Questions — OCR MEI (4301 questions)

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OCR MEI C1 Q1
1 Expand \(( 2 x + 5 ) ( x - 1 ) ( x + 3 )\), simplifying your answer.
OCR MEI C1 Q2
2 Find the discriminant of \(3 x ^ { 2 } + 5 x + 2\). Hence state the number of distinct real roots of the equation \(3 x ^ { 2 } + 5 x + 2 = 0\).
OCR MEI C1 Q3
3 Make \(x\) the subject of the formula \(y = \frac { 1 - 2 x } { x + 3 }\).
OCR MEI C1 Q4
4 Factorise \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\). Hence prove that, when \(n\) is a positive integer, \(n ^ { 3 } + 3 n ^ { 2 } + 2 n\) is always divisible by 6 .
OCR MEI C1 Q5
5 Express \(5 x ^ { 2 } + 20 x + 6\) in the form \(a ( x + b ) ^ { 2 } + c\).
OCR MEI C1 Q6
6 Rearrange the formula \(c = \sqrt { \frac { a + b } { 2 } }\) to make \(a\) the subject.
\(7 \quad\) Make \(a\) the subject of the formula \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).
OCR MEI C1 Q8
8 Prove that, when \(n\) is an integer, \(n ^ { 3 } - n\) is always even.
OCR MEI C1 Q9
9
  1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } + 6 x + 5\).
OCR MEI C1 Q10
10 Find the real roots of the equation \(x ^ { 4 } - 5 x ^ { 2 } - 36 = 0\) by considering it as a quadratic equation in \(x ^ { 2 }\).
OCR MEI C1 Q11
11 Solve the equation \(\frac { 3 x + 1 } { 2 x } = 4\).
OCR MEI C1 Q12
12 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.
OCR MEI C1 Q13
13 Rearrange \(y + 5 = x ( y + 2 )\) to make \(y\) the subject of the formula.
OCR MEI FP2 2006 June Q1
1
  1. A curve has polar equation \(r = a ( \sqrt { 2 } + 2 \cos \theta )\) for \(- \frac { 3 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
    1. Sketch the curve.
    2. Find, in an exact form, the area of the region enclosed by the curve.
    1. Find the Maclaurin series for the function \(\mathrm { f } ( x ) = \tan \left( \frac { 1 } { 4 } \pi + x \right)\), up to the term in \(x ^ { 2 }\).
    2. Use the Maclaurin series to show that, when \(h\) is small, $$\int _ { - h } ^ { h } x ^ { 2 } \tan \left( \frac { 1 } { 4 } \pi + x \right) \mathrm { d } x \approx \frac { 2 } { 3 } h ^ { 3 } + \frac { 4 } { 5 } h ^ { 5 }$$
OCR MEI FP2 2006 June Q2
2
    1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + \frac { 1 } { z ^ { n } }\) and \(z ^ { n } - \frac { 1 } { z ^ { n } }\) in simplified trigonometric form.
    2. By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), find \(A , B , C\) and \(D\) such that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
    1. Find the modulus and argument of \(4 + 4 \mathrm { j }\).
    2. Find the fifth roots of \(4 + 4 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate these fifth roots on an Argand diagram.
    3. Find integers \(p\) and \(q\) such that \(( p + q \mathrm { j } ) ^ { 5 } = 4 + 4 \mathrm { j }\).
OCR MEI FP2 2006 June Q3
3
  1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 4 & 1 & k
    3 & 2 & 5
    8 & 5 & 13 \end{array} \right)\), where \(k \neq 5\).
  2. Solve the simultaneous equations $$\begin{aligned} & 4 x + y + 7 z = 12
    & 3 x + 2 y + 5 z = m
    & 8 x + 5 y + 13 z = 0 \end{aligned}$$ giving \(x , y\) and \(z\) in terms of \(m\).
  3. Find the value of \(p\) for which the simultaneous equations $$\begin{aligned} & 4 x + y + 5 z = 12
    & 3 x + 2 y + 5 z = p
    & 8 x + 5 y + 13 z = 0 \end{aligned}$$ have solutions, and find the general solution in this case.
OCR MEI FP2 2006 June Q4
4
  1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$1 + 2 \sinh ^ { 2 } x = \cosh 2 x$$
  2. Solve the equation $$2 \cosh 2 x + \sinh x = 5 ,$$ giving the answers in an exact logarithmic form.
  3. Show that \(\int _ { 0 } ^ { \ln 3 } \sinh ^ { 2 } x \mathrm {~d} x = \frac { 10 } { 9 } - \frac { 1 } { 2 } \ln 3\).
  4. Find the exact value of \(\int _ { 3 } ^ { 5 } \sqrt { x ^ { 2 } - 9 } \mathrm {~d} x\).
OCR MEI FP2 2007 June Q1
1
  1. A curve has polar equation \(r = a ( 1 - \cos \theta )\), where \(a\) is a positive constant.
    1. Sketch the curve.
    2. Find the area of the region enclosed by the section of the curve for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) and the line \(\theta = \frac { 1 } { 2 } \pi\).
  2. Use a trigonometric substitution to show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { 1 } { 4 \sqrt { 3 } }\).
  3. In this part of the question, \(\mathrm { f } ( x ) = \arccos ( 2 x )\).
    1. Find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Use a standard series to expand \(\mathrm { f } ^ { \prime } ( x )\), and hence find the series for \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to the term in \(x ^ { 5 }\).
OCR MEI FP2 2007 June Q2
2
  1. Use de Moivre's theorem to show that \(\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta\).
    1. Find the cube roots of \(- 2 + 2 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). These cube roots are represented by points \(\mathrm { A } , \mathrm { B }\) and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M , and M represents the complex number \(w\).
    2. Draw an Argand diagram, showing the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and M .
    3. Find the modulus and argument of \(w\).
    4. Find \(w ^ { 6 }\) in the form \(a + b \mathrm { j }\).
OCR MEI FP2 2007 June Q3
3 Let \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 5 & 2
5 & 3 & - 2
2 & - 2 & - 4 \end{array} \right)\).
  1. Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } - 2 \lambda ^ { 2 } - 48 \lambda = 0\). You are given that \(\left( \begin{array} { r } 1
    - 1
    1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) corresponding to the eigenvalue 0 .
  2. Find the other two eigenvalues of \(\mathbf { M }\), and corresponding eigenvectors.
  3. Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { P } ^ { - 1 } \mathbf { M } ^ { 2 } \mathbf { P } = \mathbf { D }\).
  4. Use the Cayley-Hamilton theorem to find integers \(a\) and \(b\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M }\). Section B (18 marks)
OCR MEI FP2 2007 June Q4
4
  1. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.
    1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that \(\sinh 2 x = 2 \sinh x \cosh x\).
    2. Show that one of the stationary points on the curve $$y = 20 \cosh x - 3 \cosh 2 x$$ has coordinates \(\left( \ln 3 , \frac { 59 } { 3 } \right)\), and find the coordinates of the other two stationary points.
    3. Show that \(\int _ { - \ln 3 } ^ { \ln 3 } ( 20 \cosh x - 3 \cosh 2 x ) \mathrm { d } x = 40\).
OCR MEI FP2 2007 June Q5
5 The curve with equation \(y = \frac { x ^ { 2 } - k x + 2 k } { x + k }\) is to be investigated for different values of \(k\).
  1. Use your graphical calculator to obtain rough sketches of the curve in the cases \(k = - 2\), \(k = - 0.5\) and \(k = 1\).
  2. Show that the equation of the curve may be written as \(y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }\). Hence find the two values of \(k\) for which the curve is a straight line.
  3. When the curve is not a straight line, it is a conic.
    (A) Name the type of conic.
    (B) Write down the equations of the asymptotes.
  4. Draw a sketch to show the shape of the curve when \(1 < k < 8\). This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where \(x = 1\) and \(x = k\) respectively.
OCR MEI FP2 2008 June Q1
1
  1. A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 3 x y ^ { 2 }\).
    1. Show that the polar equation of the curve is \(r = 3 \cos \theta \sin ^ { 2 } \theta\).
    2. Hence sketch the curve.
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x\).
    1. Write down the series for \(\ln ( 1 + x )\) and the series for \(\ln ( 1 - x )\), both as far as the term in \(x ^ { 5 }\).
    2. Hence find the first three non-zero terms in the series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
    3. Use the series in part (ii) to show that \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3\).
OCR MEI FP2 2008 June Q2
2 You are given the complex numbers \(z = \sqrt { 32 } ( 1 + \mathrm { j } )\) and \(w = 8 \left( \cos \frac { 7 } { 12 } \pi + \mathrm { j } \sin \frac { 7 } { 12 } \pi \right)\).
  1. Find the modulus and argument of each of the complex numbers \(z , z ^ { * } , z w\) and \(\frac { z } { w }\).
  2. Express \(\frac { z } { w }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  3. Find the cube roots of \(z\), in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  4. Show that the cube roots of \(z\) can be written as $$k _ { 1 } w ^ { * } , \quad k _ { 2 } z ^ { * } \quad \text { and } \quad k _ { 3 } \mathrm { j } w ,$$ where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are real numbers. State the values of \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\).
OCR MEI FP2 2008 June Q3
3
  1. Given the matrix \(\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k
    1 & 0 & 1
    3 & 1 & 2 \end{array} \right)\) (where \(k \neq 3\) ), find \(\mathbf { Q } ^ { - 1 }\) in terms of \(k\).
    Show that, when \(k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1
    1 & - 8 & 2
    1 & - 5 & 1 \end{array} \right)\). The matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2
    1
    3 \end{array} \right) , \left( \begin{array} { r } - 1
    0
    1 \end{array} \right)\) and \(\left( \begin{array} { l } 4
    1
    2 \end{array} \right)\), with corresponding eigenvalues \(1 , - 1\) and 3 respectively.
  2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\), and hence find the matrix \(\mathbf { M }\).
  3. Write down the characteristic equation for \(\mathbf { M }\), and use the Cayley-Hamilton theorem to find integers \(a , b\) and \(c\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)
OCR MEI FP2 2008 June Q4
4
  1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
  2. Solve the equation \(4 \cosh ^ { 2 } x + 9 \sinh x = 13\), giving the answers in exact logarithmic form.
  3. Show that there is only one stationary point on the curve $$y = 4 \cosh ^ { 2 } x + 9 \sinh x$$ and find the \(y\)-coordinate of the stationary point.
  4. Show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 \cosh ^ { 2 } x + 9 \sinh x \right) \mathrm { d } x = 2 \ln 2 + \frac { 33 } { 8 }\).