Questions Paper 3 (332 questions)

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OCR MEI Paper 3 2021 November Q4
4 The diagram shows points \(A\) and \(B\) on the curve \(y = \left( \frac { x } { 4 } \right) ^ { - x }\).
The \(x\)-coordinate of A is 1 and the \(x\)-coordinate of B is 1.1 .
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-4_522_707_1758_278}
  1. Find the gradient of chord AB . Give your answer correct to 2 decimal places.
  2. Give the \(x\)-coordinate of a point C on the curve such that the gradient of chord AC is a better approximation to the gradient of the tangent to the curve at A .
OCR MEI Paper 3 2021 November Q5
5
  1. The diagram shows the curve \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }\).
    \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-5_574_682_315_328} On the axes in the Printed Answer Booklet, sketch graphs of
    1. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(x\),
    2. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(y\).
  2. Wolves were introduced to Yellowstone National Park in 1995. The population of wolves, \(y\), is modelled by the equation
    \(y = A e ^ { k t }\),
    where \(A\) and \(k\) are constants and \(t\) is the number of years after 1995.
    1. Give a reason why this model might be suitable for the population of wolves.
    2. When \(t = 0 , y = 21\) and when \(t = 1 , y = 51\). Find values of \(A\) and \(k\) consistent with the data.
    3. Give a reason why the model will not be a good predictor of wolf populations many years after 1995.
OCR MEI Paper 3 2021 November Q6
6 In this question you must show detailed reasoning.
Show that \(\sum _ { r = 1 } ^ { 3 } \frac { 1 } { \sqrt { r + 1 } + \sqrt { r } } = 1\).
OCR MEI Paper 3 2021 November Q7
7 Determine \(\int x \cos 2 x \mathrm {~d} x\).
OCR MEI Paper 3 2021 November Q8
8 For a particular value of \(a\), the curve \(\mathrm { y } = \frac { \mathrm { a } } { \mathrm { x } ^ { 2 } }\) passes through the point \(( 3,1 )\).
Find the coordinates of all the other points on the curve where both the \(x\)-coordinate and the \(y\)-coordinate are integers.
OCR MEI Paper 3 2021 November Q9
9 The diagram shows the curve \(\mathrm { y } = 3 - \sqrt { \mathrm { x } }\).
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-6_810_1008_1155_283}
  1. Draw the line \(\mathrm { y } = 5 \mathrm { x } - 1\) on the copy of the diagram in the Printed Answer Booklet.
  2. In this question you must show detailed reasoning. Determine the exact area of the region bounded by the curve \(y = 3 - \sqrt { x }\), the lines \(y = 5 x - 1\) and \(x = 4\) and the \(x\)-axis.
OCR MEI Paper 3 2021 November Q10
10
  1. Express \(\frac { 1 } { ( 4 x + 1 ) ( x + 1 ) }\) in partial fractions.
  2. A curve passes through the point \(( 0,2 )\) and satisfies the differential equation \(\frac { d y } { d x } = \frac { y } { ( 4 x + 1 ) ( x + 1 ) }\),
    for \(x > - \frac { 1 } { 4 }\).
    Show by integration that \(\mathrm { y } = \mathrm { A } \left( \frac { 4 \mathrm { x } + 1 } { \mathrm { x } + 1 } \right) ^ { \mathrm { B } }\) where \(A\) and \(B\) are constants to be determined.
OCR MEI Paper 3 2021 November Q12
12 Show that \(\beta = \arctan \left( \frac { 1 } { 3 } \right)\), as given in line 15 .
OCR MEI Paper 3 2021 November Q13
13
  1. Use triangle ABE in Fig. C 2 to show that \(\arctan x + \arctan \left( \frac { 1 } { x } \right) = \frac { \pi } { 2 }\), as given in line 29 .
  2. Sketch the graph of \(\mathrm { y } = \arctan \mathrm { x }\).
  3. What property of the arctan function ensures that \(\mathrm { y } > \frac { 1 } { \mathrm { x } } \Rightarrow \arctan y > \arctan \left( \frac { 1 } { \mathrm { x } } \right)\), as given in line 30 ?
OCR MEI Paper 3 2021 November Q14
14
  1. Show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) \Rightarrow \arctan \left( \frac { 1 } { 2 } \right) + \arctan \left( \frac { 1 } { 3 } \right) = \arctan 1 .$$
  2. Use the arctan addition formula in line 23 to show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) , \text { as given in line } 39 .$$
OCR MEI Paper 3 2021 November Q15
15 Prove that \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\), as given in line 41 . \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}
OCR MEI Paper 3 Specimen Q1
1 Express \(\frac { 2 } { x - 1 } + \frac { 5 } { 2 x + 1 }\) as a single fraction.
OCR MEI Paper 3 Specimen Q2
2 Find the first four terms of the binomial expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.
OCR MEI Paper 3 Specimen Q3
3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.
OCR MEI Paper 3 Specimen Q4
4 Show that \(\sum _ { r = 1 } ^ { 4 } \ln \frac { r } { r + 1 } = - \ln 5\).
OCR MEI Paper 3 Specimen Q6
6 Fig. 6 shows the curve with equation \(y = x ^ { 4 } - 6 x ^ { 2 } + 4 x + 5\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-06_869_750_370_242} \captionsetup{labelformat=empty} \caption{Fig. 6
Find the coordinates of the points of inflection.}
\end{figure}
OCR MEI Paper 3 Specimen Q7
7 By finding a counter example, disprove the following statement. If \(p\) and \(q\) are non-zero real numbers with \(p < q\), then \(\frac { 1 } { p } > \frac { 1 } { q }\).
OCR MEI Paper 3 Specimen Q8
8 In Fig. 8, OAB is a thin bent rod, with \(\mathrm { OA } = 1 \mathrm {~m} , \mathrm { AB } = 2 \mathrm {~m}\) and angle \(\mathrm { OAB } = 120 ^ { \circ }\). Angles \(\theta , \phi\) and \(h\) are as shown in Fig. 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-07_949_949_429_214} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that \(h = \sin \theta + 2 \sin \left( \theta + 60 ^ { \circ } \right)\). The rod is free to rotate about the origin so that \(\theta\) and \(\phi\) vary. You may assume that the result for \(h\) in part (a) holds for all values of \(\theta\).
  2. Find an angle \(\theta\) for which \(h = 0\).
OCR MEI Paper 3 Specimen Q9
9
  1. Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\) and \(R\) is positive and given in exact form. The function \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = \frac { 1 } { ( k + \cos \theta + 2 \sin \theta ) } , 0 \leq \theta \leq 2 \pi , k\) is a constant.
  2. The maximum value of \(\mathrm { f } ( \theta )\) is \(\frac { ( 3 + \sqrt { 5 } ) } { 4 }\). Find the value of \(k\).
OCR MEI Paper 3 Specimen Q10
10 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 4 x - 2\).
  1. Show that \(x = - 1\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Show that another root of \(\mathrm { f } ( x ) = 0\) lies between \(x = 1\) and \(x = 2\).
  3. Show that \(\mathrm { f } ( x ) = ( x + 1 ) \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } + a x + b\) and \(a\) and \(b\) are integers to be determined.
  4. Without further calculation, explain why \(\mathrm { g } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
  5. Use the Newton-Raphson formula to show that an iteration formula for finding roots of \(\mathrm { g } ( x ) = 0\) may be written $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$ Determine the root of \(\mathrm { g } ( x ) = 0\) which lies between \(x = 1\) and \(x = 2\) correct to 4 significant figures.
OCR MEI Paper 3 Specimen Q11
11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).
    1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
    2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
  2. Would the result still hold with a larger domain? Give reasons for your answer.
OCR MEI Paper 3 Specimen Q12
12 Explain why the smaller regular hexagon in Fig. C1 has perimeter 6.
OCR MEI Paper 3 Specimen Q13
13 Show that the larger regular hexagon in Fig. C1 has perimeter \(4 \sqrt { 3 }\).
OCR MEI Paper 3 Specimen Q14
14 Show that the two values of \(b\) given on line 36 are equivalent.
OCR MEI Paper 3 Specimen Q15
15 Fig. 15 shows a unit circle and the escribed regular polygon with 12 edges. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-11_839_876_356_269} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure}
  1. Show that the perimeter of the polygon is \(24 \tan 15 ^ { \circ }\).
  2. Using the formula for \(\tan ( \theta - \phi )\) show that the perimeter of the polygon is \(48 - 24 \sqrt { 3 }\).