Questions PURE (178 questions)

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Edexcel PURE 2024 October Q8
Standard +0.3
  1. Relative to a fixed origin \(O\)
  • the point \(A\) has coordinates \(( - 10,5 , - 4 )\)
  • the point \(B\) has coordinates \(( - 6,4 , - 1 )\)
The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).
  1. Find a vector equation for \(l _ { 1 }\) The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\mu\) is a scalar parameter.
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at \(B\),
  2. find the value of \(p\) and the value of \(q\). The acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
  3. Find the exact value of \(\cos \theta\) Given that the point \(C\) lies on \(l _ { 2 }\) such that \(A C\) is perpendicular to \(l _ { 2 }\)
  4. find the exact length of \(A C\), giving your answer as a surd.
Edexcel PURE 2024 October Q9
Challenging +1.2
  1. (a) Express \(\frac { 1 } { x ( 2 x - 1 ) }\) in partial fractions.
The height above ground, \(h\) metres, of a carriage on a fairground ride is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 1 } { 50 } h ( 2 h - 1 ) \cos \left( \frac { t } { 10 } \right)$$ where \(t\) seconds is the time after the start of the ride.
Given that, at the start of the ride, the carriage is 2.5 m above ground,
(b) solve the differential equation to show that, according to the model, $$h = \frac { 5 } { 10 - 8 \mathrm { e } ^ { k \sin \left( \frac { t } { 10 } \right) } }$$ where \(k\) is a constant to be found.
(c) Hence find, according to the model, the time taken for the carriage to reach its maximum height above ground for the 3rd time.
Give your answer to the nearest second.
(Solutions relying entirely on calculator technology are not acceptable.)
Edexcel PURE 2024 October Q10
Standard +0.8
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa121449-492f-4737-a9eb-a14a62ced47b-30_563_602_255_735} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve with parametric equations $$x = 3 t ^ { 2 } \quad y = \sin t \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve and the \(x\)-axis.
  1. Show that the area of \(R\) is $$k \int _ { 0 } ^ { \frac { \pi } { 2 } } t \sin ^ { 2 } t \cos t \mathrm {~d} t$$ where \(k\) is a constant to be found.
  2. Hence, using algebraic integration, find the exact area of \(R\), giving your answer in the form $$p \pi + q$$ where \(p\) and \(q\) are constants.
OCR PURE Q1
5 marks Easy -1.3
In this question you must show detailed reasoning.
  1. Express \(3^{\frac{1}{2}}\) in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [2]
  2. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
OCR PURE Q2
4 marks Easy -1.2
  1. The equation \(x^2 + 3x + k = 0\) has repeated roots. Find the value of the constant \(k\). [2]
  2. Solve the inequality \(6 + x - x^2 > 0\). [2]
OCR PURE Q3
6 marks Moderate -0.8
  1. Solve the equation \(\sin^2\theta = 0.25\) for \(0° \leq \theta < 360°\). [3]
  2. In this question you must show detailed reasoning. Solve the equation \(\tan 3\phi = \sqrt{3}\) for \(0° \leq \phi < 90°\). [3]
OCR PURE Q4
9 marks Moderate -0.8
  1. It is given that \(y = x^2 + 3x\).
    1. Find \(\frac{dy}{dx}\). [2]
    2. Find the values of \(x\) for which \(y\) is increasing. [2]
  2. Find \(\int(3 - 4\sqrt{x})dx\). [5]
OCR PURE Q5
5 marks Standard +0.8
\(N\) is an integer that is not divisible by 3. Prove that \(N^2\) is of the form \(3p + 1\), where \(p\) is an integer. [5]
OCR PURE Q6
7 marks Easy -1.2
Sketch the following curves.
  1. \(y = \frac{2}{x}\) [2]
  2. \(y = x^3 - 6x^2 + 9x\) [5]
OCR PURE Q7
7 marks Moderate -0.8
\(OABC\) is a parallelogram with \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(P\) is the midpoint of \(AC\). \includegraphics{figure_7}
  1. Find the following in terms of \(\mathbf{a}\) and \(\mathbf{c}\), simplifying your answers.
    1. \(\overrightarrow{AC}\) [1]
    2. \(\overrightarrow{OP}\) [2]
  2. Hence prove that the diagonals of a parallelogram bisect one another. [4]
OCR PURE Q8
6 marks Standard +0.8
In this question you must show detailed reasoning. The lines \(y = \frac{1}{2}x\) and \(y = -\frac{1}{2}x\) are tangents to a circle at \((2, 1)\) and \((-2, 1)\) respectively. Find the equation of the circle in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are constants. [6]
OCR PURE Q9
2 marks Easy -2.0
Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.
  1. State an advantage of using a stratified sample rather than a simple random sample. [1]
  2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK. [1]
OCR PURE Q10
5 marks Moderate -0.3
The probability distribution of a random variable \(X\) is given in the table.
\(x\)0246
P\((X = x)\)\(\frac{3}{8}\)\(\frac{5}{16}\)\(4p\)\(p\)
  1. Find the value of \(p\). [2]
  2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0. [3]
OCR PURE Q11
5 marks Moderate -0.3
The probability that Janice sees a kingfisher on any particular day is 0.3. She notes the number, \(X\), of days in a week on which she sees a kingfisher.
  1. State one necessary condition for \(X\) to have a binomial distribution. [1]
Assume now that \(X\) has a binomial distribution.
  1. Find the probability that, in a week, Janice sees a kingfisher on exactly 2 days. [1]
Each week Janice notes the number of days on which she sees a kingfisher.
  1. Find the probability that Janice sees a kingfisher on exactly 2 days in a week during at least 4 of 6 randomly chosen weeks. [3]
OCR PURE Q12
7 marks Standard +0.3
It is known that 20% of plants of a certain type suffer from a fungal disease, when grown under normal conditions. Some plants of this type are grown using a new method. A random sample of 250 of these plants is chosen, and it is found that 36 suffer from the disease. Test, at the 2% significance level, whether there is evidence that the new method reduces the proportion of plants which suffer from the disease. [7]
OCR PURE Q13
7 marks Easy -2.5
The radar diagrams illustrate some population figures from the 2011 census results. \includegraphics{figure_13} Each radius represents an age group, as follows:
Radius123456
Age group0-1718-2930-4445-5960-7475+
The distance of each dot from the centre represents the number of people in the relevant age group.
  1. The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities. [2]
  2. Approximately how many people aged 45 to 59 were there in Liverpool? [1]
  3. State the main two differences between the age profiles of the two Local Authorities. [2]
  4. James makes the following claim. "Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland." Use the radar diagrams to give a justification for this claim. [2]
OCR PURE Q1
5 marks Moderate -0.8
  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]
OCR PURE Q2
8 marks Moderate -0.3
  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]
OCR PURE Q3
4 marks Moderate -0.8
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]
OCR PURE Q4
4 marks Easy -1.2
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]
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]