Questions C3 - A-Level Maths

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 $\tan ( \alpha + \beta )$.

OCR C3 2012 January Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-3_844_837_242_621} It is given that f is a one-one function defined for all real values. The diagram shows the curve with equation $y = \mathrm { f } ( x )$. The coordinates of certain points on the curve are shown in the following table.

$x$	2	4	6	8	10	12	14
$y$	1	8	14	19	23	25	26

State the value of $\mathrm { ff } ( 6 )$ and the value of $\mathrm { f } ^ { - 1 } ( 8 )$.
On the copy of the diagram, sketch the curve $y = \mathrm { f } ^ { - 1 } ( x )$, indicating how the curves $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ are related.
Use Simpson's rule with 6 strips to find an approximation to $\int _ { 2 } ^ { 14 } \mathrm { f } ( x ) \mathrm { d } x$.

OCR C3 2012 January Q6

6
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-4_476_709_251_683} The diagram shows the curve with equation $x = \ln \left( y ^ { 3 } + 2 y \right)$. At the point $P$ on the curve, the gradient is 4 and it is given that $P$ is close to the point with coordinates (7.5,12).

Find $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.
Show that the $y$-coordinate of $P$ satisfies the equation $$y = \frac { 12 y ^ { 2 } + 8 } { y ^ { 2 } + 2 }$$
By first using an iterative process based on the equation in part (ii), find the coordinates of $P$, giving each coordinate correct to 3 decimal places.

OCR C3 2012 January Q7

Substance $A$ is decaying exponentially and its mass is recorded at regular intervals. At time $t$ years, the mass, $M$ grams, of substance $A$ is given by $$M = 40 \mathrm { e } ^ { - 0.132 t }$$ (a) Find the time taken for the mass of substance $A$ to decrease to $25 \%$ of its value when $t = 0$.
(b) Find the rate at which the mass of substance $A$ is decreasing when $t = 5$.
Substance $B$ is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance $B$ after a further year.

OCR C3 2012 January Q8

Express $\cos 4 \theta$ in terms of $\sin 2 \theta$ and hence show that $\cos 4 \theta$ can be expressed in the form $1 - k \sin ^ { 2 } \theta \cos ^ { 2 } \theta$, where $k$ is a constant to be determined.
Hence find the exact value of $\sin ^ { 2 } \left( \frac { 1 } { 24 } \pi \right) \cos ^ { 2 } \left( \frac { 1 } { 24 } \pi \right)$.
By expressing $2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$ in terms of $\cos 4 \theta$, find the greatest and least possible values of $$2 \cos ^ { 2 } 2 \theta - \frac { 8 } { 3 } \sin ^ { 2 } \theta \cos ^ { 2 } \theta$$ as $\theta$ varies.
\includegraphics[max width=\textwidth, alt={}, center]{89e54367-bb83-483a-add5-0527b71a5cac-5_606_926_267_552} The function f is defined for all real values of $x$ by $$\mathrm { f } ( x ) = k \left( x ^ { 2 } + 4 x \right) ,$$ where $k$ is a positive constant. The diagram shows the curve with equation $y = \mathrm { f } ( x )$.
The curve $y = x ^ { 2 }$ can be transformed to the curve $y = \mathrm { f } ( x )$ by the following sequence of transformations: a translation parallel to the $x$-axis,
a translation parallel to the $y$-axis,
a stretch. a translation parallel to the $x$-axis, a translation parallel to the $y$-axis, a stretch.
Give details, in terms of $k$ where appropriate, of these transformations.
Find the range of f in terms of $k$.
It is given that there are three distinct values of $x$ which satisfy the equation $| \mathrm { f } ( x ) | = 20$. Find the value of $k$ and determine exactly the three values of $x$ which satisfy the equation in this case.

OCR C3 2013 January Q1

1 For each of the following curves, find the gradient at the point with $x$-coordinate 2 .

$y = \frac { 3 x } { 2 x + 1 }$
$y = \sqrt { 4 x ^ { 2 } + 9 }$

OCR C3 2013 January Q2

2 The acute angle $A$ is such that $\tan A = 2$.

Find the exact value of $\operatorname { cosec } A$.
The angle $B$ is such that $\tan ( A + B ) = 3$. Using an appropriate identity, find the exact value of $\tan B$.

OCR C3 2013 January Q3

Given that $| t | = 3$, find the possible values of $| 2 t - 1 |$.
Solve the inequality $| x - \sqrt { 2 } | > | x + 3 \sqrt { 2 } |$.

OCR C3 2013 January Q4

4 The mass, $m$ grams, of a substance is increasing exponentially so that the mass at time $t$ hours is given by $$m = 250 \mathrm { e } ^ { 0.021 t } .$$

Find the time taken for the mass to increase to twice its initial value, and deduce the time taken for the mass to increase to 8 times its initial value.
Find the rate at which the mass is increasing at the instant when the mass is 400 grams.

OCR C3 2013 January Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-2_454_770_1628_635} The diagram shows the curve $y = \frac { 6 } { \sqrt { 3 x + 1 } }$. The shaded region is bounded by the curve and the lines $x = 2 , x = 9$ and $y = 0$.

Show that the area of the shaded region is $4 \sqrt { 7 }$ square units.
The shaded region is rotated completely about the $x$-axis. Show that the volume of the solid produced can be written in the form $k \ln 2$, where the exact value of the constant $k$ is to be determined.

OCR C3 2013 January Q7

By sketching the curves $y = \ln x$ and $y = 8 - 2 x ^ { 2 }$ on a single diagram, show that the equation $$\ln x = 8 - 2 x ^ { 2 }$$ has exactly one real root.
Explain how your diagram shows that the root is between 1 and 2 .
Use the iterative formula $$x _ { n + 1 } = \sqrt { 4 - \frac { 1 } { 2 } \ln x _ { n } } ,$$ with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places.
The curves $y = \ln x$ and $y = 8 - 2 x ^ { 2 }$ are each translated by 2 units in the positive $x$-direction and then stretched by scale factor 4 in the $y$-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places.
\includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-3_389_917_1117_557} The diagram shows the curve with equation $$x = ( y + 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the $x$-axis at $A$ and the $y$-axis at $B$.
Find an expression for $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.
Find the gradient of the curve at each of the points $A$ and $B$, giving each answer correct to 2 decimal places.

OCR C3 2013 January Q8

8 The functions f and g are defined for all real values of $x$ by $$\mathrm { f } ( x ) = x ^ { 2 } + 4 a x + a ^ { 2 } \text { and } \mathrm { g } ( x ) = 4 x - 2 a ,$$ where $a$ is a positive constant.

Find the range of f in terms of $a$.
Given that $\mathrm { fg } ( 3 ) = 69$, find the value of $a$ and hence find the value of $x$ such that $\mathrm { g } ^ { - 1 } ( x ) = x$.

OCR C3 2013 January Q9

Prove that $$\cos ^ { 2 } \left( \theta + 45 ^ { \circ } \right) - \frac { 1 } { 2 } ( \cos 2 \theta - \sin 2 \theta ) \equiv \sin ^ { 2 } \theta .$$
Hence solve the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta + 45 ^ { \circ } \right) - 3 ( \cos \theta - \sin \theta ) = 2$$ for $- 90 ^ { \circ } < \theta < 90 ^ { \circ }$.
It is given that there are two values of $\theta$, where $- 90 ^ { \circ } < \theta < 90 ^ { \circ }$, satisfying the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 3 } \theta + 45 ^ { \circ } \right) - 3 \left( \cos \frac { 2 } { 3 } \theta - \sin \frac { 2 } { 3 } \theta \right) = k ,$$ where $k$ is a constant. Find the set of possible values of $k$.