Questions H240/01 (87 questions)

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OCR H240/01 2022 June Q5
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
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
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
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
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
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
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 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
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
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
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
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.
OCR H240/01 2023 June Q5
5
  1. The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
    1. The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\). Find the values of \(a , b\) and \(c\).
    2. Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
  2. The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
    1. Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
    2. State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.
OCR H240/01 2023 June Q6
6 A curve has equation \(y = \mathrm { e } ^ { x ^ { 2 } + 3 x }\).
  1. Determine the \(x\)-coordinates of any stationary points on the curve.
  2. Show that the curve is convex for all values of \(x\).
OCR H240/01 2023 June Q7
7
  1. Use the result \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) to show that \(\cos ( A - B ) = \cos A \cos B + \sin A \sin B\). The function \(\mathrm { f } ( \theta )\) is defined as \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta - 30 ^ { \circ } \right)\), where \(\theta\) is in degrees.
  2. Show that \(f ( \theta ) = \cos ^ { 2 } \theta - \frac { 1 } { 4 }\).
    1. Determine the following.
      • The maximum value of \(\mathrm { f } ( \theta )\)
  4. The smallest positive value of \(\theta\) for which this maximum value occurs
    (ii) Determine the following.
  5. The minimum value of \(\mathrm { f } ( \theta )\)
  6. The smallest positive value of \(\theta\) for which this minimum value occurs
OCR H240/01 2023 June Q8
8
  1. Find the first three terms in the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\).
  2. State the range of values of \(x\) for which the expansion in part (a) is valid.
  3. In the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } } ( 1 + a x ) ^ { 2 }\) the coefficient of \(x ^ { 2 }\) is \(\frac { 107 } { 16 }\). Determine the possible values of the constant \(a\).
OCR H240/01 2023 June Q9
9 Conservationists are studying how the number of bees in a wildflower meadow varies according to the number of wildflower plants. The study takes place over a series of weeks in the summer. A model is suggested for the number of bees, \(B\), and the number of wildflower plants, \(F\), at time \(t\) weeks after the start of the study. In the model \(B = 20 + 2 t + \cos 3 t\) and \(F = 50 \mathrm { e } ^ { 0.1 t }\). The model assumes that \(B\) and \(F\) can be treated as continuous variables.
  1. State the meaning of \(\frac { \mathrm { d } B } { \mathrm {~d} F }\).
  2. Determine \(\frac { \mathrm { d } B } { \mathrm {~d} F }\) when \(t = 4\).
  3. Suggest a reason why this model may not be valid for values of \(t\) greater than 12 .
OCR H240/01 2023 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-07_805_775_251_242} The diagram shows part of the curve \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 4 x ^ { 2 } - 1 } + 2\). The equation \(\mathrm { f } ( x ) = 0\) has a positive root \(\alpha\) close to \(x = 0.3\).
  1. Explain why using the sign change method with \(x = 0\) and \(x = 1\) will fail to locate \(\alpha\).
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x } \right) }\).
  3. Use the iterative formula \(x _ { n + 1 } = \frac { 1 } { 4 } \sqrt { \left( 4 - 2 \mathrm { e } ^ { x _ { n } } \right) }\) with a starting value of \(x _ { 1 } = 0.3\) to find the value of \(\alpha\) correct to \(\mathbf { 4 }\) significant figures, showing the result of each iteration.
  4. An alternative iterative formula is \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where \(\mathrm { F } \left( x _ { n } \right) = \ln \left( 2 - 8 x _ { n } ^ { 2 } \right)\). By considering \(\mathrm { F } ^ { \prime } ( 0.3 )\) explain why this iterative formula will not find \(\alpha\).
OCR H240/01 2023 June Q11
11 The owners of an online shop believe that their sales can be modelled by \(S = a b ^ { t }\), where \(a\) and \(b\) are both positive constants, \(S\) is the number of items sold in a month and \(t\) is the number of complete months since starting their online shop. The sales for the first six months are recorded, and the values of \(\log _ { 10 } S\) are plotted against \(t\) in the graph below. The graph is repeated in the Printed Answer Booklet.
\includegraphics[max width=\textwidth, alt={}, center]{9473b8f7-616a-485e-963b-696c6640ae6b-08_1203_1408_552_244}
  1. Explain why the graph suggests that the given model is appropriate. The owners believe that \(a = 120\) and \(b = 1.15\) are good estimates for the parameters in the model.
  2. Show that the graph supports these estimates for the parameters.
  3. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of items sold in the seventh month after opening.
    1. Use the model \(S = 120 \times 1.15 ^ { t }\) to predict the number of months after opening when the total number of items sold after opening will first exceed 70000 .
    2. Comment on how reliable this prediction may be.
OCR H240/01 2023 June Q12
12
  1. Use the substitution \(u = \mathrm { e } ^ { x } - 2\) to show that $$\int \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = \int \frac { 7 u + 6 } { u ^ { 2 } ( u + 2 ) } \mathrm { d } u$$
  2. Hence show that $$\int _ { \ln 4 } ^ { \ln 6 } \frac { 7 \mathrm { e } ^ { x } - 8 } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } } \mathrm {~d} x = a + \ln b$$ where \(a\) and \(b\) are rational numbers to be determined. \section*{END OF QUESTION PAPER}
OCR H240/01 2020 November Q11
    1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 10 ( m + 1 ) x + 40 = 0\).
    2. Hence determine the equation of the tangent to the circle at \(A\) which passes through \(P\). [4] A second tangent is drawn from \(P\) to meet the circle at a second point \(B\). The equation of this tangent is of the form \(y = n x + 2\), where \(n\) is a constant less than 1 .
  2. Determine the exact value of \(\tan A P B\).
OCR H240/01 2019 June Q1
1 In this question you must show detailed reasoning. Solve the inequality \(10 x ^ { 2 } + x - 2 > 0\).
OCR H240/01 2019 June Q7
7 In this question you must show detailed reasoning. A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots\) is defined by \(u _ { n } = 25 \times 0.6 ^ { n }\).
Use an algebraic method to find the smallest value of \(N\) such that \(\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }\).
OCR H240/01 Q1
1 Solve the simultaneous equations. $$\begin{array} { r } x ^ { 2 } + 8 x + y ^ { 2 } = 84
x - y = 10 \end{array}$$
OCR H240/01 Q2
2 The points \(A\), \(B\) and \(C\) have position vectors \(3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } , - \mathbf { i } + 6 \mathbf { k }\) and \(7 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }\) respectively. M is the midpoint of BC .
  1. Show that the magnitude of \(\overrightarrow { O M }\) is equal to \(\sqrt { 17 }\). Point D is such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
  2. Show that position vector of the point D is \(11 \mathbf { i } - 8 \mathbf { j } - 6 \mathbf { k }\).