Questions FP1 (1491 questions)

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OCR FP1 2015 June Q5
8 marks Standard +0.3
5 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z + 2 | = 2\) and \(\arg ( z + 2 ) = \frac { 5 } { 6 } \pi\) respectively.
  1. Sketch, on a single Argand diagram, the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Find the complex number represented by the intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$
OCR FP1 2015 June Q6
7 marks Standard +0.3
6 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } 0 & 2 \\ - 1 & 0 \end{array} \right)\).
  1. The diagram in the Printed Answer Book shows the unit square \(O A B C\). The image of the unit square under the transformation represented by \(\mathbf { M }\) is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\), indicating clearly the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\).
  2. The transformation represented by \(\mathbf { M }\) is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a possible pair of transformations P and Q and state the matrices that represent them.
OCR FP1 2015 June Q7
10 marks Standard +0.3
7
  1. Use an algebraic method to find the square roots of the complex number \(5 + 12 \mathrm { i }\). You must show sufficient working to justify your answers.
  2. Hence solve the quadratic equation \(x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0\).
OCR FP1 2015 June Q8
10 marks Challenging +1.3
8
  1. Show that \(\frac { 3 } { r - 1 } - \frac { 2 } { r } - \frac { 1 } { r + 1 } \equiv \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
  2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
  3. Hence find the value of \(\sum _ { r = 4 } ^ { \infty } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
OCR FP1 2015 June Q9
10 marks Standard +0.3
9 The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4 \\ 2 & a & 3 \\ 0 & 1 & a \end{array} \right)\).
  1. Find the values of \(a\) for which \(\mathbf { D }\) is singular.
  2. Three simultaneous equations are shown below. $$\begin{array} { r } x + 3 y + 4 z = 3 \\ 2 x + a y + 3 z = 2 \\ y + a z = 0 \end{array}$$ For each of the following values of \(a\), determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
    1. \(a = 3\)
    2. \(a = 1\)
OCR FP1 2015 June Q10
10 marks Standard +0.8
10 The cubic equation \(x ^ { 3 } + 4 x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Use the substitution \(x = \sqrt { u }\) to obtain a cubic equation in \(u\).
  2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \alpha \beta \gamma\).
OCR FP1 2016 June Q1
5 marks Moderate -0.5
1 Find \(\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r - 1 )\), giving your answer in a fully factorised form.
OCR FP1 2016 June Q2
7 marks Standard +0.3
2 The complex number \(z\) has modulus \(2 \sqrt { 3 }\) and argument \(- \frac { 1 } { 3 } \pi\). Giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers, and showing clearly how you obtain them, find
  1. \(z\),
  2. \(\frac { 1 } { \left( z ^ { * } - 5 \mathrm { i } \right) ^ { 2 } }\).
OCR FP1 2016 June Q3
6 marks Standard +0.3
3 The quadratic equation \(k x ^ { 2 } + x + k = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Find the value of \(\left( \alpha + \frac { 1 } { \alpha } \right) \left( \beta + \frac { 1 } { \beta } \right)\) in terms of \(k\).
OCR FP1 2016 June Q4
6 marks Easy -1.2
4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l l } b & 0 & 5 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r } 6 \\ 4 \\ - 1 \end{array} \right)\). Find
  1. \(5 \mathbf { A } - 3 \mathbf { B }\),
  2. BC,
  3. CA .
OCR FP1 2016 June Q5
4 marks Standard +0.3
5 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 5 \text { and } u _ { n + 1 } = 3 u _ { n } + 2 \text { for } n \geqslant 1 \text {. }$$ Prove by induction that \(u _ { n } = 2 \times 3 ^ { n } - 1\).
OCR FP1 2016 June Q6
9 marks Standard +0.3
6 In an Argand diagram the points \(A\) and \(B\) represent the complex numbers \(5 + 4 \mathrm { i }\) and \(1 + 2 \mathrm { i }\) respectively.
  1. Given that \(A\) and \(B\) are the ends of a diameter of a circle \(C\), find the equation of \(C\) in complex number form. The perpendicular bisector of \(A B\) is denoted by \(l\).
  2. Sketch \(C\) and \(l\) on a single Argand diagram.
  3. Find the complex numbers represented by the points of intersection of \(C\) and \(l\).
OCR FP1 2016 June Q7
8 marks Standard +0.3
7 The matrix \(\left( \begin{array} { l l } 1 & 3 \\ 0 & 1 \end{array} \right)\) represents a transformation P .
  1. Describe fully the transformation P . The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1 \\ - 1 & 0 \end{array} \right)\).
  2. Given that \(\mathbf { M }\) represents transformation Q followed by transformation P , find the matrix that represents the transformation Q and describe fully the transformation Q .
OCR FP1 2016 June Q9
6 marks Standard +0.3
9
  1. The matrix \(\mathbf { X }\) is given by \(\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2 \\ 0 & a & 5 \\ 1 & 2 & 1 \end{array} \right)\). Show that the determinant of \(\mathbf { X }\) is \(a ^ { 2 } - 8 a + 15\).
  2. Explain briefly why the equations $$\begin{array} { r } 3 x + 3 y - 2 z = 1 \\ 3 y + 5 z = 5 \\ x + 2 y + z = 2 \end{array}$$ do not have a unique solution and determine whether these equations are consistent or inconsistent.
  3. Use an algebraic method to find the square roots of the complex number \(9 + 40 \mathrm { i }\).
  4. Show that \(9 + 40 \mathrm { i }\) is a root of the quadratic equation \(z ^ { 2 } - 18 z + 1681 = 0\).
  5. By using the substitution \(z = \frac { 1 } { u ^ { 2 } }\), find the roots of the equation \(1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
OCR MEI FP1 2009 January Q1
5 marks Easy -1.2
1
  1. Find the roots of the quadratic equation \(z ^ { 2 } - 6 z + 10 = 0\) in the form \(a + b \mathrm { j }\).
  2. Express these roots in modulus-argument form.
OCR MEI FP1 2009 January Q2
4 marks Moderate -0.8
2 Find the values of \(A , B\) and \(C\) in the identity \(2 x ^ { 2 } - 13 x + 25 \equiv A ( x - 3 ) ^ { 2 } - B ( x - 2 ) + C\).
OCR MEI FP1 2009 January Q3
5 marks Moderate -0.3
3 Fig. 3 shows the unit square, OABC , and its image, \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\), after undergoing a transformation. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-2_465_531_806_806} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Write down the matrix \(\mathbf { P }\) representing this transformation.
  2. The parallelogram \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is transformed by the matrix \(\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\). Find the coordinates of the vertices of its image, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\), following this transformation.
  3. Describe fully the transformation represented by \(\mathbf { Q P }\).
OCR MEI FP1 2009 January Q4
3 marks Moderate -0.8
4 Write down the equation of the locus represented in the Argand diagram shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-2_474_497_1932_824} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
OCR MEI FP1 2009 January Q5
6 marks Standard +0.3
5 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , - 3 \alpha\) and \(\alpha + 3\). Find the values of \(\alpha , p\) and \(q\).
OCR MEI FP1 2009 January Q6
6 marks Standard +0.3
6 Using the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) show that $$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { 1 } { 4 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$
OCR MEI FP1 2009 January Q7
7 marks Standard +0.3
7 Prove by induction that \(12 + 36 + 108 + \ldots + 4 \times 3 ^ { n } = 6 \left( 3 ^ { n } - 1 \right)\) for all positive integers \(n\).
OCR MEI FP1 2009 January Q8
12 marks Standard +0.3
8 Fig. 8 shows part of the graph of \(y = \frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) }\). Two sections of the graph have been omitted. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-3_725_1025_1160_559} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  3. Copy Fig. 8 and draw in the two missing sections.
  4. Solve the inequality \(\frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) } \leqslant 0\).
OCR MEI FP1 2009 January Q9
12 marks Standard +0.3
9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 + \mathrm { j }\) and \(\beta = 2 - \mathrm { j }\).
  1. Express \(\alpha + \beta , \alpha \alpha ^ { * }\) and \(\frac { \alpha + \beta } { \alpha }\) in the form \(a + b \mathrm { j }\).
  2. Find a quadratic equation with roots \(\alpha\) and \(\alpha ^ { * }\).
  3. \(\alpha\) and \(\beta\) are roots of a quartic equation with real coefficients. Write down the two other roots and find this quartic equation in the form \(z ^ { 4 } + A z ^ { 3 } + B z ^ { 2 } + C z + D = 0\).
OCR MEI FP1 2009 January Q10
12 marks Standard +0.3
10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 3 & 4 & - 1 \\ 1 & - 1 & k \\ - 2 & 7 & - 3 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r c } 11 & - 5 & - 7 \\ 1 & 11 & 5 + k \\ - 5 & 29 & 7 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } 42 & \alpha & 4 k - 8 \\ 10 - 5 k & - 16 + 29 k & - 12 + 6 k \\ 0 & 0 & \beta \end{array} \right)\).
  1. Show that \(\alpha = 0\) and \(\beta = 28 + 7 k\).
  2. Find \(\mathbf { A B }\) when \(k = 2\).
  3. For the case when \(k = 2\) write down the matrix \(\mathbf { A } ^ { - 1 }\).
  4. Use the result from part (iii) to solve the following simultaneous equations. $$\begin{aligned} 3 x + 4 y - z & = 1 \\ x - y + 2 z & = - 9 \\ - 2 x + 7 y - 3 z & = 26 \end{aligned}$$
OCR MEI FP1 2010 January Q1
5 marks Moderate -0.8
1 Two complex numbers are given by \(\alpha = - 3 + \mathrm { j }\) and \(\beta = 5 - 2 \mathrm { j }\).
Find \(\alpha \beta\) and \(\frac { \alpha } { \beta }\), giving your answers in the form \(a + b \mathrm { j }\), showing your working.