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1999 Paper 1 Q6
D: 1484.0 B: 1500.0
quadratics inequality optimisation linear function completing the square case analysis closed interval

  1. Find the greatest and least values of \(bx+a\) for \(-10\leqslant x \leqslant 10\), distinguishing carefully between the cases \(b>0\), \(b=0\) and \(b<0\).
  2. Find the greatest and least values of \(cx^{2}+bx+a\), where \(c\ge0\), for \(-10\leqslant x \leqslant 10\), distinguishing carefully between the cases that can arise for different values of \(b\) and \(c\).

Solution:

  1. Case \(b > 0\). Then \(bx+a\) is increasing and the greatest value is \(10b+a\), and the least value \(a-10b\) Case \(b=0\), then \(a\) is constant and the greatest and least value is \(a\) Case \(b < 0\), then \(bx+a\) is decreasing and the greatest value is \(-10b+a\) and the least value is \(10b+a\)
  2. If \(c = 0\) we have the same cases as above. If \( c > 0\) the consider \(2cx+b\). if \(b-20c > 0\) then our function is increasing on our interval and the greatest value is \(100c+10b+a\) and the least value is \(100c-10b+a\) If \(20c+b < 0\) then our function is decreasing and that calculation is reversed. If neither of these are true, then the minimum will be when \(x = - \frac{b}{2c}\) and the max at one end point.

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1994 Paper 2 Q8
D: 1600.0 B: 1516.0
curve sketching optimisation linear function inventory model profit inequality piecewise linear applied mathematics

`24 Hour Spares' stocks a small, widely used and cheap component. Every \(T\) hours \(X\) units arrive by lorry from the wholesaler, for which the owner pays a total \(\pounds (a+qX)\). It costs the owner \(\pounds b\) per hour to store one unit. If she has the units in stock she expects to sell \(r\) units per hour at \(\pounds(p+q)\) per unit. The other running costs of her business remain at \(\pounds c\) pounds an hour irrespective of whether she has stock or not. (All of the quantities \(T,X,a,b,r,q,p\) and \(c\) are greater than 0.) Explain why she should take \(X\leqslant rT\). Given that the process may be assumed continuous (the items are very small and she sells many each hour), sketch \(S(t)\) the amount of stock remaining as a function of \(t\) the time from the last delivery. Compute the total profit over each period of \(T\) hours. Show that, if \(T\) is fixed with \(T\geqslant p/b\), the business can be made profitable if \[ p^{2}>2\frac{(a+cT)b}{r}. \]

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1992 Paper 1 Q7
D: 1484.0 B: 1500.0
proof by induction polynomials integer values linear function quadratic finite differences

Let \(\mathrm{g}(x)=ax+b.\) Show that, if \(\mathrm{g}(0)\) and \(\mathrm{g}(1)\) are integers, then \(\mathrm{g}(n)\) is an integer for all integers \(n\). Let \(\mathrm{f}(x)=Ax^{2}+Bx+C.\) Show that, if \(\mathrm{f}(-1),\mathrm{f}(0)\) and \(\mathrm{f}(1)\) are integers, then \(\mathrm{f}(n)\) is an integer for all integers \(n\). Show also that, if \(\alpha\) is any real number and \(\mathrm{f}(\alpha-1),\) \(\mathrm{f}(\alpha)\) and \(\mathrm{f}(\alpha+1)\) are integers, then \(\mathrm{f}(\alpha+n)\) is an integer for all integers \(n\).

Solution: If \(g(0) \in \mathbb{Z} \Rightarrow b \in \mathbb{Z}\). If \(g(1) \in \mathbb{Z} \Rightarrow a+b \in \mathbb{Z} \Rightarrow a \in \mathbb{Z}\), therefore \(a \cdot n + b \in \mathbb{Z}\), in particular \(g(n) \in \mathbb{Z}\) for all integers \(n\). \(f(0) \in \mathbb{Z} \Rightarrow C \in \mathbb{Z}\), \(f(1) \in \mathbb{Z} = A+ B + C \in \mathbb{Z} \Rightarrow A+ B \in \mathbb{Z}\) \(f(-1) \in \mathbb{Z} = A- B + C \in \mathbb{Z} \Rightarrow A- B \in \mathbb{Z}\) \(\Rightarrow 2A, 2B \in \mathbb{Z}\) \begin{align*} f(n) &= An^2 + Bn + C \\ &= An^2-An + An+Bn + C \\ &= 2A \frac{n(n-1)}2 + (A+B)n + C \\ &\in \mathbb{Z} \end{align*} Consider \(g(x) = f(x + \alpha)\), therefore \(g(0), g(1), g(-1) \in \mathbb{Z} \Rightarrow g(n) \in \mathbb{Z} \Rightarrow f(n+\alpha) \in \mathbb{Z}\)

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