a+b)^2 = -20 + 2ab \Rightarrow ab = 12\), so we need the roots of \(x^2 +2x + 12= 0\) which clearly have different modulus.
2019 Paper 3 Q4
D: 1500.0 B: 1500.0

The \(n\)th degree polynomial P\((x)\) is said to be reflexive if:

  1. P\((x)\) is of the form \(x^n - a_1x^{n-1} + a_2x^{n-2} - \cdots + (-1)^na_n\) where \(n \geq 1\);
  2. \(a_1, a_2, \ldots, a_n\) are real;
  3. the \(n\) (not necessarily distinct) roots of the equation P\((x) = 0\) are \(a_1, a_2, \ldots, a_n\).
  1. Find all reflexive polynomials of degree less than or equal to 3.
  2. For a reflexive polynomial with \(n > 3\), show that $$2a_2 = -a_2^2 - a_3^2 - \cdots - a_n^2.$$ Deduce that, if all the coefficients of a reflexive polynomial of degree \(n\) are integers and \(a_n \neq 0\), then \(n \leq 3\).
  3. Determine all reflexive polynomials with integer coefficients.

Show Solution
  1. Suppose \(n = 1\), then all polynomials are reflexive (since \(x - a_1\) has the root \(a_1\). Suppose \(n = 2\), then we want \begin{align*} && x^2-a_1x+a_2 &= (x-a_1)(x-a_2) \\ &&&= x^2-(a_1+a_2)x+a_1a_2 \\ \Rightarrow && a_2 &= 0 \\ \end{align*} So all polynomials of the form \(x^2-a_1x\) work and no others. Suppose \(n = 3\) then we want \begin{align*} && x^3-a_1x^2+a_2x-a_3 &= (x-a_1)(x-a_2)(x-a_3) \\ &&&= x^3-(a_1+a_2+a_3)x+(a_1a_2+a_1a_3+a_2a_3)x-a_1a_2a_3 \\ \Rightarrow && a_2+a_3 &= 0 \\ && a_2a_3 &= a_2 \\ \Rightarrow && -a_2^2 &= a_2 \\ \Rightarrow && a_2 &= 0, -1 \\ && -a_1a_2^2 &= -a_2 \\ \Rightarrow && a_2 &= 0, a_2 = 1/a_1 \end{align*} So we need either \(x^3-a_1x\) or \((x+1)^2(x-1) = x^3+x^2-x-1\)
  2. Suppose \(n > 3\) then \begin{align*} && \sum a_i^2 &= \left (\sum a_i \right)^2 - 2 \sum_{i < j} a_i a_j \\ && &= a_1^2 - 2a_2 \\ \Rightarrow && 2a_2 &= a_1^2 - \sum a_i^2 \\ &&&= -a_2^2 - a_3^2 - \cdots - a_n^2 \end{align*} So \((a_2+1)^2 = 1-a_3^2 -\cdots -a_n^2\) so if \(a_n > 0\) (or any other \(a_i, i > 2\) for that matter) then we must have \(a_n = \pm 1, a_{3}, \ldots a_{n-1} = 0\), but if \(a_n = \pm 1\) \(x = 0\) is not a root. Therefore we must have \(a_0\) and \(a_i = 0\) for all \(i > 3\)
  3. The only reflexive polynomials therefore must be \(x^n - kx^{n-1}\) and \(x^{n+3}+x^{n+2}-x^{n+1}-x^n\)
2018 Paper 1 Q7
D: 1500.0 B: 1516.7

  1. In the cubic equation \(x^3-3pqx+pq(p+q)=0\,\), where \(p\) and \(q\) are distinct real numbers, use the substitution \[ x=\frac{pz+q}{z+1} \] to show that the equation reduces to \(az^3+b = 0\,\), where \(a\) and \(b\) are to be expressed in terms of \(p\) and \(q\).
  2. Show further that the equation \(x^3 - 3cx + d = 0\,\), where \(c\) and \(d\) are non-zero real numbers, can be written in the form \(x^3-3pqx+pq(p+q)=0\,\), where \(p\) and \(q\) are distinct real numbers, provided \(d^2 > 4c^3\,\).
  3. Find the real root of the cubic equation \(x^3+6x-2=0\,\).
  4. Find the roots of the equation \(x^3 - 3p^2x +2p^3=0\,\), and hence show how the equation \(x^3 - 3cx + d = 0\) can be solved in the case \(d^2 = 4c^3\,\).

Show Solution
  1. Let \(x = \frac{pz+q}{z+1}\) then \begin{align*} && 0 &= x^3-3pqx+pq(p+q) \\ &&&= \left ( \frac{pz+q}{z+1} \right)^3 - 3pq \left ( \frac{pz+q}{z+1} \right) + pq(p+q) \\ &&&= \frac{(pz+q)^3-3pq(pz+q)(z+1)^2+pq(p+q)(z+1)^3}{(z+1)^3} \\ &&&= \frac{1}{(z+1)^3} \Big ((p^3+pq(p+q)-3p^2q)z^3 + (3p^2q-6p^2q+3pq^2+3p^2q+3pq^2)z^2 + \\ &&&\qquad \qquad\quad\quad +(3pq^2-3p^2q-6pq^2+3p^2q+3qp^2)z+(q^3-3pq^2+p^2q+pq^2) \Big ) \\ &&&= \frac{(p^3+pq^2-2p^2q)z^3+(q^3+p^2q-2pq^2)}{(z+1)^3} \\ \Rightarrow && 0 &= (p^3+pq^2-2p^2q)z^3+(q^3+p^2q-2pq^2) \\ &&&= p(p-q)^2z^3 + q(p-q)^2 \\ \Rightarrow && 0 &= pz^3 + q \end{align*}
  2. We would like to find \(pq = c\) and \(pq(p+q) = d\), so \(p\) and \(q\) are roots of the quadratic \(x^2-\frac{d}{c}x + c = 0\), which has distinct real roots if \(\Delta = \frac{d^2}{c^2}-4c > 0 \Rightarrow d^2>4c^3\)
  3. Note that \(c = -2, d = -2\) so \begin{align*} && 0 &= x^3+6x-2 \\ \text{consider} && 0 &= X^2-X-2 \\ && &= (X+1)(X-2) \\ \Rightarrow && p = -1, &q = 2\\ \Rightarrow && 0 &= x^3-3\cdot 2 \cdot(-1) x + 2\cdot(-1) \cdot(-2+1) \\ \Rightarrow && 0 &= -z^3+2 \\ \Rightarrow && z &= \sqrt[3]{2} \\ \Rightarrow && \frac{-z+2}{z+1} &= \sqrt[3]{2} \\ \Rightarrow && -z+2 &= \sqrt[3]{2} z + \sqrt[3]{2} \\ \Rightarrow && z &= \frac{2-\sqrt[3]{2}}{\sqrt[3]{2}+1} \end{align*}
  4. \(\,\) \begin{align*} && 0 &= x^3 - 3p^2x + 2p^3 \\ &&&= (x-p)(x^2+px-2p^2) \\ &&&=(x-p)^2(x+2p)\\ \Rightarrow && x &= p, p, -2p \end{align*} Therefore if we have a repeated root to our associated quadratic we can find a cubic of the form \(x^3-3p^2x+2p^3\), but we know this has roots we can find.
2018 Paper 3 Q1
D: 1700.0 B: 1484.0

  1. The function \(\f\) is given by \[ \f(\beta)=\beta - \frac 1 \beta - \frac 1 {\beta^2} \ \ \ \ \ \ \ \ (\beta\ne0) \,. \] Find the stationary point of the curve \(y=\f(\beta)\,\) and sketch the curve. Sketch also the curve \(y=\g(\beta)\,\), where \[ \g(\beta) = \beta + \frac 3 \beta - \frac 1 {\beta^2} \ \ \ \ \ \ \ \ (\beta\ne0)\,. \]
  2. Let \(u\) and \(v\) be the roots of the equation \[ x^2 +\alpha x +\beta = 0 \,, \] where \(\beta\ne0\,\). Obtain expressions in terms of \(\alpha\) and \(\beta\) for \(\displaystyle u+v + \frac 1 {uv}\) and \( \displaystyle \frac 1 u + \frac 1 v + uv\,\).
  3. Given that \(\displaystyle u+v + \frac 1 {uv} = -1\,\), and that \(u\) and \(v\) are real, show that \(\displaystyle \frac 1 u+ \frac 1 v + {uv} \le -1\;\).
  4. Given instead that \(\displaystyle u+v + \frac 1 {uv} = 3 \;\), and that \(u\) and \(v\) are real, find the greatest value of \(\displaystyle \frac 1 u+ \frac 1v + {uv}\,\).

Show Solution
  1. \begin{align*} && f(\beta) &= \beta - \frac1{\beta}-\frac1{\beta^2} \\ \Rightarrow && f'(\beta) &= 1 +\frac{1}{\beta^2}+\frac{2}{\beta^3} \\ \Rightarrow && 0 &= f'(\beta) \\ &&&= 1 + \frac1{\beta^2} + \frac{2}{\beta^3} \\ \Rightarrow && 0 &= \beta^3 + \beta + 2 \\ &&&= (\beta+1)(\beta^2-\beta+2) \end{align*} Therefore the only stationary point is at \(\beta = -1, f(-1) = -1\)
    TikZ diagram
    \begin{align*} && g(\beta) &= \beta + \frac3{\beta}-\frac1{\beta^2} \\ \Rightarrow && g'(\beta) &= 1 -\frac{3}{\beta^2}+\frac{2}{\beta^3} \\ \Rightarrow && 0 &= f'(\beta) \\ &&&= 1 - \frac3{\beta^2} + \frac{2}{\beta^3} \\ \Rightarrow && 0 &= \beta^3 - 3\beta + 2 \\ &&&= (\beta-1)^2(\beta+2) \end{align*} Therefore there are stationary points at \(\beta=1,f(1) = 3, \beta=-2, f(-2) = \frac14\)
    TikZ diagram
  2. Let \(u,v\) be the roots of \(x^2 + \alpha x + \beta = 0\), then since \((x-u)(x-v) = 0\) we must have \(\alpha = -(u+v), \beta = uv\). Therefore: \begin{align*} && u+v +\frac{1}{uv} &= -\alpha + \frac{1}{\beta} \\ && \frac1u+\frac1v + uv &= \frac{u+v}{uv} + uv \\ &&&= -\frac{\alpha}{\beta} + \beta \end{align*} Given \(u+v + \frac 1 {uv} = -1\), ie \(-\alpha + \frac{1}{\beta} = -1\). Since the roots are real, we must also have that \(\alpha^2 - 4\beta \geq 0\), so \begin{align*} && -\alpha + \frac1\beta &= -1 \\ \Rightarrow && \alpha &= 1 +\frac1\beta \\ \Rightarrow && -\frac{\alpha}{\beta}+\beta &= -\frac{1}{\beta} \l1+\frac1{\beta}\r + \beta \\ &&&=\beta - \frac{1}{\beta}-\frac{1}{\beta^2} \end{align*} So we want to maximise \(f(\beta)\) subject to \(\alpha ^2 - 4\beta \geq 0\) \begin{align*} && 0 &\leq \alpha^2 -4 \beta \\ &&&= \l 1 + \frac1{\beta} \r^2 - 4\beta \\ &&&= 1+ \frac2{\beta} + \frac1{\beta^2} - 4\beta \\ \Leftrightarrow && 0 &\leq -4\beta^3+\beta^2 + 2\beta + 1 \\ &&&=-(\beta-1)(4\beta^2+3\beta+1)\\ \Leftrightarrow && \beta &\leq 1 \end{align*} But we know \(f(\beta) \leq -1\) on \((-\infty,1]\) so we're done.
  3. Given that \(-\alpha + \frac{1}{\beta} = 3\) we have \begin{align*} && -\alpha + \frac1\beta &= 3 \\ \Rightarrow && \alpha &= -3 +\frac1\beta \\ \Rightarrow && -\frac{\alpha}{\beta}+\beta &= -\frac{1}{\beta} \l-3+\frac1{\beta}\r + \beta \\ &&&=\beta + \frac{3}{\beta}-\frac{1}{\beta^2} \end{align*} which we want to maximise, subject to: \begin{align*} && 0 &\leq \alpha^2 -4 \beta \\ &&&= \l -3 + \frac1{\beta} \r^2 - 4\beta \\ &&&= 9- \frac6{\beta} + \frac1{\beta^2} - 4\beta \\ \Leftrightarrow && 0 &\leq -4\beta^3+9\beta^2 - 6\beta + 1 \\ &&&=-(\beta-1)^2(4\beta-1)\\ \Leftrightarrow && \beta &\leq \frac14 \end{align*} Therefore the maximum will either be \(f(-2) = \frac14\) or \(f(\frac14) = -\frac{15}4\). Therefore the maximum is \(\frac14\)
2017 Paper 1 Q4
D: 1500.0 B: 1516.0

  1. Let \(r\) be a real number with \(\vert r \vert<1\) and let \[ S = \sum_{n=0}^\infty r^n\,. \] You may assume without proof that \(S = \displaystyle \frac{1}{1-r}\, \). Let \(p= 1 + r +r^2\). Sketch the graph of the function \(1+r+r^2\) and deduce that \(\frac{3}{4} \le p < 3\,\). Show that, if \(1 < p < 3\), then the value of \( p\) determines \(r\), and hence \(S\), uniquely. Show also that, if \(\frac{3}{4} < p < 1\), then there are two possible values of \(S\) and these values satisfy the equation \((3-p)S^2-3S+1=0\).
  2. Let \(r\) be a real number with \(\vert r \vert<1\) and let \[ T =\sum_{n=1}^\infty nr^{n-1}\,. \] You may assume without proof that \(T = \displaystyle \dfrac{1}{(1-r)^2}\,.\) Let \( q= 1+2r+3r^2\). Find the set of values of \( q\) that determine \(T\) uniquely. Find the set of values of \(q\) for which \(T\) has two possible values. Find also a quadratic equation, with coefficients depending on \( q\), that is satisfied by these two values.

Show Solution
  1. \(\,\)
    TikZ diagram
    Notice that \(1+r+r^2\) ranges from \(\frac34\) to \(3\) over \((-1,1)\) therefore \(\frac34 \leq p < 3\) attaining its minimum but not its maximum. If \(p > 1\) we know we must be on the right branch and hence we can determine \(r\) and \(S\) uniquely. If \(\frac34 < p < 1\) then we must have two possible values for \(r\), satisfying \(r_i^2 + r+1 = p\) and so our two possible values for \(S_i = \frac{1}{1-r_i}\) or \(r_i = 1-\frac{1}{S_i}\) and so \begin{align*} && p &= \left ( 1 - \frac{1}{S} \right)^2 + 1 - \frac1S + 1 \\ \Rightarrow && pS^2 &= (S-1)^2 + S^2-S + S^2 \\ \Rightarrow && 0 &= (3-p)S^2 -3S + 1 \end{align*}
  2. \(\,\)
    TikZ diagram
    So \(\frac23 \leq q < 6\) and \(r\) and hence \(T\) is uniquely determined if \(2 \leq q < 6\). if \(\frac23
2017 Paper 3 Q3
D: 1700.0 B: 1500.0

Let \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) be the roots of the quartic equation \[ x^4 +px^3 +qx^2 +r x +s =0 \,. \] You are given that, for any such equation, \(\,\alpha \beta + \gamma\delta\,\), \(\alpha\gamma+\beta\delta\,\) and \(\,\alpha \delta + \beta\gamma\,\) satisfy a cubic equation of the form \[ y^3+Ay^2+ (pr-4s)y+ (4qs-p^2s -r^2) =0 \,. \] Determine \(A\). Now consider the quartic equation given by \(p=0\,\), \(q= 3\,\), \(r=-6\,\) and \(s=10\,\).

  1. Find the value of \(\alpha\beta + \gamma \delta\), given that it is the largest root of the corresponding cubic equation.
  2. Hence, using the values of \(q\) and \(s\), find the value of \((\alpha +\beta)(\gamma+\delta)\,\) and the value of \(\alpha\beta\) given that \(\alpha\beta >\gamma\delta\,\).
  3. Using these results, and the values of \(p\) and \(r\), solve the quartic equation.

Show Solution
\begin{align*} A &= -(\alpha \beta + \gamma\delta + \alpha\gamma+\beta\delta+\alpha \delta + \beta\gamma) \\ &= -q \end{align*}
  1. The corresponding cubic equation is: \begin{align*} && 0 &= y^3 - 3y^2-40y+(120-36) \\ &&&= y^3 -3y^2 - 40y + 84 \\ &&&= (y-7)(y-2)(y+6) \end{align*} Therefore \(\alpha\beta + \gamma \delta = 7\)
  2. \begin{align*}(\alpha+\beta)(\gamma+\delta) &= \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta \\ &= 3 -(\alpha\beta + \gamma\delta) \\ &=3-7 = -4 \end{align*} Let \(\alpha\beta\) and \(\gamma\delta\) be the roots of a quadratic; then the quadratic will be \(t^2-7t+10 = 0 \Rightarrow t = 2,5\) so \(\alpha\beta = 5\)
  3. \(\alpha\beta = 5, \gamma\delta = 2\) Consider the quadratic with roots \(\alpha+\beta\) and \(\gamma+\delta\), then \(t^2-4 = 0 \Rightarrow t = \pm 2\). Suppose \(\alpha+\beta = 2, \gamma+\delta=-2\) then \(\alpha, \beta = 1 \pm 2i, \gamma,\delta = -1 \pm i\) \(\alpha \beta \gamma + \beta\gamma\delta + \gamma\delta\alpha + \delta\alpha\beta = 5\gamma + 2\beta + 2\alpha + 5\delta = -6 \neq 6\) Suppose \(\alpha+\beta = -2, \gamma+\delta=2\) then \(\alpha, \beta = -1 \pm 2i, \gamma,\delta = 1 \pm i\) \(\alpha \beta \gamma + \beta\gamma\delta + \gamma\delta\alpha + \delta\alpha\beta = 5\gamma + 2\beta + 2\alpha + 5\delta = 6\), therefore these are there roots. (In some order): \(1 \pm i, -1 \pm 2i\)
2015 Paper 3 Q6
D: 1700.0 B: 1553.5

  1. Let \(w\) and \(z\) be complex numbers, and let \(u= w+z\) and \(v=w^2+z^2\). Prove that \(w\) and \(z\) are real if and only if \(u\) and \(v\) are real and \(u^2\le2v\).
  2. The complex numbers \(u\), \(w\) and \(z\) satisfy the equations \begin{align*} w+z-u&=0 \\ w^2+z^2 -u^2 &= - \tfrac 23 \\ w^3+z^3 -\lambda u &= -\lambda\, \end{align*} where \(\lambda \) is a positive real number. Show that for all values of \(\lambda\) except one (which you should find) there are three possible values of \(u\), all real. Are \(w\) and \(z\) necessarily real? Give a proof or counterexample.

Show Solution
  1. Notice that \(u^2 = v+2wz\), so \(w,z\) are roots of the quadratic \(t^2 -ut+\frac{u^2-v}{2}\). Therefore they are both real if \(u^2 \geq 2(u^2-v) \Rightarrow 2v \geq u^2\).
  2. \begin{align*} && w+z &= u \\ && w^2+z^2 &= u^2 - \tfrac23 \\ && w^3+z^3 &= \lambda(u-1) \\ \\ && wz &= \frac{u^2 - (u^2-\tfrac23)}{2} = \tfrac13\\ \\ && (w+z)(w^2+z^2) &= w^3+z^3+wz(w+z) \\ &&u(u^2-\tfrac23)&= \lambda(u-1)+\frac13u \\ \Rightarrow && u^3-u&= \lambda (u-1) \\ \Rightarrow && 0 &= (u-1)(u(u+1) - \lambda) \\ \Rightarrow && 0 &= (u-1)(u^2+u - \lambda) \end{align*} Therefore there will be at most 3 values for \(u\), unless \(1\) is a root of \(u^2+u-\lambda\), ie \(\lambda = 2\). Suppose \(u = 1\), then we have: \(w+z = 1, wz = 1/3 \Rightarrow w,z = \frac{-1 \pm \sqrt{-1/3}}{2}\) which are clearly complex.
2014 Paper 3 Q1
D: 1700.0 B: 1542.2

Let \(a\), \(b\) and \(c\) be real numbers such that \(a+b+c=0\) and let \[(1+ax)(1+bx)(1+cx) = 1+qx^2 +rx^3\,\] for all real \(x\). Show that \(q = bc+ca+ab\) and \(r= abc\).

  1. Show that the coefficient of \(x^n\) in the series expansion (in ascending powers of \(x\)) of \(\ln (1+qx^2+rx^3)\) is \((-1)^{n+1} S_n\) where \[S_n = \frac{a^n+b^n+c^n}{n} \,, \ \ \ \ \ \ \ \ (n\ge1).\]
  2. Find, in terms of \(q\) and \(r\), the coefficients of \(x^2\), \(x^3\) and \(x^5\) in the series expansion (in ascending powers of \(x\)) of \(\ln (1+qx^2+rx^3)\) and hence show that \(S_2S_3 =S_5\).
  3. Show that \(S_2S_5 =S_7\).
  4. Give a proof of, or find a counterexample to, the claim that \(S_2S_7=S_9\).

Show Solution
\begin{align*} (1+ax)(1+bx)(1+cx) &= (1+(a+b)x+abx^2)(1+cx) \\ &= 1+(a+b+c)x+(ab+bc+ca)x^2+abcx^3 \end{align*} Therefore by comparing coefficients, \(q = bc + ca + ab\) and \(r = abc\) as required.
  1. \begin{align*} \ln (1+qx^2 + rx^3) &= \ln(1+ax) + \ln(1+bx)+\ln(1+cx) \\ &= -\sum_{n=1}^{\infty} \frac{(-ax)^n}{n}-\sum_{n=1}^{\infty} \frac{(-bx)^n}{n}-\sum_{n=1}^{\infty} \frac{(-cx)^n}{n} \\ &= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(a^n+b^n+c^n)}{n} x^n \\ &= \sum_{n=1}^{\infty} (-1)^{n+1} S_n x^n \\ \end{align*}
  2. \begin{align*} \ln (1 + qx^2 + rx^3) &= (qx^2+rx^3) -\frac{(qx^2+rx^3)^2}{2} + O(x^6) \\ &= qx^2 + rx^3 - \frac12 q^2 x^4 - qr x^5 + O(x^6) \\ \end{align*} Comparing coefficients we see that \(S_2 = -q\) and \(S_3 = r\), we also must have \(S_5 = -qr = S_2S_3\) as required.
  3. \begin{align*} \ln (1 + qx^2 + rx^3) &= (qx^2+rx^3) -\frac{(qx^2+rx^3)^2}{2} +\frac{(qx^2+rx^3)^3}{3}+ O(x^8) \\ &= qx^2 + rx^3 - \frac12 q^2 x^4 - qr x^5 + \frac12 rx^6 + \frac13 q^3 x^6 + q^2r x^7 + O(x^8) \\ &= qx^2 + rx^3 - \frac12 q^2 x^4 - qr x^5 + \left ( \frac12 r+ \frac13 q^3 \right)x^6 + q^2r x^7 \end{align*} Comparing coefficients we see that \(S_2 = -q\) and \(S_5 =-qr\), we also must have \(S_7 = q^2r = S_2S_5\) as required.
  4. Let \(a = b = 1, c = -2\), then \(S_2 = \frac{1^2+1^2 + (-2)^2}{2} = 3, S_7 = \frac{1^2+1^2+(-2)^7}{7} = -18, S_9 = \frac{1^1+1^2+(-2)^9}{9} = \frac{2 - 512}{9} \neq 3 \cdot (-18)\)
2009 Paper 3 Q5
D: 1700.0 B: 1516.0

The numbers \(x\), \(y\) and \(z\) satisfy \begin{align*} x+y+z&= 1\\ x^2+y^2+z^2&=2\\ x^3+y^3+z^3&=3\,. \end{align*} Show that \[ yz+zx+xy=-\frac12 \,.\] Show also that \(x^2y+x^2z+y^2z+y^2x+z^2x+z^2y=-1\,\), and hence that \[ xyz=\frac16 \,.\] Let \(S_n=x^n+y^n+z^n\,\). Use the above results to find numbers \(a\), \(b\) and \(c\) such that the relation \[ S_{n+1}=aS_{n}+bS_{n-1}+cS_{n-2}\,, \] holds for all \(n\).

Show Solution
\begin{align*} && (x+y+z)^2 &= x^2 + y^2 + z^2 + 2(xy+yz+zx) \\ \Rightarrow && 1^2 &= 2 + 2(xy+yz+zx) \\ \Rightarrow && xy+yz+zx &= -\frac12 \end{align*} \begin{align*} && 1 \cdot 2 &= (x+y+z)(x^2+y^2+z^2) \\ &&&= x^3 + y^3 + z^3 + x^2y+x^2z+y^2z+y^2x+z^2x+z^2y \\ &&&= 3 + x^2y+x^2z+y^2z+y^2x+z^2x+z^2y\\ \Rightarrow && -1 &= x^2y+x^2z+y^2z+y^2x+z^2x+z^2y \end{align*} \begin{align*} && (x+y+z)^3 &= x^3 + y^3 + z^3 + \\ &&&\quad \quad 3xy^2 + 3xz^2 + \cdots + 3zx^2 + 3zy^2 + \\ &&&\quad \quad \quad 6xyz \\ \Rightarrow && 1 &= 3 + 3(-1) + 6xyz \\ \Rightarrow && xyz &= \frac16 \end{align*} Since we have \(f(t) = (t-x)(t-y)(t-z) = t^3-t^2-\frac12 t - \frac16\) is zero for \(x,y,z\) we can notice that: \(t^{n+1} = t^n +\frac12 t^{n-1} + \frac16 t^{n-2}\) is also true for \(x,y,z\) (by multiplying by \(t^{n-2}\). Therefore: \(S_{n+1} = S_n + \frac12 S_{n-1} + \frac16 S_{n-2}\)
2008 Paper 3 Q1
D: 1700.0 B: 1516.0

Find all values of \(a\), \(b\), \(x\) and \(y\) that satisfy the simultaneous equations \begin{alignat*}{3} a&+b & &=1 &\\ ax&+by & &= \tfrac13& \\ ax^2&+by^2& &=\tfrac15& \\ ax^3 &+by^3& &=\tfrac17\,.& \end{alignat*} \noindent{\bf [} {\bf Hint}: you may wish to start by multiplying the second equation by \(x+y\). {\bf ]}

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This is a second order recurrence relation, so we need to find \(m\) and \(n\) such that; \begin{align*} &&\frac15 &= m\frac13 + n \\ &&\frac17 &= m \frac15 + n\frac13 \\ \Rightarrow && m,n &= \frac67, - \frac{3}{35} \end{align*} So we now need to solve the characteristic equation: \(\lambda^2 - \frac67 \lambda + \frac{3}{35} = 0\) So \(x,y = \frac{15 \pm 2 \sqrt{30}}{35}\). We need, \begin{align*} && 1 &= a+ b \\ && \frac13 &= a \frac{15 + 2 \sqrt{30}}{35} + b \frac{15 - 2 \sqrt{30}}{35} \\ && \frac13 &= \frac{15}{35} + \frac{2 \sqrt{30}}{35}(a-b) \\ \Rightarrow && -\frac{\sqrt{30}}{18} &= a-b \\ \Rightarrow && a &= \frac{18-\sqrt{30}}{36} \\ && b &= \frac{18+\sqrt{30}}{38} \end{align*} So our two answers are: \[ (a,b,x,y) = \left (\frac{18\pm\sqrt{30}}{36} ,\frac{18\mp\sqrt{30}}{36},\frac{15 \pm 2 \sqrt{30}}{35},\frac{15 \mp 2 \sqrt{30}}{35}, \right)\]
2007 Paper 3 Q1
D: 1700.0 B: 1500.0

In this question, do not consider the special cases in which the denominators of any of your expressions are zero. Express \(\tan(\theta_1+\theta_2+\theta_3+\theta_4)\) in terms of \(t_i\), where \(t_1=\tan\theta_1\,\), etc. Given that \(\tan\theta_1\), \(\tan\theta_2\), \(\tan\theta_3\) and \(\tan\theta_4\) are the four roots of the equation \[at^4+bt^3+ct^2+dt+e=0 \] (where \(a\ne0\)), find an expression in terms of \(a\), \(b\), \(c\), \(d\) and \(e\) for \(\tan(\theta_1+\theta_2+\theta_3+\theta_4)\). The four real numbers \(\theta_1\), \(\theta_2\), \(\theta_3\) and \(\theta_4\) lie in the range \(0\le \theta_i<2\pi\) and satisfy the equation \[ p\cos2\theta+\cos(\theta-\alpha)+p=0\,,\] where \(p\) and \(\alpha\) are independent of \(\theta\). Show that \(\theta_1+\theta_2+\theta_3+\theta_4=n\pi\) for some integer \(n\).

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\begin{align*} \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &= \frac{\tan(\theta_1 + \theta_2) + \tan(\theta_3 + \theta_4)}{1 - \tan(\theta_1 +\theta_2)\tan(\theta_3+\theta_4)} \\ &= \frac{\frac{t_1+t_2}{1-t_1t_2}+\frac{t_3+t_4}{1-t_3t_4}}{1-\frac{t_1+t_2}{1-t_1t_2}\frac{t_3+t_4}{1-t_3t_4}} \\ &= \frac{(t_1+t_2)(1-t_3t_4)+(t_3+t_4)(1-t_1t_2)}{(1-t_1t_2)(1-t_3t_4)-(t_1+t_2)(t_3+t_4)} \\ &= \frac{t_1 +t_2+t_3+t_4 - (t_1t_2t_3+t_1t_2t_4+t_1t_3t_4+t_2t_3t_4)}{1-t_1t_2-t_1t_3-t_1t_4-t_2t_3-t_2t_4-t_3t_4} \end{align*} If \(t_1, t_2, t_3, t_4\) are the roots of \(at^4+bt^3+ct^2+dt+e = 0\), then \(t_1+t_2+t_3+t_4 = -\frac{b}{a}, t_1t_2+t_1t_3+t_1t_4+t_2t_3+t_2t_4+t_3t_4 = \frac{c}{a}, t_1t_2t_3+t_1t_2t_4+t_1t_3t_4+t_2t_3t_4 = -\frac{d}{a}\), therefore the expression is: \begin{align*} \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &= \frac{-\frac{b}{a}+\frac{d}{a}}{1 - \frac{c}{a}} \\ &= \frac{d-b}{a-c} \end{align*} \begin{align*} &&0 &= p \cos 2\theta + \cos (\theta - \alpha) + p \\ &&&= p (2\cos^2 \theta -1) + \cos \theta \cos \alpha - \sin \theta \sin \alpha + p \\ &&&= 2p \cos^2 \theta + \cos \theta \cos \alpha - \sin \theta \sin \alpha\\ \Rightarrow && 0 &=2p \cos \theta + \cos \alpha - \tan \theta \sin \alpha \\ \Rightarrow && -2p \cos \theta&= \cos \alpha - \tan \theta \sin \alpha \\ \Rightarrow && 4p^2 \cos^2 \theta &= \cos^2 \alpha - 2 \sin \alpha \cos \alpha \tan \theta + \sin^2 \alpha \tan^2 \theta \\ && 4p^2 \frac{1}{1 + \tan^2 \theta} &= \cos^2 \alpha - \sin 2\alpha \tan \theta + \sin^2 \alpha \tan^2 \theta \\ \Rightarrow && 4p^2 &= \cos^2 \alpha - \sin 2\alpha t+t^2-\sin2\alpha t^3+\sin^2 \alpha t^4 \\ \Rightarrow && \tan (\theta_1+\theta_2 + \theta_3+ \theta_4) &= \frac{0}{\sin^2 \alpha - 1} \\ &&&= 0 \\ \Rightarrow && \theta_1 + \theta_2 + \theta_3 + \theta_4 &= n\pi \end{align*}
1997 Paper 3 Q4
D: 1700.0 B: 1516.0

In this question, you may assume that if \(k_1,\dots,k_n\) are distinct positive real numbers, then \[\frac1n\sum_{r=1}^nk_r>\left({\prod\limits_{r=1}^n} k_r\right )^{\!\! \frac1n},\] i.e. their arithmetic mean is greater than their geometric mean. Suppose that \(a\), \(b\), \(c\) and \(d\) are positive real numbers such that the polynomial \[{\rm f}(x)=x^4-4ax^3+6b^2x^2-4c^3x+d^4\] has four distinct positive roots.

  1. Show that \(pqr,qrs,rsp\) and \(spq\) are distinct, where \(p,q,r\) and \(s\) are the roots of the polynomial \(\mathrm{f}\).
  2. By considering the relationship between the coefficients of \(\mathrm{f}\) and its roots, show that \(c > d\).
  3. Explain why the polynomial \(\mathrm{f}'(x)\) must have three distinct roots.
  4. By differentiating \(\mathrm{f}\), show that \(b > c\).
  5. Show that \(a > b\).

Show Solution
  1. Suppose \(pqr = qrs\), since the roots are positive, we can divide by \(qr\) to obtain \(p=s\) (a contradiction. Therefore all those terms are distinct.
  2. \(4c^3 = pqr+qrs+rsp+spq\), \(d^4 = pqrs\). Applying AM-GM, we obtain: \begin{align*} && c^3 = \frac{ pqr+qrs+rsp+spq}{4} & > \sqrt[4]{p^3q^3r^3s^3} = d^{3} \\ \Rightarrow && c &> d \end{align*}
  3. There must be a turning point between each root (since there are no repeated roots).
  4. \(f'(x) = 4x^3-12ax^2+12b^2-4c^3 = 4(x^3-3ax^2+3b^2-c^3)\). Letting the roots of this polynomial be \(\alpha, \beta, \gamma\) and again applying AM-GM, we must have: \begin{align*} && b^2 = \frac{\alpha\beta + \beta \gamma+\gamma \alpha}{3} &> \sqrt[3]{\alpha^2\beta^2\gamma^2} = c^2 \\ \Rightarrow && b &> c \end{align*}
  5. Again, since there are turning points between the roots of \(f'(x)\) we must have distinct roots for \(f''(x)\), ie: \(f''(x) = 3x^2-6ax+6b^2 = 3(x^2-2ax+b^2)\) has distinct real roots. But for this to occur we must have that \((2a)^2-4b^2 = 4(a^2-b^2) > 0\), ie \(a>b\)
1996 Paper 3 Q7
D: 1700.0 B: 1515.1

  1. If \(x+y+z=\alpha,\) \(xy+yz+zx=\beta\) and \(xyz=\gamma,\) find numbers \(A,B\) and \(C\) such that \[ x^{3}+y^{3}+z^{3}=A\alpha^{3}+B\alpha\beta+C\gamma. \] Solve the equations \begin{alignat*}{1} x+y+z & =1\\ x^{2}+y^{2}+z^{2} & =3\\ x^{3}+y^{3}+z^{3} & =4. \end{alignat*}
  2. The area of a triangle whose sides are \(a,b\) and \(c\) is given by the formula \[ \mathrm{area}=\sqrt{s(s-a)(s-b)(s-c)} \] where \(s\) is the semi-perimeter \(\frac{1}{2}(a+b+c).\) If \(a,b\) and \(c\) are the roots of the equation \[ x^{3}-16x^{2}+81x-128=0, \] find the area of the triangle.

Show Solution
  1. \begin{align*} (x+y+z)^3 &= x^3+y^3+z^3+ \\ &\quad 3xy^2 + 3xz^2 + 3yx^2 + \cdots + 3zy^2 \\ &\quad\quad + 6xyz \\ (x+y+z)(xy+yz+zx) &= x^2y+x^2z + \cdots + z^2 x + 3xyz \\ x^3+y^3+z^3 &= (x+y+z)^3 - 3(xy^2 + \cdots + zy^2) - 6xyz \\ &= \alpha^3 - 3(\alpha \beta - 3\gamma)-6\gamma \\ &= \alpha^3-3\alpha \beta+3\gamma \end{align*} Since \(4 = 1^3-3\cdot1\cdot(-1) + 3 \gamma \Rightarrow \gamma = 0\), therefore one of \(x,y,z = 0\). WLOG \(x = 0\), so \(y+z = 1, y^2 + z^2 = 3 \Rightarrow y^2 + (1-y)^2 = 3 \Rightarrow y^2 -y -1 = 0 \Rightarrow y = \frac{1 \pm \sqrt{5}}{2}\), so we have \((x,y,z) = (0, \frac{1 +\sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2})\) and permutations.
  2. \begin{align*} A^2 &= s(s-a)(s-b)(s-c) \\ \end{align*} Notice the second part is the same as plugging \(s= 16/2 = 8\) into our polynomial Therefore \begin{align*} A^2 &= 8 \cdot (8^3 - 16 \cdot 8^2 + 81 \cdot 8 - 128) \\ &= 8 \cdot 8 (8^2 - 16 \cdot 8 + 81- 16) \\ &= 64 (-64+81-16) \\ &= 64 \end{align*} Therefore \(A = 8\)
1992 Paper 2 Q7
D: 1600.0 B: 1500.0

The cubic equation \[ x^{3}-px^{2}+qx-r=0 \] has roots \(a,b\) and \(c\). Express \(p,q\) and \(r\) in terms of \(a,b\) and \(c\).

  1. If \(p=0\) and two of the roots are equal to each other, show that \[ 4q^{3}+27r^{2}=0. \]
  2. Show that, if two of the roots of the original equation are equal to each other, then \[ 4\left(q-\frac{p^{2}}{3}\right)^{3}+27\left(\frac{2p^{3}}{27}-\frac{pq}{3}+r\right)^{2}=0. \]

Show Solution
\(p = a+b+c, q = ab+bc+ca, r = abc\)
  1. Suppose two roots are equal to each other, this means that one of the roots is also a root of the derivative. ie \begin{align*} && 0 &= x^3+qx - r \\ && 0 &= 3x^2+q \end{align*} have a common root, but this root must satisfy \(x^2 = -\frac{q}{3}\). Then \begin{align*} &&0 &= x^3 + qx - r \\ &&&= x^3 -3x^3 - r \\ &&&= -2x^3 -r \\ \Rightarrow && r^2 &= 4x^6 \\ &&&= 4 \left ( -\frac{q}{3}\right)^3 \\ \Rightarrow && 0 &= 27r^2+4q^3 \end{align*}
  2. Consider \(x = z + \frac{p}{3}\), then the equation is: \begin{align*} x^{3}-px^{2}+qx-r &= (z + \frac{p}{3})^3 - p(z + \frac{p}{3})^2 + q(z + \frac{p}{3}) - r \\ &= z^3 + pz^2 + \frac{p^2}{3}z + \frac{p^3}{27} - \\ &\quad -pz^2-\frac{2p^2}{3}z-\frac{p^3}{9} + \\ &\quad\quad qz + \frac{pq}{3} - r \\ &= z^3+\left (\frac{p^2}{3}-\frac{2p^2}{3}+q \right)z + \left (\frac{p^3}{27}-\frac{p^3}{9}+\frac{pq}{3}-r \right) \\ &= z^3+\left (-\frac{p^2}{3}+q \right)z + \left (-\frac{2p^3}{27}+\frac{pq}{3}-r \right) \\ \end{align*} Since this equation must also have repeated roots we must have: \begin{align*} 4\left (-\frac{p^2}{3}+q \right)^3 + 27 \left (-\frac{2p^3}{27}+\frac{pq}{3}-r \right)^2 = 0 \end{align*} which is exactly our desired result
1991 Paper 3 Q10
D: 1700.0 B: 1516.0

The equation \[ x^{n}-qx^{n-1}+r=0, \] where \(n\geqslant5\) and \(q\) and \(r\) are real constants, has roots \(\alpha_{1},\alpha_{2},\ldots,\alpha_{n}.\) The sum of the products of \(m\) distinct roots is denoted by \(\Sigma_{m}\) (so that, for example, \(\Sigma_{3}=\sum\alpha_{i}\alpha_{j}\alpha_{k}\) where the sum runs over the values of \(i,j\) and \(k\) with \(n\geqslant i>j>k\geqslant1\)). The sum of \(m\)th powers of the roots is denoted by \(S_{m}\) (so that, for example, \(S_{3}=\sum\limits_{i=1}^{n}\alpha_{i}^{3}\)). Prove that \(S_{p}=q^{p}\) for \(1\leqslant p\leqslant n-1.\) You may assume that for any \(n\)th degree equation and \(1\leqslant p\leqslant n\) \[ S_{p}-S_{p-1}\Sigma_{1}+S_{p-2}\Sigma_{2}-\cdots+(-1)^{p-1}S_{1}\Sigma_{p-1}+(-1)^{p}p\Sigma_{p}=0.] \] Find expressions for \(S_{n},\) \(S_{n+1}\) and \(S_{n+2}\) in terms of \(q,r\) and \(n\). Suggest an expression for \(S_{n+m},\) where \(m < n\), and prove its validity by induction.

Show Solution
Claim: \(S_p = q^p\) for \(1 \leq p \leq n-1\) Proof: When \(p = 1\), \(S_p = \Sigma_1 = q\) as expected. Note that \(\Sigma_i = 0\) for \(i = 2, \cdots, n-1\). Using \(S_p = S_{p-1}\Sigma_{1}-S_{p-2}\Sigma_{2}+\cdots+(-1)^{p-1+1}S_{1}\Sigma_{p-1}+(-1)^{p+1}p\Sigma_{p}\), we can see that \(S_p = qS_{p-q}\) when \(1 \leq p \leq n-1\), ie \(S_p = q^p\). Note that \begin{align*} S_n &= \sum \alpha_i^n \\ &= q\sum \alpha_i^{n-1} - \sum r \\ &= qS_{n-1} - nr \\ &= q^n - nr \\ \\ S_{n+1} &= \sum \alpha_i^{n+1} \\ &= q \sum \alpha_i^{n} - r \sum \alpha_i \\ &= q^{n+1} - rq \\ \\ S_{n+2} &= \sum \alpha_i^{n+2} \\ &= q \sum \alpha_i^{n+1} - r \sum \alpha_i^2 \\ &= q^{n+2} - rq^2 \\ \end{align*} Claim: \(S_{n+m} = q^{n+m} - rq^{m}\) Proof: The obvious