Questions — OCR MEI Paper 3 (118 questions)

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OCR MEI Paper 3 2018 June Q1
1 Triangle ABC is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_451_565_520_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Find the perimeter of triangle ABC .
OCR MEI Paper 3 2018 June Q2
2 marks
2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]
OCR MEI Paper 3 2018 June Q3
3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_648_627_1507_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).
OCR MEI Paper 3 2018 June Q4
4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-5_723_844_424_612} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Determine the coordinates of the stationary points on the curve.
  2. Determine the nature of each stationary point.
  3. Write down the equation of the vertical asymptote.
  4. Deduce the set of values of \(x\) for which the curve is concave upwards.
OCR MEI Paper 3 2018 June Q5
5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.
  1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
  2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure} Find estimates of the values of \(a\) and \(k\).
  3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
  4. Give a reason why the model may not be appropriate for large values of \(t\).
OCR MEI Paper 3 2018 June Q6
6 Find the constant term in the expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 15 }\).
OCR MEI Paper 3 2018 June Q7
7 In this question you must show detailed reasoning.
Fig. 7 shows the curve \(y = 5 x - x ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-7_511_684_383_694} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} The line \(y = 4 - k x\) crosses the curve \(y = 5 x - x ^ { 2 }\) on the \(x\)-axis and at one other point.
Determine the coordinates of this other point.
OCR MEI Paper 3 2018 June Q8
8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).
  1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
  2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).
OCR MEI Paper 3 2018 June Q9
9 The function \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }\) is defined on the domain \(x \in \mathbb { R } , x \neq 0\).
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Write down the range of \(\mathrm { f } ^ { - 1 } ( x )\).
OCR MEI Paper 3 2018 June Q10
10 Point A has position vector \(\left( \begin{array} { l } a
b
0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4
2
0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2
4
2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).
  1. Show that \(a - b + 1 = 0\).
  2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
    Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.
OCR MEI Paper 3 2018 June Q11
11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.
  1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
  2. Both numbers \(a\) and \(b\) are negative.
OCR MEI Paper 3 2018 June Q12
12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.
OCR MEI Paper 3 2018 June Q13
13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
OCR MEI Paper 3 2018 June Q14
14
  1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
  2. Prove that \(h = \sqrt { a b }\), as given in line 24 .
OCR MEI Paper 3 2018 June Q15
15 It is given in lines \(31 - 32\) that the square has the smallest perimeter of all rectangles with the same area. Using this fact, prove by contradiction that among rectangles of a given perimeter, \(4 L\), the square with side \(L\) has the largest area. \section*{END OF QUESTION PAPER}
OCR MEI Paper 3 2022 June Q1
1 A curve for which \(y\) is inversely proportional to \(x\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-4_824_1125_561_242} Find the equation of the curve.
OCR MEI Paper 3 2022 June Q2
2 The function \(\mathrm { f } ( x ) = \sqrt { x }\) is defined on the domain \(x \geqslant 0\).
The function \(\mathrm { g } ( x ) = 25 - x ^ { 2 }\) is defined on the domain \(\mathbb { R }\).
  1. Write down an expression for \(\mathrm { fg } ( x )\).
    1. Find the domain of \(\mathrm { fg } ( x )\).
    2. Find the range of \(\mathrm { fg } ( x )\).
OCR MEI Paper 3 2022 June Q3
3 An infinite sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by \(a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }\), for all positive integers \(n\).
  1. Find the limit of the sequence.
  2. Prove that this is an increasing sequence.
OCR MEI Paper 3 2022 June Q4
4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
OCR MEI Paper 3 2022 June Q5
5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).
  1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
  2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.
OCR MEI Paper 3 2022 June Q6
6 A hot drink is cooling. The temperature of the drink at time \(t\) minutes is \(T ^ { \circ } \mathrm { C }\).
The rate of decrease in temperature of the drink is proportional to \(( T - 20 )\).
  1. Write down a differential equation to describe the temperature of the drink as a function of time.
  2. When \(t = 0\), the temperature of the drink is \(90 ^ { \circ } \mathrm { C }\) and the temperature is decreasing at a rate of \(4.9 ^ { \circ } \mathrm { C }\) per minute. Determine how long it takes for the drink to cool from \(90 ^ { \circ } \mathrm { C }\) to \(40 ^ { \circ } \mathrm { C }\).
OCR MEI Paper 3 2022 June Q7
7 A student is trying to find the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
She gets the first three terms as \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\).
She draws the graphs of the curves \(y = \sqrt { 1 - x ^ { 3 } } , y = 1 - \frac { x ^ { 3 } } { 2 }\) and \(y = 1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 }\) using software.
\includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-6_901_1265_516_248}
  1. Explain why \(1 - \frac { x ^ { 3 } } { 2 } + \frac { x ^ { 6 } } { 8 } \geqslant 1 - \frac { x ^ { 3 } } { 2 }\) for all values of \(x\).
  2. Explain why the graphs suggest that the student has made a mistake in the binomial expansion.
  3. Find the first four terms in the binomial expansion of \(\sqrt { 1 - x ^ { 3 } }\).
  4. State the set of values of \(x\) for which the binomial expansion in part (c) is valid.
  5. Sketch the curve \(y = 2.5 \sqrt { 1 - x ^ { 3 } }\) on the grid in the Printed Answer Booklet. \section*{(f) In this question you must show detailed reasoning.} The end of a bus shelter is modelled by the area between the curve \(\mathrm { y } = 2.5 \sqrt { 1 - x ^ { 3 } }\), the lines \(x = - 0.75 , x = 0.75\) and the \(x\)-axis. Lengths are in metres. Calculate, using your answer to part (c), an approximation for the area of the end of the bus shelter as given by this model.
OCR MEI Paper 3 2022 June Q8
8 The curves \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) and \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(\mathrm { y } = \mathrm { h } ( \mathrm { x } )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(\mathrm { y } = \mathrm { h } ^ { - 1 } ( \mathrm { x } )\) crosses the \(x\)-axis at D and the \(y\)-axis at C .
\includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-7_826_819_520_255}
  1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
  2. Determine the coordinates of A, B, C and D.
  3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(\mathrm { y } = \mathrm { mx } + c\), where \(m\) and \(c\) are constants to be determined.
  4. Points A , B , C and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\), \(b\) and \(r ^ { 2 }\) are constants to be determined.
OCR MEI Paper 3 2022 June Q9
9 Show that \(\mathrm { y } = \mathrm { x }\) has the same gradient as \(\mathrm { y } = \sin \mathrm { x }\) when \(\mathrm { x } = 0\), as stated in line 5 .
OCR MEI Paper 3 2022 June Q11
11 Show that, for the angle \(45 ^ { \circ }\), the formula \(\sin \theta \approx \frac { 4 \theta ( 180 - \theta ) } { 40500 - \theta ( 180 - \theta ) }\) given in line 28 gives the same approximation for the sine of the angle as the formula \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) given in line 23.