Questions — OCR C2 (296 questions)

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OCR C2 2008 January Q1
1 The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 11 cm . The angle \(A O B\) is 0.7 radians. Find the area of the segment shaded in the diagram.
OCR C2 2008 January Q2
2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of $$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$
OCR C2 2008 January Q3
3 Express each of the following as a single logarithm:
  1. \(\log _ { a } 2 + \log _ { a } 3\),
  2. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).
OCR C2 2008 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-2_515_713_1567_715} In the diagram, angle \(B D C = 50 ^ { \circ }\) and angle \(B C D = 62 ^ { \circ }\). It is given that \(A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}\) and \(B C = 16 \mathrm {~cm}\).
  1. Find the length of \(B D\).
  2. Find angle \(B A D\).
OCR C2 2008 January Q5
5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }\). The curve passes through the point (4,50). Find the equation of the curve.
OCR C2 2008 January Q6
6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. State what type of sequence it is.
  3. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\).
OCR C2 2008 January Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-3_579_557_858_794} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).
  1. Explain why \(\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x\) does not give the total area of the regions shaded in the diagram.
  2. Use integration to find the exact total area of the shaded regions.
OCR C2 2008 January Q8
8 The first term of a geometric progression is 10 and the common ratio is 0.8.
  1. Find the fourth term.
  2. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
  3. The sum of the first \(N\) terms is denoted by \(S _ { N }\), and the sum to infinity is denoted by \(S _ { \infty }\). Show that the inequality \(S _ { \infty } - S _ { N } < 0.01\) can be written as $$0.8 ^ { N } < 0.0002 ,$$ and use logarithms to find the smallest possible value of \(N\).
OCR C2 2008 January Q9
9
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_376_764_276_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the curve \(y = 2 \sin x\) for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). State the coordinates of the maximum and minimum points on this part of the curve.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_371_766_959_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Fig. 2 shows the curve \(y = 2 \sin x\) and the line \(y = k\). The smallest positive solution of the equation \(2 \sin x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\), and in the range \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
    (a) another solution of the equation \(2 \sin x = k\),
    (b) one solution of the equation \(2 \sin x = - k\).
  3. Find the \(x\)-coordinates of the points where the curve \(y = 2 \sin x\) intersects the curve \(y = 2 - 3 \cos ^ { 2 } x\), for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
OCR C2 2008 January Q10
10
  1. Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
  2. Hence show that \(( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }\) can be written as $$320 x ^ { 3 } + k x$$ where the value of the constant \(k\) is to be stated.
  3. Verify that \(x = 2\) is a root of the equation $$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$ and find the other possible values of \(x\).
OCR C2 2005 June Q1
1 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by $$u _ { n } = 3 n - 1 ,$$ for \(n \geqslant 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\), and state what type of sequence \(S\) is.
  2. Evaluate \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).
OCR C2 2005 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-2_579_895_817_625} A sector \(O A B\) of a circle of radius \(r \mathrm {~cm}\) has angle \(\theta\) radians. The length of the arc of the sector is 12 cm and the area of the sector is \(36 \mathrm {~cm} ^ { 2 }\) (see diagram).
  1. Write down two equations involving \(r\) and \(\theta\).
  2. Hence show that \(r = 6\), and state the value of \(\theta\).
  3. Find the area of the segment bounded by the arc \(A B\) and the chord \(A B\).
OCR C2 2005 June Q3
3
  1. Find \(\int ( 2 x + 1 ) ( x + 3 ) \mathrm { d } x\).
  2. Evaluate \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).
OCR C2 2005 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{387a37c4-0997-484c-8e28-954639169ebe-3_309_1084_269_532} In the diagram, \(A B C D\) is a quadrilateral in which \(A D\) is parallel to \(B C\). It is given that \(A B = 9 , B C = 6\), \(C A = 5\) and \(C D = 15\).
  1. Show that \(\cos B C A = - \frac { 1 } { 3 }\), and hence find the value of \(\sin B C A\).
  2. Find the angle \(A D C\) correct to the nearest \(0.1 ^ { \circ }\).
OCR C2 2005 June Q5
5 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and that the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) is 16 .
  1. Find the values of \(a\) and \(b\).
  2. Hence verify that \(\mathrm { f } ( 2 ) = 0\), and factorise \(\mathrm { f } ( x )\) completely.
OCR C2 2005 June Q6
6
  1. Find the binomial expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 }\), simplifying the terms.
  2. Hence find \(\int \left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 3 } \mathrm {~d} x\).
OCR C2 2005 June Q7
7
  1. Evaluate \(\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12\).
  2. Given that \(y = 3 \times 10 ^ { 2 x }\), show that \(x = a \log _ { 10 } ( b y )\), where the values of the constants \(a\) and \(b\) are to be found.
OCR C2 2005 June Q8
8 The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9 . The amount pumped in 2001 was 100000 barrels.
  1. Calculate the amount pumped in 2004. It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.
  2. Calculate in which year the amount pumped will fall below 5000 barrels.
  3. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production.
OCR C2 2005 June Q9
9
    1. Write down the exact values of \(\cos \frac { 1 } { 6 } \pi\) and \(\tan \frac { 1 } { 3 } \pi\) (where the angles are in radians). Hence verify that \(x = \frac { 1 } { 6 } \pi\) is a solution of the equation $$2 \cos x = \tan 2 x$$
    2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2 x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2 x$$
    1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.)
    2. State with a reason whether this approximation is an underestimate or an overestimate.
OCR C2 2006 June Q1
1 Find the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).
OCR C2 2006 June Q2
2 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = 1 - u _ { n } \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Find \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).
OCR C2 2006 June Q3
3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point (4,5). Find the equation of the curve.
OCR C2 2006 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-2_634_670_1123_740} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
  1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
  2. Use integration to find the area of the shaded region bounded by the line and the curve.
OCR C2 2006 June Q5
5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
  2. \(\sin 2 x = - \cos 2 x\).
OCR C2 2006 June Q6
6
  1. John aims to pay a certain amount of money each month into a pension fund. He plans to pay \(\pounds 100\) in the first month, and then to increase the amount paid by \(\pounds 5\) each month, i.e. paying \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
    (a) how much he will pay in the final month,
    (b) how much he will pay altogether over the whole period.
  2. Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with \(\pounds 100\) in the first month. Her monthly payments will form a geometric progression, and she will pay \(\pounds 1500\) in the final month. Calculate how much Rachel will pay altogether over the whole period.