Questions — OCR (4628 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR H240/03 2019 June Q8
10 marks Standard +0.3
8 A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after 2.4 seconds. The horizontal component of the initial velocity of \(P\) is \(\frac { 5 } { 3 } d \mathrm {~ms} ^ { - 1 }\).
  1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground.
  2. Find the vertical component of the initial velocity of \(P\). \(P\) just clears a vertical wall which is situated at a horizontal distance \(d \mathrm {~m}\) from \(O\).
  3. Find the height of the wall. The speed of \(P\) as it passes over the wall is \(16 \mathrm {~ms} ^ { - 1 }\).
  4. Find the value of \(d\) correct to 3 significant figures.
OCR H240/03 2019 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-08_362_1191_262_438} The diagram shows a small block \(B\), of mass 0.2 kg , and a particle \(P\), of mass 0.5 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac { 3 } { 4 }\). The system is released from rest. In the first 0.4 seconds of the motion \(P\) moves 0.3 m down the plane and \(B\) does not reach the pulley.
  1. Find the tension in the string during the first 0.4 seconds of the motion.
  2. Calculate the coefficient of friction between \(B\) and the horizontal surface.
OCR H240/03 2019 June Q10
13 marks Standard +0.3
10 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(R\) of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf { F }\) N. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = \left( p t ^ { 2 } - 3 t \right) \mathbf { i } + ( 8 t + q ) \mathbf { j }\), where \(p\) and \(q\) are constants and \(p < 0\).
  1. Given that when \(t = 0.5\) the magnitude of \(\mathbf { F }\) is 20 , find the value of \(p\). When \(t = 0 , R\) is at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m }\).
  2. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). When \(t = 1 , R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2 \mathbf { i } - 8 \mathbf { j }\).
  3. Find the value of \(q\). \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-09_544_1297_251_255} The diagram shows a ladder \(A B\), of length \(2 a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2 m\) is placed on the ladder at a point \(C\) where \(A C = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.
  4. Show that the normal contact force between the ladder and the wall is \(\frac { m g ( a + 2 d ) \sqrt { 3 } } { 4 h }\). It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 8 } \sqrt { 3 }\).
  5. Show that \(h = k ( a + 2 d )\), where \(k\) is a constant to be determined.
  6. Hence find, in terms of \(a\), the greatest possible value of \(d\).
  7. State one improvement that could be made to the model.
OCR H240/03 2020 November Q1
2 marks Easy -1.2
1 Triangle \(A B C\) has \(A B = 8.5 \mathrm {~cm} , B C = 6.2 \mathrm {~cm}\) and angle \(B = 35 ^ { \circ }\). Calculate the area of the triangle.
OCR H240/03 2020 November Q2
3 marks Moderate -0.8
2 A sequence of transformations maps the curve \(y = \mathrm { e } ^ { x }\) to the curve \(y = \mathrm { e } ^ { 2 x + 3 }\). Give details of these transformations.
OCR H240/03 2020 November Q3
11 marks Moderate -0.3
3 The functions f and g are defined for all real values of \(x\) by \(f ( x ) = 2 x ^ { 2 } + 6 x\) and \(g ( x ) = 3 x + 2\).
  1. Find the range of f .
  2. Give a reason why f has no inverse.
  3. Given that \(\mathrm { fg } ( - 2 ) = \mathrm { g } ^ { - 1 } ( a )\), where \(a\) is a constant, determine the value of \(a\).
  4. Determine the set of values of \(x\) for which \(\mathrm { f } ( x ) > \mathrm { g } ( x )\). Give your answer in set notation.
OCR H240/03 2020 November Q4
11 marks Standard +0.3
4 A curve has equation \(y = 2 \ln ( k - 3 x ) + x ^ { 2 } - 3 x\), where \(k\) is a positive constant.
  1. Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\). It is also given that the curve intersects the \(x\)-axis at exactly one point.
  2. Show by calculation that the \(x\)-coordinate of this point lies between 0.5 and 1.5 .
  3. Use the Newton-Raphson method, with initial value \(x _ { 0 } = 1\), to find the \(x\)-coordinate of the point where the curve intersects the \(x\)-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places.
  4. By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.
OCR H240/03 2020 November Q5
12 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-05_339_869_262_244} The diagram shows the curve \(C\) with parametric equations \(x = \frac { 3 } { t } , y = t ^ { 3 } \mathrm { e } ^ { - 2 t }\), where \(t > 0\).
The maximum point on \(C\) is denoted by \(P\).
  1. Determine the exact coordinates of \(P\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 6\).
  2. Show that the area of \(R\) is given by $$\int _ { a } ^ { b } 3 t \mathrm { e } ^ { - 2 t } \mathrm {~d} t ,$$ where \(a\) and \(b\) are constants to be determined.
  3. Hence determine the exact area of \(R\).
OCR H240/03 2020 November Q7
6 marks Moderate -0.3
7 A particle \(P\) moves with constant acceleration \(( - 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). At time \(t = 0\) seconds, \(P\) is moving with velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Determine the speed of \(P\) when \(t = 3\).
  2. Determine the change in displacement of \(P\) between \(t = 0\) and \(t = 3\).
OCR H240/03 2020 November Q8
7 marks Moderate -0.3
8 A car is travelling on a straight horizontal road. The velocity of the car, \(v \mathrm {~ms} ^ { - 1 }\), at time \(t\) seconds as it travels past three points, \(P , Q\) and \(R\), is modelled by the equation \(v = a t ^ { 2 } + b t + c\),
where \(a , b\) and \(c\) are constants.
The car passes \(P\) at time \(t = 0\) with velocity \(8 \mathrm {~ms} ^ { - 1 }\).
  1. State the value of \(c\). The car passes \(Q\) at time \(t = 5\) and at that instant its deceleration is \(0.12 \mathrm {~ms} ^ { - 2 }\). The car passes \(R\) at time \(t = 18\) with velocity \(2.96 \mathrm {~ms} ^ { - 1 }\).
  2. Determine the values of \(a\) and \(b\).
  3. Find, to the nearest metre, the distance between points \(P\) and \(R\). \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-08_469_798_251_244} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg . The other end of the string is attached to a second particle \(B\) of mass 2.5 kg . Particle \(A\) is in contact with a rough plane inclined at \(\theta\) to the horizontal, where \(\cos \theta = \frac { 4 } { 5 }\). The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Particle \(B\) hangs freely below \(P\) at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between \(A\) and the plane is \(\mu\). The system is released from rest and in the subsequent motion \(B\) hits the ground before \(A\) reaches \(P\). The speed of \(B\) at the instant that it hits the ground is \(1.2 \mathrm {~ms} ^ { - 1 }\).
OCR H240/03 2020 November Q11
13 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_474_853_264_242} A particle \(P\) moves freely under gravity in the plane of a fixed horizontal axis \(O x\), which lies on horizontal ground, and a fixed vertical axis \(O y . P\) is projected from \(O\) with a velocity whose components along \(O x\) and \(O y\) are \(U\) and \(V\), respectively. \(P\) returns to the ground at a point \(C\).
  1. Determine, in terms of \(U , V\) and \(g\), the distance \(O C\). \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_478_851_1151_244} \(P\) passes through two points \(A\) and \(B\), each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram.
  2. Write down the horizontal and vertical components of the velocity of \(P\) at \(A\).
  3. Hence determine an expression for \(d\) in terms of \(U , V , g\) and \(h\).
  4. Given that the direction of motion of \(P\) as it passes through \(A\) is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\), determine an expression for \(V\) in terms of \(g , d\) and \(h\).
OCR H240/03 2022 June Q1
3 marks Easy -1.8
1 Solve the equation \(| 2 x - 3 | = 9\).
OCR H240/03 2022 June Q2
5 marks Easy -1.2
2
  1. Give full details of the single transformation that transforms the graph of \(y = x ^ { 3 }\) to the graph of \(y = x ^ { 3 } - 8\). The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 8\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. State how the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) are related geometrically.
OCR H240/03 2022 June Q3
4 marks Moderate -0.8
3 The points \(P\) and \(Q\) have coordinates \(( 2 , - 5 )\) and \(( 3,1 )\) respectively.
Determine the equation of the circle that has \(P Q\) as a diameter. Give your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR H240/03 2022 June Q4
8 marks Standard +0.8
4 The positive integers \(x , y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4 . Also, \(x , \frac { 15 } { y }\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.
  1. Show that \(y\) satisfies the equation \(y ^ { 4 } - 16 y ^ { 2 } - 225 = 0\).
  2. Hence determine the sum to infinity of the geometric progression.
OCR H240/03 2022 June Q8
2 marks Moderate -0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-07_159_896_488_244} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15 ^ { \circ }\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N , find the tension in the rope.
OCR H240/03 2022 June Q9
6 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-07_579_848_1142_242} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm {~ms} ^ { - 1 }\), is overtaken by car \(A\) which has initial speed \(20 \mathrm {~ms} ^ { - 1 }\). From \(t = 0 \operatorname { car } A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) and the car maintains this speed in its subsequent motion.
  1. Calculate the deceleration of \(\operatorname { car } A\).
  2. Determine the value of \(t\) when \(B\) overtakes \(A\). \includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-08_293_773_354_246} A rectangular block \(B\) is at rest on a horizontal surface. A particle \(P\) of mass 2.5 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle \(Q\) of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between \(P\) and the pulley is horizontal (see diagram). The particles are released from rest with the string taut. It is given that \(B\) remains in equilibrium while \(P\) moves on the upper surface of \(B\). The tension in the string while \(P\) moves on \(B\) is 16.8 N .
OCR H240/03 2022 June Q12
13 marks Standard +0.8
12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf { F N }\). At time \(t\) seconds, where \(t \geqslant 0\), the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(P\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = ( 1 - 2 t ) \mathbf { i } + \left( 2 t ^ { 2 } + t - 13 \right) \mathbf { j }\).
  1. Show that \(P\) is never stationary.
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the acceleration of \(P\) at time \(t\). The mass of \(P\) is 0.5 kg .
  3. Determine the magnitude of \(\mathbf { F }\) when \(P\) is moving in the direction of the vector \(- 2 \mathbf { i } + \mathbf { j }\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When \(t = 1 , P\) is at the point with position vector \(\frac { 1 } { 6 } \mathbf { j }\).
  4. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\).
OCR H240/03 2022 June Q13
14 marks Standard +0.3
13 A small ball \(B\) moves in the plane of a fixed horizontal axis \(O x\), which lies on horizontal ground, and a fixed vertically upwards axis \(O y . B\) is projected from \(O\) with a velocity whose components along \(O x\) and \(O y\) are \(U \mathrm {~ms} ^ { - 1 }\) and \(V \mathrm {~ms} ^ { - 1 }\), respectively. The units of \(x\) and \(y\) are metres. \(B\) is modelled as a particle moving freely under gravity.
  1. Show that the path of \(B\) has equation \(2 U ^ { 2 } y = 2 U V x - g x ^ { 2 }\). During its motion, \(B\) just clears a vertical wall of height \(\frac { 1 } { 2 } a \mathrm {~m}\) at a horizontal distance \(a \mathrm {~m}\) from \(O\). \(B\) strikes the ground at a horizontal distance \(3 a \mathrm {~m}\) beyond the wall.
  2. Determine the angle of projection of \(B\). Give your answer in degrees correct to \(\mathbf { 3 }\) significant figures.
  3. Given that the speed of projection of \(B\) is \(54.6 \mathrm {~ms} ^ { - 1 }\), determine the value of \(a\).
  4. Hence find the maximum height of \(B\) above the ground during its motion.
  5. State one refinement of the model, other than including air resistance, that would make it more realistic. \section*{END OF QUESTION PAPER}
OCR H240/03 2023 June Q1
3 marks Moderate -0.8
1 Using logarithms, solve the equation \(4 ^ { 2 x + 1 } = 5 ^ { x }\),
giving your answer correct to \(\mathbf { 3 }\) significant figures.
OCR H240/03 2023 June Q2
5 marks Standard +0.3
2
  1. Express \(3 \sin x - 4 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to \(\mathbf { 4 }\) significant figures.
  2. Hence solve the equation \(3 \sin x - 4 \cos x = 2\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\), giving your answer correct to 3 significant figures.
OCR H240/03 2023 June Q3
8 marks Moderate -0.3
3 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + p x + q\), where \(p\) and \(q\) are constants.
    1. Given that \(\mathrm { f } ^ { \prime } ( 2 ) = 13\), find the value of \(p\).
    2. Given also that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(q\). The curve \(y = \mathrm { f } ( x )\) is translated by the vector \(\binom { 2 } { - 3 }\).
  2. Using the values from part (a), determine the equation of the curve after it has been translated. Give your answer in the form \(y = x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are integers to be found.
OCR H240/03 2023 June Q4
7 marks Standard +0.3
4 A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0\).
  1. Find the set of possible values of \(k\).
  2. It is given that \(k = - 46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac { 1 } { 2 }\).
OCR H240/03 2023 June Q5
9 marks Standard +0.8
5 A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-05_453_1200_358_242} The emblem is modelled by the region between the \(x\)-axis and the curve with parametric equations \(x = 1 + 0.2 t - \cos t , \quad y = k \sin ^ { 2 } t\), where \(k\) is a positive constant and \(0 \leqslant t \leqslant \pi\). Lengths are in metres and the area of the emblem must be \(1 \mathrm {~m} ^ { 2 }\).
  1. Show that \(k \int _ { 0 } ^ { \pi } \left( 0.2 + \sin t - 0.2 \cos ^ { 2 } t - \sin t \cos ^ { 2 } t \right) \mathrm { d } t = 1\).
  2. Determine the exact value of \(k\).
OCR H240/03 2023 June Q6
6 marks Standard +0.8
6 The first, third and fourth terms of an arithmetic progression are \(u _ { 1 } , u _ { 3 }\) and \(u _ { 4 }\) respectively, where \(u _ { 1 } = 2 \sin \theta , \quad u _ { 3 } = - \sqrt { 3 } \cos \theta , \quad u _ { 4 } = \frac { 7 } { 2 } \sin \theta\), and \(\frac { 1 } { 2 } \pi < \theta < \pi\).
  1. Determine the exact value of \(\theta\).
  2. Hence determine the value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\).