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OCR MEI FP2 2012 January Q2
18 marks Standard +0.3
2
  1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + a \cos \theta + a ^ { 2 } \cos 2 \theta + \ldots \\ & S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{aligned}$$ where \(a\) is a real number and \(| a | < 1\).
    By considering \(C + \mathrm { j } S\), show that \(C = \frac { 1 - a \cos \theta } { 1 + a ^ { 2 } - 2 a \cos \theta }\) and find a corresponding expression for \(S\).
  2. Express the complex number \(z = - 1 + \mathrm { j } \sqrt { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the 4th roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
    Show \(z\) and its 4th roots in an Argand diagram.
    Find the product of the 4th roots and mark this as a point on your Argand diagram.
OCR MEI FP2 2012 January Q3
18 marks Standard +0.3
3
  1. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 1 & 2 \\ - 4 & 3 & 2 \\ 2 & 1 & - 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } - 7 \lambda + 35 = 0\).
  2. Show that \(\lambda = 5\) is an eigenvalue of \(\mathbf { M }\), and find its other eigenvalues.
  3. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 5\). State the magnitudes and directions of the vectors \(\mathbf { M } ^ { 2 } \mathbf { v }\) and \(\mathbf { M } ^ { - 1 } \mathbf { v }\).
  4. Use the Cayley-Hamilton theorem to find the constants \(a , b , c\) such that $$\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I } .$$ Section B (18 marks)
OCR MEI FP2 2012 January Q4
18 marks Standard +0.8
4
  1. Define tanh \(t\) in terms of exponential functions. Sketch the graph of \(\tanh t\).
  2. Show that \(\operatorname { artanh } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). State the set of values of \(x\) for which this equation is valid.
  3. Differentiate the equation \(\tanh y = x\) with respect to \(x\) and hence show that the derivative of \(\operatorname { artanh } x\) is \(\frac { 1 } { 1 - x ^ { 2 } }\). Show that this result may also be obtained by differentiating the equation in part (ii).
  4. By considering \(\operatorname { artanh } x\) as \(1 \times \operatorname { artanh } x\) and using integration by parts, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \operatorname { artanh } x \mathrm {~d} x = \frac { 1 } { 4 } \ln \frac { 27 } { 16 }$$
OCR MEI FP2 2012 January Q5
18 marks Challenging +1.2
5 The points \(\mathrm { A } ( - 1,0 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { P } ( x , y )\) are such that the product of the distances PA and PB is 1 . You are given that the cartesian equation of the locus of P is $$\left( ( x + 1 ) ^ { 2 } + y ^ { 2 } \right) \left( ( x - 1 ) ^ { 2 } + y ^ { 2 } \right) = 1 .$$
  1. Show that this equation may be written in polar form as $$r ^ { 4 } + 2 r ^ { 2 } = 4 r ^ { 2 } \cos ^ { 2 } \theta$$ Show that the polar equation simplifies to $$r ^ { 2 } = 2 \cos 2 \theta$$
  2. Give a sketch of the curve, stating the values of \(\theta\) for which the curve is defined.
  3. The equation in part (i) is now to be generalised to $$r ^ { 2 } = 2 \cos 2 \theta + k$$ where \(k\) is a constant.
    (A) Give sketches of the curve in the cases \(k = 1 , k = 2\). Describe how these two curves differ at the pole.
    (B) Give a sketch of the curve in the case \(k = 4\). What happens to the shape of the curve as \(k\) tends to infinity?
  4. Sketch the curve for the case \(k = - 1\). What happens to the curve as \(k \rightarrow - 2\) ? \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI FP2 2013 January Q1
18 marks Standard +0.3
1
    1. Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
    2. By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
    3. Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
    1. A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
    2. Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.
OCR MEI FP2 2013 January Q3
18 marks Standard +0.3
3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0 \\ 3 & - 2 & - 1 \\ 0 & - 1 & 1 \end{array} \right)\).
  1. Show that the characteristic equation of \(\mathbf { M }\) is $$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
  2. Find the eigenvalues and corresponding eigenvectors of \(\mathbf { M }\).
  3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P D P } ^ { - 1 } .$$ (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).)
OCR MEI FP2 2013 January Q4
18 marks Challenging +1.8
4
  1. Show that the curve with equation $$y = 3 \sinh x - 2 \cosh x$$ has no turning points.
    Show that the curve crosses the \(x\)-axis at \(x = \frac { 1 } { 2 } \ln 5\). Show that this is also the point at which the gradient of the curve has a stationary value.
  2. Sketch the curve.
  3. Express \(( 3 \sinh x - 2 \cosh x ) ^ { 2 }\) in terms of \(\sinh 2 x\) and \(\cosh 2 x\). Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through \(360 ^ { \circ }\) about the \(x\)-axis is $$\pi \left( 3 - \frac { 5 } { 4 } \ln 5 \right) .$$ Option 2: Investigation of curves \section*{This question requires the use of a graphical calculator.}
OCR MEI FP2 2013 January Q5
18 marks Challenging +1.8
5 This question concerns the curves with polar equation $$r = \sec \theta + a \cos \theta ,$$ where \(a\) is a constant which may take any real value, and \(0 \leqslant \theta \leqslant 2 \pi\).
  1. On a single diagram, sketch the curves for \(a = 0 , a = 1 , a = 2\).
  2. On a single diagram, sketch the curves for \(a = 0 , a = - 1 , a = - 2\).
  3. Identify a feature that the curves for \(a = 1 , a = 2 , a = - 1 , a = - 2\) share.
  4. Name a distinctive feature of the curve for \(a = - 1\), and a different distinctive feature of the curve for \(a = - 2\).
  5. Show that, in cartesian coordinates, equation (*) may be written $$y ^ { 2 } = \frac { a x ^ { 2 } } { x - 1 } - x ^ { 2 }$$ Hence comment further on the feature you identified in part (iii).
  6. Show algebraically that, when \(a > 0\), the curve exists for \(1 < x < 1 + a\). Find the set of values of \(x\) for which the curve exists when \(a < 0\).
OCR MEI FP2 2014 June Q1
19 marks Standard +0.8
1
  1. Given that \(\mathrm { f } ( x ) = \arccos x\),
    1. sketch the graph of \(y = \mathrm { f } ( x )\),
    2. show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\),
    3. obtain the Maclaurin series for \(\mathrm { f } ( x )\) as far as the term in \(x ^ { 3 }\).
  2. A curve has polar equation \(r = \theta + \sin \theta , \theta \geqslant 0\).
    1. By considering \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) show that \(r\) increases as \(\theta\) increases. Sketch the curve for \(0 \leqslant \theta \leqslant 4 \pi\).
    2. You are given that \(\sin \theta \approx \theta\) for small \(\theta\). Find in terms of \(\alpha\) the approximate area bounded by the curve and the lines \(\theta = 0\) and \(\theta = \alpha\), where \(\alpha\) is small.
OCR MEI FP2 2014 June Q2
17 marks Standard +0.8
2
  1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{gathered} C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots \\ S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{gathered}$$ where \(a\) is a real number and \(| a | < 1\).
    By considering \(C + \mathrm { j } S\), show that $$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$ Find a corresponding expression for \(C\).
  2. P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number \(\sqrt { 3 } + \mathrm { j }\).
    1. Find, in the form \(x + \mathrm { j } y\), the complex numbers corresponding to the other vertices of the hexagon.
    2. The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form \(x + \mathrm { j } y\), the vertices of the new figure. Find the area of the new figure.
OCR MEI FP2 2014 June Q3
18 marks Standard +0.3
3
    1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l } 6 & - 3 \\ 4 & - 1 \end{array} \right)$$
    2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
    1. The \(3 \times 3\) matrix \(\mathbf { B }\) has characteristic equation $$\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 3 \lambda - 10 = 0$$ Show that 5 is an eigenvalue of \(\mathbf { B }\). Show that \(\mathbf { B }\) has no other real eigenvalues.
    2. An eigenvector corresponding to the eigenvalue 5 is \(\left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)\). Evaluate \(\mathbf { B } \left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)\) and \(\mathbf { B } ^ { 2 } \left( \begin{array} { r } 4 \\ - 2 \\ - 8 \end{array} \right)\).
      Solve the equation \(\mathbf { B } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 20 \\ 10 \\ 40 \end{array} \right)\) for \(x , y , z\).
    3. Show that \(\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }\).
OCR MEI FP2 2014 June Q4
18 marks Challenging +1.2
4
  1. Given that \(\sinh y = x\), show that $$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$ Differentiate (*) to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$
  2. Find \(\int \frac { 1 } { \sqrt { 25 + 4 x ^ { 2 } } } \mathrm {~d} x\), expressing your answer in logarithmic form.
  3. Use integration by substitution with \(2 x = 5 \sinh u\) to show that $$\int \sqrt { 25 + 4 x ^ { 2 } } \mathrm {~d} x = \frac { 25 } { 4 } \left( \ln \left( \frac { 2 x } { 5 } + \sqrt { 1 + \frac { 4 x ^ { 2 } } { 25 } } \right) + \frac { 2 x } { 5 } \sqrt { 1 + \frac { 4 x ^ { 2 } } { 25 } } \right) + c$$ where \(c\) is an arbitrary constant. \section*{OCR}
OCR MEI FP2 2015 June Q3
18 marks Standard +0.8
3 This question concerns the matrix \(\mathbf { M }\) where \(\mathbf { M } = \left( \begin{array} { r r r } 5 & - 1 & 3 \\ 4 & - 3 & - 2 \\ 2 & 1 & 4 \end{array} \right)\).
  1. Obtain the characteristic equation of \(\mathbf { M }\). Find the eigenvalues of \(\mathbf { M }\). These eigenvalues are denoted by \(\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }\), where \(\lambda _ { 1 } < \lambda _ { 2 } < \lambda _ { 3 }\).
  2. Verify that an eigenvector corresponding to \(\lambda _ { 1 }\) is \(\left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right)\) and that an eigenvector corresponding to \(\lambda _ { 2 }\) is \(\left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right)\). Find an eigenvector of the form \(\left( \begin{array} { l } a \\ 1 \\ c \end{array} \right)\) corresponding to \(\lambda _ { 3 }\).
  3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\). (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).) Hence write down an expression for \(\mathbf { M } ^ { 4 }\) in terms of \(\mathbf { P }\) and a diagonal matrix. You should give the elements of the diagonal matrix explicitly.
  4. Use the Cayley-Hamilton theorem to obtain an expression for \(\mathbf { M } ^ { 4 }\) as a linear combination of \(\mathbf { M }\) and \(\mathbf { M } ^ { 2 }\).
OCR MEI FP2 2015 June Q4
18 marks Standard +0.8
4
  1. Starting with the relationship \(\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1\), deduce a relationship between \(\tanh ^ { 2 } t\) and \(\operatorname { sech } ^ { 2 } t\). You are given that \(y = \operatorname { artanh } x\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 - x ^ { 2 } }\).
  3. Show, by integrating the result in part (ii), that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
  4. Show that \(\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 6 } } \frac { 1 } { 1 - 3 x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { \sqrt { 3 } } \operatorname { artanh } \frac { 1 } { 2 }\). Express this answer in logarithmic form.
  5. Use integration by parts to find \(\int \operatorname { artanh } x \mathrm {~d} x\), giving your answer in terms of logarithms. \section*{END OF QUESTION PAPER}
CAIE P3 2020 Specimen Q1
3 marks Moderate -0.5
1 Find the set of values of \(x\) for which \(3 \left( 2 ^ { 3 x + 1 } \right) < 8\). Give your answer in a simplified exact form.
CAIE P3 2020 Specimen Q2
4 marks Moderate -0.8
2
  1. Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 Specimen Q3
4 marks Easy -1.3
3
  1. Sketch the graph of \(y = | 2 x - 3 |\).
  2. Solve the inequality \(3 x - 1 > | 2 x - 3 |\).
CAIE P3 2020 Specimen Q4
9 marks Standard +0.3
4 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t - 3 } , \quad y = 4 \ln t$$ where \(t > 0\). When \(t = a\) the gradient of the curve is 2 .
  1. Show that \(a\) satisfies the equation \(a = \frac { 1 } { 2 } ( 3 - \ln a )\).
  2. Verify by calculation that this equation has a root between 1 and 2 .
  3. Use the iterative formula \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calculate \(a\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.
CAIE P3 2020 Specimen Q5
7 marks Standard +0.8
5
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
  2. Show that \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).
CAIE P3 2020 Specimen Q6
8 marks Moderate -0.5
6 The complex numbers \(1 + 3 \mathrm { i }\) and \(4 + 2 \mathrm { i }\) are denoted by \(u\) and \(v\) respectively.
  1. Find \(\frac { u } { v }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the argument of \(\frac { u } { v }\).
    In an Argand diagram, with origin \(O\), the points \(A , B\) and \(C\) represent the complex numbers \(u , v\) and \(u - v\) respectively.
  3. State fully the geometrical relationship between \(O C\) and \(B A\).
  4. Show that angle \(A O B = \frac { 1 } { 4 } \pi\) radians.
CAIE P3 2020 Specimen Q7
9 marks Standard +0.3
7
  1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P3 2020 Specimen Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-14_485_716_262_676} In the diagram, \(O A B C\) is a pyramid in which \(O A = 2\) units, \(O B = 4\) units and \(O C = 2\) units. The edge \(O C\) is vertical, the base \(O A B\) is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O B\) and \(O C\) respectively. The midpoints of \(A B\) and \(B C\) are \(M\) and \(N\) respectively.
  1. Express the vectors \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Calculate the angle between the directions of \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\).
  3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-16_307_593_269_735} The diagram shows the curve \(y = \sin ^ { 2 } 2 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \sin x\), find the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2020 Specimen Q10
11 marks Standard +0.5
10 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\).
The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x ) .$$
  2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) when \(t\) becomes large.
Edexcel AEA 2017 Specimen Q1
8 marks Challenging +1.2
1.(a)For \(| y | < 1\) ,write down the binomial series expansion of \(( 1 - y ) ^ { - 2 }\) in ascending powers of \(y\) up to and including the term in \(y ^ { 3 }\) (b)Show that when it is convergent,the series $$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$ can be written in the form \(( 1 + a x ) ^ { n }\) ,where \(a\) and \(n\) are constants to be found.
(c)Find the set of values of \(x\) for which the series in part(b)is convergent.