Questions C3 - A-Level Maths

AQA C3 2007 June Q2

2

Differentiate $( x - 1 ) ^ { 4 }$ with respect to $x$.
The diagram shows the curve with equation $y = 2 \sqrt { ( x - 1 ) ^ { 3 } }$ for $x \geqslant 1$.
\includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-02_789_1180_1190_431} The shaded region $R$ is bounded by the curve $y = 2 \sqrt { ( x - 1 ) ^ { 3 } }$, the lines $x = 2$ and $x = 4$, and the $x$-axis. Find the exact value of the volume of the solid formed when the region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
Describe a sequence of two geometrical transformations that maps the graph of $y = \sqrt { x ^ { 3 } }$ onto the graph of $y = 2 \sqrt { ( x - 1 ) ^ { 3 } }$.

AQA C3 2007 June Q3

3

Solve the equation $\operatorname { cosec } x = 2$, giving all values of $x$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.
(2 marks)
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 $0 ^ { \circ } < x < 360 ^ { \circ }$.
Solve the equation $| \operatorname { cosec } x | = 2$, giving all values of $x$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$.
(2 marks)

AQA C3 2007 June Q4

4 [Figure 1, printed on the insert, is provided for use in this question.]

Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to $\int _ { 1 } ^ { 2 } 3 ^ { x } \mathrm {~d} x$, giving your answer to three significant figures.
The curve $y = 3 ^ { x }$ intersects the line $y = x + 3$ at the point where $x = \alpha$.
1. Show that $\alpha$ lies between 0.5 and 1.5.
2. Show that the equation $3 ^ { x } = x + 3$ can be rearranged into the form $$x = \frac { \ln ( x + 3 ) } { \ln 3 }$$
3. Use the iteration $x _ { n + 1 } = \frac { \ln \left( x _ { n } + 3 \right) } { \ln 3 }$ with $x _ { 1 } = 0.5$ to find $x _ { 3 }$ to two significant figures.
4. The sketch on Figure 1 shows part of the graphs of $y = \frac { \ln ( x + 3 ) } { \ln 3 }$ and $y = x$, and the position of $x _ { 1 }$. On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.

AQA C3 2007 June Q5

5 The functions $f$ and $g$ are defined with their respective domains by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { x - 2 } \text { for } x \geqslant 2
& \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for real values of } x , x \neq 0 \end{aligned}$$

State the range of f .
1. Find fg(x).
2. Solve the equation $\operatorname { fg } ( x ) = 1$.
The inverse of f is $\mathrm { f } ^ { - 1 }$. Find $\mathrm { f } ^ { - 1 } ( x )$.

AQA C3 2007 June Q6

6

Use integration by parts to find $\int x \mathrm { e } ^ { 5 x } \mathrm {~d} x$.
1. Use the substitution $u = \sqrt { x }$ to show that $$\int \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x = \int \frac { 2 } { 1 + u } \mathrm {~d} u$$
2. Find the exact value of $\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x$.

AQA C3 2007 June Q7

7

A curve has equation $y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { x }$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
1. Find the $x$-coordinate of each of the stationary points of the curve.
2. Using your answer to part (a)(ii), determine the nature of each of the stationary points.

AQA C3 2007 June Q8

8

Write down $\int \sec ^ { 2 } x \mathrm {~d} x$.
Given that $y = \frac { \cos x } { \sin x }$, use the quotient rule to show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x$.
Prove the identity $( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x$.
Hence find $\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x$, giving your answer to two significant figures.

AQA C3 2015 June Q1

8 marks

1

Use the mid-ordinate rule with four strips to find an estimate for $\int _ { 1.5 } ^ { 5.5 } \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 ) \mathrm { d } x$, giving your answer to three decimal places.
[0pt] [4 marks]
Find the exact value of the gradient of the curve $y = \mathrm { e } ^ { 2 - x } \ln ( 3 x - 2 )$ at the point on the curve where $x = 2$.
[0pt] [4 marks]

AQA C3 2015 June Q2

4 marks

2

Sketch, on the axes below, the curve with equation $y = 4 - | 2 x + 1 |$, indicating the coordinates where the curve crosses the axes.
Solve the equation $x = 4 - | 2 x + 1 |$.
Solve the inequality $x < 4 - | 2 x + 1 |$.
Describe a sequence of two geometrical transformations that maps the graph of $y = | 2 x + 1 |$ onto the graph of $y = 4 - | 2 x + 1 |$.
[0pt] [4 marks] \section*{Answer space for question 2}
\includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-04_851_1459_1000_319}

AQA C3 2015 June Q3

3

It is given that the curves with equations $y = 6 \ln x$ and $y = 8 x - x ^ { 2 } - 3$ intersect at a single point where $x = \alpha$.
1. Show that $\alpha$ lies between 5 and 6 .
2. Show that the equation $x = 4 + \sqrt { 13 - 6 \ln x }$ can be rearranged into the form $$6 \ln x + x ^ { 2 } - 8 x + 3 = 0$$
3. Use the iterative formula $$x _ { n + 1 } = 4 + \sqrt { 13 - 6 \ln x _ { n } }$$ with $x _ { 1 } = 5$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
A curve has equation $y = \mathrm { f } ( x )$ where $\mathrm { f } ( x ) = 6 \ln x + x ^ { 2 } - 8 x + 3$.
1. Find the exact values of the coordinates of the stationary points of the curve.
2. Hence, or otherwise, find the exact values of the coordinates of the stationary points of the curve with equation $$y = 2 \mathrm { f } ( x - 4 )$$

AQA C3 2015 June Q4

3 marks

4 The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 5 - \mathrm { e } ^ { 3 x } , & \text { for all real values of } x
\mathrm {~g} ( x ) = \frac { 1 } { 2 x - 3 } , & \text { for } x \neq 1.5 \end{array}$$

Find the range of f.
The inverse of f is $\mathrm { f } ^ { - 1 }$.
1. Find $\mathrm { f } ^ { - 1 } ( x )$.
2. Solve the equation $\mathrm { f } ^ { - 1 } ( x ) = 0$.
Find an expression for $\operatorname { gg } ( x )$, giving your answer in the form $\frac { a x + b } { c x + d }$, where $a , b , c$ and $d$ are integers.
[0pt] [3 marks]

AQA C3 2015 June Q5

9 marks

5

By writing $\tan x$ as $\frac { \sin x } { \cos x }$, use the quotient rule to show that $\frac { \mathrm { d } } { \mathrm { d } x } ( \tan x ) = \sec ^ { 2 } x$.
[0pt] [2 marks]
Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.
[0pt] [4 marks]
The region bounded by the curve $y = ( 5 \sqrt { x } ) \sec x$, the $x$-axis from 0 to 1 and the line $x = 1$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid. Find the value of the volume of the solid generated, giving your answer to two significant figures.
[0pt] [3 marks]

AQA C3 2015 June Q6

6

Sketch, on the axes below, the curve with equation $y = \sin ^ { - 1 } ( 3 x )$, where $y$ is in radians. State the exact values of the coordinates of the end points of the graph.
Given that $x = \frac { 1 } { 3 } \sin y$, write down $\frac { \mathrm { d } x } { \mathrm {~d} y }$ and hence find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $y$. \section*{Answer space for question 6}
\includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-14_839_1451_813_324}

AQA C3 2015 June Q7

7 marks

7 Use the substitution $u = 6 - x ^ { 2 }$ to find the value of $\int _ { 1 } ^ { 2 } \frac { x ^ { 3 } } { \sqrt { 6 - x ^ { 2 } } } \mathrm {~d} x$, giving your answer in the form $p \sqrt { 5 } + q \sqrt { 2 }$, where $p$ and $q$ are rational numbers.
[0pt] [7 marks]

AQA C3 2015 June Q8

5 marks

8

Show that the equation $4 \operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta = k$, where $k \neq 4$, can be written in the form $$\sec ^ { 2 } \theta = \frac { k - 1 } { k - 4 }$$
Hence, or otherwise, solve the equation $$4 \operatorname { cosec } ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) - \cot ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) = 5$$ giving all values of $x$ in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-18_72_113_1055_159}

\includegraphics[max width=\textwidth, alt={}]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-20_2288_1707_221_153}

OCR C3 Q1

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

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

OCR C3 Q3

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

Find the value of $t$ for which the mass is 25 grams.
Find the rate at which the mass is decreasing when $t = 55$.

OCR C3 Q4

4

\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.
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.

OCR C3 Q5

5

Express $3 \sin \theta + 2 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
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}

OCR C3 Q8

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.

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

OCR C3 Q9

17 marks

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.

OCR C3 2010 June Q1

1 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases:

$y = x ^ { 3 } \mathrm { e } ^ { 2 x }$,
$y = \ln \left( 3 + 2 x ^ { 2 } \right)$,
$y = \frac { x } { 2 x + 1 }$.

OCR C3 2010 June Q2

2 The transformations R, S and T are defined as follows.
R : reflection in the $x$-axis
S : stretch in the $x$-direction with scale factor 3
$\mathrm { T } : \quad$ translation in the positive $x$-direction by 4 units

The curve $y = \ln x$ is transformed by R followed by T . Find the equation of the resulting curve.
Find, in terms of S and T, a sequence of transformations that transforms the curve $y = x ^ { 3 }$ to the curve $y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }$. You should make clear the order of the transformations.

OCR C3 2010 June Q3

3

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.]}

OCR C3 2010 June Q9

9 The functions f and g are defined for all real values of $x$ by $$\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x \quad \text { and } \quad \mathrm { g } ( x ) = a x + b$$ where $a$ and $b$ are non-zero constants.

Find the range of f .
Explain why the function $f$ has no inverse.
Given that $\mathrm { g } ^ { - 1 } ( x ) = \mathrm { g } ( x )$ for all values of $x$, show that $a = - 1$.
Given further that $\operatorname { gf } ( x ) < 5$ for all values of $x$, find the set of possible values of $b$.

Questions C3 (1200 questions)

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