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OCR H240/01 2021 November Q4
7 marks Standard +0.3
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
7 marks Easy -1.2
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
9 marks Standard +0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
10 marks Standard +0.3
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
11 marks Moderate -0.3
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
12 marks Challenging +1.2
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
13 marks Standard +0.3
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
6 marks Moderate -0.3
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
6 marks Easy -1.2
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
7 marks Moderate -0.8
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
8 marks Moderate -0.8
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.
OCR H240/01 2022 June Q5
8 marks Moderate -0.3
5
  1. The graph of \(y = 2 ^ { x }\) can be transformed to the graph of \(y = 2 ^ { x + 4 }\) either by a translation or by a stretch.
    1. Give full details of the translation.
    2. Give full details of the stretch.
  2. In this question you must show detailed reasoning. Solve the equation \(\log _ { 2 } ( 8 x ) = 1 - \log _ { 2 } ( 1 - x )\).
OCR H240/01 2022 June Q6
8 marks Standard +0.3
6
  1. Find the first four terms in the expansion of \(( 3 + 2 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Hence determine the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 3 + 2 y + 4 y ^ { 2 } \right) ^ { 5 }\).
OCR H240/01 2022 June Q7
8 marks Standard +0.8
7 A curve has equation \(2 x ^ { 3 } + 6 x y - 3 y ^ { 2 } = 2\).
Show that there are no points on this curve where the tangent is parallel to \(y = x\).
OCR H240/01 2022 June Q8
9 marks Moderate -0.3
8
  1. Substance \(A\) is decaying exponentially such that its mass is \(m\) grams at time \(t\) minutes. Find the missing values of \(m\) and \(t\) in the following table.
    \(t\)01050
    \(m\)1250750450
  2. Substance \(B\) is also decaying exponentially, according to the model \(m = 160 \mathrm { e } ^ { - 0.055 t }\), where \(m\) grams is its mass after \(t\) minutes.
    1. Determine the value of \(t\) for which the mass of substance \(B\) is half of its original mass.
    2. Determine the rate of decay of substance \(B\) when \(t = 15\).
  3. State whether substance \(A\) or substance \(B\) is decaying at a faster rate, giving a reason for your answer.
OCR H240/01 2022 June Q9
7 marks Standard +0.8
9 Use the substitution \(x = 2 \sin \theta\) to show that \(\int _ { 1 } ^ { \sqrt { 3 } } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 3 } \pi\).
OCR H240/01 2022 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{38b515c2-4764-4b51-a1f5-9b48d46610f0-7_545_659_255_244} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(O A\). The angle \(A O B\) is \(\theta\) radians. \(M\) is the mid-point of \(O A\). The ratio of areas \(O M B : M A B\) is 2:3.
  1. Show that \(\theta = 1.25 \sin \theta\). The equation \(\theta = 1.25 \sin \theta\) has only one root for \(\theta > 0\).
  2. This root can be found by using the iterative formula \(\theta _ { n + 1 } = 1.25 \sin \theta _ { n }\) with a starting value of \(\theta _ { 1 } = 0.5\).
    • Write down the values of \(\theta _ { 2 } , \theta _ { 3 }\) and \(\theta _ { 4 }\).
    • Hence find the value of this root correct to \(\mathbf { 3 }\) significant figures.
    • The diagram in the Printed Answer Booklet shows the graph of \(y = 1.25 \sin \theta\), for \(0 \leqslant \theta \leqslant \pi\).
    • Use this diagram to show how the iterative process used in (b) converges to this root.
    • State the type of convergence.
    • Draw a suitable diagram to show why using an iterative process with the formula \(\theta _ { n + 1 } = \sin ^ { - 1 } \left( 0.8 \theta _ { n } \right)\) does not converge to the root found in (b).
OCR H240/01 2022 June Q11
9 marks Standard +0.3
11 The gradient function of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } \ln x } { \mathrm { e } ^ { 3 y } }\).
The curve passes through the point (e, 1).
  1. Find the equation of this curve, giving your answer in the form \(\mathrm { e } ^ { 3 y } = \mathrm { f } ( x )\).
  2. Show that, when \(x = \mathrm { e } ^ { 2 }\), the \(y\)-coordinate of this curve can be written as \(y = a + \frac { 1 } { 3 } \ln \left( b \mathrm { e } ^ { 3 } + c \right)\), where \(a , b\) and \(c\) are constants to be determined.
OCR H240/01 2022 June Q12
12 marks Standard +0.3
12 A curve has parametric equations \(x = \frac { 1 } { t } , y = 2 t\). The point \(P\) is \(\left( \frac { 1 } { p } , 2 p \right)\).
  1. Show that the equation of the tangent at \(P\) can be written as \(y = - 2 p ^ { 2 } x + 4 p\). The tangent to this curve at \(P\) crosses the \(x\)-axis at the point \(A\) and the normal to this curve at \(P\) crosses the \(x\)-axis at the point \(B\).
  2. Show that the ratio \(P A : P B\) is \(1 : 2 p ^ { 2 }\). \section*{END OF QUESTION PAPER}
OCR H240/01 2023 June Q1
5 marks Standard +0.8
1 In the triangle \(A B C\), the length \(A B = 6 \mathrm {~cm}\), the length \(A C = 15 \mathrm {~cm}\) and the angle \(B A C = 30 ^ { \circ }\).
  1. Calculate the length \(B C\). \(D\) is the point on \(A C\) such that the length \(B D = 4 \mathrm {~cm}\).
  2. Calculate the possible values of the angle \(A D B\).
OCR H240/01 2023 June Q2
8 marks Moderate -0.8
2
    1. Show that \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }\) can be written in the form \(\frac { a } { b + c x }\), where \(a , b\) and \(c\) are constants to be determined.
    2. Hence solve the equation \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2\).
  2. In this question you must show detailed reasoning. Solve the equation \(2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0\).
OCR H240/01 2023 June Q3
7 marks Moderate -0.8
3
  1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } + 2 x\), use differentiation from first principles to show that \(\mathrm { f } ^ { \prime } ( x ) = 2 x + 2\).
  2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x + 2\) and the curve passes through the point \(( - 1,5 )\). Find the equation of the curve.
OCR H240/01 2023 June Q4
8 marks Moderate -0.8
4 It is given that \(A B C D\) is a quadrilateral. The position vector of \(A\) is \(\mathbf { i } + \mathbf { j }\), and the position vector of \(B\) is \(3 \mathbf { i } + 5 \mathbf { j }\).
  1. Find the length \(A B\).
  2. The position vector of \(C\) is \(p \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant greater than 1 . Given that the length \(A B\) is equal to the length \(B C\), determine the position vector of \(C\).
  3. The point \(M\) is the midpoint of \(A C\). Given that \(\overrightarrow { M D } = 2 \overrightarrow { B M }\), determine the position vector of \(D\).
  4. State the name of the quadrilateral \(A B C D\), giving a reason for your answer.