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OCR H240/01 2018 June Q13
13 marks Standard +0.8
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
8 marks Moderate -0.8
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
7 marks Standard +0.3
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
7 marks Moderate -0.8
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
8 marks Standard +0.3
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 marks Moderate -0.3
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 marks Standard +0.3
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
11 marks Standard +0.3
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
11 marks Standard +0.3
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
10 marks Standard +0.3
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 marks Standard +0.3
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
5 marks Moderate -0.3
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
8 marks Easy -1.3
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. \(7 \times 9 ^ { 3 c } - 4 \times 27 ^ { 2 c }\)
OCR H240/01 2020 November Q3
10 marks Standard +0.3
3 A cylindrical metal tin of radius \(r \mathrm {~cm}\) is closed at both ends. It has a volume of \(16000 \pi \mathrm {~cm} ^ { 3 }\).
  1. Show that its total surface area, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 2 \pi r ^ { 2 } + 32000 \pi r ^ { - 1 }\).
  2. Use calculus to determine the minimum total surface area of the tin. You should justify that it is a minimum.
OCR H240/01 2020 November Q4
3 marks Moderate -0.8
4 Prove by contradiction that there is no greatest multiple of 5 .
OCR H240/01 2020 November Q5
8 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-5_424_583_255_244} The diagram shows points \(A\) and \(B\), which have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to an origin \(O\). \(P\) is the point on \(O B\) such that \(O P : P B = 3 : 1\) and \(Q\) is the midpoint of \(A B\).
  1. Find \(\overrightarrow { P Q }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\). The line \(O A\) is extended to a point \(R\), so that \(P Q R\) is a straight line.
  2. Explain why \(\overrightarrow { P R } = k ( 2 \mathbf { a } - \mathbf { b } )\), where \(k\) is a constant.
  3. Hence determine the ratio \(O A : A R\).
OCR H240/01 2020 November Q6
9 marks Moderate -0.3
6 A mobile phone company records their annual sales on \(31 ^ { \text {st } }\) December every year.
Paul thinks that the annual sales, \(S\) million, can be modelled by the equation \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants and \(t\) is the number of years since \(31 ^ { \text {st } }\) December 2015. Paul tests his theory by using the annual sales figures from \(31 ^ { \text {st } }\) December 2015 to \(31 { } ^ { \text {st } }\) December 2019. He plots these results on a graph, with \(t\) on the horizontal axis and \(\log _ { 10 } S\) on the vertical axis.
  1. Explain why, if Paul's model is correct, the results should lie on a straight line of best fit on his graph. The results lie on a straight line of best fit which has a gradient of 0.146 and an intercept on the vertical axis of 0.583 .
  2. Use these values to obtain estimates for \(a\) and \(b\), correct to 2 significant figures.
  3. Use this model to predict the year in which, on the \(31 { } ^ { \text {st } }\) December, the annual sales would first be recorded as greater than 200 million.
  4. Give a reason why this prediction may not be reliable.
OCR H240/01 2020 November Q7
11 marks Standard +0.3
7 Two students, Anna and Ben, are starting a revision programme. They will both revise for 30 minutes on Day 1. Anna will increase her revision time by 15 minutes for every subsequent day. Ben will increase his revision time by \(10 \%\) for every subsequent day.
  1. Verify that on Day 10 Anna does 94 minutes more revision than Ben, correct to the nearest minute. Let Day \(X\) be the first day on which Ben does more revision than Anna.
  2. Show that \(X\) satisfies the inequality \(X > \log _ { 1.1 } ( 0.5 X + 0.5 ) + 1\).
  3. Use the iterative formula \(x _ { n + 1 } = \log _ { 1.1 } \left( 0.5 x _ { n } + 0.5 \right) + 1\) with \(x _ { 1 } = 10\) to find the value of \(X\). You should show the result of each iteration.
    1. Give a reason why Anna's revision programme may not be realistic.
    2. Give a different reason why Ben's revision programme may not be realistic.
OCR H240/01 2020 November Q8
7 marks Moderate -0.8
8
  1. Differentiate \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) with respect to \(x\).
  2. Hence show that \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) is increasing for all values of \(x\).
OCR H240/01 2020 November Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-6_391_606_1672_244} The diagram shows the graph of \(y = | 2 x - 3 |\).
  1. State the coordinates of the points of intersection with the axes.
  2. Given that the graphs of \(y = | 2 x - 3 |\) and \(y = a x + 2\) have two distinct points of intersection, determine
    1. the set of possible values of \(a\),
    2. the \(x\)-coordinates of the points of intersection of these graphs, giving your answers in terms of \(a\).
OCR H240/01 2020 November Q10
11 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-7_352_545_258_239} The diagram shows the curve \(y = \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right)\), for \(1 \leqslant x \leqslant 2\).
  1. Use rectangles of width 0.25 to find upper and lower bounds for \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x\). Give your answers correct to 3 significant figures.
    1. Use the substitution \(t = \sqrt { x - 1 }\) to show that \(\int \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = \int 2 t \sin \left( \frac { 1 } { 2 } t \right) \mathrm { d } t\).
    2. Hence show that \(\int _ { 1 } ^ { 2 } \sin \left( \frac { 1 } { 2 } \sqrt { x - 1 } \right) \mathrm { d } x = 8 \sin \frac { 1 } { 2 } - 4 \cos \frac { 1 } { 2 }\).
OCR H240/01 2020 November Q12
9 marks Standard +0.3
12 Find the general solution of the differential equation \(\left( 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 20 x - 35 )\).
Give your answer in the form \(y = \mathrm { f } ( x )\). \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}
OCR H240/01 2021 November Q1
4 marks Moderate -0.8
1 Determine the set of values of \(k\) such that the equation \(x ^ { 2 } + 4 x + ( k + 3 ) = 0\) has two distinct real roots.
OCR H240/01 2021 November Q2
4 marks Easy -1.3
2 Alex is comparing the cost of mobile phone contracts. Contract \(\boldsymbol { A }\) has a set-up cost of \(\pounds 40\) and then costs 4 p per minute. Contract \(\boldsymbol { B }\) has no set-up cost, does not charge for the first 100 minutes and then costs 6 p per minute.
  1. Find an expression for the cost of each of the contracts in terms of \(m\), where \(m\) is the number of minutes for which the phone is used and \(m > 100\).
  2. Hence find the value of \(m\) for which both contracts would cost the same.
OCR H240/01 2021 November Q3
4 marks Easy -1.2
3 It is given that \(x\) is proportional to the product of the square of \(y\) and the positive square root of \(z\). When \(y = 2\) and \(z = 9 , x = 30\).
  1. Write an equation for \(x\) in terms of \(y\) and \(z\).
  2. Find the value of \(x\) when \(y = 3\) and \(z = 25\).