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OCR Further Additional Pure 2018 March Q1
5 marks Standard +0.3
1 Determine the solution of the simultaneous linear congruences $$x \equiv 4 ( \bmod 7 ) , \quad x \equiv 25 ( \bmod 41 ) .$$
OCR Further Additional Pure 2018 March Q2
5 marks Standard +0.8
2 Four points \(A , B , C\) and \(D\) have coordinates \(( 1,2,5 ) , ( 3,4 , - 4 ) , ( 6,2,3 )\) and \(( 0,3,7 )\) respectively. Find the volume of tetrahedron \(A B C D\).
OCR Further Additional Pure 2018 March Q3
10 marks Challenging +1.2
3 The surface \(S\) has equation \(z = \frac { x } { y } \sin y + \frac { y } { x } \cos x\) where \(0 < x \leqslant \pi\) and \(0 < y \leqslant \pi\).
  1. Find
OCR Further Additional Pure 2018 March Q4
12 marks Hard +2.3
4
  1. (a) Find all the quadratic residues modulo 11.
    (b) Prove that the equation \(y ^ { 5 } = x ^ { 2 } + 2017\) has no solution in integers \(x\) and \(y\).
  2. In this question you must show detailed reasoning. The numbers \(M\) and \(N\) are given by $$M = 11 ^ { 12 } - 1 \text { and } N = 3 ^ { 2 } \times 5 \times 7 \times 13 \times 61$$ Prove that \(M\) is divisible by \(N\).
OCR Further Additional Pure 2018 March Q5
15 marks Challenging +1.8
5
  1. (a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that \(X _ { 0 } = 12\) and \(X _ { 1 } = 1\).
    (b) Show that the sequence \(\left\{ X _ { n } \right\}\) approaches a geometric sequence as \(n\) increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that \(X _ { n }\) represents the profit (in \(\pounds\) ) at the end of year \(n\).
  2. (a) Determine the number of years taken for the projected profit to exceed one million pounds.
    (b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
  3. (a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
    (b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
    (c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next \(X _ { n }\) in the process.
OCR Further Additional Pure 2018 March Q6
14 marks Challenging +1.8
6 In this question you must show detailed reasoning. It is given that \(I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t\) for integers \(n \geqslant 0\).
  1. Show that \(I _ { 1 } = \frac { 7 } { 3 }\).
  2. Prove that, for \(n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }\). The curve \(C\) is defined parametrically by $$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$ When the curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
  3. Determine
OCR Further Additional Pure 2018 March Q7
14 marks Challenging +1.8
7 The set \(M\) contains all matrices of the form \(\mathbf { X } ^ { n }\), where \(\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)\) and \(n\) is a positive integer.
  1. Show that \(M\) contains exactly 12 elements.
  2. Deduce that \(M\), together with the operation of matrix multiplication, form a cyclic group \(G\).
  3. Determine all the proper subgroups of \(G\). \section*{END OF QUESTION PAPER}
OCR FP1 AS 2018 March Q1
6 marks Moderate -0.8
1
  1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
  2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.
OCR FP1 AS 2018 March Q2
5 marks Moderate -0.5
2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find the values of the following expressions.
    1. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
    2. \(\alpha ^ { 2 } + \beta ^ { 2 }\) \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 3 \\ 3 \\ - 5 \end{array} \right) \end{aligned}$$
OCR FP1 AS 2018 March Q3
6 marks Standard +0.3
$$\begin{aligned} l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4 \\ 2 \\ 5 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 3 \\ - 2 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ a \\ 1 \end{array} \right) + \mu \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) \end{aligned}$$
  1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Determine the value of \(a\).
OCR FP1 AS 2018 March Q4
6 marks Standard +0.8
4 Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geqslant 9\).
OCR FP1 AS 2018 March Q5
7 marks Moderate -0.8
5 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & a ^ { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c c } 0.6 & b & 0 \\ - b & 0.6 & 0 \\ 0 & 0 & 1 \end{array} \right)\).
  1. \(\mathbf { A }\) represents a reflection. Write down the value of \(\operatorname { det } \mathbf { A }\).
  2. Hence find the possible values of \(a\).
  3. \(\mathbf { r }\) is the position vector of a point \(R\). Given that \(\mathbf { A r } = \mathbf { r }\) describe the location of \(R\).
  4. \(\mathbf { B }\) represents a rotation. Write down the value of \(\operatorname { det } \mathbf { B }\).
  5. Hence find the possible values of \(b\).
OCR FP1 AS 2018 March Q6
10 marks Standard +0.3
6 The matrix \(\mathbf { A }\) is given by \(\left( \begin{array} { l l } 1 & 2 \\ 1 & a \end{array} \right)\) and the matrix \(\mathbf { B }\) is given by \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & b \end{array} \right)\).
  1. Find the matrix \(\mathbf { A B }\).
  2. State the conditions on \(a\) and \(b\) for \(\mathbf { A B }\) to be a singular matrix. \(P Q R S\) is a quadrilateral and the vertices \(P , Q , R\) and \(S\) are in clockwise order. A transformation, T , is represented by the matrix \(\mathbf { A B }\).
  3. State the effect on both the area and also the orientation of the image of \(P Q R S\) under T in each of the following cases.
    1. \(\quad a = 1\) and \(b = 1\)
    2. \(\quad a = 2\) and \(b = 3\)
OCR FP1 AS 2018 March Q7
9 marks Standard +0.3
7 In this question you must show detailed reasoning.
  1. Find the square roots of the number \(528 + 46 \mathrm { i }\) giving your answers in the form \(a + b \mathrm { i }\).
  2. \(\quad 3 + 2 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - a x + 78 = 0\), where \(a\) is a real number. Find the value of \(a\).
OCR FP1 AS 2018 March Q8
11 marks Challenging +1.3
8 In this question you must show detailed reasoning. A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by
  • \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\)
  • \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3 \\ 1 \\ 2 \end{array} \right)\).
    1. Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\). for all integers \(n \geqslant 1\).
    2. Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).
\section*{END OF QUESTION PAPER}
OCR FS1 AS 2018 March Q1
7 marks Easy -1.2
1 A learner driver keeps taking the driving test until she passes. The number of attempts taken, up to and including the pass, is denoted by \(X\).
  1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X \sim \operatorname { Geo } ( 0.4 )\).
  2. Find \(\mathrm { P } ( X < 6 )\).
  3. Find \(\mathrm { E } ( X )\).
  4. Find \(\operatorname { Var } ( X )\).
OCR FS1 AS 2018 March Q2
6 marks Standard +0.3
2 The number of calls received by a customer service department in 30 minutes is denoted by \(W\). It is known that \(\mathrm { E } ( W ) = 6.5\).
  1. It is given that \(W\) has a Poisson distribution.
    1. Write down the standard deviation of \(W\).
    2. Find the probability that the total number of calls received in a randomly chosen period of 2 hours is less than 30 .
    3. It is given instead that \(W\) has a uniform distribution on \([ 1 , N ]\). Calculate the value of \(\mathrm { P } ( W > 3 )\).
OCR FS1 AS 2018 March Q3
8 marks Standard +0.8
3 A pack of 40 cards consists of 10 cards in each of four colours: red, yellow, blue and green. The pack is dealt at random into four "hands", each of 10 cards. The hands are labelled North, South, East and West.
  1. Find the probability that West has exactly 3 red cards.
  2. Find the probability that West has exactly 3 red cards, given that East and West have between them a total of exactly 5 red cards.
  3. South has 5 red cards and 5 blue cards. These cards are placed in a row in a random order. Find the probability that the colour of each card is different from the colour of the preceding card.
OCR FS1 AS 2018 March Q4
9 marks Moderate -0.3
4 A spinner has edges numbered \(1,2,3,4\) and 5 . When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table.
Score, \(N\)12345
Probability0.30.20.2\(x\)\(y\)
It is known that \(\mathrm { E } ( N ) = 2.55\).
  1. Find \(\operatorname { Var } ( N )\).
  2. Find \(\mathrm { E } ( 3 N + 2 )\).
  3. Find \(\operatorname { Var } ( 3 N + 2 )\).
OCR FS1 AS 2018 March Q5
4 marks Easy -1.8
5 The speed \(v \mathrm {~ms} ^ { - 1 }\) of a car at time \(t\) seconds after it starts to accelerate was measured at 1 -second intervals. The results are shown in the following diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5843350-52f9-4fed-adf4-86ceb958033f-3_661_1186_1078_443}
  1. State whether \(t\) or \(v\) or neither is a controlled variable. The value of the product moment correlation coefficient \(r\) for the data is 0.987 correct to 3 significant figures.
  2. The speed of the car is converted to miles per hour and the time to minutes. State the value of \(r\) for the converted data.
  3. State the value of Spearman's rank correlation coefficient \(r _ { s }\) for the data.
  4. What information does \(r\) give about the data that is not given by \(r _ { s }\) ?
OCR FS1 AS 2018 March Q6
7 marks Challenging +1.2
6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]
OCR FS1 AS 2018 March Q7
11 marks Standard +0.3
7 The numbers of students taking A levels in three subjects at a school were classified by the year in which they entered the school as follows.
\cline { 2 - 5 } \multicolumn{1}{c|}{}SubjectMathematicsEnglishPhysics
\multirow{3}{*}{
Year of
Entry
}		Year 717167
\cline { 2 - 5 }Year 121325
The Head of the school carries out a significance test at the \(10 \%\) level to test whether subjects taken are independent of year of entry.
  1. Show that in carrying out the test it is necessary to combine columns.
  2. Suggest a reason why it is more sensible to combine the columns for Mathematics and Physics than the columns for Physics and English.
  3. Carry out the test.
  4. State which cell gives the largest contribution to the test statistic.
  5. Interpret your answer to part (iv).
OCR FS1 AS 2018 March Q8
8 marks Challenging +1.2
8 In a competition, entrants have to give ranks from 1 to 7 to each of seven resorts. The correct ranks for the resorts are decided by an expert.
  1. One competitor chooses his ranks randomly. By considering all the possible rankings, find the probability that the value of Spearman's rank correlation coefficient \(r _ { s }\) between the competitor's ranks and the expert's ranks is at least \(\frac { 27 } { 28 }\).
  2. Another competitor ranks the seven resorts. A significance test is carried out to test whether there is evidence that this competitor is merely guessing the rank order of the seven resorts. The critical region is \(r _ { s } \geqslant \frac { 27 } { 28 }\). State the significance level of the test. \section*{END OF QUESTION PAPER}
OCR FM1 AS 2018 March Q1
7 marks Easy -1.2
1 A particle \(P\) of mass 2.4 kg is attached to one end of a light inextensible string of length 1.4 m . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal table. \(P\) moves on the table at constant speed along a circular path with \(O\) at its centre. The magnitude of the tension in the string is 21 N .
  1. (a) Find the magnitude of the acceleration of \(P\).
    (b) State the direction of the acceleration of \(P\).
  2. Find the speed of \(P\).
  3. Find the time taken for \(P\) to complete a single revolution.
OCR FM1 AS 2018 March Q2
5 marks Standard +0.3
2 A pump is pumping still water from the base of a well at a constant rate of 300 kg per minute. The well is 4.5 m deep and water is released from the pump at ground level in a horizontal jet with a speed of \(6.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Ignoring any energy losses due to resistance, calculate the power generated by the pump.