Questions — OCR MEI FP1 (190 questions)

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OCR MEI FP1 2009 January Q4
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
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 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 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
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
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
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. $$\begin{aligned} 3 x + 4 y - z & = 1
    x - y + 2 z & = - 9
    - 2 x + 7 y - 3 z & = 26 \end{aligned}$$ \footnotetext{OCR
    RECOGNISING ACHIEVEMENT
OCR MEI FP1 2010 January Q1
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.
OCR MEI FP1 2010 January Q2
2 You are given that \(\mathbf { A } = \left( \begin{array} { r } 4
- 2
4 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 5 & 1
2 & - 3 \end{array} \right) , \mathbf { C } = \left( \begin{array} { l l l } 5 & 1 & 8 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { r r } - 2 & 0
4 & 1 \end{array} \right)\).
  1. Calculate, where they exist, \(\mathbf { A B } , \mathbf { C A } , \mathbf { B } + \mathbf { D }\) and \(\mathbf { A C }\) and indicate any that do not exist.
  2. Matrices \(\mathbf { B }\) and \(\mathbf { D }\) represent transformations B and D respectively. Find the single matrix that represents transformation B followed by transformation D.
OCR MEI FP1 2010 January Q3
3 The roots of the cubic equation \(4 x ^ { 3 } - 12 x ^ { 2 } + k x - 3 = 0\) may be written \(a - d , a\) and \(a + d\). Find the roots and the value of \(k\).
OCR MEI FP1 2010 January Q4
4 You are given that if \(\mathbf { M } = \left( \begin{array} { r r r } 4 & 0 & 1
- 6 & 1 & 1
5 & 2 & 5 \end{array} \right)\) then \(\mathbf { M } ^ { - 1 } = \frac { 1 } { k } \left( \begin{array} { r r r } - 3 & - 2 & 1
- 35 & - 15 & 10
17 & 8 & - 4 \end{array} \right)\).
Find the value of \(k\). Hence solve the following simultaneous equations. $$\begin{aligned} 4 x + z & = 9
- 6 x + y + z & = 32
5 x + 2 y + 5 z & = 81 \end{aligned}$$
OCR MEI FP1 2010 January Q5
5 Use standard series formulae to show that \(\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r - 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } - 19 \right)\).
OCR MEI FP1 2010 January Q6
6 Prove by induction that \(1 \times 2 + 2 \times 3 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }\) for all positive integers \(n\). Section B (36 marks)
OCR MEI FP1 2010 January Q7
7 A curve has equation \(y = \frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) }\).
  1. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your answers.
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0\).
OCR MEI FP1 2010 January Q8
8
  1. Fig. 8 shows an Argand diagram. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{df275813-15de-496f-9742-427a9e03f431-3_892_899_1048_664} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
    1. Write down the equation of the locus represented by the circumference of circle B.
    2. Write down the two inequalities that define the shaded region between, but not including, circles A and B.
    1. Draw an Argand diagram to show the region where $$\frac { \pi } { 4 } < \arg ( z - ( 2 + \mathrm { j } ) ) < \frac { 3 \pi } { 4 }$$
    2. Determine whether the point \(43 + 47 \mathrm { j }\) lies within this region.
OCR MEI FP1 2010 January Q9
9
  1. Verify that \(\frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 2 } { r } - \frac { 3 } { r + 1 } + \frac { 1 } { r + 2 }\).
  2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 3 } { 2 } - \frac { 2 } { n + 1 } + \frac { 1 } { n + 2 } .$$
  3. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\) converges as \(n\) tends to infinity.
  4. Find \(\sum _ { r = 50 } ^ { 100 } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\), giving your answer to 3 significant figures.
OCR MEI FP1 2011 January Q1
1 Find the values of \(P , Q , R\) and \(S\) in the identity \(3 x ^ { 3 } + 18 x ^ { 2 } + P x + 31 \equiv Q ( x + R ) ^ { 3 } + S\).
OCR MEI FP1 2011 January Q2
2 You are given that \(\mathbf { M } = \left( \begin{array} { r r } 4 & 0
- 1 & 3 \end{array} \right)\).
  1. The transformation associated with \(\mathbf { M }\) is applied to a figure of area 3 square units. Find the area of the transformed figure.
  2. Find \(\mathbf { M } ^ { - 1 }\) and \(\operatorname { det } \mathbf { M } ^ { - 1 }\).
  3. Explain the significance of \(\operatorname { det } \mathbf { M } \times \operatorname { det } \mathbf { M } ^ { - 1 }\) in terms of transformations.
OCR MEI FP1 2011 January Q3
3 The roots of the cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + 8 x + 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).
OCR MEI FP1 2011 January Q4
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 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 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
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
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
\(\mathbf { 9 }\) 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? $$\begin{aligned} - 2 x + y - 5 z & = p
    3 x - 8 y + z & = q
    x - y + 2 z & = r \end{aligned}$$