Questions — OCR (4619 questions)

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OCR PURE 2023 May Q5
6 marks Standard +0.3
5 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = x ^ { 3 } - 4 x\).
\includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-05_499_695_404_251} Determine the total area enclosed by the curve and the \(x\)-axis.
OCR PURE 2023 May Q8
7 marks Challenging +1.2
8 In this question you must show detailed reasoning. A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 12 = 0\). Two tangents to this circle pass through the point \(( 0,1 )\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same.
Find the angle between these two tangents.
OCR PURE 2018 May Q8
9 marks Standard +0.3
8 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }\). The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 8\) and \(x = a\), where \(a > 8\).
\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of \(a\).
OCR PURE 2066 Q1
3 marks Easy -1.2
1 In this question you must show detailed reasoning. Solve the equation \(x ( 3 - \sqrt { 5 } ) = 24\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
OCR PURE 2020 October Q3
7 marks Moderate -0.8
3 In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4 \sqrt { x } - 3 x + 1\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR PURE 2020 October Q4
9 marks Standard +0.3
4 In this question you must show detailed reasoning. The cubic polynomial \(6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
  1. Use the factor theorem to show that \(k = - 37\).
  2. Using this value of \(k\), factorise \(\mathrm { f } ( x )\) completely.
    1. Hence find the three values of \(t\) satisfying the equation \(6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0\).
    2. Express the sum of the three values found in part (c)(i) as a single logarithm.
OCR PURE 2020 October Q6
6 marks Moderate -0.3
6 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{7fc02f90-8f8b-4153-bba1-dc0807124e96-4_650_661_1765_242}
The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.
OCR PURE 2022 June Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning. Solve the equation \(3 x + 1 = 4 \sqrt { x }\).
OCR PURE 2023 May Q2
4 marks Moderate -0.3
2 In this question you must show detailed reasoning. Solve the equation \(x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x\). Give your answer in the form \(a \sqrt { 5 } + b\), where \(a\) and \(b\) are integers to be determined.
OCR PURE 2023 May Q8
7 marks Moderate -0.3
8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).
OCR Further Statistics AS 2018 June Q6
5 marks
6 In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(\mathrm { E } ( T ) = 5.625\) and the standard deviation of \(T\) is 1.875 . Find the values of the parameters of the distribution.
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 2 2021 November Q2
8 marks Moderate -0.3
2 In this question you must show detailed reasoning. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 3 - 7 \mathrm { i }\) and \(z _ { 2 } = 2 + 4 \mathrm { i }\).
  1. Express each of the following as exact numbers in the form \(a + b \mathrm { i }\).
    1. \(3 z _ { 1 } + 4 z _ { 2 }\)
    2. \(z _ { 1 } z _ { 2 }\)
    3. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
  2. Write \(z _ { 1 }\) in modulus-argument form giving the modulus in exact form and the argument correct to \(\mathbf { 3 }\) significant figures.
OCR Further Statistics 2022 June Q3
8 marks Standard +0.8
3 In this question you must show detailed reasoning. A discrete random variable \(V\) has the following probability distribution, where \(p\) and \(q\) are constants.
\(v\)0123
\(\mathrm { P } ( \mathrm { V } = \mathrm { v } )\)\(p\)\(q\)0.120.2
It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).
OCR Further Pure Core AS 2022 June Q3
6 marks Challenging +1.2
3 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
OCR Further Pure Core AS 2024 June Q9
8 marks Challenging +1.8
9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation
\(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).
OCR Further Pure Core AS 2020 November Q1
6 marks Moderate -0.3
1 In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\).
\(2 \mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D.
P is a reflection in the \(x\)-axis. \(\mathbf { A }\) is the matrix that represents P .
  1. Write down the matrix \(\mathbf { A }\). Q is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). \(\mathbf { B }\) is the
    matrix that represents Q . matrix that represents Q.
  2. Find the matrix \(\mathbf { B }\). T is P followed by Q. C is the matrix that represents T.
  3. Determine the matrix \(\mathbf { C }\).
    \(L\) is the line whose equation is \(y = x\).
  4. Explain whether or not \(L\) is a line of invariant points under \(T\). An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
  5. Explain what the value of the determinant of \(\mathbf { C }\) means about
    • the area of \(N\) compared to the area of \(M\),
    • the orientation of \(N\) compared to the orientation of \(M\).
OCR Further Pure Core AS 2020 November Q3
12 marks Moderate -0.3
3 In this question you must show detailed reasoning. The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3 z - 4 z ^ { * }\)
    2. \(( z + 1 - 3 i ) ^ { 2 }\)
    3. \(\frac { z + 1 } { z - 1 }\)
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
  3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.
    • \(| \omega |\)
    • \(\arg ( \omega )\), giving your answer correct to 3 significant figures
OCR Further Pure Core AS 2020 November Q5
7 marks Challenging +1.2
5 In this question you must show detailed reasoning. The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
OCR Further Mechanics 2022 June Q5
9 marks Challenging +1.8
5 In this question you must show detailed reasoning. The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = 4\) and the curve with equation \(\mathrm { y } = \frac { 15 } { \sqrt { \mathrm { x } ^ { 2 } + 9 } }\) is occupied by a uniform lamina. The centre of mass of the lamina is at the point \(G ( \bar { x } , \bar { y } )\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-4_944_954_598_228}
  1. Show that \(\bar { x } = \frac { 2 } { \ln 3 }\).
  2. Determine the value of \(\bar { y }\). Give your answer correct to \(\mathbf { 3 }\) significant figures.
    \(P\) is the point on the curved edge of the lamina where \(x = 3\). The lamina is freely suspended from \(P\) and hangs in equilibrium in a vertical plane.
  3. Determine the acute angle that the longest straight edge of the lamina makes with the vertical.
OCR Further Additional Pure 2019 June Q8
11 marks Hard +2.3
8 In this question you must show detailed reasoning.
  1. Prove that \(2 ( p - 2 ) ^ { p - 2 } \equiv - 1 ( \bmod p )\), where \(p\) is an odd prime.
  2. Find two odd prime factors of the number \(N = 2 \times 34 ^ { 34 } - 2 ^ { 15 }\). \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Further Additional Pure 2021 November Q1
3 marks Moderate -0.8
1 In this question you must show detailed reasoning. Express the number \(\mathbf { 4 1 7 2 3 } _ { 10 }\) in hexadecimal (base 16).
OCR Further Pure Core 2 2019 June Q3
5 marks Standard +0.3
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
OCR Further Pure Core 2 2022 June Q3
6 marks Challenging +1.2
3 In this question you must show detailed reasoning. The roots of the equation \(4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
OCR Further Pure Core 2 2022 June Q9
9 marks Challenging +1.2
9 In this question you must show detailed reasoning.
  1. Show that \(\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta\), where \(a\) is an integer to be determined.
  2. Hence show that \(\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }\), where \(b\) and \(c\) are integers to be determined.