Questions — OCR (4619 questions)

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OCR Further Pure Core 1 2018 March Q4
7 marks Standard +0.8
4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).
OCR Further Pure Core 1 2018 March Q5
6 marks Challenging +1.2
5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$
OCR Further Pure Core 1 2018 March Q6
6 marks Moderate -0.5
6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point \(O\). Initially the mass hangs at rest vertically below \(O\). The mass is then pulled to one side with the string taut and released from rest. \(\theta\) is the angle, in radians, that the string makes with the vertical through \(O\) at time \(t\) seconds and \(\theta\) may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$
  1. Write down the general solution to this differential equation.
  2. Initially the pendulum is released from rest at an angle of \(\theta _ { 0 }\). Find the particular solution to the equation in this case.
  3. State any limitations on the model.
OCR Further Pure Core 1 2018 March Q7
8 marks Challenging +1.2
7
  1. Using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$ \section*{(ii) In this question you must show detailed reasoning.} By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3 .$$ Give your answer in the form \(\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }\) where \(a , b\) and \(c\) are integers.
OCR Further Pure Core 1 2018 March Q8
7 marks Standard +0.8
8 You are given that \(\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }\) where \(a\) and \(b\) are positive constants. The first three terms in the Maclaurin expansion of \(\mathrm { f } ( x )\) are \(1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }\).
  1. Find the value of \(a\) and the value of \(b\).
  2. Explain if there is any restriction on the value of \(x\) in order for the expansion to be valid.
OCR Further Pure Core 1 2018 March Q9
8 marks Standard +0.8
9 In an experiment, at time \(t\) minutes there is \(Q\) grams of substance present.
It is known that the substance decays at a rate that is proportional to \(1 + Q ^ { 2 }\). Initially there are 100 grams of the substance present and after 100 minutes there are 50 grams present. Find the amount of the substance present after 400 minutes.
OCR Further Pure Core 1 2018 March Q10
14 marks Challenging +1.8
10
  1. (a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form $$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$ The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
    (b) Explain why the curve is symmetrical in the \(x\)-axis.
    (c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
  2. The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid formed. \section*{END OF QUESTION PAPER}
OCR Further Pure Core 2 2018 March Q1
8 marks Standard +0.3
1 Plane \(\Pi\) has equation \(3 x - y + 2 z = 33\). Line \(l\) has the following vector equation. $$l : \quad \mathbf { r } = \left( \begin{array} { l } 1
0
5 \end{array} \right) + \lambda \left( \begin{array} { l }
OCR Further Pure Core 2 2018 March Q3
3 marks Moderate -0.5
3 \end{array} \right)$$
  1. Find the acute angle between \(\Pi\) and \(l\).
  2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
  3. \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\). 2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
  4. Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
  5. Plot the following on the Argand diagram in the Printed Answer Booklet.
    • \(z\)
    • \(z ^ { 2 }\)
    • State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
    • State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
    3 In this question you must show detailed reasoning. Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\).
OCR Further Pure Core 2 2018 March Q5
5 marks Moderate -0.3
5 \end{array} \right) + \lambda \left( \begin{array} { l } 2
2
3 \end{array} \right)$$
  1. Find the acute angle between \(\Pi\) and \(l\).
  2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
  3. \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\). 2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
  4. Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
  5. Plot the following on the Argand diagram in the Printed Answer Booklet.
    • \(z\)
    • \(z ^ { 2 }\)
    • State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
    • State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
    3 In this question you must show detailed reasoning. Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\). 4 You are given that the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x + 4 = 0\) has three roots, \(\alpha , \beta\) and \(\gamma\).
    By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\). 5 In this question you must show detailed reasoning.
    An ant starts from a fixed point \(O\) and walks in a straight line for 1.5 s . Its velocity, \(v \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), can be modelled by \(v = \frac { 1 } { \sqrt { 9 - t ^ { 2 } } }\). By finding the mean value of \(v\) in \(0 \leqslant t \leqslant 1.5\), deduce the average velocity of the ant.
OCR Further Pure Core 2 2018 March Q6
12 marks Standard +0.8
6 In this question you must show detailed reasoning.
  1. Find the coordinates of all stationary points on the graph of \(y = 6 \sinh ^ { 2 } x - 13 \cosh x\), giving your answers in an exact, simplified form.
  2. By finding the second derivative, classify the stationary points found in part (i).
OCR Further Pure Core 2 2018 March Q7
12 marks
7 In the following set of simultaneous equations, \(a\) and \(b\) are constants. $$\begin{aligned} 3 x + 2 y - z & = 5 \\ 2 x - 4 y + 7 z & = 60 \\ a x + 20 y - 25 z & = b \end{aligned}$$
  1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\).
  2. Determine the value of \(a\) for which there is no unique solution for \(x , y\) and \(z\).
  3. (a) Find the values of \(\alpha\) and \(\beta\) for which \(\alpha ( 2 y - z ) + \beta ( - 4 y + 7 z ) = 20 y - 25 z\) for any \(y\) and \(z\).
    (b) Hence, for the case where there is no unique solution for \(x , y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions.
    (c) When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations.
OCR Further Pure Core 2 2018 March Q8
12 marks Challenging +1.8
8 In this question you must show detailed reasoning.
Show that \(\int _ { 0 } ^ { 2 } \frac { 2 x ^ { 2 } + 3 x - 1 } { x ^ { 3 } - 3 x ^ { 2 } + 4 x - 12 } \mathrm {~d} x = \frac { 3 } { 8 } \pi - \ln 9\).
OCR Further Pure Core 2 2018 March Q9
14 marks Challenging +1.2
9 In this question you must show detailed reasoning.
  1. Show that \(\mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } = 2 \mathrm { i } \sin \theta\).
  2. Hence, show that \(\frac { 2 } { \mathrm { e } ^ { 2 \mathrm { i } \theta } - 1 } = - ( 1 + \mathrm { i } \cot \theta )\).
  3. Two series, \(C\) and \(S\), are defined as follows. $$\begin{aligned} & C = 2 + 2 \cos \frac { \pi } { 10 } + 2 \cos \frac { \pi } { 5 } + 2 \cos \frac { 3 \pi } { 10 } + 2 \cos \frac { 2 \pi } { 5 } \\ & S = 2 \sin \frac { \pi } { 10 } + 2 \sin \frac { \pi } { 5 } + 2 \sin \frac { 3 \pi } { 10 } + 2 \sin \frac { 2 \pi } { 5 } \end{aligned}$$ By considering \(C + \mathrm { i } S\), find a simplified expression for \(C\) in terms of only integers and \(\cot \frac { \pi } { 20 }\).
  4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. \section*{END OF QUESTION PAPER}
OCR Further Statistics 2018 March Q1
6 marks Standard +0.3
1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).
  1. Find
    (a) \(\mathrm { P } ( X + Y ) > 40\),
    (b) \(\operatorname { Var } ( 2 X - Y )\).
  2. State a necessary assumption for your calculations in part (i) to be valid.
OCR Further Statistics 2018 March Q2
5 marks Challenging +1.2
2 A continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find \(E ( \sqrt { } X )\).
OCR Further Statistics 2018 March Q3
9 marks Standard +0.8
3 Adila has a pack of 50 cards.
  1. Each of the 50 cards is numbered with a different integer from 1 to 50 . Adila selects 5 cards at random without replacement.
    (a) Find the probability that exactly 3 of the 5 cards have numbers which are 10 or less.
    (b) Adila arranges the 5 cards in a line in a random order. Find the probability that the 5 cards are arranged in numerically increasing order. 10 of the 50 cards are blue and the rest are green.
  2. Adila randomly selects three sets of 10 cards each, without replacement. The sets are labelled \(A , B\) and \(C\). Given that \(A\) contains 3 blue cards and 7 green cards, find the probability that \(B\) contains exactly 2 blue cards and \(C\) contains exactly 3 blue cards.
OCR Further Statistics 2018 March Q4
9 marks Moderate -0.8
4 Sheena travels to school by bus. She records the number of minutes, \(T\), that her bus is late on each of 32 days. She believes that on average \(T\) is greater than 5, and she carries out a significance test at the \(5 \%\) level.
  1. State a condition needed for a Wilcoxon test to be valid in this case. Assume now that this condition is satisfied.
  2. State an advantage of using a Wilcoxon test rather than a sign test.
  3. Calculate the critical region for the test, in terms of a variable which should be defined.
OCR Further Statistics 2018 March Q5
8 marks Standard +0.3
5 A spinner has 5 edges. Each edge is numbered with a different integer from 1 to 5 . When the spinner is spun, it is equally likely to come to rest on any one of the edges. The spinner is spun 100 times. The number of times on which the spinner comes to rest on the edge numbered 5 is denoted by \(X\).
  1. \(\mathrm { E } ( X )\),
  2. \(\operatorname { Var } ( X )\).
    1. Write down
    2. Use a normal distribution with the same mean and variance as in your answers to part (i) to estimate the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\).
    3. Use the binomial distribution to find exactly the smallest value of \(n\) such that \(\mathrm { P } ( X \geqslant n ) < 0.02\). Show the values of all relevant calculations.
OCR Further Statistics 2018 March Q6
10 marks Standard +0.3
6 The captain of a sports team analyses the team's results according to the weather conditions, classified as "sunny" and "not sunny". The frequencies are shown in the following table.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Results
\cline { 3 - 5 } \multicolumn{2}{c|}{}WinDrawLose
\multirow{2}{*}{Weather}Sunny1235
\cline { 2 - 5 }Not sunny81210
  1. Test at the \(5 \%\) significance level whether the team's performances are associated with weather conditions.
  2. (a) Identify the cell that gives the largest contribution to the test statistic.
    (b) Interpret your answer to part (ii)(a).
OCR Further Statistics 2018 March Q7
9 marks Standard +0.3
7 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } x \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(\mathrm { f } ( x )\) satisfies the conditions for a probability density function.
  2. Find the value of \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 15 } { 16 }\).
OCR Further Statistics 2018 March Q8
11 marks Challenging +1.2
8 At a wine-tasting competition, two judges give marks out of 100 to 7 wines as follows.
Wine\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Judge I86.387.587.688.889.489.990.5
Judge II85.388.182.787.789.089.491.5
  1. A spectator claims that there is a high level of agreement between the rank orders of the marks given by the two judges. Test the spectator's claim at the \(1 \%\) significance level.
  2. A competitor ranks the wines in a random order. The value of Spearman's rank correlation coefficient between the competitor and Judge I is \(r _ { s }\).
    (a) Find the probability that \(r _ { s } = 1\).
    (b) Show that \(r _ { s }\) cannot take the value \(\frac { 55 } { 56 }\).
OCR Further Statistics 2018 March Q9
8 marks Challenging +1.2
9 The values of a set of bivariate data \(\left( x _ { i } , y _ { i } \right)\) can be summarised by $$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$ Ten independent observations of \(Y\) are obtained, all corresponding to \(x = 20\). It may be assumed that the variance of \(Y\) is 1.9 , independently of the value of \(x\). Find a \(95 \%\) confidence interval for the mean \(\bar { Y }\) of the 10 observations of \(Y\). \section*{END OF QUESTION PAPER}
OCR Further Mechanics 2018 March Q1
6 marks Standard +0.3
1 A particle \(P\) of mass 4.2 kg is free to move along the \(x\)-axis which is horizontal. \(P\) is projected from the origin, \(O\), in the positive \(x\) direction with a speed of \(2 \mathrm {~ms} ^ { - 1 }\). As \(P\) moves between \(O\) and the point \(A\) where \(x = 4\), it is acted upon by a variable force of magnitude \(\left( 12 x - 3 x ^ { 2 } \right) \mathrm { N }\) acting in the direction \(O A\).
  1. Calculate the work done by the force as \(P\) moves from \(O\) to \(A\).
  2. Hence, assuming that no other force acts on \(P\), calculate the speed of \(P\) at \(A\).
OCR Further Mechanics 2018 March Q2
10 marks Standard +0.8
2 The region bounded by the \(x\)-axis and the curve \(y = a x ( 2 - x )\), where \(a\) is a constant, is occupied by a uniform lamina \(L _ { 1 }\) (see Fig. 1). Units on the axes are metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a8c9d007-e67f-4637-9e74-630ba9a91442-2_385_349_906_849} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Write down the value of the \(x\)-coordinate of the centre of mass of \(L _ { 1 }\).
  2. Show that the \(y\)-coordinate of the centre of mass of \(L _ { 1 }\) is \(\frac { 2 } { 5 } a\). The mass of \(L _ { 1 }\) is \(M \mathrm {~kg}\). A uniform rectangular lamina of width 2 m and height \(a \mathrm {~m}\) is made from a different material from that of \(L _ { 1 }\) and has a mass of \(2 M \mathrm {~kg}\). A new lamina, \(L _ { 2 }\), is formed by joining the straight edge of \(L _ { 1 }\) to an edge of the rectangular lamina of length 2 m (see Fig. 2). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a8c9d007-e67f-4637-9e74-630ba9a91442-2_547_273_1772_890} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(L _ { 2 }\) is freely suspended from one of its right-angled corners and hangs in equilibrium with its edge of length 2 m making an angle of \(20 ^ { \circ }\) with the horizontal.
  3. Find the value of \(a\), giving your answer correct to 3 significant figures.