Question 3 - A-Level Maths

OCR C3 2010 June — Question 3

Exam Board	OCR
Module	C3 (Core Mathematics 3)
Year	2010
Session	June
Topic	Trig Equations

Express the equation $\operatorname { cosec } \theta ( 3 \cos 2 \theta + 7 ) + 11 = 0$ in the form $a \sin ^ { 2 } \theta + b \sin \theta + c = 0$, where $a , b$ and $c$ are constants.
Hence solve, for $- 180 ^ { \circ } < \theta < 180 ^ { \circ }$, the equation $\operatorname { cosec } \theta ( 3 \cos 2 \theta + 7 ) + 11 = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{cd1bde44-ab7e-45e6-ac22-346145eba3a0-2_648_951_1530_598} The diagram shows part of the curve $y = \frac { k } { x }$, where $k$ is a positive constant. The points $A$ and $B$ on the curve have $x$-coordinates 2 and 6 respectively. Lines through $A$ and $B$ parallel to the axes as shown meet at the point $C$. The region $R$ is bounded by the curve and the lines $x = 2 , x = 6$ and $y = 0$. The region $S$ is bounded by the curve and the lines $A C$ and $B C$. It is given that the area of the region $R$ is $\ln 81$.
Show that $k = 4$.
Find the exact volume of the solid produced when the region $S$ is rotated completely about the $x$-axis.
Solve the inequality $| 2 x + 1 | \leqslant | x - 3 |$.
Given that $x$ satisfies the inequality $| 2 x + 1 | \leqslant | x - 3 |$, find the greatest possible value of $| x + 2 |$.
Show by calculation that the equation $$\tan ^ { 2 } x - x - 2 = 0$$ where $x$ is measured in radians, has a root between 1.0 and 1.1.
Use the iteration formula $x _ { n + 1 } = \tan ^ { - 1 } \sqrt { 2 + x _ { n } }$ with a suitable starting value to find this root correct to 5 decimal places. You should show the outcome of each step of the process.
Deduce a root of the equation $$\sec ^ { 2 } 2 x - 2 x - 3 = 0$$
\includegraphics[max width=\textwidth, alt={}]{cd1bde44-ab7e-45e6-ac22-346145eba3a0-3_771_1087_1128_529}
The diagram shows the curve with equation $y = ( 3 x - 1 ) ^ { 4 }$. The point $P$ on the curve has coordinates $( 1,16 )$ and the tangent to the curve at $P$ meets the $x$-axis at the point $Q$. The shaded region is bounded by $P Q$, the $x$-axis and that part of the curve for which $\frac { 1 } { 3 } \leqslant x \leqslant 1$. Find the exact area of this shaded region.
Express $3 \cos x + 3 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
The expression $\mathrm { T } ( x )$ is defined by $\mathrm { T } ( x ) = \frac { 8 } { 3 \cos x + 3 \sin x }$.
(a) Determine a value of $x$ for which $\mathrm { T } ( x )$ is not defined.
(b) Find the smallest positive value of $x$ satisfying $\mathrm { T } ( 3 x ) = \frac { 8 } { 9 } \sqrt { 6 }$, giving your answer in an exact form. \section*{[Question 9 is printed overleaf.]}

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