Questions — OCR MEI (4333 questions)

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OCR MEI FP1 2006 January Q4
5 marks Moderate -0.8
4 The matrix equation \(\left( \begin{array} { r r } 6 & - 2 \\ - 3 & 1 \end{array} \right) \binom { x } { y } = \binom { a } { b }\) represents two simultaneous linear equations in \(x\) and \(y\).
  1. Write down the two equations.
  2. Evaluate the determinant of \(\left( \begin{array} { r r } 6 & - 2 \\ - 3 & 1 \end{array} \right)\). What does this value tell you about the solution of the equations in part (i)?
OCR MEI FP1 2006 January Q5
6 marks Standard +0.3
5 The cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
  2. Find the cubic equation with roots \(2 \alpha , 2 \beta\) and \(2 \gamma\), simplifying your answer as far as possible.
OCR MEI FP1 2006 January Q6
7 marks Standard +0.3
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }\).
OCR MEI FP1 2006 January Q7
13 marks Standard +0.3
7 A curve has equation \(y = \frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } }\).
  1. Show that \(y\) can never be zero.
  2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
  3. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
  4. Sketch the curve.
  5. Solve the inequality \(\frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } } \leqslant - 2\).
OCR MEI FP1 2006 January Q8
11 marks Standard +0.3
8 You are given that the complex number \(\alpha = 1 + \mathrm { j }\) satisfies the equation \(z ^ { 3 } + 3 z ^ { 2 } + p z + q = 0\), where \(p\) and \(q\) are real constants.
  1. Find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\) in the form \(a + b \mathrm { j }\). Hence show that \(p = - 8\) and \(q = 10\).
  2. Find the other two roots of the equation.
  3. Represent the three roots on an Argand diagram.
OCR MEI FP1 2006 January Q9
12 marks Standard +0.3
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 \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4048c232-6a4e-4baa-9262-93428f375203-4_821_837_475_612} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Write down the image of the point \(( 10,50 )\) under transformation T .
  2. P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
  3. All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
  4. In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
  5. For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
  6. Find the \(2 \times 2\) matrix which represents the transformation.
  7. Show that this matrix is singular. Relate this result to the transformation.
OCR MEI FP1 2007 January Q1
2 marks Easy -1.2
1 Is the following statement true or false? Justify your answer. $$x ^ { 2 } = 4 \text { if and only if } x = 2$$
OCR MEI FP1 2007 January Q2
6 marks Easy -1.2
2
  1. Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 7 = 0\), simplifying your answers as far as possible.
  2. Represent these roots on an Argand diagram.
OCR MEI FP1 2007 January Q3
7 marks Easy -1.2
3 The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 3 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively under the transformation represented by the matrix \(\mathbf { M } = \left( \begin{array} { l l } 2 & 0 \\ 0 & \frac { 1 } { 2 } \end{array} \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4a339746-195f-477a-952e-02fbdfd9cce5-2_446_444_1046_808} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Draw a diagram showing the image of the triangle after the transformation, labelling the image of each point clearly.
  2. Describe fully the transformation represented by the matrix \(\mathbf { M }\).
OCR MEI FP1 2007 January Q4
6 marks Moderate -0.5
4 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } + 1 \right)\), factorising your answer as far as possible.
OCR MEI FP1 2007 January Q5
7 marks Standard +0.8
5 The roots of the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find the cubic equation whose roots are \(2 \alpha + 1,2 \beta + 1\) and \(2 \gamma + 1\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2007 January Q6
8 marks Moderate -0.5
6 Prove by induction that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\).
OCR MEI FP1 2007 January Q7
12 marks Moderate -0.3
7 A curve has equation \(y = \frac { 5 } { ( x + 2 ) ( 4 - x ) }\).
  1. Write down the value of \(y\) when \(x = 0\).
  2. Write down the equations of the three asymptotes.
  3. Sketch the curve.
  4. Find the values of \(x\) for which \(\frac { 5 } { ( x + 2 ) ( 4 - x ) } = 1\) and hence solve the inequality $$\frac { 5 } { ( x + 2 ) ( 4 - x ) } < 1 .$$
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
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.