Questions FP1 (1385 questions)

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OCR MEI FP1 2005 June Q5
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
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
7 Find \(\sum _ { r = 1 } ^ { n } 3 r ( r - 1 )\), expressing your answer in a fully factorised form.
OCR MEI FP1 2005 June Q8
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
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
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 2006 June Q1
1
  1. State the transformation represented by the matrix \(\left( \begin{array} { r r } 1 & 0
    0 & - 1 \end{array} \right)\).
  2. Write down the \(2 \times 2\) matrix for rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
  3. Find the \(2 \times 2\) matrix for rotation through \(90 ^ { \circ }\) anticlockwise about the origin, followed by reflection in the \(x\)-axis.
OCR MEI FP1 2006 June Q2
2 Find the values of \(A\), \(B\), \(C\) and \(D\) in the identity $$2 x ^ { 3 } - 3 x ^ { 2 } + x - 2 \equiv ( x + 2 ) \left( A x ^ { 2 } + B x + C \right) + D .$$
OCR MEI FP1 2006 June Q3
3 The cubic equation \(z ^ { 3 } + 4 z ^ { 2 } - 3 z + 1 = 0\) 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. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 22\).
OCR MEI FP1 2006 June Q4
4 Indicate, on separate Argand diagrams,
  1. the set of points \(z\) for which \(| z - ( 3 - \mathrm { j } ) | \leqslant 3\),
  2. the set of points \(z\) for which \(1 < | z - ( 3 - \mathrm { j } ) | \leqslant 3\),
  3. the set of points \(z\) for which \(\arg ( z - ( 3 - \mathrm { j } ) ) = \frac { 1 } { 4 } \pi\).
OCR MEI FP1 2006 June Q5
5
  1. The matrix \(\mathbf { S } = \left( \begin{array} { l l } - 1 & 2
    - 3 & 4 \end{array} \right)\) represents a transformation.
    (A) Show that the point \(( 1,1 )\) is invariant under this transformation.
    (B) Calculate \(\mathbf { S } ^ { - 1 }\).
    (C) Verify that \(( 1,1 )\) is also invariant under the transformation represented by \(\mathbf { S } ^ { - 1 }\).
  2. Part (i) may be generalised as follows. If \(( x , y )\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf { T }\), it is also invariant under the transformation represented by \(\mathbf { T } ^ { - 1 }\). Starting with \(\mathbf { T } \binom { x } { y } = \binom { x } { y }\), or otherwise, prove this result.
OCR MEI FP1 2006 June Q6
6 Prove by induction that \(3 + 6 + 12 + \ldots + 3 \times 2 ^ { n - 1 } = 3 \left( 2 ^ { n } - 1 \right)\) for all positive integers \(n\).
OCR MEI FP1 2006 June Q9
9
  1. Show that \(r ( r + 1 ) ( r + 2 ) - ( r - 1 ) r ( r + 1 ) \equiv 3 r ( r + 1 )\).
  2. Hence use the method of differences to find an expression for \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\).
  3. Show that you can obtain the same expression for \(\sum _ { r = 1 } ^ { n } r ( r + 1 )\) using the standard formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\).
OCR MEI FP1 2007 June Q1
1 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 2 & - 1
4 & 3 \end{array} \right)\).
  1. Find the inverse of \(\mathbf { M }\).
  2. A triangle of area 2 square units undergoes the transformation represented by the matrix \(\mathbf { M }\). Find the area of the image of the triangle following this transformation.
OCR MEI FP1 2007 June Q2
2 Write down the equation of the locus represented by the circle in the Argand diagram shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d7e20cfb-da04-4d7b-bcda-53f99f6faec4-2_581_600_872_737} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
OCR MEI FP1 2007 June Q3
3 Find the values of the constants \(A\), \(B\), \(C\) and \(D\) in the identity $$x ^ { 3 } - 4 \equiv ( x - 1 ) \left( A x ^ { 2 } + B x + C \right) + D .$$
OCR MEI FP1 2007 June Q4
4 Two complex numbers, \(\alpha\) and \(\beta\), are given by \(\alpha = 1 - 2 \mathrm { j }\) and \(\beta = - 2 - \mathrm { j }\).
  1. Represent \(\beta\) and its complex conjugate \(\beta ^ { * }\) on an Argand diagram.
  2. Express \(\alpha \beta\) in the form \(a + b \mathrm { j }\).
  3. Express \(\frac { \alpha + \beta } { \beta }\) in the form \(a + b \mathrm { j }\).
OCR MEI FP1 2007 June Q5
5 The roots of the cubic equation \(x ^ { 3 } + 3 x ^ { 2 } - 7 x + 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha , 3 \beta\) and \(3 \gamma\), expressing your answer in a form with integer coefficients.
  1. Show that \(\frac { 1 } { r + 2 } - \frac { 1 } { r + 3 } = \frac { 1 } { ( r + 2 ) ( r + 3 ) }\).
  2. Hence use the method of differences to find \(\frac { 1 } { 3 \times 4 } + \frac { 1 } { 4 \times 5 } + \frac { 1 } { 5 \times 6 } + \ldots + \frac { 1 } { 52 \times 53 }\).
OCR MEI FP1 2007 June Q7
7 Prove by induction that \(\sum _ { r = 1 } ^ { n } 3 ^ { r - 1 } = \frac { 3 ^ { n } - 1 } { 2 }\).
OCR MEI FP1 2007 June Q8
8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( x - 3 ) ( x + 1 ) ( x - 1 ) }\).
  1. Write down 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. Determine whether the curve approaches the horizontal asymptote from above or below for
    (A) large positive values of \(x\),
    (B) large negative values of \(x\).
  4. Sketch the curve.
OCR MEI FP1 2007 June Q9
9 The cubic equation \(x ^ { 3 } + A x ^ { 2 } + B x + 15 = 0\), where \(A\) and \(B\) are real numbers, has a root \(x = 1 + 2 \mathrm { j }\).
  1. Write down the other complex root.
  2. Explain why the equation must have a real root.
  3. Find the value of the real root and the values of \(A\) and \(B\).
OCR MEI FP1 2007 June Q10
10 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & k
2 & 1 & 2
3 & 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r c c } - 5 & - 2 + 2 k & - 4 - k
8 & - 1 - 3 k & - 2 + 2 k
1 & - 8 & 5 \end{array} \right)\) and that \(\mathbf { A B }\) is of the form \(\mathbf { A B } = \left( \begin{array} { c c c } k - n & 0 & 0
0 & k - n & 0
0 & 0 & k - n \end{array} \right)\).
  1. Find the value of \(n\).
  2. Write down the inverse matrix \(\mathbf { A } ^ { - 1 }\) and state the condition on \(k\) for this inverse to exist.
  3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. $$\begin{aligned} x - 2 y + z = & 1
    2 x + y + 2 z = & 12
    3 x + 2 y - z = & 3 \end{aligned}$$
OCR MEI FP1 2008 June Q1
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
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 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1
2 & 2 \end{array} \right)\).