Questions - OCR C3 - A-Level Maths

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OCR C3 Q4

9 marks Moderate -0.3

(i) Use the identity for $\cos ( A + B )$ to prove that

$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for $\cos x \neq 0$, $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of $x$ in the interval $0 \leq x \leq 180 ^ { \circ }$ for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$

OCR C3 Q5

9 marks Standard +0.8

(i) Show that the equation

$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of $\pi$ the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for $x$ in the interval $0 \leq x \leq \pi$.

OCR C3 Q6

10 marks Standard +0.8

6. \includegraphics[max width=\textwidth, alt={}, center]{687756c0-2038-4077-8c5c-fe0ca0f6ce65-2_444_825_1571_516} The diagram shows the curve with equation $y = \sqrt { \frac { x } { x + 1 } }$.
The shaded region is bounded by the curve, the $x$-axis and the line $x = 3$.

Use Simpson's rule with six strips to estimate the area of the shaded region. The shaded region is rotated through four right angles about the $x$-axis.
Show that the volume of the solid formed is $\pi ( 3 - \ln 4 )$.

OCR C3 Q7

11 marks Standard +0.8

7. (i) Sketch on the same diagram the graphs of $y = 4 a ^ { 2 } - x ^ { 2 }$ and $y = | 2 x - a |$, where $a$ is a positive constant. Show, in terms of $a$, the coordinates of any points where each graph meets the coordinate axes.
(ii) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

OCR C3 Q8

12 marks Standard +0.3

A curve has the equation $y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , x \neq 0$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
  [0pt]
2. Show that the curve has a stationary point in the interval [1.3,1.4].
The point $A$ on the curve has $x$-coordinate 2 .
Show that the tangent to the curve at $A$ passes through the origin. The tangent to the curve at $A$ intersects the curve again at the point $B$.
The $x$-coordinate of $B$ is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with $x _ { 0 } = - 1$.
Find $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ to 7 significant figures and hence state the $x$-coordinate of $B$ to 5 significant figures.

OCR C3 2009 January Q1

5 marks Moderate -0.8

1 Find

$\int 8 \mathrm { e } ^ { - 2 x } \mathrm {~d} x$,
$\int ( 4 x + 5 ) ^ { 6 } \mathrm {~d} x$.

OCR C3 2009 January Q2

5 marks Moderate -0.5

Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
Deduce an approximation to $\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x$.

OCR C3 2009 January Q3

7 marks Moderate -0.3

Express $2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta }$ in terms of $\sec \theta$.
Hence solve, for $0 ^ { \circ } < \theta < 360 ^ { \circ }$, the equation $$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$

OCR C3 2009 January Q4

7 marks Moderate -0.3

4 For each of the following curves, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and determine the exact $x$-coordinate of the stationary point:

$y = \left( 4 x ^ { 2 } + 1 \right) ^ { 5 }$,
$y = \frac { x ^ { 2 } } { \ln x }$.

OCR C3 2009 January Q5

8 marks Moderate -0.3

5 The mass, $M$ grams, of a certain substance is increasing exponentially so that, at time $t$ hours, the mass is given by $$M = 40 \mathrm { e } ^ { k t }$$ where $k$ is a constant. The following table shows certain values of $t$ and $M$.

$t$	0	21	63
$M$		80

In either order,
1. find the values missing from the table,
2. determine the value of $k$.
3. Find the rate at which the mass is increasing when $t = 21$.

OCR C3 2009 January Q6

9 marks Standard +0.3

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

9 marks Standard +0.3

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

10 marks Standard +0.8

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

12 marks Standard +0.8

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 $90 ^ { \circ }$.

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 Standard +0.3

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

9 marks Standard +0.8

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

12 marks Standard +0.3

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

12 marks Standard +0.3

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

13 marks Standard +0.3

The function f is defined for all real values of $x$ by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$
1. Show that $\mathrm { f } ^ { \prime } ( x ) > 0$ for all $x$.
2. 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.
3. \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

Moderate -0.8

8 (b) (i)




8 (b) (ii)
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9 (a)
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9

\section*{RECOGNISING ACHIEVEMENT} RECOGNISING ACHIEVEMENT

OCR C3 2012 January Q1

3 marks Moderate -0.8

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

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

Questions — OCR C3 (339 questions)

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OCR C3 Q4

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