Questions - OCR C3 - A-Level Maths

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 $y = \mathrm { f } ( x )$.

A sequence of transformations maps the curve $y = \sqrt { } x$ to the curve $y = \mathrm { f } ( x )$. Give details of these transformations.
Explain how you can tell that f is a one-one function and find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
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.
State the range of f .
Find the value of $\mathrm { ff } ( 4 )$.
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 $$Y = 80 \mathrm { e } ^ { - 0.02 t } .$$
Find the value of $t$ for which $Y = 20$, giving your answer correct to 2 significant figures.
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 $y = \cos ^ { - 1 } x$.
Sketch the curve with equation $y = 3 \cos ^ { - 1 } ( x - 1 )$, showing the coordinates of the points where the curve meets the axes.
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.
Show by calculation that the root of the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$ lies between 1.8 and 1.9.
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 $x = 0$.
Find the equation of the tangent to the curve at $P$.
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.
Deduce an approximation to the area of the region $B$. 9
By first writing $\sin 3 \theta$ as $\sin ( 2 \theta + \theta )$, show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
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.
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.
Show by calculation that this real root lies between 2 and 3 .
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 $y = 5 ^ { x - 1 }$.
Show that $x = 1 + \frac { \ln y } { \ln 5 }$.
Find an expression for $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.
Hence find the exact value of the gradient of the curve $y = 5 ^ { x - 1 }$ at the point (3, 25). 5
Write down the identity expressing $\sin 2 \theta$ in terms of $\sin \theta$ and $\cos \theta$.
Given that $\sin \alpha = \frac { 1 } { 4 }$ and $\alpha$ is acute, show that $\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }$.
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 $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
Evaluate $\mathrm { ff } ( - 3 )$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
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 $\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)$.
Express $5 \cos x + 12 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
Hence give details of a pair of transformations which transforms the curve $y = \cos x$ to the curve $y = 5 \cos x + 12 \sin x$.
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.
Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the solid is given by $$V = \pi \left( \mathrm { e } ^ { p } + 4 \mathrm { e } ^ { \frac { 1 } { 2 } p } + p - 5 \right) .$$
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
$\cot \theta$,
$\cos 2 \theta$. 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
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 $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the value of $\frac { \mathrm { d } y } { \mathrm {~d} t }$ when $t = 4$, giving your answer correct to 3 significant figures. 5
Express $4 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
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 $y = 0$.
Find the exact area of the shaded region.
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.
Give full details of the translation and stretch involved.
Sketch the graph of $y = \ln \left( \frac { 1 } { 2 } x - a \right)$.
Sketch, on another diagram, the graph of $y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|$.
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 $P Q$.
Show by differentiation that the $x$-coordinate of $Q$ is 2 .
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.
Deduce an approximation to the area of region $B$. 9 Functions $f$ and $g$ are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi ,
\mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$
State the range of f and the range of g .
Show that $\operatorname { gf } ( 0.5 ) = 2.16$, correct to 3 significant figures, and explain why $\mathrm { fg } ( 0.5 )$ is not defined.
Find the set of values of $x$ for which $\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )$ is not defined.

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