Questions — OCR (4907 questions)

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OCR PURE Q5
6 marks Standard +0.3
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]
OCR PURE Q6
11 marks Moderate -0.3
  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]
OCR PURE Q7
5 marks Standard +0.3
  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]
OCR PURE Q8
7 marks Challenging +1.2
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]
OCR PURE Q9
4 marks Easy -1.8
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]
OCR PURE Q10
9 marks Easy -1.2
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]
    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]
OCR PURE Q11
8 marks Moderate -0.3
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]
    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]
OCR PURE Q12
4 marks Easy -2.3
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]
OCR Further Pure Core AS 2020 November Q1
6 marks Standard +0.8
In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(-77 - 36\text{i}\). [6]
OCR Further Pure Core AS 2020 November Q2
10 marks Moderate -0.8
P, Q and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P.
  1. Write down the matrix A. [1]
Q is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). B is the matrix that represents Q.
  1. Find the matrix B. [2]
T is P followed by Q. C is the matrix that represents T.
  1. Determine the matrix C. [2]
\(L\) is the line whose equation is \(y = x\).
  1. Explain whether or not \(L\) is a line of invariant points under T. [2]
An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
  1. Explain what the value of the determinant of C means about
    [3]
OCR Further Pure Core AS 2020 November Q3
12 marks Moderate -0.3
In this question you must show detailed reasoning. The complex number \(7 - 4\text{i}\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b\text{i}\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3z - 4z^*\) [2]
    2. \((z + 1 - 3\text{i})^2\) [2]
    3. \(\frac{z + 1}{z - 1}\) [2]
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]
  3. The complex number \(\omega\) is such that \(z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))\). Find the following.
    [3]
OCR Further Pure Core AS 2020 November Q4
6 marks Standard +0.3
You are given the system of equations $$a^2x - 2y = 1$$ $$x + b^2y = 3$$ where \(a\) and \(b\) are real numbers.
  1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\). [4]
  2. Explain why the method used in part (a) works for all values of \(a\) and \(b\). [2]
OCR Further Pure Core AS 2020 November Q5
7 marks Challenging +1.8
In this question you must show detailed reasoning. The cubic equation \(5x^3 + 3x^2 - 4x + 7 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta\), \(\beta + \gamma\) and \(\gamma + \alpha\). [7]
OCR Further Pure Core AS 2020 November Q6
5 marks Challenging +1.2
Prove that \(n! > 2^{2n}\) for all integers \(n \geq 9\). [5]
OCR Further Pure Core AS 2020 November Q7
6 marks Standard +0.3
The equations of two intersecting lines are $$\mathbf{r} = \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} -3 \\ 1 \\ -1 \end{pmatrix}$$ where \(a\) is a constant.
  1. Find a vector, \(\mathbf{b}\), which is perpendicular to both lines. [2]
  2. Show that \(\mathbf{b} \cdot \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} = \mathbf{b} \cdot \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix}\). [2]
  3. Hence, or otherwise, find the value of \(a\). [2]
OCR Further Pure Core AS 2020 November Q8
8 marks Challenging +1.8
Two loci, \(C_1\) and \(C_2\), are defined by $$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$ $$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$ where \(d\) is a real number.
  1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C_1\) and \(C_2\). [You may assume that \(C_1 \cap C_2 \neq \emptyset\).] [6]
  2. Explain why the solution found in part (a) is not valid when \(d = 3\). [2]
OCR Further Statistics AS Specimen Q1
5 marks Moderate -0.8
Two music critics, \(P\) and \(Q\), give scores to seven concerts as follows.
Concert1234567
Score by critic \(P\)1211613171614
Score by critic \(Q\)913814181620
  1. Calculate Spearman's rank correlation coefficient, \(r_s\), for these scores. [4]
  2. Without carrying out a hypothesis test, state what your answer tells you about the views of the two critics. [1]
OCR Further Statistics AS Specimen Q2
7 marks Standard +0.8
The probability distribution of a discrete random variable \(W\) is given in the table.
\(w\)0123
\(\mathrm{P}(W = w)\)0.190.18\(x\)\(y\)
Given that \(\mathrm{E}(W) = 1.61\), find the value of \(\mathrm{Var}(3W + 2)\). [7]
OCR Further Statistics AS Specimen Q3
8 marks Standard +0.3
Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below.
BlackNon-black
Men6971
Women3055
Test at the 5\% significance level whether Carl's belief is justified. [8]
OCR Further Statistics AS Specimen Q4
6 marks Challenging +1.2
  1. Four men and four women stand in a random order in a straight line. Determine the probability that no one is standing next to a person of the same gender. [3]
  2. \(x\) men, including Mr Adam, and \(x\) women, including Mrs Adam, are arranged at random in a straight line. Show that the probability that Mr Adam is standing next to Mrs Adam is \(\frac{1}{x}\). [3]
OCR Further Statistics AS Specimen Q5
7 marks Standard +0.3
  1. The random variable \(X\) has the distribution \(\mathrm{Geo}(0.6)\).
    1. Find \(\mathrm{P}(X \geq 8)\). [2]
    2. Find the value of \(\mathrm{E}(X)\). [1]
    3. Find the value of \(\mathrm{Var}(X)\). [1]
  2. The random variable \(Y\) has the distribution \(\mathrm{Geo}(p)\). It is given that \(\mathrm{P}(Y < 4) = 0.986\) correct to 3 significant figures. Use an algebraic method to find the value of \(p\). [3]
OCR Further Statistics AS Specimen Q6
13 marks Moderate -0.3
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
  1. State these two assumptions. [2]
  2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\mathrm{Po}(0.8)\).
    1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
    2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
  2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
  3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\mathrm{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]
OCR Further Statistics AS Specimen Q7
4 marks Standard +0.3
The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.
\(x\)012
Observed frequency\(N - 1\)\(N - 1\)\(N + 2\)
The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]
OCR Further Statistics AS Specimen Q8
10 marks Standard +0.3
The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.
\(x\)9.39.79.79.79.910.210.511.010.610.6
\(y\)480501540552547622655701712708
source: www.tylervigen.com
  1. Find the equation of the regression line of \(y\) on \(x\). [2]
You are given that the product moment correlation coefficient is 0.959.
  1. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. [1]
It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.
  1. Assume that the data is a random sample of all years.
    1. Carry out the test at the 10\% significance level. [6]
    2. Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. [1]
OCR Further Mechanics AS Specimen Q1
6 marks Moderate -0.8
A roundabout in a playground can be modeled as a horizontal circular platform with centre \(O\). The roundabout is free to rotate about a vertical axis through \(O\). A child sits without slipping on the roundabout at a horizontal distance of 1.5 m from \(O\) and completes one revolution in 2.4 seconds.
  1. Calculate the speed of the child. [3]
  2. Find the magnitude and direction of the acceleration of the child. [3]