Questions - OCR C3 - A-Level Maths

$t$	0	21	63
$M$		80

In either order,
(a) find the values missing from the table,
(b) determine the value of $k$.
Find the rate at which the mass is increasing when $t = 21$.

OCR C3 2009 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_627_689_264_726} The function f is defined for all real values of $x$ by $$f ( x ) = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$ The graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ meet at the point $P$, and the graph of $y = \mathrm { f } ^ { - 1 } ( x )$ meets the $x$-axis at $Q$ (see diagram).

Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and determine the $x$-coordinate of the point $Q$.
State how the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ are related geometrically, and hence show that the $x$-coordinate of the point $P$ is the root of the equation $$x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$$
Use an iterative process, based on the equation $x = \sqrt [ 3 ] { \frac { 1 } { 2 } x + 2 }$, to find the $x$-coordinate of $P$, giving your answer correct to 2 decimal places.

OCR C3 2009 January Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-3_419_700_1809_721} The diagram shows the curve $y = \mathrm { e } ^ { k x } - a$, where $k$ and $a$ are constants.

Give details of the pair of transformations which transforms the curve $y = \mathrm { e } ^ { x }$ to the curve $y = \mathrm { e } ^ { k x } - a$.
Sketch the curve $y = \left| \mathrm { e } ^ { k x } - a \right|$.
Given that the curve $y = \left| \mathrm { e } ^ { k x } - a \right|$ passes through the points $( 0,13 )$ and $( \ln 3,13 )$, find the values of $k$ and $a$.

OCR C3 2009 January Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-4_538_702_264_719} The diagram shows the curve with equation $$y = \frac { 6 } { \sqrt { x } } - 3$$ The point $P$ has coordinates $( 0 , p )$. The shaded region is bounded by the curve and the lines $x = 0$, $y = 0$ and $y = p$. The shaded region is rotated completely about the $y$-axis to form a solid of volume $V$.

Show that $V = 16 \pi \left( 1 - \frac { 27 } { ( p + 3 ) ^ { 3 } } \right)$.
It is given that $P$ is moving along the $y$-axis in such a way that, at time $t$, the variables $p$ and $t$ are related by $$\frac { \mathrm { d } p } { \mathrm {~d} t } = \frac { 1 } { 3 } p + 1 .$$ Find the value of $\frac { \mathrm { d } V } { \mathrm {~d} t }$ at the instant when $p = 9$.

OCR C3 2009 January Q9

By first expanding $\cos ( 2 \theta + \theta )$, prove that $$\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta$$
Hence prove that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
Show that the only solutions of the equation $$1 + \cos 6 \theta = 18 \cos ^ { 2 } \theta$$ are odd multiples of $90 ^ { \circ }$.

OCR C3 2010 January Q1

1 Find $\int \frac { 10 } { ( 2 x - 7 ) ^ { 2 } } \mathrm {~d} x$.

OCR C3 2010 January Q2

2 The angle $\theta$ is such that $0 ^ { \circ } < \theta < 90 ^ { \circ }$.

Given that $\theta$ satisfies the equation $6 \sin 2 \theta = 5 \cos \theta$, find the exact value of $\sin \theta$.
Given instead that $\theta$ satisfies the equation $8 \cos \theta \operatorname { cosec } ^ { 2 } \theta = 3$, find the exact value of $\cos \theta$.

OCR C3 2010 January Q3

Find, in simplified form, the exact value of $\int _ { 10 } ^ { 20 } \frac { 60 } { x } \mathrm {~d} x$.
Use Simpson's rule with two strips to find an approximation to $\int _ { 10 } ^ { 20 } \frac { 60 } { x } \mathrm {~d} x$.
Use your answers to parts (i) and (ii) to show that $\ln 2 \approx \frac { 25 } { 36 }$.

OCR C3 2010 January Q4

4
\includegraphics[max width=\textwidth, alt={}, center]{18ad8019-1dc8-44bb-8a25-eaf5e05465ab-2_444_1249_1233_447} The function f is defined for all real values of $x$ by $$\mathrm { f } ( x ) = 2 - \sqrt [ 3 ] { x + 1 }$$ The diagram shows the graph of $y = \mathrm { f } ( x )$.

Evaluate $\mathrm { ff } ( - 126 )$.
Find the set of values of $x$ for which $\mathrm { f } ( x ) = | \mathrm { f } ( x ) |$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
State how the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ are related geometrically. The equation of a curve is $y = \left( x ^ { 2 } + 1 \right) ^ { 8 }$.

OCR C3 2010 January Q6

6 Given that $$\int _ { 0 } ^ { \ln 4 } \left( k \mathrm { e } ^ { 3 x } + ( k - 2 ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x = 185$$ find the value of the constant $k$.

Leaking oil is forming a circular patch on the surface of the sea. The area of the patch is increasing at a rate of 250 square metres per hour. Find the rate at which the radius of the patch is increasing at the instant when the area of the patch is 1900 square metres. Give your answer correct to 2 significant figures.
The mass of a substance is decreasing exponentially. Its mass now is 150 grams and its mass, $m$ grams, at a time $t$ years from now is given by $$m = 150 \mathrm { e } ^ { - k t } ,$$ where $k$ is a positive constant. Find, in terms of $k$, the number of years from now at which the mass will be decreasing at a rate of 3 grams per year.
1. The curve $y = \sqrt { x }$ can be transformed to the curve $y = \sqrt { 2 x + 3 }$ by means of a stretch parallel to the $y$-axis followed by a translation. State the scale factor of the stretch and give details of the translation.
2. It is given that $N$ is a positive integer. By sketching on a single diagram the graphs of $y = \sqrt { 2 x + 3 }$ and $y = \frac { N } { x ^ { 3 } }$, show that the equation $$\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }$$ has exactly one real root.
3. A sequence $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ has the property that $$x _ { n + 1 } = N ^ { \frac { 1 } { 3 } } \left( 2 x _ { n } + 3 \right) ^ { - \frac { 1 } { 6 } }$$ For certain values of $x _ { 1 }$ and $N$, it is given that the sequence converges to the root of the equation $\sqrt { 2 x + 3 } = \frac { N } { x ^ { 3 } }$.

OCR C3 2010 January Q9

9 The value of $\tan 10 ^ { \circ }$ is denoted by $p$. Find, in terms of $p$, the value of

$\tan 55 ^ { \circ }$,
$\tan 5 ^ { \circ }$,
$\tan \theta$, where $\theta$ satisfies the equation $3 \sin \left( \theta + 10 ^ { \circ } \right) = 7 \cos \left( \theta - 10 ^ { \circ } \right)$.

OCR C3 2011 January Q1

1 Solve the equation $| 3 x + 4 a | = 5 a$, where $a$ is a positive constant.

OCR C3 2011 January Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_538_1061_388_541} The diagram shows the curve with equation $y = \mathrm { f } ( x )$. It is given that $\mathrm { f } ( - 7 ) = 0$ and that there are stationary points at $( - 2 , - 6 )$ and $( 0,0 )$. Sketch the curve with equation $y = - 4 \mathrm { f } ( x + 3 )$, indicating the coordinates of the stationary points.

OCR C3 2011 January Q3

3 A giant spherical balloon is being inflated in a theme park. The radius of the balloon is increasing at a rate of 12 cm per hour. Find the rate at which the surface area of the balloon is increasing at the instant when the radius is 150 cm . Give your answer in $\mathrm { cm } ^ { 2 }$ per hour correct to 2 significant figures.
[0pt] [Surface area of sphere $= 4 \pi r ^ { 2 }$.]

OCR C3 2011 January Q4

Express $24 \sin \theta + 7 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
Hence solve the equation $24 \sin \theta + 7 \cos \theta = 12$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.

OCR C3 2011 January Q5

9 marks

5
\includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_559_1191_1749_479} The diagram shows the curve with equation $y = \frac { 6 } { \sqrt { 3 x - 2 } }$. The region $R$, shaded in the diagram, is bounded by the curve and the lines $x = 1 , x = a$ and $y = 0$, where $a$ is a constant greater than 1 . It is given that the area of $R$ is 16 square units. Find the value of $a$ and hence find the exact volume of the solid formed when $R$ is rotated completely about the $x$-axis.
[0pt] [9]

OCR C3 2011 January Q6

6 The curve with equation $y = \frac { 3 x + 4 } { x ^ { 3 } - 4 x ^ { 2 } + 2 }$ has a stationary point at $P$. It is given that $P$ is close to the point with coordinates $( 2.4 , - 1.6 )$.

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that the $x$-coordinate of $P$ satisfies the equation $$x = \sqrt [ 3 ] { \frac { 16 } { 3 } x + 1 }$$
By first using an iterative process based on the equation in part (i), find the coordinates of $P$, giving each coordinate correct to 3 decimal places.

OCR C3 2011 January Q7

7 The function f is defined for $x > 0$ by $\mathrm { f } ( x ) = \ln x$ and the function g is defined for all real values of $x$ by $\mathrm { g } ( x ) = x ^ { 2 } + 8$.

Find the exact, positive value of $x$ which satisfies the equation $\operatorname { fg } ( x ) = 8$.
State which one of f and g has an inverse and define that inverse function.
Find the exact value of the gradient of the curve $y = \operatorname { gf } ( x )$ at the point with $x$-coordinate $\mathrm { e } ^ { 3 }$.
Use Simpson's rule with four strips to find an approximate value of $$\int _ { - 4 } ^ { 4 } \mathrm { fg } ( x ) \mathrm { d } x$$ giving your answer correct to 3 significant figures.

OCR C3 2011 January Q8

1. Sketch the graph of $y = \operatorname { cosec } x$ for $0 < x < 4 \pi$.
2. It is given that $\operatorname { cosec } \alpha = \operatorname { cosec } \beta$, where $\frac { 1 } { 2 } \pi < \alpha < \pi$ and $2 \pi < \beta < \frac { 5 } { 2 } \pi$. By using your sketch, or otherwise, express $\beta$ in terms of $\alpha$.
1. Write down the identity giving $\tan 2 \theta$ in terms of $\tan \theta$.
2. Given that $\cot \phi = 4$, find the exact value of $\tan \phi \cot 2 \phi \tan 4 \phi$, showing all your working.

OCR C3 2011 January Q9

The function f is defined for all real values of $x$ by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$ (a) Show that $\mathrm { f } ^ { \prime } ( x ) > 0$ for all $x$.
(b) Show that the set of values of $x$ for which $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ is the same as the set of values of $x$ for which $\mathrm { f } ( x ) > 0$, and state what this set of values is.
\includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-04_634_830_641_699} The function g is defined for all real values of $x$ by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x } ,$$ where $k$ is a constant greater than 1 . The graph of $y = \mathrm { g } ( x )$ is shown above. Find the range of g , giving your answer in simplified form.

OCR C3 2011 January Q10

8 (b) (i)




8 (b) (ii)
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9

\section*{RECOGNISING ACHIEVEMENT} RECOGNISING ACHIEVEMENT

OCR C3 2012 January Q1

1 Show that $\int _ { \sqrt { 2 } } ^ { \sqrt { 6 } } \frac { 2 } { x } \mathrm {~d} x = \ln 3$.

OCR C3 2012 January Q2

2
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-2_490_713_447_660} The diagram shows part of the curve $y = \frac { 6 } { ( 2 x + 1 ) ^ { 2 } }$. The shaded region is bounded by the curve and the lines $x = 0 , x = 1$ and $y = 0$. Find the exact volume of the solid produced when this shaded region is rotated completely about the $x$-axis.

OCR C3 2012 January Q3

3 Find the equation of the normal to the curve $y = \frac { x ^ { 2 } + 4 } { x + 2 }$ at the point $\left( 1 , \frac { 5 } { 3 } \right)$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

OCR C3 2012 January Q4

4 The acute angles $\alpha$ and $\beta$ are such that $$2 \cot \alpha = 1 \text { and } 24 + \sec ^ { 2 } \beta = 10 \tan \beta \text {. }$$

State the value of $\tan \alpha$ and determine the value of $\tan \beta$.
Hence find the exact value of $\tan ( \alpha + \beta )$.

\(t\)	0	21	63
\(M\)		80

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