Questions — OCR MEI FP1 (195 questions)

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OCR MEI FP1 2007 January Q8
11 marks Moderate -0.3
8 It is given that \(m = - 4 + 2 \mathrm { j }\).
  1. Express \(\frac { 1 } { m }\) in the form \(a + b \mathrm { j }\).
  2. Express \(m\) in modulus-argument form.
  3. Represent the following loci on separate Argand diagrams.
    (A) \(\arg ( z - m ) = \frac { \pi } { 4 }\) (B) \(0 < \arg ( z - m ) < \frac { \pi } { 4 }\)
OCR MEI FP1 2007 January Q9
13 marks Standard +0.3
9 Matrices \(\mathbf { M }\) and \(\mathbf { N }\) are given by \(\mathbf { M } = \left( \begin{array} { l l } 3 & 2 \\ 0 & 1 \end{array} \right)\) and \(\mathbf { N } = \left( \begin{array} { r r } 1 & - 3 \\ 1 & 4 \end{array} \right)\).
  1. Find \(\mathbf { M } ^ { - 1 }\) and \(\mathbf { N } ^ { - 1 }\).
  2. Find \(\mathbf { M N }\) and \(( \mathbf { M N } ) ^ { - \mathbf { 1 } }\). Verify that \(( \mathbf { M N } ) ^ { - 1 } = \mathbf { N } ^ { - 1 } \mathbf { M } ^ { - 1 }\).
  3. The result \(( \mathbf { P Q } ) ^ { - 1 } = \mathbf { Q } ^ { - 1 } \mathbf { P } ^ { - 1 }\) is true for any two \(2 \times 2\), non-singular matrices \(\mathbf { P }\) and \(\mathbf { Q }\). The first two lines of a proof of this general result are given below. Beginning with these two lines, complete the general proof. $$\begin{aligned} & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q } = \mathbf { I } \\ \Rightarrow & ( \mathbf { P Q } ) ^ { - 1 } \mathbf { P Q Q } \mathbf { Q } ^ { - 1 } = \mathbf { I Q } ^ { - 1 } \end{aligned}$$
OCR MEI FP1 2008 January Q1
5 marks Moderate -0.8
1 You are given that matrix \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 1 \\ 0 & 3 \end{array} \right)\) and matrix \(\mathbf { B } = \left( \begin{array} { r r } 3 & 1 \\ - 2 & 4 \end{array} \right)\).
  1. Find BA.
  2. A plane shape of area 3 square units is transformed using matrix \(\mathbf { A }\). The image is transformed using matrix B. What is the area of the resulting shape?
OCR MEI FP1 2008 January Q2
5 marks Moderate -0.8
2 You are given that \(\alpha = - 3 + 4 \mathrm { j }\).
  1. Calculate \(\alpha ^ { 2 }\).
  2. Express \(\alpha\) in modulus-argument form.
OCR MEI FP1 2008 January Q3
7 marks Moderate -0.8
3
  1. Show that \(z = 3\) is a root of the cubic equation \(z ^ { 3 } + z ^ { 2 } - 7 z - 15 = 0\) and find the other roots.
  2. Show the roots on an Argand diagram.
OCR MEI FP1 2008 January Q4
6 marks Moderate -0.8
4 Using the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { n } [ ( r + 1 ) ( r - 2 ) ] = \frac { 1 } { 3 } n \left( n ^ { 2 } - 7 \right)\).
OCR MEI FP1 2008 January Q5
5 marks Standard +0.3
5 The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha , \beta\) and \(\gamma\), where $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha \beta \gamma & = - 7 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 13 \end{aligned}$$
  1. Write down the values of \(p\) and \(r\).
  2. Find the value of \(q\).
OCR MEI FP1 2008 January Q6
8 marks Standard +0.3
6 A sequence is defined by \(a _ { 1 } = 7\) and \(a _ { k + 1 } = 7 a _ { k } - 3\).
  1. Calculate the value of the third term, \(a _ { 3 }\).
  2. Prove by induction that \(a _ { n } = \frac { \left( 13 \times 7 ^ { n - 1 } \right) + 1 } { 2 }\).
OCR MEI FP1 2008 January Q7
11 marks Standard +0.3
7 The sketch below shows part of the graph of \(y = \frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) }\). One section of the graph has been omitted. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-3_842_1198_477_552} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Find the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
  3. Copy the sketch and draw in the missing section.
  4. Solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) } \geqslant 0\).
OCR MEI FP1 2008 January Q8
12 marks Standard +0.3
8
  1. On a single Argand diagram, sketch the locus of points for which
    (A) \(| z - 3 \mathrm { j } | = 2\),
    (B) \(\quad \arg ( z + 1 ) = \frac { 1 } { 4 } \pi\).
  2. Indicate clearly on your Argand diagram the set of points for which $$| z - 3 \mathrm { j } | \leqslant 2 \quad \text { and } \quad \arg ( z + 1 ) \leqslant \frac { 1 } { 4 } \pi .$$
  3. (A) By drawing an appropriate line through the origin, indicate on your Argand diagram the point for which \(| z - 3 j | = 2\) and \(\arg z\) has its minimum possible value.
    (B) Calculate the value of \(\arg z\) at this point.
OCR MEI FP1 2008 January Q9
13 marks Moderate -0.8
9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = x\) and has the same \(x\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-4_807_825_402_660} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the image of the point ( \(- 3,7\) ) under transformation T .
  2. Write down the image of the point \(( x , y )\) under transformation T .
  3. Find the \(2 \times 2\) matrix which represents the transformation.
  4. Describe the transformation M represented by the matrix \(\left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).
  5. Find the matrix representing the composite transformation of T followed by M .
  6. Find the image of the point \(( x , y )\) under this composite transformation. State the equation of the line on which all of these images lie.
OCR MEI FP1 2005 June Q1
5 marks Moderate -0.8
1
  1. Find the inverse of the matrix \(\mathbf { A } = \left( \begin{array} { l l } 4 & 3 \\ 1 & 2 \end{array} \right)\).
  2. Use this inverse to solve the simultaneous equations $$\begin{aligned} 4 x + 3 y & = 5 \\ x + 2 y & = - 4 \end{aligned}$$ showing your working clearly.
OCR MEI FP1 2005 June Q2
5 marks Moderate -0.8
2 Find the roots of the quadratic equation \(x ^ { 2 } - 8 x + 17 = 0\) in the form \(a + b \mathrm { j }\).
Express these roots in modulus-argument form.
OCR MEI FP1 2005 June Q3
3 marks Moderate -0.3
3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).
OCR MEI FP1 2005 June Q4
5 marks Standard +0.3
4 The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots \(2 \alpha\) and \(2 \beta\).
OCR MEI FP1 2005 June Q5
5 marks Standard +0.3
5
  1. Sketch the locus \(| z - ( 3 + 4 j ) | = 2\) on an Argand diagram.
  2. On the same diagram, sketch the locus \(\arg ( z - 4 ) = \frac { 1 } { 2 } \pi\).
  3. Indicate clearly on your sketch the points which satisfy both $$| z - ( 3 + 4 j ) | = 2 \quad \text { and } \quad \arg ( z - 4 ) = \frac { 1 } { 2 } \pi$$
OCR MEI FP1 2005 June Q6
7 marks Moderate -0.5
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\).
OCR MEI FP1 2005 June Q7
6 marks Moderate -0.5
7 Find \(\sum _ { r = 1 } ^ { n } 3 r ( r - 1 )\), expressing your answer in a fully factorised form.
OCR MEI FP1 2005 June Q8
14 marks Challenging +1.2
8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } }\).
  1. Find the equations of the asymptotes.
  2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
  3. Sketch the curve.
  4. Solve the inequality \(\frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } } \geqslant - 1\).
OCR MEI FP1 2005 June Q9
10 marks Standard +0.3
9 The quartic equation \(x ^ { 4 } + A x ^ { 3 } + B x ^ { 2 } + C x + D = 0\), where \(A , B , C\) and \(D\) are real numbers, has roots \(2 + \mathrm { j }\) and - 2 j .
  1. Write down the other roots of the equation.
  2. Find the values of \(A , B , C\) and \(D\).
OCR MEI FP1 2005 June Q10
12 marks Standard +0.3
10
  1. You are given that $$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$ Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
  2. Hence find the sum of the infinite series $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$
OCR MEI FP1 2008 June Q1
4 marks Easy -1.2
1
  1. Write down the matrix for reflection in the \(y\)-axis.
  2. Write down the matrix for enlargement, scale factor 3, centred on the origin.
  3. Find the matrix for reflection in the \(y\)-axis, followed by enlargement, scale factor 3 , centred on the origin.
OCR MEI FP1 2008 June Q2
7 marks Standard +0.3
2 Indicate on a single Argand diagram
  1. the set of points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\),
  2. the set of points for which \(\arg ( z - 2 \mathrm { j } ) = \pi\),
  3. the two points for which \(| z - ( - 3 + 2 \mathrm { j } ) | = 2\) and \(\arg ( z - 2 \mathrm { j } ) = \pi\).
OCR MEI FP1 2008 June Q3
3 marks Moderate -0.3
3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1 \\ 2 & 2 \end{array} \right)\).
OCR MEI FP1 2008 June Q4
5 marks Moderate -0.8
4 Find the values of \(A , B , C\) and \(D\) in the identity \(3 x ^ { 3 } - x ^ { 2 } + 2 \equiv A ( x - 1 ) ^ { 3 } + \left( x ^ { 3 } + B x ^ { 2 } + C x + D \right)\).