Questions — OCR MEI (4456 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
OCR MEI FP2 2016 June Q1
Standard +0.3
1
    1. Given that \(\mathrm { f } ( x ) = \arctan x\), write down an expression for \(\mathrm { f } ^ { \prime } ( x )\). Assuming that \(x\) is small, use a binomial expansion to express \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\).
    2. Hence express \(\arctan x\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).
  2. Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
  3. A curve has polar equation \(r = \frac { a } { \sqrt { \theta } }\) where \(a > 0\).
    1. Sketch the curve for \(\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi\).
    2. State what happens to \(r\) as \(\theta\) tends to zero.
    3. Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines \(\theta = \frac { \pi } { 4 }\) and \(\theta = 2 \pi\). Give your answer in an exact simplified form.
      1. (i) Express \(2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)\) in terms of \(z\) where \(z = \cos \theta + \mathrm { j } \sin \theta\).
        (ii) The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\ S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta \end{aligned}$$ Show that $$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$ Hence show that, for even values of \(n\), $$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
      2. Write the complex number \(z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), expressing \(r\) and \(\theta\) as simply as possible. Hence find the cube roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Show the points representing \(z\) and its cube roots on an Argand diagram.
        1. Find the eigenvalues and eigenvectors of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 2 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right)$$ Hence express \(\mathbf { M }\) in the form \(\mathbf { P D P } ^ { - 1 }\) where \(\mathbf { D }\) is a diagonal matrix.
        2. Write down an equation for \(\mathbf { M } ^ { n }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence obtain expressions for the elements of \(\mathbf { M } ^ { n }\).
          Show that \(\mathbf { M } ^ { n }\) tends to a limit as \(n\) tends to infinity. Find that limit.
        3. Express \(\mathbf { M } ^ { - 1 }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence determine whether or not \(\left( \mathbf { M } ^ { - 1 } \right) ^ { n }\) tends to a limit as \(n\) tends to infinity. Section B (18 marks)
          1. Given that \(y = \cosh x\), use the definition of \(\cosh x\) in terms of exponential functions to prove that $$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
          2. Solve the equation $$\cosh x + \cosh 2 x = 5$$ giving the roots in an exact logarithmic form.
          3. Sketch the curve with equation \(y = \cosh x + \cosh 2 x\). Show on your sketch the line \(y = 5\). Find the area of the finite region bounded by the curve and the line \(y = 5\). Give your answer in an exact form that does not involve hyperbolic functions. \section*{END OF QUESTION PAPER}
OCR MEI S1 Q2
Easy -1.2
2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.
Number correct123
Frequency123
  1. Draw a vertical line chart to illustrate the data.
  2. State the type of skewness shown by your diagram.
  3. Calculate the mean and the mean squared deviation of the data.
  4. How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?
OCR MEI FP1 Q9
Standard +0.3
9 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 0.8 & 0.6 \\ 0.6 & - 0.8 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 }\). You are now given that the matrix \(M\) represents a reflection in a line through the origin.
  2. Explain how your answer to part (i) relates to this information.
  3. By investigating the invariant points of the reflection, find the equation of the mirror line.
  4. Describe fully the transformation represented by the matrix \(\mathbf { P } = \left( \begin{array} { c c } 0.8 & - 0.6 \\ 0.6 & 0.8 \end{array} \right)\).
  5. A composite transformation is formed by the transformation represented by \(\mathbf { P }\) followed by the transformation represented by \(\mathbf { M }\). Find the single matrix that represents this composite transformation.
  6. The composite transformation described in part (v) is equivalent to a single reflection. What is the equation of the mirror line of this reflection? \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS
    4755
    \textbackslash section*\{Further Concepts For Advanced Mathematics (FP1)\}}
    Tuesday 7 JUNE 2005Afternoon1 hour 30 minutes
    Additional materials:
    Answer booklet
    Graph paper
    MEI Examination Formulae and Tables (MF2)
    TIME 1 hour 30 minutes
OCR MEI FP1 Q10
Standard +0.3
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 $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$ RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS} Further Concepts For Advanced Mathematics (FP1)
    Wednesday 18 JANUARY 2006 Afternoon ..... 1 hour 30 minutes
    Additional materials:
    8 page answer booklet
    Graph paper
    MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes
OCR MEI C1 Q1
3 marks Easy -1.2
Solve the inequality \(2(x - 3) < 6x + 15\). [3]
OCR MEI C1 Q2
3 marks Easy -1.2
Make \(r\) the subject of \(V = \frac{4}{3}\pi r^3\). [3]
OCR MEI C1 Q3
2 marks Easy -1.8
In each case, choose one of the statements $$P \Rightarrow Q \quad\quad P \Leftarrow Q \quad\quad P \Leftrightarrow Q$$ to describe the complete relationship between P and Q.
  1. For \(n\) an integer: P: \(n\) is an even number Q: \(n\) is a multiple of 4 [1]
  2. For triangle ABC: P: B is a right-angle Q: \(AB^2 + BC^2 = AC^2\) [1]
OCR MEI C1 Q4
4 marks Moderate -0.8
Find the coefficient of \(x^3\) in the expansion of \((2 + 3x)^5\). [4]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of the following.
  1. \(\left(\frac{1}{3}\right)^{-2}\) [2]
  2. \(16^{\frac{1}{4}}\) [2]
OCR MEI C1 Q6
5 marks Moderate -0.8
The line \(L\) is parallel to \(y = -2x + 1\) and passes through the point \((5, 2)\). Find the coordinates of the points of intersection of \(L\) with the axes. [5]
OCR MEI C1 Q7
5 marks Easy -1.2
Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]
OCR MEI C1 Q8
5 marks Moderate -0.8
Find, in the form \(y = mx + c\), the equation of the line passing through A\((3, 7)\) and B\((5, -1)\). Show that the midpoint of AB lies on the line \(x + 2y = 10\). [5]
OCR MEI C1 Q9
5 marks Moderate -0.8
Simplify \((3 + \sqrt{2})(3 - \sqrt{2})\). Express \(\frac{1 + \sqrt{2}}{3 - \sqrt{2}}\) in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational. [5]
OCR MEI C1 Q10
12 marks Moderate -0.8
\includegraphics{figure_10} Fig. 10 shows a circle with centre C\((2, 1)\) and radius 5.
  1. Show that the equation of the circle may be written as $$x^2 + y^2 - 4x - 2y - 20 = 0.$$ [3]
  2. Find the coordinates of the points P and Q where the circle cuts the \(y\)-axis. Leave your answers in the form \(a \pm \sqrt{b}\). [3]
  3. Verify that the point A\((5, -3)\) lies on the circle. Show that the tangent to the circle at A has equation \(4y = 3x - 27\). [6]
OCR MEI C1 Q11
12 marks Moderate -0.3
A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).
  1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
  2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). Write down an equation for the resulting graph. You need not simplify your answer. Find also the intercept on the \(y\)-axis of the resulting graph. [5]
OCR MEI C1 Q12
12 marks Moderate -0.3
  1. Show that the graph of \(y = x^2 - 3x + 11\) is above the \(x\)-axis for all values of \(x\). [3]
  2. Find the set of values of \(x\) for which the graph of \(y = 2x^2 + x - 10\) is above the \(x\)-axis. [4]
  3. Find algebraically the coordinates of the points of intersection of the graphs of $$y = x^2 - 3x + 11 \quad\text{and}\quad y = 2x^2 + x - 10.$$ [5]
OCR MEI C1 2006 January Q1
2 marks Easy -1.2
\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]
OCR MEI C1 2006 January Q2
4 marks Moderate -0.8
\includegraphics{figure_2} Fig. 2 shows graphs \(A\) and \(B\).
  1. State the transformation which maps graph \(A\) onto graph \(B\). [2]
  2. The equation of graph \(A\) is \(y = f(x)\). Which one of the following is the equation of graph \(B\)? \(y = f(x) + 2\) \quad \(y = f(x) - 2\) \quad \(y = f(x + 2)\) \quad \(y = f(x - 2)\) \(y = 2f(x)\) \quad \(y = f(x + 3)\) \quad \(y = f(x - 3)\) \quad \(y = 3f(x)\) [2]
OCR MEI C1 2006 January Q3
4 marks Easy -1.8
Find the binomial expansion of \((2 + x)^4\), writing each term as simply as possible. [4]
OCR MEI C1 2006 January Q4
4 marks Easy -1.8
Solve the inequality \(\frac{3(2x + 1)}{4} > -6\). [4]
OCR MEI C1 2006 January Q5
4 marks Moderate -0.8
Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]
OCR MEI C1 2006 January Q6
3 marks Easy -1.2
When \(x^3 + 3x + k\) is divided by \(x - 1\), the remainder is 6. Find the value of \(k\). [3]
OCR MEI C1 2006 January Q7
5 marks Moderate -0.8
\includegraphics{figure_7} The line AB has equation \(y = 4x - 5\) and passes through the point B(2, 3), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C. Find the equation of the line BC and the \(x\)-coordinate of C. [5]
OCR MEI C1 2006 January Q8
5 marks Easy -1.3
  1. Simplify \(5\sqrt{8} + 4\sqrt{50}\). Express your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
  2. Express \(\frac{\sqrt{3}}{6 - \sqrt{3}}\) in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are rational. [3]
OCR MEI C1 2006 January Q9
5 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
  2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]