Questions — OCR MEI (4333 questions)

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OCR MEI FP1 2011 January Q4
6 marks Moderate -0.8
4 Represent on an Argand diagram the region defined by \(2 < | z - ( 3 + 2 \mathrm { j } ) | \leqslant 3\).
OCR MEI FP1 2011 January Q5
5 marks Moderate -0.3
5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 3 - 4 r ) = \frac { 1 } { 2 } n ( n + 1 ) \left( 1 - 2 n ^ { 2 } \right)\).
OCR MEI FP1 2011 January Q6
6 marks Standard +0.3
6 A sequence is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 ^ { n + 1 }\). Prove by induction that \(u _ { n } = 2 ^ { n + 1 } + 1\).
OCR MEI FP1 2011 January Q7
12 marks Standard +0.3
7 Fig. 7 shows part of the curve with equation \(y = \frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d91a83d-971e-48ca-aa1a-09f2c1a8093a-3_894_890_447_625} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the coordinates of the two points where the curve crosses the axes.
  2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  3. Determine how the curve approaches the horizontal asymptote for large positive and large negative values of \(x\).
  4. On the copy of Fig. 7, sketch the rest of the curve.
  5. Solve the inequality \(\frac { x + 5 } { ( 2 x - 5 ) ( 3 x + 8 ) } < 0\).
OCR MEI FP1 2011 January Q8
12 marks Standard +0.3
8 The function \(\mathrm { f } ( z ) = z ^ { 4 } - z ^ { 3 } + a z ^ { 2 } + b z + c\) has real coefficients. The equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha , \beta\), \(\gamma\) and \(\delta\) where \(\alpha = 1\) and \(\beta = 1 + \mathrm { j }\).
  1. Write down the other complex root and explain why the equation must have a second real root.
  2. Write down the value of \(\alpha + \beta + \gamma + \delta\) and find the second real root.
  3. Find the values of \(a , b\) and \(c\).
  4. Write down \(\mathrm { f } ( - z )\) and the roots of \(\mathrm { f } ( - z ) = 0\).
OCR MEI FP1 2011 January Q9
12 marks Standard +0.3
\(\mathbf { 9 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 5 \\ 3 & a & 1 \\ 1 & - 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 2 a + 1 & 3 & 1 + 5 a \\ - 5 & 1 & - 13 \\ - 3 - a & - 1 & - 2 a - 3 \end{array} \right)\).
  1. Show that \(\mathbf { A B } = ( 8 + a ) \mathbf { I }\).
  2. State the value of \(a\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Write down \(\mathbf { A } ^ { - 1 }\) in terms of \(a\), when \(\mathbf { A } ^ { - 1 }\) exists.
  3. Use \(\mathbf { A } ^ { - 1 }\) to solve the following simultaneous equations. $$\begin{aligned} - 2 x + y - 5 z & = - 55 \\ 3 x + 4 y + z & = - 9 \\ x - y + 2 z & = 26 \end{aligned}$$
  4. What can you say about the solutions of the following simultaneous equations? $$\begin{aligned} - 2 x + y - 5 z & = p \\ 3 x - 8 y + z & = q \\ x - y + 2 z & = r \end{aligned}$$
OCR MEI FP1 2012 January Q1
5 marks Moderate -0.8
\(\mathbf { 1 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & p & - 4 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 0 & q \\ 2 & - 2 \\ 1 & - 3 \end{array} \right)\).
  1. Find \(\mathbf { A B }\).
  2. Hence prove that matrix multiplication is not commutative.
OCR MEI FP1 2012 January Q2
5 marks Easy -1.2
2 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x ^ { 3 } - 3 \equiv ( x + 3 ) \left( A x ^ { 2 } + B x + C \right) + D\).
OCR MEI FP1 2012 January Q3
6 marks Moderate -0.3
3 Given that \(z = 6\) is a root of the cubic equation \(z ^ { 3 } - 10 z ^ { 2 } + 37 z + p = 0\), find the value of \(p\) and the other roots.
OCR MEI FP1 2012 January Q4
6 marks Moderate -0.3
4 Using the standard summation formulae, find \(\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 )\). Give your answer in a fully factorised form.
OCR MEI FP1 2012 January Q5
6 marks Standard +0.3
5 The equation \(z ^ { 3 } - 5 z ^ { 2 } + 3 z - 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(\frac { \alpha } { 2 } + 1 , \frac { \beta } { 2 } + 1\), \(\frac { \gamma } { 2 } + 1\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2012 January Q6
8 marks Standard +0.8
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r 3 ^ { r - 1 } = \frac { 1 } { 4 } \left[ 3 ^ { n } ( 2 n - 1 ) + 1 \right]\). Section B (36 marks)
OCR MEI FP1 2012 January Q7
14 marks Standard +0.8
7 A curve has equation \(y = \frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 }\).
  1. Find the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the three asymptotes.
  3. Determine whether the curve approaches the horizontal asymptote from above or from below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
  5. Solve the inequality \(\frac { ( x + 1 ) ( 2 x - 1 ) } { x ^ { 2 } - 3 } < 2\).
OCR MEI FP1 2012 January Q8
10 marks Standard +0.8
8
  1. Sketch on an Argand diagram the locus, \(C\), of points for which \(| z - 4 | = 3\).
  2. By drawing appropriate lines through the origin, indicate on your Argand diagram the point A on the locus \(C\) where \(\arg z\) has its maximum value. Indicate also the point B on the locus \(C\) where \(\arg z\) has its minimum value.
  3. Given that \(\arg z = \alpha\) at A and \(\arg z = \beta\) at B , indicate on your Argand diagram the set of points for which \(\beta \leqslant \arg z \leqslant \alpha\) and \(| z - 4 | \geqslant 3\).
  4. Calculate the value of \(\alpha\) and the value of \(\beta\).
OCR MEI FP1 2012 January Q9
12 marks Standard +0.3
9 The matrix \(\mathbf { R }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
  1. Explain in terms of transformations why \(\mathbf { R } ^ { 4 } = \mathbf { I }\).
  2. Describe the transformation represented by \(\mathbf { R } ^ { - 1 }\) and write down the matrix \(\mathbf { R } ^ { - 1 }\).
  3. \(\mathbf { S }\) is the matrix representing rotation through \(60 ^ { \circ }\) anticlockwise about the origin. Find \(\mathbf { S }\).
  4. Write down the smallest positive integers \(m\) and \(n\) such that \(\mathbf { S } ^ { m } = \mathbf { R } ^ { n }\), explaining your answer in terms of transformations.
  5. Find \(\mathbf { R S }\) and explain in terms of transformations why \(\mathbf { R S } = \mathbf { S R }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR MEI FP1 2013 January Q1
5 marks Moderate -0.8
1 Transformation A is represented by matrix \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and transformation B is represented by matrix \(\mathbf { B } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 3 \end{array} \right)\).
  1. Describe transformations A and B .
  2. Find the matrix for the composite transformation A followed by B .
OCR MEI FP1 2013 January Q2
4 marks Moderate -0.3
2 Given that \(z = a + b \mathrm { j }\), find \(\operatorname { Re } \left( \frac { z } { z ^ { * } } \right)\) and \(\operatorname { Im } \left( \frac { z } { z ^ { * } } \right)\).
OCR MEI FP1 2013 January Q3
6 marks Standard +0.3
3 You are given that \(z = 2 + \mathrm { j }\) is a root of the cubic equation \(2 z ^ { 3 } + p z ^ { 2 } + 22 z - 15 = 0\), where \(p\) is real. Find the other roots and the value of \(p\).
OCR MEI FP1 2013 January Q4
7 marks Standard +0.3
4
  1. Show that \(x ^ { 2 } - x + 2 > 0\) for all real \(x\).
  2. Solve the inequality \(\frac { 2 x } { x ^ { 2 } - x + 2 } > x\).
OCR MEI FP1 2013 January Q5
6 marks Standard +0.3
5 You are given that \(\frac { 3 } { ( 5 + 3 x ) ( 2 + 3 x ) } \equiv \frac { 1 } { 2 + 3 x } - \frac { 1 } { 5 + 3 x }\).
  1. Use this result to find \(\sum _ { r = 1 } ^ { 100 } \frac { 1 } { ( 5 + 3 r ) ( 2 + 3 r ) }\), giving your answer as an exact fraction.
  2. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 + 3 r ) ( 2 + 3 r ) }\) converges as \(n\) tends to infinity.
OCR MEI FP1 2013 January Q6
8 marks Standard +0.3
6 Prove by induction that \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots + ( - 1 ) ^ { n - 1 } n ^ { 2 } = ( - 1 ) ^ { n - 1 } \frac { n ( n + 1 ) } { 2 }\).
OCR MEI FP1 2013 January Q7
13 marks Standard +0.8
7 Fig. 7 shows a sketch of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e47c6fb-574b-4eee-81c8-4031fee9e2ba-3_696_975_406_529} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down the equations of the three asymptotes and the coordinates of the points where the curve crosses the axes. Hence write down the solution of the inequality \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } > 0\).
  2. The equation \(\frac { x - 4 } { ( x - 5 ) ( x - 8 ) } = k\) has no real solutions. Show that \(- 1 < k < - \frac { 1 } { 9 }\). Relate this result to the graph of \(y = \frac { x - 4 } { ( x - 5 ) ( x - 8 ) }\).
OCR MEI FP1 2013 January Q8
11 marks Challenging +1.2
8
  1. Indicate on an Argand diagram the set of points \(z\) for which \(| z - ( - 8 + 15 \mathrm { j } ) | < 10\).
  2. Using the diagram, show that \(7 < | z | < 27\).
  3. Mark on your Argand diagram the point, \(P\), at which \(| z - ( - 8 + 15 \mathrm { j } ) | = 10\) and \(\arg z\) takes its maximum value. Find the modulus and argument of \(z\) at \(P\).
OCR MEI FP1 2013 January Q9
12 marks Standard +0.3
9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 8 & - 7 & - 12 \\ - 10 & 5 & 15 \\ - 9 & 6 & 6 \end{array} \right)\) and \(\mathbf { A } ^ { - 1 } = k \left( \begin{array} { r r r } 4 & 2 & 3 \\ 5 & 4 & 0 \\ 1 & - 1 & 2 \end{array} \right)\).
  1. Find the exact value of \(k\).
  2. Using your answer to part (i), solve the following simultaneous equations. $$\begin{aligned} 8 x - 7 y - 12 z & = 14 \\ - 10 x + 5 y + 15 z & = - 25 \\ - 9 x + 6 y + 6 z & = 3 \end{aligned}$$ You are also given that \(\mathbf { B } = \left( \begin{array} { r r r } - 7 & 5 & 15 \\ a & - 8 & - 21 \\ 2 & - 1 & - 3 \end{array} \right)\) and \(\mathbf { B } ^ { - 1 } = \frac { 1 } { 3 } \left( \begin{array} { r r r } 1 & 0 & 5 \\ - 4 & - 3 & 1 \\ 2 & 1 & b \end{array} \right)\).
  3. Find the values of \(a\) and \(b\).
  4. Write down an expression for \(( \mathbf { A B } ) ^ { - 1 }\) in terms of \(\mathbf { A } ^ { - 1 }\) and \(\mathbf { B } ^ { - 1 }\). Hence find \(( \mathbf { A B } ) ^ { - 1 }\).
OCR MEI FP1 2009 June Q1
5 marks Moderate -0.8
1
  1. Find the inverse of the matrix \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 1 \\ 3 & 2 \end{array} \right)\).
  2. Use this inverse to solve the simultaneous equations $$\begin{aligned} & 4 x - y = 49 \\ & 3 x + 2 y = 100 \end{aligned}$$ showing your working clearly.