Questions — OCR MEI (4301 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI AS Paper 2 2021 November Q10
  1. Show that PQ is perpendicular to QR . A circle passes through \(\mathrm { P } , \mathrm { Q }\) and R .
  2. Determine the coordinates of the centre of the circle.
OCR MEI Paper 2 2024 June Q5
  1. In the Printed Answer Booklet, complete the copy of the two-way table.
  2. Calculate the probability that an A-level student selected at random does not study chemistry given that they do not study mathematics.
OCR MEI Paper 2 2020 November Q12
  1. Given that \(q < 2 p\), determine the values of \(p\) and \(q\).
  2. The spinner is spun 10 times. Calculate the probability that exactly one 5 is obtained. Elaine's teacher believes that the probability that the spinner shows a 1 is greater than 0.2 . The spinner is spun 100 times and gives a score of 1 on 28 occasions.
  3. Conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the probability of obtaining a score of 1 is greater than 0.2 .
OCR MEI Further Pure Core AS 2023 June Q7
  1. By expanding \(( \sqrt { 3 } + \mathrm { i } ) ^ { 5 }\), express \(z ^ { 5 }\) in the form \(\mathrm { a } +\) bi where \(a\) and \(b\) are real and exact.
    1. Express \(z\) in modulus-argument form.
    2. Hence find \(z ^ { 5 }\) in modulus-argument form.
    3. Use this result to verify your answers to part (a).
OCR MEI Further Mechanics B AS Specimen Q5
  1. Find an expression for the stiffness of the spring, \(k \mathrm { Nm } ^ { - 1 }\), in terms of \(m , h\) and \(g\). The particle is pushed down a further distance from the equilibrium position and released from rest. At time \(t\) seconds, the displacement of the particle from the equilibrium position of the system is \(y \mathrm {~m}\) in the downward direction, as shown in Fig. 5.3. You are given that \(| y | \leq h\).
  2. Show that the motion of the particle is modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { g y } { h } = 0\).
  3. Find an expression for the period of the motion of the particle.
  4. Would the model for the motion of the particle be valid for large values of \(m\) ? Justify your answer.
OCR MEI Further Pure Core 2020 November Q10
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
OCR MEI Further Pure Core Specimen Q7
  1. Use the Maclaurin series for \(\ln ( 1 + x )\) up to the term in \(x ^ { 3 }\) to obtain an approximation to \(\ln 1.5\).
  2. (A) Find the error in the approximation in part (i).
    (B) Explain why the Maclaurin series in part (i), with \(x = 2\), should not be used to find an approximation to \(\ln 3\).
  3. Find a cubic approximation to \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
  4. (A) Use the approximation in part (iii) to find approximations to
    • ln 1.5 and
    • \(\quad \ln 3\).
      (B) Comment on your answers to part (iv) (A).
OCR MEI Further Pure Core Specimen Q14
  1. Starting with the result $$\mathrm { e } ^ { \mathrm { i } \theta } = \cos \theta + \mathrm { i } \sin \theta$$ show that
    (A) \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta\)
    (B) \(\cos \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right)\).
  2. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity
  3. Using the result in part (i) (B), obtain the values of the constants \(P , Q , R\) and \(S\) in the identity
  4. Show that \(\cos \frac { \pi } { 12 } = \left( \frac { 26 + 15 \sqrt { 3 } } { 64 } \right) ^ { \frac { 1 } { 6 } }\).
  5. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity $$\cos 6 \theta \equiv a \cos ^ { 6 } \theta + b \cos ^ { 4 } \theta + c \cos ^ { 2 } \theta + d$$ $$\cos ^ { 6 } \theta \equiv P \cos 6 \theta + Q \cos 4 \theta + R \cos 2 \theta + S$$
OCR MEI Further Mechanics Major 2023 June Q9
  1. Determine the following, in either order.
    • The components of the velocity of P , parallel and perpendicular to the plane, immediately before P hits the plane at A .
    • The distance OA.
    After P hits the plane at A it continues to move away from O . Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.
  2. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree.
    \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-08_615_759_251_244} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O . The point A lies on the rim of the bowl such that AO makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A , with speed \(u\), where \(\mathrm { u } > \sqrt { \frac { \mathrm { gr } } { 2 } }\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A . The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.
  3. Determine, in terms of \(m , g , u , r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B . The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4 m g\).
  4. Determine, in terms of \(r\), the vertical distance of B above the floor. It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.
  5. Show that \(\mathrm { u } ^ { 2 } > 2 \mathrm { gr }\).
OCR MEI Further Mechanics Major 2020 November Q9
  1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4 a\).
  2. Determine the coefficient of friction between the bar and the ring. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-07_850_835_258_255} \captionsetup{labelformat=empty} \caption{Fig. 10}
    \end{figure} Fig. 10 shows a small bead P of mass \(m\) which is threaded on a smooth thin wire. The wire is in the form of a circle of radius \(a\) and centre O . The wire is fixed in a vertical plane. The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by \(\theta\).
  3. Determine the value of \(\theta\) when the magnitude of the reaction of the wire on the bead is \(\frac { 7 } { 2 } m g\).
  4. Show that the angular velocity of P when OP makes an angle \(\theta\) with the downward vertical is given by \(k \sqrt { \frac { g } { a } } \cos \left( \frac { \theta } { 2 } \right)\), stating the value of the constant \(k\).
  5. Hence determine, in terms of \(g\) and \(a\), the angular acceleration of P when \(\theta\) takes the value found in part (a).
OCR MEI Further Mechanics Major 2021 November Q10
  1. Determine the magnitude of the normal reaction of the wire on P in terms of \(m , g , a , u\) and \(\theta\), when P is between B and C . P collides with a fixed barrier at C . The coefficient of restitution between P and the fixed barrier is \(e\). After this collision P moves back towards B . On the straight portion BA , the motion of P is resisted by a constant horizontal force \(F\).
  2. Show that P will reach A if $$F b \leqslant \frac { 1 } { 2 } m \left[ e ^ { 2 } u ^ { 2 } + k \left( 1 - e ^ { 2 } \right) g a \right] ,$$ where \(k\) is an integer to be determined.
OCR MEI Further Statistics Major 2022 June Q6
  1. Determine a 95\% confidence interval for the mean weight of liquid paraffin in a tub.
  2. Explain whether the confidence interval supports the researcher's belief.
  3. Explain why the sample has to be random in order to construct the confidence interval.
    [0pt]
  4. A 95\% confidence interval for the mean weight in grams of another ingredient in the skin cream is [1.202, 1.398]. This confidence interval is based on a large sample and the unbiased estimate of the population variance calculated from the sample is 0.25 . Find each of the following.
    • The mean of the sample
    • The size of the sample
OCR MEI Further Extra Pure 2020 November Q5
  1. Show that \(\mathbf { f }\) is also an eigenvector of \(\mathbf { A }\).
  2. State the eigenvalue associated with \(\mathbf { f }\). You are now given that \(\mathbf { A }\) represents a reflection in 3-D space.
  3. Explain the significance of \(\mathbf { e }\) and \(\mathbf { f }\) in relation to the transformation that \(\mathbf { A }\) represents.
  4. State the cartesian equation of the plane of reflection of the transformation represented by \(\mathbf { A }\).
OCR MEI Further Pure Core AS 2024 June Q4
4 In this question you must show detailed reasoning. The roots of the cubic equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\) are \(\alpha , \beta\) and \(\gamma\).
  1. Find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { 2 } ( \alpha - 1 ) , \frac { 1 } { 2 } ( \beta - 1 )\) and \(\frac { 1 } { 2 } ( \gamma - 1 )\).
  2. Hence or otherwise solve the equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).
OCR MEI Further Pure Core AS 2024 June Q9
9 In this question you must show detailed reasoning. Find a vector \(\mathbf { v }\) which has the following properties.
  • It is a unit vector.
  • It is parallel to the plane \(2 x + 2 y + z = 10\).
  • It makes an angle of \(45 ^ { \circ }\) with the normal to the plane \(\mathrm { x } + \mathrm { z } = 5\).
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OCR MEI Paper 2 2018 June Q12
12 You must show detailed reasoning in this question. In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English. Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.
OCR MEI Paper 2 2019 June Q15
15 You must show detailed reasoning in this question. The screenshot in Fig. 15 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_387_954_1599_260} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure} The distribution is symmetrical about the line \(x = 35\) and there is a point of inflection at \(x = 31\).
Fifty independent readings of \(X\) are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .
OCR MEI Paper 2 2023 June Q15
15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.
OCR MEI Paper 2 2023 June Q17
17 In this question you must show detailed reasoning. Solve the equation \(2 \sin x + \sec x = 4 \cos x\), where \(- \pi < x < \pi\).
OCR MEI Paper 2 2024 June Q16
16 In this question you must show detailed reasoning. Find the particular solution of the differential equation $$\frac { d y } { d x } = \frac { 9 y } { ( x - 1 ) ( x + 2 ) }$$ given that \(x = 2\) when \(y = 16\). \section*{END OF QUESTION PAPER}
OCR MEI Paper 2 2020 November Q10
10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$
  1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
  2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]
    \(n\)01234567
    \(\mathrm { x } _ { \mathrm { n } }\)10.9585090.9500840.9482610.947860.9477720.9477530.947748
    \captionsetup{labelformat=empty} \caption{Fig. 10.1}
    \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]
    \(x\)\(y\)
    0.9477475\(- 7.79967 \mathrm { E } - 07\)
    0.9477485\(- 2.90821 \mathrm { E } - 06\)
    \(x\)\(y\)
    0.947745\(4.54066 \mathrm { E } - 06\)
    0.947755\(- 1.67417 \mathrm { E } - 05\)
    \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{table}
  3. Consider the information in Fig. 10.1 and Fig. 10.2.
    • Write 4.54066E-06 in standard mathematical notation.
    • State the value of the root as accurately as you can, justifying your answer.
OCR MEI Paper 2 2020 November Q14
14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-16_647_898_404_233} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.
OCR MEI Paper 2 Specimen Q1
1 In this question you must show detailed reasoning. Find the coordinates of the points of intersection of the curve \(y = x ^ { 2 } + x\) and the line \(2 x + y = 4\).
OCR MEI Paper 3 2022 June Q10
10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve \(\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }\) has a stationary point near \(x = 3\).
  • Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
  • Show that this stationary point is a maximum turning point.
OCR MEI Paper 3 2024 June Q3
3 In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = x ^ { 5 }\) and the square \(O A B C\) where the points \(A , B\) and \(C\) have coordinates \(( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\) respectively. The curve cuts the square into two parts.
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-04_658_780_1318_230} Show that the relationship between the areas of the two parts of the square is
\(\frac { \text { Area to left of curve } } { \text { Area below curve } } = 5\).