Questions — OCR PURE (139 questions)

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OCR PURE Q7
7 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-5_944_938_260_244} The diagram shows the curve \(C\) with equation \(y = 4 x ^ { 2 } - 10 x + 7\) and two straight lines, \(l _ { 1 }\) and \(l _ { 2 }\). The line \(l _ { 1 }\) is the normal to \(C\) at the point \(\left( \frac { 1 } { 2 } , 3 \right)\). The line \(l _ { 2 }\) is the normal to \(C\) at the minimum point of \(C\).
  1. Determine the equation of \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined. The shaded region shown in the diagram is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\).
  2. Determine the inequalities that define the shaded region, including its boundaries.
OCR PURE Q9
4 marks Moderate -0.8
9 A cyclist travels along a straight horizontal road between house \(A\) and house \(B\). The cyclist starts from rest at \(A\) and moves with constant acceleration for 20 seconds, reaching a velocity of \(15 \mathrm {~ms} ^ { - 1 }\). The cyclist then moves at this constant velocity before decelerating at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).
  1. Find the time, in seconds, for which the cyclist is decelerating.
  2. Sketch a velocity-time graph for the motion of the cyclist between \(A\) and \(B\). [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between \(A\) and \(B\) is 1950 m .
  3. Find the time, in seconds, for which the cyclist is moving at constant velocity.
OCR PURE Q10
8 marks Standard +0.3
10 A particle \(P\) is moving in a straight line. At time \(t\) seconds, where \(t \geqslant 0 , P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) and acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a = 4 t - 9\). It is given that \(v = 2\) when \(t = 1\).
  1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) is instantaneously at rest when \(t = t _ { 1 }\) and \(t = t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
  2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
  3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).
OCR PURE Q11
13 marks Challenging +1.2
11 Two balls \(P\) and \(Q\) have masses 0.6 kg and 0.4 kg respectively. The balls are attached to the ends of a string. The string passes over a pulley which is fixed at the edge of a rough horizontal surface. Ball \(P\) is held at rest on the surface 2 m from the pulley. Ball \(Q\) hangs vertically below the pulley. Ball \(Q\) is attached to a third ball \(R\) of mass \(m \mathrm {~kg}\) by another string and \(R\) hangs vertically below \(Q\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-7_419_945_493_246} The system is released from rest with the strings taut. Ball \(P\) moves towards the pulley with acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\) and a constant frictional force of magnitude 4.5 N opposes the motion of \(P\). The balls are modelled as particles, the pulley is modelled as being small and smooth, and the strings are modelled as being light and inextensible.
  1. By considering the motion of \(P\), find the tension in the string connecting \(P\) and \(Q\).
  2. Hence determine the value of \(m\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When the balls have been in motion for 0.4 seconds the string connecting \(Q\) and \(R\) breaks.
  3. Show that, according to the model, \(P\) does not reach the pulley. It is given that in fact ball \(P\) does reach the pulley.
  4. Identify one factor in the modelling that could account for this difference.
OCR PURE Q6
7 marks Standard +0.3
  1. Show that the equation \(6 \cos ^ { 2 } \theta = \tan \theta \cos \theta + 4\) can be expressed in the form \(6 \sin ^ { 2 } \theta + \sin \theta - 2 = 0\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-4_446_1150_1119_338} The diagram shows parts of the curves \(y = 6 \cos ^ { 2 } \theta\) and \(y = \tan \theta \cos \theta + 4\), where \(\theta\) is in degrees. Solve the inequality \(6 \cos ^ { 2 } \theta > \tan \theta \cos \theta + 4\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
OCR PURE Q11
4 marks Challenging +1.2
11 In this question you must show detailed reasoning. A biased four-sided spinner has edges numbered \(1,2,3,4\). When the spinner is spun, the probability that it will land on the edge numbered \(X\) is given by \(P ( X = x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 10 } x & x = 1,2,3,4 , \\ 0 & \text { otherwise } . \end{cases}\)
  1. Draw a table showing the probability distribution of \(X\). The spinner is spun three times and the value of \(X\) is noted each time.
  2. Find the probability that the third value of \(X\) is greater than the sum of the first two values of \(X\).
OCR PURE 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 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 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 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 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 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 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 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 PURE Q1
5 marks Easy -1.3
In this question you must show detailed reasoning.
  1. Express \(3^{\frac{1}{2}}\) in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [2]
  2. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
OCR PURE Q2
4 marks Easy -1.2
  1. The equation \(x^2 + 3x + k = 0\) has repeated roots. Find the value of the constant \(k\). [2]
  2. Solve the inequality \(6 + x - x^2 > 0\). [2]
OCR PURE Q3
6 marks Moderate -0.8
  1. Solve the equation \(\sin^2\theta = 0.25\) for \(0° \leq \theta < 360°\). [3]
  2. In this question you must show detailed reasoning. Solve the equation \(\tan 3\phi = \sqrt{3}\) for \(0° \leq \phi < 90°\). [3]
OCR PURE Q4
9 marks Moderate -0.8
  1. It is given that \(y = x^2 + 3x\).
    1. Find \(\frac{dy}{dx}\). [2]
    2. Find the values of \(x\) for which \(y\) is increasing. [2]
  2. Find \(\int(3 - 4\sqrt{x})dx\). [5]
OCR PURE Q5
5 marks Standard +0.8
\(N\) is an integer that is not divisible by 3. Prove that \(N^2\) is of the form \(3p + 1\), where \(p\) is an integer. [5]
OCR PURE Q6
7 marks Easy -1.2
Sketch the following curves.
  1. \(y = \frac{2}{x}\) [2]
  2. \(y = x^3 - 6x^2 + 9x\) [5]
OCR PURE Q7
7 marks Moderate -0.8
\(OABC\) is a parallelogram with \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(P\) is the midpoint of \(AC\). \includegraphics{figure_7}
  1. Find the following in terms of \(\mathbf{a}\) and \(\mathbf{c}\), simplifying your answers.
    1. \(\overrightarrow{AC}\) [1]
    2. \(\overrightarrow{OP}\) [2]
  2. Hence prove that the diagonals of a parallelogram bisect one another. [4]
OCR PURE Q8
6 marks Standard +0.8
In this question you must show detailed reasoning. The lines \(y = \frac{1}{2}x\) and \(y = -\frac{1}{2}x\) are tangents to a circle at \((2, 1)\) and \((-2, 1)\) respectively. Find the equation of the circle in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are constants. [6]
OCR PURE Q9
2 marks Easy -2.0
Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.
  1. State an advantage of using a stratified sample rather than a simple random sample. [1]
  2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK. [1]
OCR PURE Q10
5 marks Moderate -0.3
The probability distribution of a random variable \(X\) is given in the table.
\(x\)0246
P\((X = x)\)\(\frac{3}{8}\)\(\frac{5}{16}\)\(4p\)\(p\)
  1. Find the value of \(p\). [2]
  2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0. [3]
OCR PURE Q11
5 marks Moderate -0.3
The probability that Janice sees a kingfisher on any particular day is 0.3. She notes the number, \(X\), of days in a week on which she sees a kingfisher.
  1. State one necessary condition for \(X\) to have a binomial distribution. [1]
Assume now that \(X\) has a binomial distribution.
  1. Find the probability that, in a week, Janice sees a kingfisher on exactly 2 days. [1]
Each week Janice notes the number of days on which she sees a kingfisher.
  1. Find the probability that Janice sees a kingfisher on exactly 2 days in a week during at least 4 of 6 randomly chosen weeks. [3]