Questions — OCR Further Pure Core 1 (134 questions)

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OCR Further Pure Core 1 2018 March Q2
2 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)\).
  1. Find the value of \(a\) such that \(\mathbf { A B } = \mathbf { B A }\).
  2. Prove by counter example that matrix multiplication for \(2 \times 2\) matrices is not commutative.
  3. A triangle of area 4 square units is transformed by the matrix B. Find the area of the image of the triangle following this transformation.
  4. Find the equations of the invariant lines of the form \(y = m x\) for the transformation represented by matrix \(\mathbf { B }\).
OCR Further Pure Core 1 2018 March Q3
3 Prove by mathematical induction that, for all integers \(n \geqslant 1 , n ^ { 5 } - n\) is divisible by 5 .
OCR Further Pure Core 1 2018 March Q4
4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).
OCR Further Pure Core 1 2018 March Q5
5 By using a suitable substitution, which should be stated, show that $$\int _ { \frac { 3 } { 2 } } ^ { \frac { 5 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 13 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln ( 1 + \sqrt { 2 } )$$
OCR Further Pure Core 1 2018 March Q6
6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point \(O\). Initially the mass hangs at rest vertically below \(O\). The mass is then pulled to one side with the string taut and released from rest. \(\theta\) is the angle, in radians, that the string makes with the vertical through \(O\) at time \(t\) seconds and \(\theta\) may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$
  1. Write down the general solution to this differential equation.
  2. Initially the pendulum is released from rest at an angle of \(\theta _ { 0 }\). Find the particular solution to the equation in this case.
  3. State any limitations on the model.
OCR Further Pure Core 1 2018 March Q7
7
  1. Using the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$4 \sinh ^ { 3 } x = \sinh 3 x - 3 \sinh x$$ \section*{(ii) In this question you must show detailed reasoning.} By making a suitable substitution, find the real root of the equation $$16 u ^ { 3 } + 12 u = 3 .$$ Give your answer in the form \(\frac { \left( a ^ { \frac { 1 } { b } } - a ^ { - \frac { 1 } { b } } \right) } { c }\) where \(a , b\) and \(c\) are integers.
OCR Further Pure Core 1 2018 March Q8
8 You are given that \(\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }\) where \(a\) and \(b\) are positive constants. The first three terms in the Maclaurin expansion of \(\mathrm { f } ( x )\) are \(1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }\).
  1. Find the value of \(a\) and the value of \(b\).
  2. Explain if there is any restriction on the value of \(x\) in order for the expansion to be valid.
OCR Further Pure Core 1 2018 March Q9
9 In an experiment, at time \(t\) minutes there is \(Q\) grams of substance present.
It is known that the substance decays at a rate that is proportional to \(1 + Q ^ { 2 }\). Initially there are 100 grams of the substance present and after 100 minutes there are 50 grams present. Find the amount of the substance present after 400 minutes.
OCR Further Pure Core 1 2018 March Q10
10
  1. (a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form $$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$ The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
    (b) Explain why the curve is symmetrical in the \(x\)-axis.
    (c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
  2. The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. Find the exact volume of the solid formed. \section*{END OF QUESTION PAPER}
OCR Further Pure Core 1 2018 September Q1
1 In this question you must show detailed reasoning.
For the complex number \(z\) it is given that \(| z | = 2\) and \(\arg z = \frac { 1 } { 6 } \pi\).
Find the following in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact numbers.
  1. \(z\)
  2. \(z ^ { 2 }\)
  3. \(\frac { z } { z ^ { * } }\)
OCR Further Pure Core 1 2018 September Q2
2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - 1 | = 5\) and \(\arg ( z + 4 + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\) respectively.
  1. Sketch on a single Argand diagram the loci \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Indicate by shading on your Argand diagram the following set of points. $$\{ z : | z - 1 | \leqslant 5 \} \cap \left\{ z : 0 \leqslant \arg ( z + 4 + 4 i ) \leqslant \frac { 1 } { 4 } \pi \right\}$$
OCR Further Pure Core 1 2018 September Q3
3 A sequence is defined by \(a _ { 1 } = 6\) and \(a _ { n + 1 } = 5 a _ { n } - 2\) for \(n \geqslant 1\).
Prove by induction that for all integers \(n \geqslant 1 , a _ { n } = \frac { 11 \times 5 ^ { n - 1 } + 1 } { 2 }\).
OCR Further Pure Core 1 2018 September Q4
4 In this question you must show detailed reasoning.
Find the exact value of each of the following.
  1. \(\int _ { 1 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x + 10 } \mathrm {~d} x\)
  2. The mean value of \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) in the interval \([ 0,0.5 ]\)
OCR Further Pure Core 1 2018 September Q5
5 Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), have equations \(3 x + 2 y + z = 4\) and \(2 x + y + z = 3\) respectively.
  1. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(L\) has equation \(x = 1 - y = 2 - z\).
  2. Show that \(L\) lies in both planes.
OCR Further Pure Core 1 2018 September Q6
6
  1. Find as a single algebraic fraction an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
  2. Determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
OCR Further Pure Core 1 2018 September Q7
7 In this question you must show detailed reasoning.
Find \(\int _ { 2 } ^ { 3 } \frac { x + 1 } { x ^ { 3 } - x ^ { 2 } + x - 1 } \mathrm {~d} x\), expressing your answer in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers.
OCR Further Pure Core 1 2018 September Q8
8
  1. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\sinh 2 x = 2 \sinh x \cosh x\). You are given the function \(\mathrm { f } ( x ) = a \cosh x - \cosh 2 x\), where \(a\) is a positive constant.
  2. Verify that, for any value of \(a\), the curve \(y = \mathrm { f } ( x )\) has a stationary point on the \(y\)-axis.
  3. Find the coordinates of the stationary point found in part (ii).
  4. Determine the maximum value of \(a\) for which the stationary point found in part (ii) is the only stationary point on the curve \(y = \mathrm { f } ( x )\). You are given that for any value of \(a\) greater than the value found in part (iv) there are three stationary points, the one found in part (ii) and two others, one of which satisfies \(x > 0\).
  5. Find the coordinates of this point when \(a = 6\). Give your answer in the form \(\left( \cosh ^ { - 1 } p , q \right)\).
OCR Further Pure Core 1 2018 September Q9
9 The diagram below shows the curve \(r = 4 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{c03cae53-eb00-496b-948f-ccff676bc03c-3_311_775_1713_644}
  1. On the diagram in your Printed Answer Booklet, shade the region \(R\) for which $$r \leqslant 4 \sin 3 \theta \text { and } 0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi .$$ In this question you must show detailed reasoning.
  2. Find the exact area of the region \(R\).
OCR Further Pure Core 1 2018 September Q10
10
  1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
  2. Find the range of \(x\) for which the expansion is valid.
  3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$
OCR Further Pure Core 1 2018 September Q11
11 A particular radioactive substance decays over time.
A scientist models the amount of substance, \(x\) grams, at time \(t\) hours by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } + \frac { 1 } { 10 } x = \mathrm { e } ^ { - 0.1 t } \cos t .$$
  1. Solve the differential equation to find the general solution for \(x\) in terms of \(t\). Initially there was 10 g of the substance.
  2. Find the particular solution of the differential equation.
  3. Find to 6 significant figures the amount of substance that would be predicted by the model at
    (a) 6 hours,
    (b) 6.25 hours.
  4. Comment on the appropriateness of the model for predicting the amount of substance over time. \section*{END OF QUESTION PAPER}
OCR Further Pure Core 1 2018 December Q1
1 Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.
  1. Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
  2. Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\).
OCR Further Pure Core 1 2018 December Q2
2 The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{8315a796-0e7d-464f-8604-9fe3ab7af359-2_470_657_913_319}
  1. The greatest value of \(r\) on the curve occurs at the point \(P\).
    1. Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
    2. Find the value of \(r\) at the point \(P\).
    3. Mark the point \(P\) on the copy of the graph in the Printed Answer Booklet.
  2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve.
OCR Further Pure Core 1 2018 December Q3
3 You are given that \(\mathrm { f } ( x ) = \ln ( 2 + x )\).
  1. Determine the exact value of \(\mathrm { f } ^ { \prime } ( 0 )\).
  2. Show that \(\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }\).
  3. Hence write down the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\).
OCR Further Pure Core 1 2018 December Q4
4 In this question you must show detailed reasoning.
You are given that \(z = \sqrt { 3 } + \mathrm { i }\).
\(n\) is the smallest positive whole number such that \(z ^ { n }\) is a positive whole number.
  1. Determine the value of \(n\).
  2. Find the value of \(z ^ { n }\).
OCR Further Pure Core 1 2018 December Q5
5 You are given that \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 2 & 1
2 & 5 & 2
3 & - 2 & - 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 1 & 0 & 1
- 8 & 4 & 0
19 & - 8 & - 1 \end{array} \right)\).
  1. Find \(\mathbf { A B }\).
  2. Hence write down \(\mathbf { A } ^ { - 1 }\).
  3. You are given three simultaneous equations $$\begin{array} { r } x + 2 y + z = 0
    2 x + 5 y + 2 z = 1
    3 x - 2 y - z = 4 \end{array}$$
    1. Explain how you can tell, without solving them, that there is a unique solution to these equations.
    2. Find this unique solution.