OCR C2 (Core Mathematics 2)

2
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-02_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 \(A B\).

4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-03_314_1089_267_529} 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 \(0.1 ^ { \circ }\).

7

1. Evaluate \(\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12\).
2. Given that \(y = 3 \times 10 ^ { 2 x }\), show that \(x = a \log _ { 10 } (\) by \()\), where the values of the constants \(a\) and \(b\) are to be found.

9

1. 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 $$2 \cos x = \tan 2 x$$
2. 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. 1 The 20th term of an arithmetic progression is 10 and the 50th term is 70.
   3. Find the first term and the common difference.
   4. Show that the sum of the first 29 terms is zero. 2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate
   5. the area of the triangle,
   6. the length of \(C A\),
   7. the size of angle \(C\). 3
   8. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
   9. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 }$$ 4
      \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-05_657_803_1283_671} 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 \(D B\).
   10. Find the perimeter of the shaded region.
   11. Find the area of the shaded region. 5 In a geometric progression, the first term is 5 and the second term is 4.8.
   12. Show that the sum to infinity is 125 .
   13. 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 \(n\). 6
3. Find \(\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x\).
4. 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 \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\). 7
   3. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
5. \(\log _ { 10 } \left( \frac { x } { y } \right)\)
6. \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
   (ii) 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. 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 \(\mathrm { f } ( x )\).
   (i) Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
   (ii) Find \(\int _ { - 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).
   (iii) 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 \(- 1 \leqslant x \leqslant 2\). 9 (i) Sketch, on a single diagram showing values of \(x\) from \(- 180 ^ { \circ }\) to \(+ 180 ^ { \circ }\), the graphs of \(y = \tan x\) and \(y = 4 \cos x\). The equation $$\tan x = 4 \cos x$$ has two roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). These are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
   (ii) Show \(\alpha\) and \(\beta\) on your sketch, and express \(\beta\) in terms of \(\alpha\).
   (iii) Show that the equation \(\tan x = 4 \cos x\) may be written as $$4 \sin ^ { 2 } x + \sin x - 4 = 0$$ Hence find the value of \(\beta - \alpha\), correct to the nearest degree. 1 Find the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\). 2 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = 1 - u _ { n } \text { for } n \geqslant 1 .$$ (i) Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
   (ii) Find \(\sum _ { n = 1 } ^ { 100 } u _ { n }\). 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. 4
   \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-08_636_670_1123_740} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).
   (i) Find the \(x\)-coordinates of the points of intersection of the curve and the line.
   (ii) Use integration to find the area of the shaded region bounded by the line and the curve. 5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
   (i) \(2 \sin ^ { 2 } x = 1 + \cos x\).
   (ii) \(\sin 2 x = - \cos 2 x\). 6 (i) 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
7. how much he will pay in the final month,
8. how much he will pay altogether over the whole period.
   (ii) 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. 7
   \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-09_488_1027_995_559} The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the arc \(D C\).
   (i) Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
   (ii) Find the area of the shaded segment.
   (iii) Find the perimeter of the shaded segment. 8 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 10\) is denoted by \(\mathrm { f } ( x )\). It is given that, when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 12 . It is also given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
   (i) Find the values of \(a\) and \(b\).
   (ii) Divide \(\mathrm { f } ( x )\) by \(( x + 2 )\) to find the quotient and the remainder.
   (i) Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
   (ii) Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
   (iii) The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$ 1 In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms. 2 The diagram shows a sector \(O A B\) of a circle, centre \(O\) and radius 8 cm . The angle \(A O B\) is \(46 ^ { \circ }\).
   (i) Express \(46 ^ { \circ }\) in radians, correct to 3 significant figures.
   (ii) Find the length of the arc \(A B\).
   (iii) Find the area of the sector \(O A B\). 3 (i) Find \(\int ( 4 x - 5 ) \mathrm { d } x\).
   (ii) 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. 4 In a triangle \(A B C , A B = 5 \sqrt { 2 } \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and angle \(B = 60 ^ { \circ }\).
   (i) Find the exact area of the triangle, giving your answer as simply as possible.
   (ii) Find the length of \(A C\), correct to 3 significant figures. 5
9. 1. Express \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x\) as a single logarithm.
   2. Hence solve the equation \(\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x = 2\).
10. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int _ { 3 } ^ { 9 } \log _ { 10 } x d x$$ giving your answer correct to 3 significant figures.
    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.
11. Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
    (ii) Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(- 180 ^ { \circ }\) and \(180 ^ { \circ }\). 8 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 7 x + 33\).
    (i) Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
    (ii) Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
    (iii) 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.
    (i) Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures.
    (ii) There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02 ^ { N } \leqslant 1.52 .$$ (iii) Hence, by using logarithms, find the greatest number of trips possible. \section*{Jan 2007}

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\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-13_696_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 \(a\). June 2007 1 A geometric progression \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 \text {. }$$
3. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
4. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\). 2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms. 3 Use logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures. 4
   \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-14_543_855_1155_646} The diagram shows the curve \(\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }\).
5. Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve \(\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }\), the x -axis, and the lines \(\mathrm { x } = 1\) and \(\mathrm { x } = 3\). Give your answer correct to 3 significant figures.
6. State with a reason whether this approximation is an under-estimate or an over-estimate. 5
7. Show that the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1$$ can be expressed in the form $$3 \sin ^ { 2 } \theta + \sin \theta - 2 = 0 .$$
8. Hence solve the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1 ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). 6 (a) (i) Find \(\int x \left( x ^ { 2 } - 4 \right) d x\)
9. Hence evaluate \(\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x\).
   (b) Find \(\int \frac { 6 } { x ^ { 3 } } d x\) 7 (a) In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
   (b) In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio. 8
   \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-15_305_744_1043_703} The diagram shows a triangle ABC , where angle BAC is 0.9 radians. BAD is a sector of the circle with centre \(A\) and radius \(A B\).
10. The area of the sector \(B A D\) is \(16.2 \mathrm {~cm} ^ { 2 }\). Show that the length of \(A B\) is 6 cm .
11. The area of triangle \(A B C\) is twice the area of sector \(B A D\). Find the length of \(A C\).
12. Find the perimeter of the region \(B C D\). 9 The polynomial \(f ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$
13. (a) Show that \(( x + 1 )\) is a factor of \(f ( x )\).
    (b) Hence find the exact roots of the equation \(f ( x ) = 0\).
14. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( \mathrm { x } ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root. \section*{Jan 2008} 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. 2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of $$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$ 3 Express each of the following as a single logarithm:
15. \(\log _ { a } 2 + \log _ { a } 3\),
16. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\). 4
    \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-16_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 \(B C = 16 \mathrm {~cm}\).
17. Find the length of \(B D\).
18. Find angle \(B A D\). 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. 6 A sequence of terms \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1$$
19. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
20. State what type of sequence it is.
21. Given that \(\sum _ { n = 1 } ^ { N } u _ { n } = 2200\), find the value of \(N\). 7
    \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-17_588_569_854_788} The diagram shows part of the curve \(y = x ^ { 2 } - 3 x\) and the line \(x = 5\).
22. 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.
23. Use integration to find the exact total area of the shaded regions. 8 The first term of a geometric progression is 10 and the common ratio is 0.8.
24. Find the fourth term.
25. Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
26. 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 \(N\). 9
27. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_770_274_731} \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.
28. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_771_954_730} \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 \(2 \sin x = - k\).
29. 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 \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). 10
30. Find the binomial expansion of \(( 2 x + 5 ) ^ { 4 }\), simplifying the terms.
31. 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.
32. 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 \(x\).