Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 2007 January Q2
2 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.
OCR MEI C2 2007 January Q3
3 Given that \(\cos \theta = \frac { 1 } { 3 }\) and \(\theta\) is acute, find the exact value of \(\tan \theta\).
OCR MEI C2 2007 January Q4
4 Sequences \(\mathrm { A } , \mathrm { B }\) and C are shown below. They each continue in the pattern established by the given terms. $$\begin{array} { l l l l l l l l l } \text { A: } & 1 , & 2 , & 4 , & 16 , & 32 , & \ldots &
\text { B: } & 20 , & - 10 , & 5 , & - 2.5 , & 1.25 , & - 0.625 , & \ldots
\text { C: } & 20 , & 5 , & 1 , & 20 , & 5 , & 1 , & \ldots \end{array}$$
  1. Which of these sequences is periodic?
  2. Which of these sequences is convergent?
  3. Find, in terms of \(n\), the \(n\)th term of sequence A .
OCR MEI C2 2007 January Q5
5 A is the point \(( 2,1 )\) on the curve \(y = \frac { 4 } { x ^ { 2 } }\).
B is the point on the same curve with \(x\)-coordinate 2.1.
  1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places.
  2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A .
  3. Use calculus to find the gradient of the curve at A .
OCR MEI C2 2007 January Q6
6 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI C2 2007 January Q7
7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\).
OCR MEI C2 2007 January Q8
8 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.
OCR MEI C2 2007 January Q9
9 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 8 x\). The curve passes through the point \(( 1,5 )\). Find the equation of the curve.
OCR MEI C2 2007 January Q10
10
  1. Express \(\log _ { a } x ^ { 4 } + \log _ { a } \left( \frac { 1 } { x } \right)\) as a multiple of \(\log _ { a } x\).
  2. Given that \(\log _ { 10 } b + \log _ { 10 } c = 3\), find \(b\) in terms of \(c\).
OCR MEI C2 2007 January Q11
11 Fig. 11.1 shows a village green which is bordered by 3 straight roads \(A B , B C\) and \(C A\). The road AC runs due North and the measurements shown are in metres. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_460_1143_486_591} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
\end{figure}
  1. Calculate the bearing of B from C , giving your answer to the nearest \(0.1 ^ { \circ }\).
  2. Calculate the area of the village green. The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-4_440_1002_1436_737} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} The new road is an arc of a circle with centre O and radius 130 m .
  3. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures.
    (B) Show that the area of land added to the village green is \(5300 \mathrm {~m} ^ { 2 }\) correct to 2 significant figures.
OCR MEI C2 2007 January Q12
12 Fig. 12 is a sketch of the curve \(y = 2 x ^ { 2 } - 11 x + 12\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4c0b4b0-f13c-49a9-9f98-f86f28d1f577-5_478_951_333_792} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure}
  1. Show that the curve intersects the \(x\)-axis at \(( 4,0 )\) and find the coordinates of the other point of intersection of the curve and the \(x\)-axis.
  2. Find the equation of the normal to the curve at the point \(( 4,0 )\). Show also that the area of the triangle bounded by this normal and the axes is 1.6 units \(^ { 2 }\).
  3. Find the area of the region bounded by the curve and the \(x\)-axis.
OCR MEI C2 2008 January Q1
1 Differentiate \(10 x ^ { 4 } + 12\).
OCR MEI C2 2008 January Q2
2 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 48th term of this sequence.
  2. Find the sum of the first 48 terms of this sequence.
OCR MEI C2 2008 January Q3
3 You are given that \(\tan \theta = \frac { 1 } { 2 }\) and the angle \(\theta\) is acute. Show, without using a calculator, that \(\cos ^ { 2 } \theta = \frac { 4 } { 5 }\).
OCR MEI C2 2008 January Q4
4 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-2_625_869_1155_639} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { A } , \mathrm { B }\) and C .
  1. \(y = 2 \mathrm { f } ( x )\)
  2. \(y = \mathrm { f } ( x + 3 )\)
OCR MEI C2 2008 January Q5
5 Find \(\int \left( 12 x ^ { 5 } + \sqrt [ 3 ] { x } + 7 \right) \mathrm { d } x\).
OCR MEI C2 2008 January Q6
6
  1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\). Give your answers in the form \(k \pi\).
OCR MEI C2 2008 January Q7
7
  1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
  2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
  3. Sketch the curve with equation \(y = 2 ^ { x }\).
OCR MEI C2 2008 January Q8
8 The second term of a geometric progression is 18 and the fourth term is 2 . The common ratio is positive. Find the sum to infinity of this progression.
OCR MEI C2 2008 January Q9
9 You are given that \(\log _ { 10 } y = 3 x + 2\).
  1. Find the value of \(x\) when \(y = 500\), giving your answer correct to 2 decimal places.
  2. Find the value of \(y\) when \(x = - 1\).
  3. Express \(\log _ { 10 } \left( y ^ { 4 } \right)\) in terms of \(x\).
  4. Find an expression for \(y\) in terms of \(x\). Section B (36 marks)
OCR MEI C2 2008 January Q10
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-3_501_493_1434_826} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a solid cuboid with square base of side \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\). Its volume is \(120 \mathrm {~cm} ^ { 3 }\).
  1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the cuboid is given by \(A = 2 x ^ { 2 } + \frac { 480 } { x }\).
  2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} x ^ { 2 } }\).
  3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case.
OCR MEI C2 2008 January Q11
11
  1. The course for a yacht race is a triangle, as shown in Fig. 11.1. The yachts start at A , then travel to B , then to C and finally back to A . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_661_869_404_680} \captionsetup{labelformat=empty} \caption{Fig. 11.1}
    \end{figure} (A) Calculate the total length of the course for this race.
    (B) Given that the bearing of the first stage, AB , is \(175 ^ { \circ }\), calculate the bearing of the second stage, BC.
  2. Fig. 11.2 shows the course of another yacht race. The course follows the arc of a circle from P to \(Q\), then a straight line back to \(P\). The circle has radius 120 m and centre \(O\); angle \(P O Q = 136 ^ { \circ }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-4_709_821_1603_703} \captionsetup{labelformat=empty} \caption{Fig. 11.2}
    \end{figure} Calculate the total length of the course for this race.
OCR MEI C2 2008 January Q12
12
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-5_652_764_269_733} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  2. You are given that \(\mathrm { f } ( x ) = x ^ { 4 }\).
    (A) Complete this identity for \(\mathrm { f } ( x + h )\). $$f ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$ (B) Simplify \(\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (C) Find \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\).
    (D) State what this limit represents.
OCR MEI C2 2005 June Q1
1 Differentiate \(x + \sqrt { x ^ { 3 } }\).
OCR MEI C2 2005 June Q2
2 The \(n\)th term of an arithmetic progression is \(6 + 5 n\). Find the sum of the first 20 terms.