Questions — OCR C2 (296 questions)

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OCR C2 2005 January Q1
1 Simplify \(( 3 + 2 x ) ^ { 3 } - ( 3 - 2 x ) ^ { 3 }\).
OCR C2 2005 January Q2
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = \frac { 1 } { 1 - u _ { n } } \text { for } n \geqslant 1 .$$
  1. Write down the values of \(u _ { 2 } , u _ { 3 } , u _ { 4 }\) and \(u _ { 5 }\).
  2. Deduce the value of \(u _ { 200 }\), showing your reasoning.
OCR C2 2005 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_488_604_895_769} A landmark \(L\) is observed by a surveyor from three points \(A , B\) and \(C\) on a straight horizontal road, where \(A B = B C = 200 \mathrm {~m}\). Angles \(L A B\) and \(L B A\) are \(65 ^ { \circ }\) and \(80 ^ { \circ }\) respectively (see diagram). Calculate
  1. the shortest distance from \(L\) to the road,
  2. the distance \(L C\).
OCR C2 2005 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{608720b6-5b18-45e9-8838-c94b347ab3b7-2_547_511_1813_817} The diagram shows a sketch of parts of the curves \(y = \frac { 16 } { x ^ { 2 } }\) and \(y = 17 - x ^ { 2 }\).
  1. Verify that these curves intersect at the points \(( 1,16 )\) and \(( 4,1 )\).
  2. Calculate the exact area of the shaded region between the curves.
OCR C2 2005 January Q5
5
  1. Prove that the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ can be expressed in the form $$2 \cos ^ { 2 } \theta + \cos \theta - 1 = 0$$
  2. Hence solve the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 2005 January Q6
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
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
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.
    (a) \(y = a ^ { x }\), where \(a\) is a constant such that \(a > 1\).
    (b) \(y = 2 b ^ { x }\), where \(b\) is a constant such that \(0 < b < 1\).
  2. 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
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
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
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
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
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
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
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 \(a\).