OCR C3 (Core Mathematics 3)

1 The function f is defined for all real values of \(x\) by $$f ( x ) = 10 - ( x + 3 ) ^ { 2 }$$

1. State the range of f .
2. Find the value of \(\mathrm { ff } ( - 1 )\).

3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t }$$

1. Find the value of \(t\) for which the mass is 25 grams.
2. Find the rate at which the mass is decreasing when \(t = 55\).

4

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-02_586_793_1274_717} The diagram shows the curve \(y = \frac { 2 } { \sqrt { } x }\). The region \(R\), shaded in the diagram, is bounded by the curve and by the lines \(x = 1 , x = 5\) and \(y = 0\). The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed.
2. Use Simpson's rule, with 4 strips, to find an approximate value for $$\int _ { 1 } ^ { 5 } \sqrt { } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer correct to 3 decimal places.

5

1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac { 7 } { 2 }\), giving all solutions for which \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). \section*{June 2005}

8
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-03_579_901_959_623} The diagram shows part of each of the curves \(y = e ^ { \frac { 1 } { 5 } x }\) and \(y = \sqrt [ 3 ] { } ( 3 x + 8 )\). The curves meet, as shown in the diagram, at the point \(P\). The region \(R\), shaded in the diagram, is bounded by the two curves and by the \(y\)-axis.

1. Show by calculation that the \(x\)-coordinate of \(P\) lies between 5.2 and 5.3.
2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 5 } { 3 } \ln ( 3 x + 8 )\).
3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(P\) correct to 2 decimal places.
4. Use integration, and your answer to part (iii), to find an approximate value of the area of the region \(R\).

9
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-04_629_647_262_749} The function f is defined by \(\mathrm { f } ( x ) = \sqrt { } ( m x + 7 ) - 4\), where \(x \geqslant - \frac { 7 } { m }\) and \(m\) is a positive constant. The diagram shows the curve \(y = \mathrm { f } ( x )\).

1. A sequence of transformations maps the curve \(y = \sqrt { } x\) to the curve \(y = \mathrm { f } ( x )\). Give details of these transformations.
2. Explain how you can tell that f is a one-one function and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. It is given that the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) do not meet. Explain how it can be deduced that neither curve meets the line \(y = x\), and hence determine the set of possible values of \(m\). [5] Jan 2006
   1 Show that \(\int _ { 2 } ^ { 8 } \frac { 3 } { x } \mathrm {~d} x = \ln 64\). 2 Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(\sec ^ { 2 } \theta = 4 \tan \theta - 2\). 3 (a) Differentiate \(x ^ { 2 } ( x + 1 ) ^ { 6 }\) with respect to \(x\).
   (b) Find the gradient of the curve \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 3 }\) at the point where \(x = 1\). 4
   \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-05_531_737_884_705} The function f is defined by \(\mathrm { f } ( x ) = 2 - \sqrt { x }\) for \(x \geqslant 0\). The graph of \(y = \mathrm { f } ( x )\) is shown above.
4. State the range of f .
5. Find the value of \(\mathrm { ff } ( 4 )\).
6. Given that the equation \(| \mathrm { f } ( x ) | = k\) has two distinct roots, determine the possible values of the constant \(k\). 5
   \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-05_490_750_1966_701} The diagram shows the curves \(y = ( 1 - 2 x ) ^ { 5 }\) and \(y = \mathrm { e } ^ { 2 x - 1 } - 1\). The curves meet at the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve. 6 (a)

   \(t\)	0	10	20
   \(X\)	275	440

   The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\).
   (b) The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80 \mathrm { e } ^ { - 0.02 t } .$$
7. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures.
8. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures. 7
   \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-06_461_737_1123_705} The diagram shows the curve with equation \(y = \cos ^ { - 1 } x\).
9. Sketch the curve with equation \(y = 3 \cos ^ { - 1 } ( x - 1 )\), showing the coordinates of the points where the curve meets the axes.
10. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) has exactly one root.
11. Show by calculation that the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) lies between 1.8 and 1.9.
12. The sequence defined by $$x _ { 1 } = 2 , \quad x _ { n + 1 } = 1 + \cos \left( \frac { 1 } { 3 } x _ { n } \right)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\). \section*{[Questions 8 and 9 are printed overleaf.]} 8
    \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-07_790_748_264_699} The diagram shows part of the curve \(y = \ln \left( 5 - x ^ { 2 } \right)\) which meets the \(x\)-axis at the point \(P\) with coordinates ( 2,0 ). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(P Q\) and \(x = 0\).
13. Find the equation of the tangent to the curve at \(P\).
14. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures.
15. Deduce an approximation to the area of the region \(B\). 9
16. By first writing \(\sin 3 \theta\) as \(\sin ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
17. Determine the greatest possible value of $$9 \sin \left( \frac { 10 } { 3 } \alpha \right) - 12 \sin ^ { 3 } \left( \frac { 10 } { 3 } \alpha \right)$$ and find the smallest positive value of \(\alpha\) (in degrees) for which that greatest value occurs.
18. Solve, for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\), the equation \(3 \sin 6 \beta \operatorname { cosec } 2 \beta = 4\). \section*{June 2006} 1 Find the equation of the tangent to the curve \(y = \sqrt { 4 x + 1 }\) at the point ( 2,3 ). 2 Solve the inequality \(| 2 x - 3 | < | x + 1 |\). 3 The equation \(2 x ^ { 3 } + 4 x - 35 = 0\) has one real root.
19. Show by calculation that this real root lies between 2 and 3 .
20. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { 17.5 - 2 x _ { n } }$$ with a suitable starting value, to find the real root of the equation \(2 x ^ { 3 } + 4 x - 35 = 0\) correct to 2 decimal places. You should show the result of each iteration. 4 It is given that \(y = 5 ^ { x - 1 }\).
21. Show that \(x = 1 + \frac { \ln y } { \ln 5 }\).
22. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
23. Hence find the exact value of the gradient of the curve \(y = 5 ^ { x - 1 }\) at the point (3, 25). 5
24. Write down the identity expressing \(\sin 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
25. Given that \(\sin \alpha = \frac { 1 } { 4 }\) and \(\alpha\) is acute, show that \(\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }\).
26. Solve, for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\), the equation \(5 \sin 2 \beta \sec \beta = 3\). \section*{June 2006} 6
    \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-09_570_591_264_776} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
27. Evaluate \(\mathrm { ff } ( - 3 )\).
28. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
29. Sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\). Indicate the coordinates of the points where the graph meets the axes. 7 (a) Find the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 } { ( 4 x - 1 ) ^ { 2 } } \mathrm {~d} x\).
    (b)
    \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-09_570_761_1676_731} The diagram shows part of the curve \(y = \frac { 1 } { x }\). The point \(P\) has coordinates \(\left( a , \frac { 1 } { a } \right)\) and the point \(Q\) has coordinates \(\left( 2 a , \frac { 1 } { 2 a } \right)\), where \(a\) is a positive constant. The point \(R\) is such that \(P R\) is parallel to the \(x\)-axis and \(Q R\) is parallel to the \(y\)-axis. The region shaded in the diagram is bounded by the curve and by the lines \(P R\) and \(Q R\). Show that the area of this shaded region is \(\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\).
30. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
31. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\).
32. Solve, for \(0 ^ { \circ } < x < 360 ^ { \circ }\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1 ^ { \circ }\). 9
    \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-10_565_725_671_712} The diagram shows the curve with equation \(y = 2 \ln ( x - 1 )\). The point \(P\) has coordinates ( \(0 , p\) ). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0 , y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(\boldsymbol { y }\)-axis to form a solid.
33. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the solid is given by $$V = \pi \left( \mathrm { e } ^ { p } + 4 \mathrm { e } ^ { \frac { 1 } { 2 } p } + p - 5 \right) .$$
34. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \mathrm {~cm} \mathrm {~min} ^ { - 1 }\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures. 1 Find the equation of the tangent to the curve \(y = \frac { 2 x + 1 } { 3 x - 1 }\) at the point \(\left( 1 , \frac { 3 } { 2 } \right)\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. 2 It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac { 12 } { 13 }\). Find the exact value of
35. \(\cot \theta\),
36. \(\cos 2 \theta\). 3 (a) It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$ on the same diagram, show that the equation $$x ^ { 5 } + b x - a = 0$$ has exactly one real root.
    (b) Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }\), with a suitable starting value, to find the real root of the equation \(x ^ { 5 } + 2 x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places. 4
37. Given that \(x = ( 4 t + 9 ) ^ { \frac { 1 } { 2 } }\) and \(y = 6 \mathrm { e } ^ { \frac { 1 } { 2 } x + 1 }\), find expressions for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
38. Hence find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) when \(t = 4\), giving your answer correct to 3 significant figures. 5
39. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
40. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\). \section*{Jan 2007} 6
    \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-12_485_960_262_589} The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 2 } }\). The shaded region is bounded by the curve and the lines \(x = 0 , x = 2\) and \(y = 0\).
41. Find the exact area of the shaded region.
42. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid formed, simplifying your answer. 7 The curve \(y = \ln x\) is transformed to the curve \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.
43. Give full details of the translation and stretch involved.
44. Sketch the graph of \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\).
45. Sketch, on another diagram, the graph of \(y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|\).
46. State, in terms of \(a\), the set of values of \(x\) for which \(\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)\). 8
    \includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-13_528_1435_267_354} The diagram shows the curve with equation \(y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve and the line \(P Q\).
47. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2 .
48. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places.
49. Deduce an approximation to the area of region \(B\). 9 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi ,
    \mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$
50. State the range of f and the range of g .
51. Show that \(\operatorname { gf } ( 0.5 ) = 2.16\), correct to 3 significant figures, and explain why \(\mathrm { fg } ( 0.5 )\) is not defined.
52. Find the set of values of \(x\) for which \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\) is not defined.