Equation with preliminary work

6

1. Given that \(x > 0\), find the two smallest values of \(x\), in radians, for which \(3 \tan ( 2 x + 1 ) = 1\). Show all necessary working.
2. The function f : \(x \mapsto 3 \cos ^ { 2 } x - 2 \sin ^ { 2 } x\) is defined for \(0 \leqslant x \leqslant \pi\).
   1. Express \(\mathrm { f } ( x )\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
   2. Find the range of \(f\).

9. Solutions based entirely on graphical or numerical methods are not acceptable in this question.

1. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$3 \sin \left( 2 \theta - 10 ^ { \circ } \right) = 1$$ giving your answers to one decimal place.
2. The first three terms of an arithmetic sequence are $$\sin \alpha , \frac { 1 } { \tan \alpha } \text { and } 2 \sin \alpha$$ where \(\alpha\) is a constant.
   (a) Show that \(2 \cos \alpha = 3 \sin ^ { 2 } \alpha\) Given that \(\pi < \alpha < 2 \pi\),
   (b) find, showing all working, the value of \(\alpha\) to 3 decimal places.

8

1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

7

1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

1

1. Starting with an equilateral triangle, prove that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).

4

1. Starting with an equilateral triangle, prove that \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ee79022b-b9a6-4076-8db7-67b9788ac28a-3_1032_1113_264_466} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} Fig. 5 shows the graph of \(y = 2 ^ { x }\).

7

1. Show that, when \(x\) is an acute angle, \(\tan x \sqrt { 1 - \sin ^ { 2 } x } = \sin x\).
2. Solve \(4 \sin ^ { 2 } y = \sin y\) for \(0 ^ { \circ } \leqslant y \leqslant 360 ^ { \circ }\).

1. In this question you should show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Solve, for \(0 < \theta \leqslant 450 ^ { \circ }\), the equation $$5 \cos ^ { 2 } \theta = 6 \sin \theta$$ giving your answers to one decimal place.
2. (a) A student's attempt to solve the question
   "Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(3 \tan x - 5 \sin x = 0\) " is set out below. $$\begin{gathered} 3 \tan x - 5 \sin x = 0 \\ 3 \frac { \sin x } { \cos x } - 5 \sin x = 0 \\ 3 \sin x - 5 \sin x \cos x = 0 \\ 3 - 5 \cos x = 0 \\ \cos x = \frac { 3 } { 5 } \\ x = 53.1 ^ { \circ } \end{gathered}$$ Identify two errors or omissions made by this student, giving a brief explanation of each. The first four positive solutions, in order of size, of the equation $$\cos \left( 5 \alpha + 40 ^ { \circ } \right) = \frac { 3 } { 5 }$$ are \(\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }\) and \(\alpha _ { 4 }\) (b) Find, to the nearest degree, the value of \(\alpha _ { 4 }\)

1. (i) Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation,

$$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
(ii) (a) A student's attempt at the question
"Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x \\ 7 \times \frac { \sin x } { \cos x } = 8 \sin x \\ 7 \sin x = 8 \sin x \cos x \\ 7 = 8 \cos x \\ \cos x = \frac { 7 } { 8 } \\ x = 29.0 ^ { \circ } \text { (to } 3 \text { sf) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
(b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$

3 In this question you must show detailed reasoning.

1. Solve the equation \(4 \sin ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
2. Prove that \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).

4

1. On the axes in the Printed Answer Booklet, sketch the graph of \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve the equation \(\sin 2 \theta = - \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\). \(5 M\) is the event that an A-level student selected at random studies mathematics. \(C\) is the event that an A-level student selected at random studies chemistry.
   You are given that \(\mathrm { P } ( M ) = 0.42 , \mathrm { P } ( C ) = 0.36\) and \(\mathrm { P } ( \mathrm { M }\) and \(\mathrm { C } ) = 0.24\). These probabilities are shown in the two-way table below.

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(M\)	\(M ^ { \prime }\)	Total
   \(C\)	0.24		0.36
   \(C ^ { \prime }\)
   Total	0.42		1

6

1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
   1. \(y = 2 \sin x\);
   2. \(y = - \sin x\);
   3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

6

1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
   1. \(y = 2 \sin x\);
   2. \(y = - \sin x\);
   3. \(y = \sin \left( x - 30 ^ { \circ } \right)\).
2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

8

1. Describe the single geometrical transformation by which the curve with equation \(y = \tan \frac { 1 } { 2 } x\) can be obtained from the curve \(y = \tan x\).
2. Solve the equation \(\tan \frac { 1 } { 2 } x = 3\) in the interval \(\mathbf { 0 } < \boldsymbol { x } < \mathbf { 4 } \boldsymbol { \pi }\), giving your answers in radians to three significant figures.
3. Solve the equation $$\cos \theta ( \sin \theta - 3 \cos \theta ) = 0$$ in the interval \(0 < \theta < 2 \pi\), giving your answers in radians to three significant figures.
   (5 marks)

8

1. 1. Given that \(4 \sin x + 5 \cos x = 0\), find the value of \(\tan x\).
   2. Hence solve the equation \(( 1 - \tan x ) ( 4 \sin x + 5 \cos x ) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your values of \(x\) to the nearest degree.
2. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
   [0pt] [4 marks]

4. (a) Sketch, for \(0 \leq x \leq 360 ^ { \circ }\), the graph of \(y = \sin \left( x + 30 ^ { \circ } \right)\).
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation $$\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }$$

9. (i) Solve, for \(0 ^ { \circ } < x < 180 ^ { \circ }\), the equation \(\sin \left( 2 x + 50 ^ { \circ } \right) = 0.6\), giving your answers to 1 d. p.
(ii) In the triangle \(A B C , A C = 18 \mathrm {~cm} , \angle A B C = 60 ^ { \circ }\) and \(\sin A = \frac { 1 } { 3 }\).

1. Use the sine rule to show that \(B C = 4 \sqrt { } 3\).
2. Find the exact value of \(\cos A\). L

4. (a) Sketch, for \(0 \leq x \leq 360 ^ { \circ }\), the graph of \(y = \sin \left( x + 30 ^ { \circ } \right)\).
(b) Write down the coordinates of the points at which the graph meets the axes.
(c) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(\sin \left( x + 30 ^ { \circ } \right) = - \frac { 1 } { 2 }\).

4 In this question you must show detailed reasoning.

1. Show that \(\cos A + \sin A \tan A = \sec A\).
2. Solve the equation \(\tan 2 \theta = 3 \tan \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8

1. Solve the equation \(\cos x = 0.3\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
2. The diagram shows the graph of \(y = \cos x\) for \(0 \leqslant x \leqslant 2 \pi\) and the line \(y = k\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648} The line \(y = k\) intersects the curve \(y = \cos x , 0 \leqslant x \leqslant 2 \pi\), at the points \(P\) and \(Q\). The point \(M\) is the minimum point of the curve.
   1. Write down the coordinates of the point \(M\).
   2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of \(Q\) in terms of \(\pi\) and \(\alpha\).
3. Describe the geometrical transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
4. Solve the equation \(\cos 2 x = \cos \frac { 4 \pi } { 5 }\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving the values of \(x\) in terms of \(\pi\).
   (4 marks)

7

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Write down the two solutions of the equation \(\tan x = \tan 61 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. 1. Given that \(\sin \theta + \cos \theta = 0\), show that \(\tan \theta = - 1\).
   2. Hence solve the equation \(\sin \left( x - 20 ^ { \circ } \right) + \cos \left( x - 20 ^ { \circ } \right) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
4. Describe the single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x - 20 ^ { \circ } \right)\).
5. The curve \(y = \tan x\) is stretched in the \(x\)-direction with scale factor \(\frac { 1 } { 4 }\) to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

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}

3

1. Solve the equation \(\operatorname { cosec } x = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)
2. The diagram shows the graph of \(y = \operatorname { cosec } x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-03_609_1045_559_479}
   1. The point \(A\) on the curve is where \(x = 90 ^ { \circ }\). State the \(y\)-coordinate of \(A\).
   2. Sketch the graph of \(y = | \operatorname { cosec } x |\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Solve the equation \(| \operatorname { cosec } x | = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)