Questions — OCR (4907 questions)

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OCR C1 2007 January Q9
12 marks Moderate -0.3
  1. Find the equation of the line through \(A\) parallel to the line \(y = 4 x - 5\), giving your answer in the form \(y = m x + c\).
  2. Calculate the length of \(A B\), giving your answer in simplified surd form.
  3. Find the equation of the line which passes through the mid-point of \(A B\) and which is perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2009 June Q9
8 marks Moderate -0.8
  1. Calculate the length of \(A B\).
  2. Find the coordinates of the mid-point of \(A B\).
  3. Find the equation of the line through \(( 1,3 )\) which is parallel to \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 Q9
10 marks Standard +0.3
  1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The straight line \(m\) has gradient 8 and passes through the origin, \(O\).
  2. Write down an equation for \(m\). The lines \(l\) and \(m\) intersect at the point \(R\).
  3. Show that \(O P = O R\).
OCR C2 Q5
8 marks Moderate -0.3
  1. Find the number of sit-ups that Habib will do in the fifth week.
  2. Show that he will do a total of 1512 sit-ups during the first eight weeks. In the \(n\)th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
  3. Find the value of \(n\).
OCR C2 Q7
10 marks Standard +0.3
  1. Show that the common difference is 5 .
  2. Find the 12th term. Another arithmetic sequence has first term -12 and common difference 7 .
    Given that the sums of the first \(n\) terms of these two sequences are equal,
  3. find the value of \(n\).
OCR C4 2008 January Q7
8 marks Standard +0.3
  1. Given that $$A ( \sin \theta + \cos \theta ) + B ( \cos \theta - \sin \theta ) \equiv 4 \sin \theta$$ find the values of the constants \(A\) and \(B\).
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 4 \sin \theta } { \sin \theta + \cos \theta } \mathrm { d } \theta$$ giving your answer in the form \(a \pi - \ln b\).
OCR C4 2008 June Q7
8 marks Moderate -0.3
  1. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
  2. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\).
OCR C4 2009 June Q7
9 marks Moderate -0.3
  1. The vector \(\mathbf { u } = \frac { 3 } { 13 } \mathbf { i } + b \mathbf { j } + c \mathbf { k }\) is perpendicular to the vector \(4 \mathbf { i } + \mathbf { k }\) and to the vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find the values of \(b\) and \(c\), and show that \(\mathbf { u }\) is a unit vector.
  2. Calculate, to the nearest degree, the angle between the vectors \(4 \mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\).
OCR S1 2011 January Q5
10 marks Moderate -0.8
  1. The number of free gifts that Jan receives in a week is denoted by \(X\). Name a suitable probability distribution with which to model \(X\), giving the value(s) of any parameter(s). State any assumption(s) necessary for the distribution to be a valid model. Assume now that your model is valid.
  2. Find
    1. \(\mathrm { P } ( X \leqslant 2 )\),
    2. \(\mathrm { P } ( X = 2 )\).
    3. Find the probability that, in the next 7 weeks, there are exactly 3 weeks in which Jan receives exactly 2 free gifts. 6
    4. The diagram shows 7 cards, each with a digit printed on it. The digits form a 7 -digit number. {}
      3
      2
    7. \multirow[t]{21}{*}{3
    8. }
      9. \multirow[t]{4}{*}{3
    9. }
      10. 3
    10. {}
OCR S1 2012 January Q7
8 marks Moderate -0.8
  1. State a suitable distribution that can be used as a model for \(X\), giving the value(s) of any parameter(s). State also any necessary condition(s) for this distribution to be a good model. Use the distribution stated in part (i) to find
  2. \(\mathrm { P } ( X = 4 )\),
  3. \(\mathrm { P } ( X \geqslant 4 )\).
OCR S2 2007 June Q4
6 marks Moderate -0.3
  1. State two conditions needed for \(X\) to be well modelled by a normal distribution.
  2. It is given that \(X \sim \mathrm {~N} \left( 50.0,8 ^ { 2 } \right)\). The mean of 20 random observations of \(X\) is denoted by \(\bar { X }\). Find \(\mathrm { P } ( \bar { X } > 47.0 )\). 5 The number of system failures per month in a large network is a random variable with the distribution \(\operatorname { Po } ( \lambda )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 2.5\) is carried out by counting \(R\), the number of system failures in a period of 6 months. The result of the test is that \(\mathrm { H } _ { 0 }\) is rejected if \(R > 23\) but is not rejected if \(R \leqslant 23\).
  3. State the alternative hypothesis.
  4. Find the significance level of the test.
  5. Given that \(\mathrm { P } ( R > 23 ) < 0.1\), use tables to find the largest possible actual value of \(\lambda\). You should show the values of any relevant probabilities. 6 In a rearrangement code, the letters of a message are rearranged so that the frequency with which any particular letter appears is the same as in the original message. In ordinary German the letter \(e\) appears \(19 \%\) of the time. A certain encoded message of 20 letters contains one letter \(e\).
  6. Using an exact binomial distribution, test at the \(10 \%\) significance level whether there is evidence that the proportion of the letter \(e\) in the language from which this message is a sample is less than in German, i.e., less than \(19 \%\).
  7. Give a reason why a binomial distribution might not be an appropriate model in this context. 7 Two continuous random variables \(S\) and \(T\) have probability density functions as follows. $$\begin{array} { l l } S : & f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \\ T : & g ( x ) = \begin{cases} \frac { 3 } { 2 } x ^ { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases} \end{array}$$
  8. Sketch on the same axes the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). [You should not use graph paper or attempt to plot points exactly.]
  9. Explain in everyday terms the difference between the two random variables.
  10. Find the value of \(t\) such that \(\mathrm { P } ( T > t ) = 0.2\). 8 A random variable \(Y\) is normally distributed with mean \(\mu\) and variance 12.25. Two statisticians carry out significance tests of the hypotheses \(\mathrm { H } _ { 0 } : \mu = 63.0 , \mathrm { H } _ { 1 } : \mu > 63.0\).
  11. Statistician \(A\) uses the mean \(\bar { Y }\) of a sample of size 23, and the critical region for his test is \(\bar { Y } > 64.20\). Find the significance level for \(A\) 's test.
  12. Statistician \(B\) uses the mean of a sample of size 50 and a significance level of \(5 \%\).
    1. Find the critical region for \(B\) 's test.
    2. Given that \(\mu = 65.0\), find the probability that \(B\) 's test results in a Type II error.
    3. Given that, when \(\mu = 65.0\), the probability that \(A\) 's test results in a Type II error is 0.1365 , state with a reason which test is better. 9 (a) The random variable \(G\) has the distribution \(\mathrm { B } ( n , 0.75 )\). Find the set of values of \(n\) for which the distribution of \(G\) can be well approximated by a normal distribution.
      (b) The random variable \(H\) has the distribution \(\mathrm { B } ( n , p )\). It is given that, using a normal approximation, \(\mathrm { P } ( H \geqslant 71 ) = 0.0401\) and \(\mathrm { P } ( H \leqslant 46 ) = 0.0122\).
      1. Find the mean and standard deviation of the approximating normal distribution.
      2. Hence find the values of \(n\) and \(p\).
OCR S4 2010 June Q3
7 marks Challenging +1.8
  1. Assuming that all rankings are equally likely, show that \(\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }\). The marks of 5 randomly chosen students from School \(A\) and 6 randomly chosen students from School \(B\), who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.
    Rank1234567891011
    School\(A\)\(A\)\(A\)\(B\)\(A\)\(A\)\(B\)\(B\)\(B\)\(B\)\(B\)
  2. For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.
OCR M1 2010 January Q5
11 marks Moderate -0.3
  1. Find the value of \(t\) when \(A\) and \(B\) have the same speed.
  2. Calculate the value of \(t\) when \(B\) overtakes \(A\).
  3. On a single diagram, sketch the \(( t , x )\) graphs for the two cyclists for the time from \(t = 0\) until after \(B\) has overtaken \(A\).
OCR M1 2015 June Q3
8 marks Standard +0.3
  1. Calculate the distance \(A\) cycles, and hence find the period of time for which \(B\) walks before finding the bicycle.
  2. Find \(T\).
  3. Calculate the distance \(A\) and \(B\) each travel.
OCR M2 2008 January Q6
11 marks Standard +0.3
  1. Show that the tension in the string is 4.16 N , correct to 3 significant figures.
  2. Calculate \(\omega\).
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_510_417_1238_904} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lower part of the string is now attached to a point \(R\), vertically below \(P\). \(P B\) makes an angle \(30 ^ { \circ }\) with the vertical and \(R B\) makes an angle \(60 ^ { \circ }\) with the vertical. The bead \(B\) now moves in a horizontal circle of radius 1.5 m with constant speed \(v _ { \mathrm { m } } \mathrm { m } ^ { - 1 }\) (see Fig. 2).
    1. Calculate the tension in the string.
    2. Calculate \(v\).
OCR M2 2006 June Q6
11 marks Standard +0.3
  1. Calculate the tension in the string and hence find the angular speed of \(Q\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_489_1358_1286_392} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The particle \(Q\) on the plane is now fixed to a point 0.2 m from the hole at \(A\) and the particle \(P\) rotates in a horizontal circle of radius 0.2 m (see Fig. 2).
  2. Calculate the tension in the string.
  3. Calculate the speed of \(P\).
OCR M2 2008 June Q5
8 marks Standard +0.3
  1. Show that the distance from the ball to the centre of mass of the toy is 10.7 cm , correct to 1 decimal place.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_312_1051_1509_587} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The toy lies on horizontal ground in a position such that the ball is touching the ground (see Fig. 2). Determine whether the toy is lying in equilibrium or whether it will move to a position where the rod is vertical.
OCR M2 2009 June Q5
11 marks Standard +0.3
  1. Fig. 1 Fig. 1 shows a uniform lamina \(B C D\) in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from \(B\) is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina \(A B D E\) has dimensions \(A B = 12 \mathrm {~cm}\) and \(A E = 6 \mathrm {~cm}\). A single plane object is formed by attaching the rectangular lamina to the lamina \(B C D\) along \(B D\) (see Fig. 2). The mass of \(A B D E\) is 3 kg and the mass of \(B C D\) is 2 kg . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at \(C\) and rests in equilibrium.
  3. Calculate the angle that \(A C\) makes with the vertical.
OCR M2 2015 June Q7
11 marks Standard +0.8
  1. Show that \(\mu = \frac { 2 } { 3 }\). A small object of weight \(a W \mathrm {~N}\) is placed on the ladder at its mid-point and the object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at its lowest point \(A\).
  2. Given that the system is in equilibrium, find the set of possible values of \(a\).
OCR M3 2009 January Q4
10 marks Standard +0.8
  1. Show that \(v ^ { 2 } = 9 + 9.8 \sin \theta\).
  2. Find, in terms of \(\theta\), the radial and tangential components of the acceleration of \(P\).
  3. Show that the tension in the string is \(( 3.6 + 5.88 \sin \theta ) \mathrm { N }\) and hence find the value of \(\theta\) at the instant when the string becomes slack, giving your answer correct to 1 decimal place.
OCR M3 2010 January Q6
13 marks Challenging +1.2
  1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
  2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
  3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).
OCR M3 2007 June Q6
13 marks Challenging +1.8
  1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
  2. Verify that \(P\) and \(Q\) together just reach the safety net.
  3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
  4. State two additional modelling assumptions made when answering this question.
OCR M4 2008 June Q6
15 marks Challenging +1.3
  1. Show that the moment of inertia of the lamina about the axis through \(X\) is \(\frac { 4 } { 3 } m a ^ { 2 }\).
  2. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(\omega ^ { 2 } = \frac { 6 g } { 5 a }\).
  3. At an instant when \(\cos \theta = \frac { 3 } { 5 }\), show that \(R = 0\), and given also that \(\sin \theta = \frac { 4 } { 5 }\) find \(S\) in terms of \(m\) and \(g\).
OCR M4 2010 June Q6
12 marks Challenging +1.2
  1. Show that \(\theta = 0\) is a position of stable equilibrium.
  2. Show that the kinetic energy of the system is \(4 m a ^ { 2 } \dot { \theta } ^ { 2 }\).
  3. By differentiating the energy equation, then making suitable approximations for \(\sin \theta\) and \(\cos \theta\), find the approximate period of small oscillations about the equilibrium position \(\theta = 0\). \section*{[Question 7 is printed overleaf.]}
OCR M4 2013 June Q5
14 marks Standard +0.8
  1. Find the magnitude and bearing of the velocity of \(U\) relative to \(P\).
  2. Find the shortest distance between \(P\) and \(U\) in the subsequent motion.
    (ii) Plane \(Q\) is flying with constant velocity \(160 \mathrm {~ms} ^ { - 1 }\) in the direction which brings it as close as possible to \(U\).
    1. Find the bearing of the direction in which \(Q\) is flying.
    2. Find the shortest distance between \(Q\) and \(U\) in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-3_771_769_262_646} A square frame \(A B C D\) consists of four uniform rods \(A B , B C , C D , D A\), rigidly joined at \(A , B , C , D\). Each rod has mass 0.6 kg and length 1.5 m . The frame rotates freely in a vertical plane about a fixed horizontal axis passing through the mid-point \(O\) of \(A D\). At time \(t\) seconds the angle between \(A D\) and the horizontal, measured anticlockwise, is \(\theta\) radians (see diagram).
      1. Show that the moment of inertia of the frame about the axis through \(O\) is \(3.15 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
      2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 5.6 \sin \theta\).
      3. Deduce that the frame can make small oscillations which are approximately simple harmonic, and find the period of these oscillations. The frame is at rest with \(A D\) horizontal. A couple of constant moment 25 Nm about the axis is then applied to the frame.
      4. Find the angular speed of the frame when it has rotated through 1.2 radians.