OCR MEI Paper 3 (Paper 3) 2018 June

1 Triangle ABC is shown in Fig. 1. \begin{figure}[h] \end{figure} Find the perimeter of triangle ABC .

2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]

3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h] \end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).

4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Determine the coordinates of the stationary points on the curve.
2. Determine the nature of each stationary point.
3. Write down the equation of the vertical asymptote.
4. Deduce the set of values of \(x\) for which the curve is concave upwards.

5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.

1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} Find estimates of the values of \(a\) and \(k\).
3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
4. Give a reason why the model may not be appropriate for large values of \(t\).

6 Find the constant term in the expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 15 }\).

7 In this question you must show detailed reasoning.
Fig. 7 shows the curve \(y = 5 x - x ^ { 2 }\). \begin{figure}[h] \end{figure} The line \(y = 4 - k x\) crosses the curve \(y = 5 x - x ^ { 2 }\) on the \(x\)-axis and at one other point.
Determine the coordinates of this other point.

8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).

1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).

9 The function \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }\) is defined on the domain \(x \in \mathbb { R } , x \neq 0\).

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Write down the range of \(\mathrm { f } ^ { - 1 } ( x )\).

10 Point A has position vector \(\left( \begin{array} { l } a
b
0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4
2
0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2
4
2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).

1. Show that \(a - b + 1 = 0\).
2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.

1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
2. Both numbers \(a\) and \(b\) are negative.

12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.

13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.

14

1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
2. Prove that \(h = \sqrt { a b }\), as given in line 24 .

15 It is given in lines \(31 - 32\) that the square has the smallest perimeter of all rectangles with the same area. Using this fact, prove by contradiction that among rectangles of a given perimeter, \(4 L\), the square with side \(L\) has the largest area. \section*{END OF QUESTION PAPER}