Questions C2 (1410 questions)

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OCR C2 2006 January Q6
6
  1. Find \(\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x\).
    1. Find the value, in terms of \(a\), of \(\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x\), where \(a\) is a constant greater than 1 .
    2. Deduce the value of \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\).
OCR C2 2006 January Q7
7
  1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
    (a) \(\log _ { 10 } \left( \frac { x } { y } \right)\)
    (b) \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
  2. Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of \(y\) correct to 3 decimal places.
OCR C2 2006 January Q8
8 The cubic polynomial \(2 x ^ { 3 } + k x ^ { 2 } - x + 6\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  1. Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
  2. Find \(\int _ { - 1 } ^ { 2 } f ( x ) \mathrm { d } x\).
  3. Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 1 \leqslant x \leqslant 2\). \section*{[Question 9 is printed overleaf.]}
OCR C2 2007 January Q1
1 In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms.
OCR C2 2007 January Q2
2 The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 8 cm . The angle \(A O B\) is \(46 ^ { \circ }\).
  1. Express \(46 ^ { \circ }\) in radians, correct to 3 significant figures.
  2. Find the length of the arc \(A B\).
  3. Find the area of the sector \(O A B\).
OCR C2 2007 January Q3
3
  1. Find \(\int ( 4 x - 5 ) \mathrm { d } x\).
  2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x - 5\). The curve passes through the point (3,7). Find the equation of the curve.
OCR C2 2007 January Q4
4 In a triangle \(A B C , A B = 5 \sqrt { 2 } \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and angle \(B = 60 ^ { \circ }\).
  1. Find the exact area of the triangle, giving your answer as simply as possible.
  2. Find the length of \(A C\), correct to 3 significant figures.
OCR C2 2007 January Q5
5
    1. Express \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x\) as a single logarithm.
    2. Hence solve the equation \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x = 2\).
  2. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int _ { 3 } ^ { 9 } \log _ { 10 } x \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.
OCR C2 2007 January Q6
6
  1. Find and simplify the first four terms in the expansion of \(( 1 + 4 x ) ^ { 7 }\) in ascending powers of \(x\).
  2. In the expansion of $$( 3 + a x ) ( 1 + 4 x ) ^ { 7 }$$ the coefficient of \(x ^ { 2 }\) is 1001 . Find the value of \(a\).
  3. (a) Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), indicating the coordinates of any points where the curve meets the axes.
    (b) Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
  4. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(- 180 ^ { \circ }\) and \(180 ^ { \circ }\).
OCR C2 2007 January Q8
8 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 7 x + 33\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
  2. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form as simply as possible. On its first trip between Malby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses \(2 \%\) more coal than the previous trip.
OCR C2 2007 January Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{dd199f4d-8cf3-4b1e-92aa-d54e9e94da57-4_693_931_269_607} The diagram shows the graph of \(y = 1 - 3 x ^ { - \frac { 1 } { 2 } }\).
  1. Verify that the curve intersects the \(x\)-axis at \(( 9,0 )\).
  2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\) ). Given that the area of the shaded region is 4 square units, find the value of \(a\).
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 \(0.1 ^ { \circ }\).