Questions Further Pure Core 1 (136 questions)

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OCR Further Pure Core 1 Specimen Q1
2 marks Easy -1.8
1 Show that \(\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }\).
OCR Further Pure Core 1 Specimen Q2
5 marks Standard +0.3
2 In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).
  1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
  2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).
OCR Further Pure Core 1 Specimen Q3
6 marks Challenging +1.2
3 In this question you must show detailed reasoning. The diagram below shows the curve \(r = 2 \cos 4 \theta\) for \(- k \pi \leq \theta \leq k \pi\) where \(k\) is a constant to be determined. Calculate the exact area enclosed by the curve.
OCR Further Pure Core 1 Specimen Q4
3 marks Standard +0.3
4 Draw the region in an Argand diagram for which \(| z | \leq 2\) and \(| z | > | z - 3 i |\).
OCR Further Pure Core 1 Specimen Q5
5 marks Standard +0.8
5
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }\).
  2. Find \(\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x\).
OCR Further Pure Core 1 Specimen Q6
5 marks Standard +0.8
6 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
The equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\) has roots \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
Find the values of \(p , q\) and \(r\).
OCR Further Pure Core 1 Specimen Q7
7 marks Challenging +1.2
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 3 } { 1 } = \frac { y - 5 } { 2 } = \frac { z + 2 } { - 3 }\) and \(\frac { x - 4 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z - 7 } { 4 }\).
  1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
OCR Further Pure Core 1 Specimen Q8
8 marks Standard +0.3
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. $$\begin{array} { r r r l } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$
OCR Further Pure Core 1 Specimen Q9
5 marks Standard +0.8
9 Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$
OCR Further Pure Core 1 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]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}
  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 Specimen Q11
19 marks Challenging +1.2
11 A company is required to weigh any goods before exporting them overseas. When a crate is placed on a set of weighing scales, the mass displayed takes time to settle down to its final value. The company wishes to model the mass, \(m \mathrm {~kg}\), which is displayed \(t\) seconds after a crate X is placed on the scales.
For the displayed mass it is assumed that the rate of change of the quantity \(\left( 0.5 \frac { \mathrm {~d} m } { \mathrm {~d} t } + m \right)\) with respect to time is proportional to \(( 80 - m )\).
  1. Show that \(\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 2 \mathrm {~km} = 160 \mathrm { k }\), where \(k\) is a real constant. It is given that the complementary function for the differential equation in part (i) is \(\mathrm { e } ^ { \lambda t } ( A \cos 2 t + B \sin 2 t )\), where \(A\) and \(B\) are arbitrary constants.
  2. Show that \(k = \frac { 5 } { 2 }\) and state the value of the constant \(\lambda\). When X is initially placed on the scales the displayed mass is zero and the rate of increase of the displayed mass is \(160 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\).
  3. Find \(m\) in terms of \(t\).
  4. Describe the long term behaviour of \(m\).
  5. With reference to your answer to part (iv), comment on a limitation of the model.
  6. (a) Find the value of \(m\) that corresponds to the stationary point on the curve \(m = \mathrm { f } ( t )\) with the smallest positive value of \(t\).
    (b) Interpret this value of \(m\) in the context of the model.
  7. Adapt the differential equation \(\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 5 m = 400\) to model the mass displayed \(t\) seconds after a crate Y , of mass 100 kg , is placed on the scales. \section*{END OF QUESTION PAPER} \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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OCR Further Pure Core 1 2023 June Q9
14 marks Challenging +1.8
9 In this question you must show detailed reasoning.
  1. Use de Moivre's theorem to determine constants \(A\), \(B\) and \(C\) such that $$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$ The function f is defined by \(\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1\).
  2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\). \includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260} The diagram shows the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\) for \(0 \leqslant x < 1\) and the asymptote \(x = 1\). The region \(R\) is the unbounded region between the curve, the \(x\)-axis, the line \(x = 0\) and the line \(x = 1\). You are given that the area of \(R\) is finite.
  3. Determine the exact area of \(R\).
OCR Further Pure Core 1 2023 June Q6
4 marks Standard +0.8
6 In this question you must show detailed reasoning. The power output, \(p\) watts, of a machine at time \(t\) hours after it is switched on can be modelled by the equation \(\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )\) for \(t \geqslant 0\). Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to \(\mathbf { 2 }\) decimal places.
OCR Further Pure Core 1 2022 June Q1
6 marks Standard +0.8
1 In this question you must show detailed reasoning.
  1. Show that \(\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }\). The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \sqrt { \operatorname { sinhx } }\), the \(x\)-axis and the line with equation \(x = 2 \ln 3\) (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242} A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating \(R\) completely about the \(x\)-axis.
  2. Determine, according to the model, the exact volume of one chocolate piece.
OCR Further Pure Core 1 2018 March Q1
5 marks Moderate -0.3
1 In this question you must show detailed reasoning.
Find the square roots of \(24 + 10 \mathrm { i }\), giving your answers in the form \(a + b \mathrm { i }\).
OCR Further Pure Core 1 2018 March Q2
10 marks Standard +0.3
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
4 marks Moderate -0.3
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
7 marks Standard +0.8
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
6 marks Challenging +1.2
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 marks Moderate -0.5
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
8 marks Challenging +1.2
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
7 marks Standard +0.8
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
8 marks Standard +0.8
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
14 marks Challenging +1.8
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
7 marks Moderate -0.8
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. \(\frac { z } { z ^ { * } }\)