OCR PURE

1. Prove that \(\cos x + \sin x \tan x \equiv \frac{1}{\cos x}\) (where \(x \neq \frac{1}{2}n\pi\) for any odd integer \(n\)). [3]
2. Solve the equation \(2\sin^2 x = \cos^2 x\) for \(0° \leqslant x \leqslant 180°\). [2]

1. The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\), \(\begin{pmatrix} -3 \\ 6 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 12 \end{pmatrix}\) respectively.
   1. Show that \(B\) lies on \(AC\). [2]
   2. Find the ratio \(AB : BC\). [1]
2. The diagram shows the line \(x + y = 6\) and the point \(P(2, 4)\) that lies on the line. A copy of the diagram is given in the Printed Answer Booklet. \includegraphics{figure_1} The distinct point \(Q\) also lies on the line \(x + y = 6\) and is such that \(|\overrightarrow{OQ}| = |\overrightarrow{OP}|\), where \(O\) is the origin. Find the magnitude and direction of the vector \(\overrightarrow{PQ}\). [3]
3. The point \(R\) is such that \(\overrightarrow{PR}\) is perpendicular to \(\overrightarrow{OP}\) and \(|\overrightarrow{PR}| = \frac{1}{2}|\overrightarrow{OP}|\). Write down the position vectors of the two possible positions of the point \(R\). [2]

The diagram shows the graph of \(y = f(x)\), where \(f(x)\) is a quadratic function of \(x\). A copy of the diagram is given in the Printed Answer Booklet. \includegraphics{figure_2}

1. On the copy of the diagram in the Printed Answer Booklet, draw a possible graph of the gradient function \(y = f'(x)\). [3]
2. State the gradient of the graph of \(y = f''(x)\). [1]

A curve has equation \(y = e^{3x}\).

1. Determine the value of \(x\) when \(y = 10\). [2]
2. Determine the gradient of the tangent to the curve at the point where \(x = 2\). [2]

In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = x^3 - 4x\). \includegraphics{figure_3} Determine the total area enclosed by the curve and the \(x\)-axis. [6]

1. Determine the two real roots of the equation \(8x^6 + 7x^3 - 1 = 0\). [3]
2. Determine the coordinates of the stationary points on the curve \(y = 8x^7 + \frac{49}{4}x^4 - 7x\). [4]
3. For each of the stationary points, use the value of \(\frac{d^2y}{dx^2}\) to determine whether it is a maximum or a minimum. [4]

1. Two real numbers are denoted by \(a\) and \(b\).
   1. Write down expressions for the following.
   2. Prove that the mean of the squares of \(a\) and \(b\) is greater than or equal to the square of their mean. [3]
2. You are given that the result in part (a)(ii) is true for any two or more real numbers. Explain what this result shows about the variance of a set of data. [1]

In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]

In a survey, 50 people were asked whether they had passed A-level English and whether they had passed A-level Mathematics. The numbers of people in various categories are shown in the Venn diagram. \includegraphics{figure_4}

1. A person is chosen at random from the 50 people. Find the probability that this person has passed A-level Mathematics. [1]
2. Two people are chosen at random, without replacement, from those who have passed A-level in at least one of the two subjects. Determine the probability that both of these people have passed A-level Mathematics. [3]

The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram. \includegraphics{figure_5}

1. Calculate the number of boulders with masses between 60 and 65 kg. [2]
2. 1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample. [3]
   2. Explain why your answers are only estimates. [1]
3. Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution. [2]
4. Give one advantage of using a histogram rather than a pie chart in this context. [1]

Casey and Riley attend a large school. They are discussing the music preferences of the students at their school. Casey believes that the favourite band of 75% of the students is Blue Rocking. Riley believes that the true figure is greater than 75%. They plan to carry out a hypothesis test at the 5% significance level, using the hypotheses \(H_0: p = 0.75\) and \(H_1: p > 0.75\). They choose a random sample of 60 students from the school, and note the number, \(X\), who say that their favourite band is Blue Rocking. They find that \(X = 50\).

1. Assuming the null hypothesis to be true, Riley correctly calculates that \(P(X = 50) = 0.0407\), correct to 3 significant figures. Riley says that, because this value is less than 0.05, the null hypothesis should be rejected. Explain why this statement is incorrect. [1]
2. Carry out the test. [5]
3. 1. State which mathematical model is used in the calculation in part (b), including the value(s) of any parameter(s). [1]
   2. The random sample was chosen without replacement. Explain whether this invalidates the model used in part (b). [1]

This question deals with information about the populations of Local Authorities (LAs) in the North of England, taken from the 2011 census. \includegraphics{figure_6} Fig. 1 and Fig. 2 both show strong correlation, but of two different kinds.

1. For each diagram, use a single word to describe the kind of correlation shown. [1]
2. For each diagram, suggest a reason, in context, why the correlation is of the particular kind described in part (a). [2]

Fig. 3 is the same as Fig. 2 but with the point \(A\) marked. Fig. 4 shows information about the same LAs as Fig. 2 and Fig. 3. \includegraphics{figure_7}

1. Point \(A\) in Fig. 3 and point \(B\) in Fig. 4 represent the same LA. Explain how you can tell that this LA has a large population. [1]