Questions — OCR (4907 questions)

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OCR Further Pure Core 2 Specimen Q1
4 marks Standard +0.3
Find \(\sum_{r=1}^{n}(r+1)(r+5)\). Give your answer in a fully factorised form. [4]
OCR Further Pure Core 2 Specimen Q2
4 marks Standard +0.8
In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac{8}{\sqrt{16+x^3}}\), the \(x\)-axis and the lines \(x=0\) and \(x=4\). Region \(R\) is rotated through \(360°\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
OCR Further Pure Core 2 Specimen Q3
4 marks Standard +0.3
\begin{enumerate}[label=(\roman*)] \item Find \(\sum_{r=1}^{n}\left(\frac{1}{r}-\frac{1}{r+2}\right)\). [3] \item What does the sum in part (i) tend to as \(n \to \infty\)? Justify your answer. [1]
OCR Further Pure Core 2 Specimen Q4
5 marks Challenging +1.2
It is given that \(\frac{5x^2+x+12}{x^2+kx} = \frac{A}{x} + \frac{Bx+C}{x^2+k}\) where \(k\), \(A\), \(B\) and \(C\) are positive integers. Determine the set of possible values of \(k\). [5]
OCR Further Pure Core 2 Specimen Q5
4 marks Standard +0.8
In this question you must show detailed reasoning. Evaluate \(\int_0^{\infty} 2xe^{-x} dx\). [You may use the result \(\lim_{x \to \infty} xe^{-x} = 0\).] [4]
OCR Further Pure Core 2 Specimen Q6
8 marks Standard +0.3
The equation of a plane \(\Pi\) is \(x-2y-z=30\). \begin{enumerate}[label=(\roman*)] \item Find the acute angle between the line \(\mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} -5 \\ 3 \\ 2 \end{pmatrix}\) and \(\Pi\). [4] \item Determine the geometrical relationship between the line \(\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}\) and \(\Pi\). [4]
OCR Further Pure Core 2 Specimen Q7
7 marks Challenging +1.8
\begin{enumerate}[label=(\roman*)] \item Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2x \sin 4x\) up to and including the term in \(x^3\). [4] \item Hence find, in exact surd form, an approximation to the least positive root of the equation \(2\sin x \sin 2x \sin 4x = x\). [3]
OCR Further Pure Core 2 Specimen Q8
8 marks Challenging +1.2
The equation of a curve is \(y = \cosh^2 x - 3\sinh x\). Show that \(\left(\ln\left(\frac{3+\sqrt{13}}{2}\right), -\frac{5}{4}\right)\) is the only stationary point on the curve. [8]
OCR Further Pure Core 2 Specimen Q9
6 marks Standard +0.8
A curve has equation \(x^4 + y^4 = x^2 + y^2\), where \(x\) and \(y\) are not both zero. \begin{enumerate}[label=(\roman*)] \item Show that the equation of the curve in polar coordinates is \(r^2 = \frac{2}{2-\sin^2 2\theta}\). [4] \item Deduce that no point on the curve \(x^4 + y^4 = x^2 + y^2\) is further than \(\sqrt{2}\) from the origin. [2]
OCR Further Pure Core 2 Specimen Q10
8 marks Hard +2.3
Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos r\theta\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos 10\theta\). [8]
OCR Further Pure Core 2 Specimen Q11
17 marks Challenging +1.2
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x + 0.2y \text{ and } \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substances \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
    1. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). [2]
    2. Comment on this result in the context of the industrial process. [2]
  3. Express \(y\) in terms of \(t\). [5]
  4. Determine the maximum amount of substance \(Y\) present during the process. [3]
  5. How long does it take to produce 9 kg of substance \(Z\)? [2]
OCR Further Statistics 2020 November Q1
4 marks Moderate -0.8
The continuous random variable \(X\) has the distribution \(\text{N}(\mu, 30)\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a 95\% confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures. [4]
OCR Further Statistics 2020 November Q2
8 marks Standard +0.3
A book collector compared the prices of some books, \(£x\), when new in 1972 and the prices of copies of the same books, \(£y\), on a second-hand website in 2018. The results are shown in Table 1 and are summarised below the table.
BookABCDEFGHIJKL
\(x\)0.950.650.700.900.551.401.500.501.150.350.200.35
\(y\)6.067.002.005.874.005.367.192.503.008.291.372.00
Table 1 \(n = 12, \Sigma x = 9.20, \Sigma y = 54.64, \Sigma x^2 = 8.9950, \Sigma y^2 = 310.4572, \Sigma xy = 46.0545\)
  1. It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381, correct to 3 significant figures.
    1. State what this information tells you about a scatter diagram illustrating the data. [1]
    2. Test at the 5\% significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018. [5]
  2. The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books. [2]
OCR Further Statistics 2020 November Q3
9 marks Challenging +1.2
Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B. She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219.
  1. Test at the 10\% significance level whether route A has shorter journey times than route B. [8]
  2. State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid. [1]
OCR Further Statistics 2020 November Q4
7 marks Standard +0.8
The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\text{E}(3X) = 30\) and \(\text{Var}(3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]
OCR Further Statistics 2020 November Q5
11 marks Challenging +1.2
26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.
  1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
  2. All 26 cards are arranged in a line, in random order.
    1. Show that the probability that all the vowels are next to one another is \(\frac{1}{2990}\). [3]
    2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]
OCR Further Statistics 2020 November Q6
11 marks Standard +0.3
The numbers of CD players sold in a shop on three consecutive weekends were 7, 6 and 2. It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).
  1. How appropriate is the Poisson distribution as a model for \(X\)? [2]
Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).
  1. Find
    1. P\((X = 6)\), [2]
    2. P\((X \geqslant 8)\). [2]
The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).
  1. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive. [3]
  2. State an assumption needed for your answer to part (c) to be valid. [1]
  3. Give a reason why the assumption in part (d) may not be valid in practice. [1]
OCR Further Statistics 2020 November Q7
10 marks Standard +0.3
A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, \(X\), up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency \(f\) with which each value of \(X\) is obtained. His results are shown in Table 1, together with the values of \(xf\).
\(x\)123456\(\geqslant 7\)Total
Frequency \(f\)2015913101023100
\(xf\)203027525060161400
Table 1
  1. State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s). [3]
Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.
\(x\)123456\(\geqslant 7\)
Observed frequency \(O\)2015913101023
Expected frequency \(E\)2518.7514.06310.5477.9105.93317.798
\((O - E)^2/E\)10.751.8230.5710.5522.7891.520
Table 2
  1. Show how the expected frequency corresponding to \(x \geqslant 7\) was obtained. [2]
  2. Carry out the test. [5]
OCR Further Statistics 2020 November Q8
15 marks Standard +0.8
The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{k}{x^n} & x \geqslant 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1.
  1. Find \(k\) in terms of \(n\). [3]
    1. When \(n = 4\), find the cumulative distribution function of \(X\). [3]
    2. Hence determine P\((X > 7 | X > 5)\) when \(n = 4\). [4]
  3. Determine the values of \(n\) for which Var\((X)\) is not defined. [5]
OCR Further Mechanics 2023 June Q1
8 marks Standard +0.3
One end of a light inextensible string of length \(0.8\) m is attached to a particle \(P\) of mass \(m\) kg. The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3\) m s\(^{-1}\) so that it moves in a vertical circular path with centre \(O\) (see diagram). \includegraphics{figure_1} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac{1}{3}\pi\) radians with the downward vertical through \(O\).
  1. Show that at this instant the speed of \(P\) is \(4.5\) m s\(^{-1}\). [3]
  2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant. [3]
  3. Find the magnitude of the tangential acceleration of \(P\) at this instant. [2]
OCR Further Mechanics 2023 June Q2
11 marks Standard +0.3
Materials have a measurable property known as the Young's Modulus, \(E\). If a force is applied to one face of a block of the material then the material is stretched by a distance called the extension. Young's modulus is defined as the ratio \(\frac{\text{Stress}}{\text{Strain}}\) where Stress is defined as the force per unit area and Strain is the ratio of the extension of the block to the length of the block.
  1. Show that Strain is a dimensionless quantity. [1]
  2. By considering the dimensions of both Stress and Strain determine the dimensions of \(E\). [2]
It is suggested that the speed of sound in a material, \(c\), depends only upon the value of Young's modulus for the material, \(E\), the volume of the material, \(V\), and the density (or mass per unit volume) of the material, \(\rho\).
  1. Use dimensional analysis to suggest a formula for \(c\) in terms of \(E\), \(V\) and \(\rho\). [5]
  2. The speed of sound in a certain material is \(500\) m s\(^{-1}\).
    1. Use your formula from part (c) to predict the speed of sound in the material if the value of Young's modulus is doubled but all other conditions are unchanged. [1]
    2. With reference to your formula from part (c), comment on the effect on the speed of sound in the material if the volume is doubled but all other conditions are unchanged. [1]
  3. Suggest one possible limitation caused by using dimensional analysis to set up the model in part (c). [1]
OCR Further Mechanics 2023 June Q3
7 marks Challenging +1.2
Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is \(5\) kg and the mass of \(B\) is \(3\) kg. At the instant before they collide, • the velocity of \(A\) is \(4\) m s\(^{-1}\) at an angle of \(60°\) to the line of centres, • the velocity of \(B\) is \(6\) m s\(^{-1}\) along the line of centres (see diagram). \includegraphics{figure_3} The coefficient of restitution for collisions between the two discs is \(\frac{3}{4}\). Determine the angle that the velocity of \(A\) makes with the line of centres after the collision. [7]
OCR Further Mechanics 2023 June Q4
9 marks Standard +0.3
\(ABCD\) is a uniform lamina in the shape of a kite with \(BA = BC = 0.37\) m, \(DA = DC = 0.91\) m and \(AC = 0.7\) m (see diagram). The centre of mass of \(ABCD\) is \(G\). \includegraphics{figure_4}
  1. Explain why \(G\) lies on \(BD\). [1]
  2. Show that the distance of \(G\) from \(B\) is \(0.36\) m. [4]
The lamina \(ABCD\) is freely suspended from the point \(A\).
  1. Determine the acute angle that \(CD\) makes with the horizontal, stating which of \(C\) or \(D\) is higher. [4]
OCR Further Mechanics 2023 June Q5
13 marks Challenging +1.3
A particle \(P\) of mass \(2\) kg moves along the \(x\)-axis. At time \(t = 0\), \(P\) passes through the origin \(O\) with speed \(3\) m s\(^{-1}\). At time \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) m s\(^{-1}\), where \(t \geqslant 0\), \(x \geqslant 0\) and \(v \geqslant 0\). While \(P\) is in motion the only force acting on \(P\) is a resistive force \(F\) of magnitude \((v^2 + 1)\) N acting in the negative \(x\)-direction.
  1. Find an expression for \(v\) in terms of \(x\). [5]
  2. Determine the distance travelled by \(P\) while its speed drops from \(3\) m s\(^{-1}\) to \(2\) m s\(^{-1}\). [2]
Particle \(Q\) is identical to particle \(P\). At a different time, \(Q\) is moving along the \(x\)-axis under the influence of a single constant resistive force of magnitude \(1\) N. When \(t' = 0\), \(Q\) is at the origin and its speed is \(3\) m s\(^{-1}\).
  1. By comparing the motion of \(P\) with the motion of \(Q\) explain why \(P\) must come to rest at some finite time when \(t < 6\) with \(x < 9\). [3]
  2. Sketch the velocity-time graph for \(P\). You do not need to indicate any values on your sketch. [1]
  3. Determine the maximum displacement of \(P\) from \(O\) during \(P\)'s motion. [2]
OCR Further Mechanics 2023 June Q6
12 marks Challenging +1.2
A particle \(P\) of mass \(3\) kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = (32\sinh(2t))\mathbf{i} + (32\cosh(2t) - 257)\mathbf{j}.$$
    1. By considering kinetic energy, determine the work done by \(\mathbf{F}\) over the interval \(0 \leqslant t \leqslant \ln 2\). [5]
    2. Explain the significance of the sign of the answer to part (a)(i). [1]
  2. Determine the rate at which \(\mathbf{F}\) is working at the instant when \(P\) is moving parallel to the \(\mathbf{i}\)-direction. [6]