Questions — OCR C2 (296 questions)

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OCR C2 2010 January Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_646_839_255_653} The diagram shows parts of the curves \(y = x ^ { 2 } + 1\) and \(y = 11 - \frac { 9 } { x ^ { 2 } }\), which intersect at \(( 1,2 )\) and \(( 3,10 )\). Use integration to find the exact area of the shaded region enclosed between the two curves.
OCR C2 2010 January Q6
6 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + a x ^ { 2 } + b x + 15$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 3\) ) is a factor of \(\mathrm { f } ( x )\) and that, when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ), the remainder is 35 .
  1. Find the values of \(a\) and \(b\).
  2. Using these values of \(a\) and \(b\), divide \(\mathrm { f } ( x )\) by ( \(x + 3\) ).
OCR C2 2010 January Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_469_673_1720_737} The diagram shows triangle \(A B C\), with \(A B = 10 \mathrm {~cm} , B C = 13 \mathrm {~cm}\) and \(C A = 14 \mathrm {~cm} . E\) and \(F\) are points on \(A B\) and \(A C\) respectively such that \(A E = A F = 4 \mathrm {~cm}\). The sector \(A E F\) of a circle with centre \(A\) is removed to leave the shaded region \(E B C F\).
  1. Show that angle \(C A B\) is 1.10 radians, correct to 3 significant figures.
  2. Find the perimeter of the shaded region \(E B C F\).
  3. Find the area of the shaded region \(E B C F\).
OCR C2 2010 January Q8
8 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 8 \quad \text { and } \quad u _ { n + 1 } = u _ { n } + 3 .$$
  1. Show that \(u _ { 5 } = 20\).
  2. The \(n\)th term of the sequence can be written in the form \(u _ { n } = p n + q\). State the values of \(p\) and \(q\).
  3. State what type of sequence it is.
  4. Find the value of \(N\) such that \(\sum _ { n = 1 } ^ { 2 N } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } = 1256\).
OCR C2 2010 January Q9
9
  1. Sketch the curve \(y = 6 \times 5 ^ { x }\), stating the coordinates of any points of intersection with the axes.
  2. The point \(P\) on the curve \(y = 9 ^ { x }\) has \(y\)-coordinate equal to 150 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
  3. The curves \(y = 6 \times 5 ^ { x }\) and \(y = 9 ^ { x }\) intersect at the point \(Q\). Show that the \(x\)-coordinate of \(Q\) can be written as \(x = \frac { 1 + \log _ { 3 } 2 } { 2 - \log _ { 3 } 5 }\).
OCR C2 2011 January Q1
1
  1. Find and simplify the first three terms, in ascending powers of \(x\), in the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 - 5 x ) ( 1 + 2 x ) ^ { 7 }\).
OCR C2 2011 January Q2
2 A sequence \(S\) has terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) defined by \(u _ { n } = 3 n + 2\) for \(n \geqslant 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. State what type of sequence \(S\) is.
  3. Find \(\sum _ { n = 101 } ^ { 200 } u _ { n }\).
OCR C2 2011 January Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-02_510_791_991_678} The diagram shows the curve \(y = \sqrt { x - 3 }\).
  1. Use the trapezium rule, with 4 strips each of width 0.5 , to find an approximate value for the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 5\). Give your answer correct to 3 significant figures.
  2. State, with a reason, whether this approximation is an underestimate or an overestimate.
OCR C2 2011 January Q4
4
  1. Use logarithms to solve the equation \(5 ^ { x - 1 } = 120\), giving your answer correct to 3 significant figures.
  2. Solve the equation \(\log _ { 2 } x + 2 \log _ { 2 } 3 = \log _ { 2 } ( x + 5 )\).
OCR C2 2011 January Q5
5 In a geometric progression, the sum to infinity is four times the first term.
  1. Show that the common ratio is \(\frac { 3 } { 4 }\).
  2. Given that the third term is 9 , find the first term.
  3. Find the sum of the first twenty terms.
OCR C2 2011 January Q6
6
  1. Find \(\int \frac { x ^ { 3 } + 3 x ^ { \frac { 1 } { 2 } } } { x } \mathrm {~d} x\).
    1. Find, in terms of \(a\), the value of \(\int _ { 2 } ^ { a } 6 x ^ { - 4 } \mathrm {~d} x\), where \(a\) is a constant greater than 2 .
    2. Deduce the value of \(\int _ { 2 } ^ { \infty } 6 x ^ { - 4 } \mathrm {~d} x\).
OCR C2 2011 January Q7
7 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  1. \(3 \tan 2 x = 1\)
  2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)
OCR C2 2011 January Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-03_420_729_1027_708} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 5 cm . Angle \(A O B\) is \(\theta\) radians. The area of triangle \(A O B\) is \(8 \mathrm {~cm} ^ { 2 }\).
  1. Given that the angle \(\theta\) is obtuse, find \(\theta\). The shaded segment in the diagram is bounded by the chord \(A B\) and the arc \(A B\).
  2. Find the area of the segment, giving your answer correct to 3 significant figures.
  3. Find the perimeter of the segment, giving your answer correct to 3 significant figures.
OCR C2 2011 January Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{c52fe7e9-0442-4b3e-b924-2e5e4b3e98f5-04_584_785_255_680} The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = - 4 x ^ { 3 } + 9 x ^ { 2 } + 10 x - 3\).
  1. Verify that the curve crosses the \(x\)-axis at ( 3,0 ) and hence state a factor of \(\mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  3. Hence find the other two points of intersection of the curve with the \(x\)-axis.
  4. The region enclosed by the curve and the \(x\)-axis is shaded in the diagram. Use integration to find the total area of this region.
OCR C2 2012 January Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_319_454_246_810} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 12 cm . The reflex angle \(A O B\) is 4.2 radians.
  1. Find the perimeter of the sector.
  2. Find the area of the sector.
OCR C2 2012 January Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_536_917_1016_577} The diagram shows the curve \(y = \log _ { 10 } ( 2 x + 1 )\).
  1. Use the trapezium rule with 4 strips each of width 1.5 to find an approximation to the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 4\) and \(x = 10\). Give your answer correct to 3 significant figures.
  2. Explain why this approximation is an under-estimate.
OCR C2 2012 January Q3
3 One of the terms in the binomial expansion of \(( 4 + a x ) ^ { 6 }\) is \(160 x ^ { 3 }\).
  1. Find the value of \(a\).
  2. Using this value of \(a\), find the first two terms in the expansion of \(( 4 + a x ) ^ { 6 }\) in ascending powers of \(x\).
OCR C2 2012 January Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-3_622_513_244_776} The diagram shows two points \(A\) and \(B\) on a straight coastline, with \(A\) being 2.4 km due north of \(B\). A stationary ship is at point \(C\), on a bearing of \(040 ^ { \circ }\) and at a distance of 2 km from \(B\).
  1. Find the distance \(A C\), giving your answer correct to 3 significant figures.
  2. Find the bearing of \(C\) from \(A\).
  3. Find the shortest distance from the ship to the coastline.
OCR C2 2012 January Q5
5 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\).
  2. Given that \(\mathrm { f } ( 2 ) = 0\), express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
  3. Determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.
OCR C2 2012 January Q6
6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
  3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
  4. Find the sum to infinity of the geometric progression in part (iii).
OCR C2 2012 January Q7
7
  1. Find \(\int \left( x ^ { 2 } + 4 \right) ( x - 6 ) \mathrm { d } x\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-4_449_551_349_758} The diagram shows the curve \(y = 6 x ^ { \frac { 3 } { 2 } }\) and part of the curve \(y = \frac { 8 } { x ^ { 2 } } - 2\), which intersect at the point \(( 1,6 )\). Use integration to find the area of the shaded region enclosed by the two curves and the \(x\)-axis.
OCR C2 2012 January Q8
8
  1. Use logarithms to solve the equation \(7 ^ { w - 3 } - 4 = 180\), giving your answer correct to 3 significant figures.
  2. Solve the simultaneous equations $$\log _ { 10 } x + \log _ { 10 } y = \log _ { 10 } 3 , \quad \log _ { 10 } ( 3 x + y ) = 1$$
OCR C2 2012 January Q9
9
  1. Sketch the graph of \(y = \tan \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\) on the axes provided.
    On the same axes, sketch the graph of \(y = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\), indicating the point of intersection with the \(y\)-axis.
  2. Show that the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) can be expressed in the form $$3 \sin ^ { 2 } \left( \frac { 1 } { 2 } x \right) + \sin \left( \frac { 1 } { 2 } x \right) - 3 = 0$$ Hence solve the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).
OCR C2 2013 January Q1
1 The diagram shows triangle \(A B C\), with \(A C = 14 \mathrm {~cm} , B C = 10 \mathrm {~cm}\) and angle \(A B C = 63 ^ { \circ }\).
  1. Find angle \(C A B\).
  2. Find the length of \(A B\).
OCR C2 2013 January Q2
2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 7 \text { and } u _ { n + 1 } = u _ { n } + 4 \text { for } n \geqslant 1 .$$
  1. Show that \(u _ { 17 } = 71\).
  2. Show that \(\sum _ { n = 1 } ^ { 35 } u _ { n } = \sum _ { n = 36 } ^ { 50 } u _ { n }\).