Questions — OCR MEI (4333 questions)

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OCR MEI FP1 2014 June Q7
12 marks Standard +0.8
7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.
  1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
  2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
  3. Sketch the curve.
  4. Find the set of values of \(x\) for which \(y > 0\).
OCR MEI FP1 2014 June Q8
12 marks Standard +0.8
8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).
  1. Express \(w\) in modulus-argument form.
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).
OCR MEI FP1 2014 June Q9
12 marks Standard +0.3
9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & 3 & - 1 \\ - 1 & \alpha & - 1 \\ - 2 & - 1 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } 3 \alpha - 1 & - 8 & \alpha - 3 \\ 5 & 1 & 2 \\ 2 \alpha + 1 & - 5 & \alpha + 3 \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } \gamma & 0 & 0 \\ \beta & \gamma & 0 \\ 0 & 0 & \gamma \end{array} \right)\).
  1. Show that \(\beta = 0\).
  2. Find \(\gamma\) in terms of \(\alpha\).
  3. Write down \(\mathbf { A } ^ { - 1 }\) for the case when \(\alpha = 2\). State the value of \(\alpha\) for which \(\mathbf { A } ^ { - 1 }\) does not exist.
  4. Use your answer to part (iii) to solve the following simultaneous equations. $$\begin{aligned} x + 3 y - z & = 25 \\ - x + 2 y - z & = 11 \\ - 2 x - y + 3 z & = - 23 \end{aligned}$$
OCR MEI FP1 2015 June Q1
6 marks Moderate -0.8
1 Given that \(\mathbf { M } \binom { x } { y } = \binom { 1 } { 3 }\), where \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 3 \\ 8 & 21 \end{array} \right)\), find \(x\) and \(y\).
OCR MEI FP1 2015 June Q2
5 marks Moderate -0.8
2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.
OCR MEI FP1 2015 June Q3
6 marks Moderate -0.3
3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).
OCR MEI FP1 2015 June Q4
6 marks Standard +0.3
4 Indicate, on a single Argand diagram
  1. the set of points for which \(\arg ( z - ( - 1 - \mathrm { j } ) ) = \frac { \pi } { 4 }\),
  2. the set of points for which \(| z - ( 1 + 2 j ) | = 2\),
  3. the set of points for which \(| z - ( 1 + 2 j ) | \geqslant 2\) and \(0 \leqslant \arg ( z - ( - 1 - j ) ) \leqslant \frac { \pi } { 4 }\).
OCR MEI FP1 2015 June Q5
7 marks Moderate -0.3
5
  1. Show that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) = \mathrm { n } ^ { 2 }\).
  2. Show that \(\frac { \sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( 2 \mathrm { r } - 1 ) } { \sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } ( 2 \mathrm { r } - 1 ) } = \mathrm { k }\), where \(k\) is a constant to be determined.
OCR MEI FP1 2015 June Q6
6 marks Standard +0.3
6 A sequence is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 5\). Prove by induction that \(u _ { n } = \frac { 3 ^ { n - 1 } + 5 } { 2 }\). Section B (36 marks)
OCR MEI FP1 2015 June Q7
12 marks Standard +0.8
7 A curve has equation \(\mathrm { y } = \frac { ( 3 \mathrm { x } + 2 ) ( \mathrm { x } - 3 ) } { ( \mathrm { x } - 2 ) ( \mathrm { x } + 1 ) }\).
  1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes.
  2. Sketch the curve, justifying how it approaches the horizontal asymptote.
  3. Find the set of values of \(x\) for which \(y \geqslant 3\).
OCR MEI FP1 2015 June Q8
12 marks Standard +0.3
8 The complex number \(5 + 4 \mathrm { j }\) is denoted by \(\alpha\).
  1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\), showing your working.
  2. The real numbers \(q\) and \(r\) are such that \(\alpha ^ { 3 } + \mathrm { q } \alpha ^ { 2 } + 11 \alpha + \mathrm { r } = 0\). Find \(q\) and \(r\). Let \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + \mathrm { qz } ^ { 2 } + 11 \mathrm { z } + \mathrm { r }\), where \(q\) and \(r\) are as in part (ii).
  3. Solve the equation \(\mathrm { f } ( z ) = 0\).
  4. Solve the equation \(z ^ { 4 } + q z ^ { 3 } + 11 z ^ { 2 } + r z = z ^ { 3 } + q z ^ { 2 } + 11 z + r\).
OCR MEI FP1 2015 June Q9
12 marks Moderate -0.3
9 The triangle ABC has vertices at \(\mathrm { A } ( 0,0 ) , \mathrm { B } ( 0,2 )\) and \(\mathrm { C } ( 4,1 )\). The matrix \(\left( \begin{array} { r r } 1 & - 2 \\ 3 & 0 \end{array} \right)\) represents a transformation T .
  1. The transformation \(T\) maps triangle \(A B C\) onto triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Find the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\). Triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\) is now mapped onto triangle \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\) using the matrix \(\mathbf { M } = \left( \begin{array} { l l } 4 & 0 \\ 0 & 2 \end{array} \right)\).
  2. Describe fully the transformation represented by \(\mathbf { M }\).
  3. Triangle \(\mathrm { A } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\) is now mapped back onto ABC by a single transformation. Find the matrix representing this transformation.
  4. Calculate the area of \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\).
OCR MEI FP1 2016 June Q1
4 marks Moderate -0.3
1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 8 & - 2 \\ p & 1 \end{array} \right)\), where \(p \neq - 4\).
  1. Find the inverse of \(\mathbf { M }\) in terms of \(p\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{578345cb-e7a1-41fd-abf8-a977912965e8-2_1086_885_584_587} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} The triangle shown in Fig. 1 undergoes the transformation represented by the matrix \(\left( \begin{array} { c c } 8 & - 2 \\ 3 & 1 \end{array} \right)\). Find the area of the image of the triangle following this transformation.
OCR MEI FP1 2016 June Q2
6 marks Standard +0.3
2 The complex number \(z _ { 1 }\) is \(2 - 5 \mathrm { j }\) and the complex number \(z _ { 2 }\) is \(( a - 1 ) + ( 2 - b ) \mathrm { j }\), where \(a\) and \(b\) are real.
  1. Express \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } }\) in the form \(x + y \mathrm { j }\), giving \(x\) and \(y\) in exact form. You must show clearly how you obtain your
    answer.
  2. Given that \(\frac { z _ { 1 } { } ^ { * } } { z _ { 1 } } = z _ { 2 }\), find the exact values of \(a\) and \(b\).
OCR MEI FP1 2016 June Q3
6 marks Standard +0.3
3 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } \lambda & 6 & - 4 \\ 2 & 5 & - 1 \\ - 1 & 4 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } - 19 & 34 & - 14 \\ 5 & - 5 & 5 \\ - 13 & 18 & - 3 \end{array} \right)\) and \(\mathbf { A B } = \mu \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity
matrix.
  1. Find the values of \(\lambda\) and \(\mu\).
  2. Hence find \(\mathbf { B } ^ { - 1 }\).
OCR MEI FP1 2016 June Q4
6 marks Standard +0.3
4
  1. Use standard series to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - p ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + ( 3 - 2 p ) n - p \right) ,$$ where \(p\) is a constant.
  2. Given that the coefficients of \(n ^ { 3 }\) and \(n ^ { 4 }\) in the expression for \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - p )\) are equal, find the value of \(p\).
OCR MEI FP1 2016 June Q5
8 marks Standard +0.3
5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 3 - 4 \mathrm { j } | = 5\) and arg \(( z + 3 - 6 \mathrm { j } ) = \frac { 1 } { 2 } \pi\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Write down the complex number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading on your sketch, the region satisfying $$| z + 3 - 4 \mathrm { j } | \geqslant 5 \text { and } \frac { 1 } { 2 } \pi \leqslant \arg ( z + 3 - 6 \mathrm { j } ) \leqslant \frac { 3 } { 4 } \pi .$$
OCR MEI FP1 2016 June Q6
6 marks Standard +0.8
6 A sequence is defined by \(u _ { 1 } = 8\) and \(u _ { n + 1 } = 3 u _ { n } + 2 n + 5\). Prove by induction that \(u _ { n } = 4 \left( 3 ^ { n } \right) - n - 3\).
OCR MEI FP1 2016 June Q7
13 marks Standard +0.8
7 The function \(\mathrm { f } ( z ) = 2 z ^ { 4 } - 9 z ^ { 3 } + A z ^ { 2 } + B z - 26\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has two real roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\), and two complex roots, \(\gamma\) and \(\delta\), where \(\gamma = 3 + 2 \mathrm { j }\).
  1. Show that \(\alpha + \beta = - \frac { 3 } { 2 }\) and find the value of \(\alpha \beta\).
  2. Hence find the two real roots \(\alpha\) and \(\beta\).
  3. Find the values of \(A\) and \(B\).
  4. Write down the roots of the equation \(\mathrm { f } \left( \frac { w } { \mathrm { j } } \right) = 0\).
OCR MEI FP1 2016 June Q8
12 marks Standard +0.8
8 A curve has equation \(y = \frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 }\).
  1. Find the equations of the two vertical asymptotes and the one horizontal asymptote of this curve.
  2. State, with justification, how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 3 x ^ { 2 } - 9 } { x ^ { 2 } + 3 x - 4 } \geqslant 0\).
OCR MEI FP1 2016 June Q9
11 marks Challenging +1.2
9 You are given that \(\frac { 3 } { 4 ( 2 r - 1 ) } - \frac { 1 } { 2 r + 1 } + \frac { 1 } { 4 ( 2 r + 3 ) } = \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) }\).
  1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 } { 3 } - \frac { 3 } { 4 ( 2 n + 1 ) } + \frac { 1 } { 4 ( 2 n + 3 ) } .$$
  2. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) }\) converges as \(n\) tends to infinity.
  3. Find the sum of the finite series $$\frac { 45 } { 39 \times 41 \times 43 } + \frac { 47 } { 41 \times 43 \times 45 } + \frac { 49 } { 43 \times 45 \times 47 } + \ldots + \frac { 105 } { 99 \times 101 \times 103 } ,$$ giving your answer to 3 significant figures. \section*{END OF QUESTION PAPER}
OCR MEI S1 2009 January Q1
7 marks Easy -1.2
1 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money\(\pounds 0\)\(\pounds 10\)\(\pounds 100\)\(\pounds 5000\)
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket.
  2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.
OCR MEI S1 2009 January Q2
5 marks Moderate -0.8
2 Thomas has six tiles, each with a different letter of his name on it.
  1. Thomas arranges these letters in a random order. Find the probability that he arranges them in the correct order to spell his name.
  2. On another occasion, Thomas picks three of the six letters at random. Find the probability that he picks the letters T, O and M (in any order).
OCR MEI S1 2009 January Q3
8 marks Easy -1.3
3 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.
OCR MEI S1 2009 January Q4
8 marks Moderate -0.8
4 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.
  1. Find the probability that in a batch of 50 there is
    (A) exactly one faulty teapot,
    (B) more than one faulty teapot.
  2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.