Questions — OCR MEI FP1 (190 questions)

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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 \(\mathbf { M } = \left( \begin{array} { r r } - 1 & - 1
2 & 2 \end{array} \right)\).
OCR MEI FP1 2008 June Q4
4 Find the values of \(A , B , C\) and \(D\) in the identity \(3 x ^ { 3 } - x ^ { 2 } + 2 \equiv A ( x - 1 ) ^ { 3 } + \left( x ^ { 3 } + B x ^ { 2 } + C x + D \right)\).
OCR MEI FP1 2008 June Q5
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
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
  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
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
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
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
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
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
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 \(\mathbf { Q P }\).