Questions H240/01 (87 questions)

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OCR H240/01 2020 November Q3
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
4 Prove by contradiction that there is no greatest multiple of 5 .
OCR H240/01 2020 November Q5
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
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
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
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
\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
4 marks
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
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
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
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
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 \(z = 25\).
OCR H240/01 2021 November Q4
4 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 11 x + 6\).
  1. Use the factor theorem to show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in fully factorised form.
  3. Hence solve the equation \(2 \times 8 ^ { y } - 3 \times 4 ^ { y } - 11 \times 2 ^ { y } + 6 = 0\).
OCR H240/01 2021 November Q5
5
  1. The graph of the function \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to \(P\) on each of the following curves.
    1. \(\quad y = \mathrm { f } ( x ) + 3\)
    2. \(\quad y = 2 \mathrm { f } ( 3 x - 1 )\)
    3. \(y = \mathrm { f } ^ { - 1 } ( x )\)

  2. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-5_494_739_806_333} The diagram shows part of the graph of \(y = \mathrm { g } ^ { \prime } ( x )\). This is the graph of the gradient function of \(y = \mathrm { g } ( x )\). The graph intersects the \(x\)-axis at \(x = - 2\) and \(x = 4\).
    1. State the \(x\)-coordinate of any stationary points on the graph of \(y = \mathrm { g } ( x )\).
    2. State the set of values of \(x\) for which \(y = \mathrm { g } ( x )\) is a decreasing function.
    3. State the \(x\)-coordinate of any points of inflection on the graph of \(y = \mathrm { g } ( x )\).
OCR H240/01 2021 November Q6
6
  1. Find the first three terms in the expansion of \(( 8 - 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
  2. State the range of values of \(x\) for which the expansion in part (a) is valid.
  3. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 8 - 3 x ) ^ { \frac { 1 } { 3 } } } { ( 1 + 2 x ) ^ { 2 } }\).
OCR H240/01 2021 November Q7
7 The curve \(y = \left( x ^ { 2 } - 2 \right) \ln x\) has one stationary point which is close to \(x = 1\).
  1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(2 x ^ { 2 } \ln x + x ^ { 2 } - 2 = 0\).
  2. Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form \(x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } \ln x _ { n } + 3 x _ { n } ^ { 2 } + 2 } { 4 x _ { n } \left( \ln x _ { n } + 1 \right) }\).
  3. Apply the Newton-Raphson formula with initial value \(x _ { 1 } = 1\) to find \(x _ { 2 }\) and \(x _ { 3 }\).
  4. Find the coordinates of this stationary point, giving each coordinate correct to \(\mathbf { 3 }\) decimal places.
OCR H240/01 2021 November Q8
8 Functions f and g are defined for \(0 \leqslant x \leqslant 2 \pi\) by \(\mathrm { f } ( x ) = 2 \tan x\) and \(\mathrm { g } ( x ) = \sec x\).
    1. State the range of f .
    2. State the range of \(g\).
    1. Show that \(\operatorname { fg } ( 0.6 ) = 5.33\), correct to 3 significant figures.
    2. Explain why \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( 0.6 )\) is not defined.
  3. In this question you must show detailed reasoning. Solve the equation \(( \mathrm { f } ( x ) ) ^ { 2 } + 6 \mathrm {~g} ( x ) = 0\).
OCR H240/01 2021 November Q9
9 A particle moves in the \(x - y\) plane so that at time \(t\) seconds, where \(t \geqslant 0\), its coordinates are given by \(x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }\).
  1. Explain why the path of the particle never crosses the \(x\)-axis.
  2. Determine the exact values of \(t\) when the path of the particle intersects the \(y\)-axis.
  3. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }\).
  4. Hence find the coordinates of the particle when its path is parallel to the \(y\)-axis.
OCR H240/01 2021 November Q10
10

  1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
    1. Explain why the length of \(C D\) can be written as \(a \cos y\).
    2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
    3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
  2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).
OCR H240/01 2021 November Q11
11
  1. Use the substitution \(u ^ { 2 } = x ^ { 2 } + 3\) to show that \(\int \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 4 } { 3 } \left( x ^ { 2 } - 6 \right) \sqrt { x ^ { 2 } + 3 } + c\).
  2. In this question you must show detailed reasoning.
    \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_620_951_1836_317} The graph shows part of the curve \(y = \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 2 } }\).
    Find the exact area enclosed by the curve \(y = \frac { 4 x ^ { 3 } } { \sqrt { x ^ { 2 } + 3 } }\), the normal to this curve at the point \(( 1,2 )\) and the \(x\)-axis.
OCR H240/01 2021 November Q12
12 A cake is cooling so that, \(t\) minutes after it is removed from an oven, its temperature is \(\theta ^ { \circ } \mathrm { C }\). When the cake is removed from the oven, its temperature is \(160 ^ { \circ } \mathrm { C }\). After 10 minutes its temperature has fallen to \(125 ^ { \circ } \mathrm { C }\).
  1. In a simple model, the rate of decrease of the temperature of the cake is assumed to be constant.
    1. Write down a differential equation for this model.
    2. Solve this differential equation to find \(\theta\) in terms of \(t\).
    3. State one limitation of this model.
  2. In a revised model, the rate of decrease of the temperature of the cake is proportional to the difference between the temperature of the cake and the temperature of the room. The temperature of the room is a constant \(20 ^ { \circ } \mathrm { C }\).
    1. Write down a differential equation for this revised model.
    2. Solve this differential equation to find \(\theta\) in terms of \(t\).
  3. The cake can be decorated when its temperature is \(25 ^ { \circ } \mathrm { C }\). Find the difference in time between when the two models would predict that the cake can be decorated, giving your answer correct to the nearest minute. \section*{END OF QUESTION PAPER}
OCR H240/01 2022 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-4_303_451_358_242} The diagram shows part of the curve \(y = \sqrt { x ^ { 2 } - 1 }\).
  1. Use the trapezium rule with 4 intervals to find an estimate for \(\int _ { 1 } ^ { 3 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x\). Give your answer correct to \(\mathbf { 3 }\) significant figures.
  2. State whether the value from part (a) is an under-estimate or an over-estimate, giving a reason for your answer.
  3. Explain how the trapezium rule could be used to obtain a more accurate estimate.
OCR H240/01 2022 June Q2
2
  1. Given that \(a\) and \(b\) are real numbers, find a counterexample to disprove the statement that, if \(a > b\), then \(a ^ { 2 } > b ^ { 2 }\).
  2. A student writes the statement that \(\sin x ^ { \circ } = 0.5 \Longleftrightarrow x ^ { \circ } = 30 ^ { \circ }\).
    1. Explain why this statement is incorrect.
    2. Write a corrected version of this statement.
  3. Prove that the sum of four consecutive multiples of 4 is always a multiple of 8 .
OCR H240/01 2022 June Q3
1 marks
3
  1. In this question you must show detailed reasoning.
    Find the coordinates of the points of intersection of the curves with equations \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\).
  2. The diagram shows the curves \(y = x ^ { 2 } - 2 x + 1\) and \(y = - x ^ { 2 } + 6 x - 5\). This diagram is repeated in the Printed Answer Booklet.
    \includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-5_377_542_603_322} On the diagram in the Printed Answer Booklet, draw the line \(y = 2 x - 2\).
  3. Show on your diagram in the Printed Answer Booklet the region of the \(x - y\) plane within which all three of the following inequalities are satisfied.
    \(y \geqslant x ^ { 2 } - 2 x + 1 \quad y \leqslant - x ^ { 2 } + 6 x - 5 \quad y \leqslant 2 x - 2\)
    You should indicate the region for which all the inequalities hold by labelling the region \(R\).[1]
OCR H240/01 2022 June Q4
4
  1. Write \(2 x ^ { 2 } + 6 x + 7\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are constants.
  2. State the coordinates of the minimum point on the graph of \(y = 2 x ^ { 2 } + 6 x + 7\).
  3. Hence deduce
    • the minimum value of \(2 \tan ^ { 2 } \theta + 6 \tan \theta + 7\),
    • the smallest positive value of \(\theta\), in degrees, for which the minimum value occurs.