Questions C2 (1550 questions)

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OCR C2 2005 January Q6
9 marks Moderate -0.8
6
  1. Find \(\int x \left( x ^ { 2 } + 2 \right) \mathrm { d } x\).
    1. Find \(\int \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).
    2. The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } x }\). Find the equation of the curve, given that it passes through the point \(( 4,0 )\).
OCR C2 2005 January Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-3_563_639_1379_753} The diagram shows an equilateral triangle \(A B C\) with sides of length 12 cm . The mid-point of \(B C\) is \(O\), and a circular arc with centre \(O\) joins \(D\) and \(E\), the mid-points of \(A B\) and \(A C\).
  1. Find the length of the arc \(D E\), and show that the area of the sector \(O D E\) is \(6 \pi \mathrm {~cm} ^ { 2 }\).
  2. Find the exact area of the shaded region.
OCR C2 2005 January Q8
9 marks Standard +0.3
8
  1. On a single diagram, sketch the curves with the following equations. In each case state the coordinates of any points of intersection with the axes.
    1. \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
    2. \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
    3. The curves in part (i) intersect at the point \(P\). Prove that the \(x\)-coordinate of \(P\) is $$\frac { 1 } { \log _ { 2 } a - \log _ { 2 } b } .$$
OCR C2 2005 January Q9
11 marks Standard +0.3
9 A geometric progression has first term \(a\), where \(a \neq 0\), and common ratio \(r\), where \(r \neq 1\). The difference between the fourth term and the first term is equal to four times the difference between the third term and the second term.
  1. Show that \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1 = 0\).
  2. Show that \(r - 1\) is a factor of \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\). Hence factorise \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\).
  3. Hence find the two possible values for the ratio of the geometric progression. Give your answers in an exact form.
  4. For the value of \(r\) for which the progression is convergent, prove that the sum to infinity is \(\frac { 1 } { 2 } a ( 1 + \sqrt { } 5 )\).
OCR C2 2006 January Q1
6 marks Moderate -0.8
1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .
  1. Find the first term and the common difference.
  2. Show that the sum of the first 29 terms is zero.
OCR C2 2006 January Q2
6 marks Moderate -0.8
2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate
  1. the area of the triangle,
  2. the length of \(C A\),
  3. the size of angle \(C\).
OCR C2 2006 January Q3
6 marks Moderate -0.8
3
  1. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$
OCR C2 2006 January Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The angle \(A O B\) is 1.8 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. It is given that \(O A = O B = 20 \mathrm {~cm}\) and \(O C = O D = 15 \mathrm {~cm}\). The shaded region is bounded by the arcs \(A B\) and \(C D\) and by the lines \(C A\) and \(D B\).
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
OCR C2 2006 January Q5
8 marks Standard +0.3
5 In a geometric progression, the first term is 5 and the second term is 4.8 .
  1. Show that the sum to infinity is 125 .
  2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).
OCR C2 2006 January Q6
8 marks Standard +0.3
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
8 marks Moderate -0.5
7
  1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
    1. \(\log _ { 10 } \left( \frac { x } { y } \right)\)
    2. \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
    3. 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
12 marks Moderate -0.3
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 2008 January Q1
4 marks Moderate -0.3
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
4 marks Easy -1.2
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
4 marks Easy -1.8
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
5 marks Moderate -0.3
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
6 marks Easy -1.2
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
8 marks Easy -1.3
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
8 marks Standard +0.3
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
11 marks Standard +0.3
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 marks Moderate -0.8
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 \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
    1. another solution of the equation \(2 \sin x = k\),
    2. 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
12 marks Moderate -0.3
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
6 marks Easy -1.3
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
7 marks Standard +0.3
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
7 marks Moderate -0.8
3
  1. Find \(\int ( 2 x + 1 ) ( x + 3 ) \mathrm { d } x\).
  2. Evaluate \(\int _ { 0 } ^ { 9 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\).