Questions — OCR MEI FP1 (195 questions)

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OCR MEI FP1 2008 June Q5
5 marks Moderate -0.5
5 You are given that \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 4 \\ 3 & 2 & 5 \\ 4 & 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 1 & 0 & 2 \\ 14 & - 14 & 7 \\ - 5 & 7 & - 4 \end{array} \right)\).
  1. Calculate AB.
  2. Write down \(\mathbf { A } ^ { - 1 }\).
OCR MEI FP1 2008 June Q6
5 marks Moderate -0.3
6 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 3 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(2 \alpha , 2 \beta\) and \(2 \gamma\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2008 June Q7
7 marks Standard +0.3
7
  1. Show that \(\frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 } \equiv \frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) }\) for all integers \(r\).
  2. Hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\). Section B (36 marks)
OCR MEI FP1 2008 June Q8
12 marks Standard +0.3
8 A curve has equation \(y = \frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) }\).
  1. Write down the equations of the three asymptotes.
  2. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  3. Sketch the curve.
  4. Solve the inequality \(\frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) } < 0\).
OCR MEI FP1 2008 June Q9
11 marks Moderate -0.3
9 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 2 - 2 \mathrm { j }\) and \(\beta = - 1 + \mathrm { j }\). \(\alpha\) and \(\beta\) are both roots of a quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers.
  1. Write down the other two roots.
  2. Represent these four roots on an Argand diagram.
  3. Find the values of \(A , B , C\) and \(D\).
OCR MEI FP1 2008 June Q10
13 marks Standard +0.3
10
  1. Using the standard formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\), prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( 3 n + 1 )$$
  2. Prove the same result by mathematical induction.
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.
OCR MEI FP1 2010 January Q2
7 marks Moderate -0.8
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
6 marks Standard +0.3
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
6 marks Moderate -0.3
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
6 marks Moderate -0.8
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 marks Standard +0.3
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
12 marks Standard +0.3
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
12 marks Standard +0.3
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
12 marks Standard +0.3
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.