Questions — OCR MEI FP1 (190 questions)

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OCR MEI FP1 2007 January Q8
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
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
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
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
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
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 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
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
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
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
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
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
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 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
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
  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 \(\arg ( z - ( 3 - \mathrm { j } ) ) = \frac { 1 } { 4 } \pi\).