Questions Further Pure Core 1 (136 questions)

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OCR Further Pure Core 1 2018 September Q2
6 marks Standard +0.3
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
8 marks Standard +0.3
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
5 marks Standard +0.8
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
6 marks Challenging +1.2
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
13 marks Challenging +1.2
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
5 marks Standard +0.8
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
6 marks Standard +0.3
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
8 marks Standard +0.8
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
    1. 6 hours,
    2. 6.25 hours.
    3. 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
5 marks Standard +0.3
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
9 marks Standard +0.8
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
7 marks Standard +0.3
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 marks Standard +0.3
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
6 marks Standard +0.3
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.
OCR Further Pure Core 1 2018 December Q6
5 marks Moderate -0.3
6 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4 .
OCR Further Pure Core 1 2018 December Q7
7 marks Challenging +1.2
7
  1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
  2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
OCR Further Pure Core 1 2018 December Q8
9 marks Standard +0.8
8
  1. Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
  2. Solve the equation \(4 \tanh ^ { 2 } x + \tanh x - 3 = 0\), giving the solution in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers to be determined.
  3. Explain why the equation in part (b) has only one root.
OCR Further Pure Core 1 2018 December Q9
7 marks Standard +0.8
9 In this question you must show detailed reasoning. Find \(\int _ { - 1 } ^ { 11 } \frac { 1 } { \sqrt { x ^ { 2 } + 6 x + 13 } } \mathrm {~d} x\) giving your answer in the form \(\ln ( p + q \sqrt { 2 } )\) where \(p\) and \(q\) are integers to be determined.
OCR Further Pure Core 1 2018 December Q10
16 marks Standard +0.8
10 In a predator-prey environment the population, at time \(t\) years, of predators is \(x\) and prey is \(y\). The populations of predators and prey are measured in hundreds. The populations are modelled by the following simultaneous differential equations. $$\frac { \mathrm { d } x } { \mathrm {~d} t } = y \quad \frac { \mathrm {~d} y } { \mathrm {~d} t } = 2 y - 5 x$$
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 5 x\).
    1. Find the general solution for \(x\).
    2. Find the equivalent general solution for \(y\). Initially there are 100 predators and 300 prey.
  3. Find the particular solutions for \(x\) and \(y\).
  4. Determine whether the model predicts that the predators will die out before the prey.
OCR Further Pure Core 1 2017 Specimen Q8
8 marks Standard +0.8
8
  1. Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3 \\ 2 x + 4 y + 5 z = & 9 \\ 7 x + 11 y + 12 z = & 20 \end{array}$$
  2. Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r r } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$
OCR Further Pure Core 1 2017 Specimen Q10
10 marks Standard +0.3
10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}
  1. Justify each of the following statements.
    1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
    2. \(A B C D E\) is a regular pentagon.
    3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
    4. \(b ^ { * } = e\)
    5. \(a + b + c + d + e = 0\)
    6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).
OCR Further Pure Core 1 2021 June Q2
4 marks Standard +0.3
2 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 2021 June Q3
5 marks Moderate -0.3
3 Prove by induction that, for all positive integers \(n , 7 ^ { n } + 3 ^ { n - 1 }\) is a multiple of 4.
OCR Further Pure Core 1 2021 June Q4
7 marks Challenging +1.2
4
  1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
  2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).