Questions — OCR H240/01 (87 questions)

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OCR H240/01 2018 June Q1
1 The points \(A\) and \(B\) have coordinates \(( 1,5 )\) and \(( 4,17 )\) respectively. Find the equation of the straight line which passes through the point \(( 2,8 )\) and is perpendicular to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are constants.
OCR H240/01 2018 June Q2
2
  1. Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate.
OCR H240/01 2018 June Q3
3 In this question you must show detailed reasoning.
Find the two real roots of the equation \(x ^ { 4 } - 5 = 4 x ^ { 2 }\). Give the roots in an exact form.
OCR H240/01 2018 June Q4
4 Prove algebraically that \(n ^ { 3 } + 3 n - 1\) is odd for all positive integers \(n\).
OCR H240/01 2018 June Q5
5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
  1. Find the centre and radius of the circle.
  2. Find the coordinates of any points where the line \(y = 2 x - 3\) meets the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
  3. State what can be deduced from the answer to part (ii) about the line \(y = 2 x - 3\) and the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
OCR H240/01 2018 June Q6
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).
  1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
  3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
  4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.
OCR H240/01 2018 June Q7
7 Chris runs half marathons, and is following a training programme to improve his times. His time for his first half marathon is 150 minutes. His time for his second half marathon is 147 minutes. Chris believes that his times can be modelled by a geometric progression.
  1. Chris sets himself a target of completing a half marathon in less than 120 minutes. Show that this model predicts that Chris will achieve his target on his thirteenth half marathon.
  2. After twelve months Chris has spent a total of 2974 minutes, to the nearest minute, running half marathons. Use this model to find how many half marathons he has run.
  3. Give two reasons why this model may not be appropriate when predicting the time for a half marathon.
OCR H240/01 2018 June Q8
8
  1. Find the first three terms in the expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\).
  2. The expansion of \(\frac { a + b x } { \sqrt { 4 - x } }\) is \(16 - x \ldots\). Find the values of the constants \(a\) and \(b\).
OCR H240/01 2018 June Q9
9 The function f is defined for all real values of \(x\) as \(\mathrm { f } ( x ) = c + 8 x - x ^ { 2 }\), where \(c\) is a constant.
  1. Given that the range of f is \(\mathrm { f } ( x ) \leqslant 19\), find the value of \(c\).
  2. Given instead that \(\mathrm { ff } ( 2 ) = 8\), find the possible values of \(c\).
OCR H240/01 2018 June Q10
10 A curve has parametric equations \(x = t + \frac { 2 } { t }\) and \(y = t - \frac { 2 } { t }\), for \(t \neq 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Explain why the curve has no stationary points.
  3. By considering \(x + y\), or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.
OCR H240/01 2018 June Q11
11 In a science experiment a substance is decaying exponentially. Its mass, \(M\) grams, at time \(t\) minutes is given by \(M = 300 e ^ { - 0.05 t }\).
  1. Find the time taken for the mass to decrease to half of its original value. A second substance is also decaying exponentially. Initially its mass was 400 grams and, after 10 minutes, its mass was 320 grams.
  2. Find the time at which both substances are decaying at the same rate.
OCR H240/01 2018 June Q12
10 marks
12 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}, center]{1ba9fa5f-310f-4429-9bd1-4004852d5b3e-6_716_479_292_794} The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\).
[0pt] [10]
OCR H240/01 2018 June Q13
13 A scientist is attempting to model the number of insects, \(N\), present in a colony at time \(t\) weeks. When \(t = 0\) there are 400 insects and when \(t = 1\) there are 440 insects.
  1. A scientist assumes that the rate of increase of the number of insects is inversely proportional to the number of insects present at time \(t\).
    (a) Write down a differential equation to model this situation.
    (b) Solve this differential equation to find \(N\) in terms of \(t\).
  2. In a revised model it is assumed that \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ^ { 2 } } { 3988 \mathrm { e } ^ { 0.2 t } }\). Solve this differential equation to find \(N\) in terms of \(t\).
  3. Compare the long-term behaviour of the two models.
OCR H240/01 2019 June Q2
2 The point \(A\) is such that the magnitude of \(\overrightarrow { O A }\) is 8 and the direction of \(\overrightarrow { O A }\) is \(240 ^ { \circ }\).
    1. Show the point \(A\) on the axes provided in the Printed Answer Booklet.
    2. Find the position vector of point \(A\). Give your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\). The point \(B\) has position vector \(6 \mathbf { i }\).
  2. Find the exact area of triangle \(A O B\). The point \(C\) is such that \(O A B C\) is a parallelogram.
  3. Find the position vector of \(C\). Give your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
OCR H240/01 2019 June Q3
3 The function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 17\) for \(x \geqslant k\), where \(k\) is a constant.
  1. Given that \(\mathrm { f } ^ { - 1 } ( x )\) exists, state the least possible value of \(k\).
  2. Evaluate \(\mathrm { ff } ( 5 )\).
  3. Solve the equation \(\mathrm { f } ( x ) = x\).
  4. Explain why your solution to part (c) is also the solution to the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\).
OCR H240/01 2019 June Q4
4 Sam starts a job with an annual salary of \(\pounds 16000\). It is promised that the salary will go up by the same amount every year. In the second year Sam is paid \(\pounds 17200\).
  1. Find Sam's salary in the tenth year.
  2. Find the number of complete years needed for Sam's total salary to first exceed \(\pounds 500000\).
  3. Comment on how realistic this model may be in the long term.
OCR H240/01 2019 June Q5
5 A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
  2. Find the equation of the normal to the curve at the point ( 1,2 ).
OCR H240/01 2019 June Q6
6 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x\). Use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } + 3\).
OCR H240/01 2019 June Q8
8 A cylindrical tank is initially full of water. There is a small hole at the base of the tank out of which the water leaks. The height of water in the tank is \(x \mathrm {~m}\) at time \(t\) seconds. The rate of change of the height of water may be modelled by the assumption that it is proportional to the square root of the height of water. When \(t = 100 , x = 0.64\) and, at this instant, the height is decreasing at a rate of \(0.0032 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = - 0.004 \sqrt { x }\).
  2. Find an expression for \(x\) in terms of \(t\).
  3. Hence determine at what time, according to this model, the tank will be empty.
OCR H240/01 2019 June Q9
9
  1. Express \(3 \cos 3 x + 7 \sin 3 x\) in the form \(R \cos ( 3 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. Give full details of a sequence of three transformations needed to transform the curve \(y = \cos x\) to the curve \(y = 3 \cos 3 x + 7 \sin 3 x\).
  3. Determine the greatest value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
  4. Determine the least value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
OCR H240/01 2019 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-6_405_661_251_703} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 6 cm .
The angle \(A O B\) is \(\theta\) radians.
The area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\) is \(7.2 \mathrm {~cm} ^ { 2 }\).
  1. Show that \(\theta = 0.4 + \sin \theta\).
  2. Let \(\mathrm { F } ( \theta ) = 0.4 + \sin \theta\). By considering the value of \(\mathrm { F } ^ { \prime } ( \theta )\) where \(\theta = 1.2\), explain why using an iterative method based on the equation in part (a) will converge to the root, assuming that 1.2 is sufficiently close to the root.
  3. Use the iterative formula \(\theta _ { n + 1 } = 0.4 + \sin \theta _ { n }\) with a starting value of 1.2 to find the value of \(\theta\) correct to 4 significant figures.
    You should show the result of each iteration.
  4. Use a change of sign method to show that the value of \(\theta\) found in part (c) is correct to 4 significant figures.
OCR H240/01 2019 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{05bec6d6-b526-4b6f-86f3-39aa38cbf5c6-7_540_734_260_667} The diagram shows part of the curve \(y = \ln ( x - 4 )\).
  1. Use integration by parts to show that \(\int \ln ( x - 4 ) \mathrm { d } x = ( x - 4 ) \ln | x - 4 | - x + c\).
  2. State the equation of the vertical asymptote to the curve \(y = \ln ( x - 4 )\).
  3. Find the total area enclosed by the curve \(y = \ln ( x - 4 )\), the \(x\)-axis and the lines \(x = 4.5\) and \(x = 7\). Give your answer in the form \(a \ln 3 + b \ln 2 + c\) where \(a , b\) and \(c\) are constants to be found.
OCR H240/01 2019 June Q12
12 A curve has equation \(y = a ^ { 3 x ^ { 2 } }\), where \(a\) is a constant greater than 1 .
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x a ^ { 3 x ^ { 2 } } \ln a\).
  2. The tangent at the point \(\left( 1 , a ^ { 3 } \right)\) passes through the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the value of \(a\), giving your answer in an exact form.
  3. By considering \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) show that the curve is convex for all values of \(x\). \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR H240/01 2020 November Q1
1
  1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta\).
  2. Hence determine an approximate solution to \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta\).
OCR H240/01 2020 November Q2
2 Simplify fully.
  1. \(\sqrt { 12 a } \times \sqrt { 3 a ^ { 5 } }\)
  2. \(\left( 64 b ^ { 3 } \right) ^ { \frac { 1 } { 3 } } \times \left( 4 b ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\)
  3. \(7 \times 9 ^ { 3 c } - 4 \times 27 ^ { 2 c }\)