Questions — OCR MEI FP1 (190 questions)

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OCR MEI FP1 2012 January Q1
\(\mathbf { 1 }\) 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
OCR MEI FP1 2009 June Q2
2 Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.
OCR MEI FP1 2009 June Q3
3
  1. Sketch the graph of \(y = \frac { 2 } { x + 4 }\).
  2. Solve the inequality $$\frac { 2 } { x + 4 } \leqslant x + 3$$ showing your working clearly.
OCR MEI FP1 2009 June Q4
4 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } + p x + q = 0\) are \(2 w , - 6 w\) and \(3 w\). Find the values of the roots and the values of \(p\) and \(q\).
OCR MEI FP1 2009 June Q5
5
  1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } \equiv \frac { 5 } { ( 5 r - 2 ) ( 5 r + 3 ) }\) for all integers \(r\).
  2. Hence use the method of differences to show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }\).
OCR MEI FP1 2009 June Q6
6 Prove by induction that \(3 + 10 + 17 + \ldots + ( 7 n - 4 ) = \frac { 1 } { 2 } n ( 7 n - 1 )\) for all positive integers \(n\). Section B (36 marks)
OCR MEI FP1 2009 June Q7
7 A curve has equation \(y = \frac { ( x + 2 ) ( 3 x - 5 ) } { ( 2 x + 1 ) ( x - 1 ) }\).
  1. Write down 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 below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.