Questions — OCR MEI FP1 (190 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR MEI FP1 2012 June Q6
6 A sequence is defined by \(a _ { 1 } = 1\) and \(a _ { k + 1 } = 3 \left( a _ { k } + 1 \right)\).
  1. Calculate the value of the third term, \(a _ { 3 }\).
  2. Prove by induction that \(a _ { n } = \frac { 5 \times 3 ^ { n - 1 } - 3 } { 2 }\).
OCR MEI FP1 2012 June Q7
7 A curve has equation \(y = \frac { x ^ { 2 } - 25 } { ( x - 3 ) ( x + 4 ) ( 3 x + 2 ) }\).
  1. Write down the coordinates of the points where the curve crosses the axes.
  2. Write down the equations of the asymptotes.
  3. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\).
  4. Sketch the curve.
OCR MEI FP1 2012 June Q8
8
  1. Verify that \(1 + 3 \mathrm { j }\) is a root of the equation \(3 z ^ { 3 } - 2 z ^ { 2 } + 22 z + 40 = 0\), showing your working.
  2. Explain why the equation must have exactly one real root.
  3. Find the other roots of the equation.
OCR MEI FP1 2012 June Q9
9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 3 & - 4 & 1
2 & 1 & k
7 & - 1 & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r c } - 4 & - 5 & 11
- 19 & - 4 & - 7
- 9 & - 31 & 2 - k \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } 79 & 0 & - 3 - k
- 9 k - 27 & - 31 k - 14 & q
p & 0 & 82 + k \end{array} \right)\) where \(p\) and \(q\) are to be determined.
  1. Show that \(p = 0\) and \(q = 15 + 2 k - k ^ { 2 }\). It is now given that \(k = - 3\).
  2. Find \(\mathbf { A B }\) and hence write down the inverse matrix \(\mathbf { A } ^ { - 1 }\).
  3. Use a matrix method to find the values of \(x , y\) and \(z\) that satisfy the equation \(\mathbf { A } \left( \begin{array} { l } x
    y
    z \end{array} \right) = \left( \begin{array} { r } 14
    - 23
    9 \end{array} \right)\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR MEI FP1 2013 June Q1
1 Find the values of \(A , B , C\) and \(D\) in the identity \(2 x \left( x ^ { 2 } - 5 \right) \equiv ( x - 2 ) \left( A x ^ { 2 } + B x + C \right) + D\).
OCR MEI FP1 2013 June Q2
2 You are given that \(z = \frac { 3 } { 2 }\) is a root of the cubic equation \(2 z ^ { 3 } + 9 z ^ { 2 } + 2 z - 30 = 0\). Find the other two roots.
OCR MEI FP1 2013 June Q3
3 You are given that \(\mathbf { N } = \left( \begin{array} { r r r } - 9 & - 2 & - 4
3 & 2 & 2
5 & 1 & 2 \end{array} \right)\) and \(\mathbf { N } ^ { - 1 } = \left( \begin{array} { r r r } 1 & 0 & 2
2 & 1 & 3
- \frac { 7 } { 2 } & p & - 6 \end{array} \right)\).
  1. Find the value of \(p\).
  2. Solve the equation \(\mathbf { N } \left( \begin{array} { c } x
    y
    z \end{array} \right) = \left( \begin{array} { r } - 39
    5
    22 \end{array} \right)\).
OCR MEI FP1 2013 June Q4
4 The complex number \(z _ { 1 }\) is \(3 - 2 \mathrm { j }\) and the complex number \(z _ { 2 }\) has modulus 5 and argument \(\frac { \pi } { 4 }\).
  1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { j }\), giving \(a\) and \(b\) in exact form.
  2. Represent \(z _ { 1 } , z _ { 2 } , z _ { 1 } + z _ { 2 }\) and \(z _ { 1 } - z _ { 2 }\) on a single Argand diagram.
OCR MEI FP1 2013 June Q5
5 You are given that \(\frac { 4 } { ( 4 n - 3 ) ( 4 n + 1 ) } \equiv \frac { 1 } { 4 n - 3 } - \frac { 1 } { 4 n + 1 }\). Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 4 r - 3 ) ( 4 r + 1 ) } = \frac { n } { 4 n + 1 }$$
OCR MEI FP1 2013 June Q6
6 The cubic equation \(x ^ { 3 } - 5 x ^ { 2 } + 3 x - 6 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with roots \(\frac { \alpha } { 3 } + 1 , \frac { \beta } { 3 } + 1\) and \(\frac { \gamma } { 3 } + 1\), simplifying your answer as far as possible.
OCR MEI FP1 2013 June Q7
7 Fig. 7 shows an incomplete sketch of \(y = \frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) }\) where \(a , b\) and \(c\) are integers. The asymptotes of the curve are also shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{597abea9-6d00-416e-9203-d5bce9bd1af1-3_928_996_493_535} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Determine the values of \(a , b\) and \(c\). Use these values of \(a , b\) and \(c\) throughout the rest of the question.
  2. Determine how the curve approaches the horizontal asymptote for large positive values of \(x\), and for large negative values of \(x\), justifying your answer. On the copy of Fig. 7, sketch the rest of the curve.
  3. Find the \(x\) coordinates of the points on the curve where \(y = 1\). Write down the solution to the inequality \(\frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) } < 1\).
OCR MEI FP1 2013 June Q8
8
  1. Use standard series formulae to show that $$\sum _ { r = 1 } ^ { n } [ r ( r - 1 ) - 1 ] = \frac { 1 } { 3 } n ( n + 2 ) ( n - 2 )$$
  2. Prove (*) by mathematical induction.
OCR MEI FP1 2013 June Q9
9
  1. Describe fully the transformation Q , represented by the matrix \(\mathbf { Q }\), where \(\mathbf { Q } = \left( \begin{array} { r l } 0 & 1
    - 1 & 0 \end{array} \right)\). The transformation M is represented by the matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { r r } 0 & - 1
    0 & 1 \end{array} \right)\).
  2. M maps all points on the line \(y = 2\) onto a single point, P. Find the coordinates of P.
  3. M maps all points on the plane onto a single line, \(l\). Find the equation of \(l\).
  4. M maps all points on the line \(n\) onto the point ( - 6 , 6). Find the equation of \(n\).
  5. Show that \(\mathbf { M }\) is singular. Relate this to the transformation it represents.
  6. R is the composite transformation M followed by Q . R maps all points on the plane onto the line \(q\). Find the equation of \(q\).
OCR MEI FP1 2014 June Q1
1 Use standard series formulae to find \(\sum _ { r = 1 } ^ { n } r ( r - 2 )\), factorising your answer as far as possible.
OCR MEI FP1 2014 June Q2
2 Fig. 2 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]{3df020b0-fb7b-454b-b354-36cc2b8df5f6-2_595_739_571_664} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Write down the matrix \(\mathbf { T }\) representing this transformation. The quadrilateral \(\mathrm { OA } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }\) is reflected in the \(x\)-axis to give a new quadrilateral, \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
  2. Write down the matrix representing reflection in the \(x\)-axis.
  3. Find the single matrix that will transform OABC onto \(\mathrm { OA } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\).
OCR MEI FP1 2014 June Q3
3 You are given that \(z = 2 + 3 \mathrm { j }\) is a root of the quartic equation \(z ^ { 4 } - 5 z ^ { 3 } + 15 z ^ { 2 } - 5 z - 26 = 0\). Find the other roots.
OCR MEI FP1 2014 June Q4
4 Use the identity \(\frac { 1 } { 2 r + 3 } - \frac { 1 } { 2 r + 5 } \equiv \frac { 2 } { ( 2 r + 3 ) ( 2 r + 5 ) }\) and the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 3 ) ( 2 r + 5 ) }\), expressing your answer as a single fraction.
OCR MEI FP1 2014 June Q5
5 The roots of the cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the cubic equation whose roots are \(3 \alpha - 1,3 \beta - 1\) and \(3 \gamma - 1\), expressing your answer in a form with integer coefficients.
OCR MEI FP1 2014 June Q6
6 Prove by induction that \(\frac { 1 } { 1 \times 3 } + \frac { 1 } { 3 \times 5 } + \frac { 1 } { 5 \times 7 } + \ldots + \frac { 1 } { ( 2 n - 1 ) ( 2 n + 1 ) } = \frac { n } { 2 n + 1 }\).
OCR MEI FP1 2014 June Q7
7 A curve has equation \(y = \frac { x ^ { 2 } - 5 } { ( x + 3 ) ( x - 2 ) ( a x - 1 ) }\), where \(a\) is a constant.
  1. Find the coordinates of the points where the curve crosses the \(x\)-axis and the \(y\)-axis.
  2. You are given that the curve has a vertical asymptote at \(x = \frac { 1 } { 2 }\). Write down the value of \(a\) and the equations of the other asymptotes.
  3. Sketch the curve.
  4. Find the set of values of \(x\) for which \(y > 0\).
OCR MEI FP1 2014 June Q8
8 You are given the complex number \(w = 2 + 2 \sqrt { 3 } \mathrm { j }\).
  1. Express \(w\) in modulus-argument form.
  2. Indicate on an Argand diagram the set of points, \(z\), which satisfy both of the following inequalities. $$- \frac { \pi } { 2 } \leqslant \arg z \leqslant \frac { \pi } { 3 } \text { and } | z | \leqslant 4$$ Mark \(w\) on your Argand diagram and find the greatest value of \(| z - w |\).
OCR MEI FP1 2014 June Q9
9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & 3 & - 1
- 1 & \alpha & - 1
- 2 & - 1 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } 3 \alpha - 1 & - 8 & \alpha - 3
5 & 1 & 2
2 \alpha + 1 & - 5 & \alpha + 3 \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } \gamma & 0 & 0
\beta & \gamma & 0
0 & 0 & \gamma \end{array} \right)\).
  1. Show that \(\beta = 0\).
  2. Find \(\gamma\) in terms of \(\alpha\).
  3. Write down \(\mathbf { A } ^ { - 1 }\) for the case when \(\alpha = 2\). State the value of \(\alpha\) for which \(\mathbf { A } ^ { - 1 }\) does not exist.
  4. Use your answer to part (iii) to solve the following simultaneous equations. $$\begin{aligned} x + 3 y - z & = 25
    - x + 2 y - z & = 11
    - 2 x - y + 3 z & = - 23 \end{aligned}$$
OCR MEI FP1 2015 June Q1
1 Given that \(\mathbf { M } \binom { x } { y } = \binom { 1 } { 3 }\), where \(\mathbf { M } = \left( \begin{array} { r r } 4 & - 3
8 & 21 \end{array} \right)\), find \(x\) and \(y\).
OCR MEI FP1 2015 June Q2
2 Find the roots of the quadratic equation \(z ^ { 2 } - 4 z + 13 = 0\).
Find the modulus and argument of each root.
OCR MEI FP1 2015 June Q3
3 The equation \(2 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } + \mathrm { qx } + \mathrm { r } = 0\) has a root at \(x = 4\). The sum of the roots is 6 and the product of the roots is - 10 . Find \(p , q\) and \(r\).