Questions - OCR MEI C3 - A-Level Maths

An initial mass of 100 grams of the substance decays to 50 grams in 1500 years. Find $A$ and $k$.
The substance becomes safe when $99 \%$ of its initial mass has decayed. Find how long it will take before the substance becomes safe.

OCR MEI C3 2009 June Q3

3 Sketch the curve $y = 2 \arccos x$ for $- 1 \leqslant x \leqslant 1$.

OCR MEI C3 2009 June Q4

4 Fig. 4 shows a sketch of the graph of $y = 2 | x - 1 |$. It meets the $x$ - and $y$-axes at ( $a , 0$ ) and ( $0 , b$ ) respectively. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1167a0e5-48c8-48e0-b2d1-76a50bad03ad-2_478_556_1247_792} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} Find the values of $a$ and $b$.

OCR MEI C3 2009 June Q5

5 The equation of a curve is given by $\mathrm { e } ^ { 2 y } = 1 + \sin x$.

By differentiating implicitly, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.
Find an expression for $y$ in terms of $x$, and differentiate it to verify the result in part (i).

OCR MEI C3 2009 June Q6

6 Given that $\mathrm { f } ( x ) = \frac { x + 1 } { x - 1 }$, show that $\mathrm { ff } ( x ) = x$.
Hence write down the inverse function $\mathrm { f } ^ { - 1 } ( x )$. What can you deduce about the symmetry of the curve $y = \mathrm { f } ( x )$ ?

OCR MEI C3 2009 June Q7

Show that
(A) $( x - y ) \left( x ^ { 2 } + x y + y ^ { 2 } \right) = x ^ { 3 } - y ^ { 3 }$,
(B) $\left( x + \frac { 1 } { 2 } y \right) ^ { 2 } + \frac { 3 } { 4 } y ^ { 2 } = x ^ { 2 } + x y + y ^ { 2 }$.
Hence prove that, for all real numbers $x$ and $y$, if $x > y$ then $x ^ { 3 } > y ^ { 3 }$. Section B (36 marks)

OCR MEI C3 2009 June Q8

8 Fig. 8 shows the line $y = x$ and parts of the curves $y = \mathrm { f } ( x )$ and $y = \mathrm { g } ( x )$, where $$\mathrm { f } ( x ) = \mathrm { e } ^ { x - 1 } , \quad \mathrm {~g} ( x ) = 1 + \ln x$$ The curves intersect the axes at the points A and B , as shown. The curves and the line $y = x$ meet at the point C . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1167a0e5-48c8-48e0-b2d1-76a50bad03ad-3_807_897_1016_625} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Find the exact coordinates of A and B . Verify that the coordinates of C are $( 1,1 )$.
Prove algebraically that $\mathrm { g } ( x )$ is the inverse of $\mathrm { f } ( x )$.
Evaluate $\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$, giving your answer in terms of e.
Use integration by parts to find $\int \ln x \mathrm {~d} x$. Hence show that $\int _ { \mathrm { e } ^ { - 1 } } ^ { 1 } \mathrm {~g} ( x ) \mathrm { d } x = \frac { 1 } { \mathrm { e } }$.
Find the area of the region enclosed by the lines OA and OB , and the arcs AC and BC .

OCR MEI C3 2009 June Q9

9 Fig. 9 shows the curve $y = \frac { x ^ { 2 } } { 3 x - 1 }$.
P is a turning point, and the curve has a vertical asymptote $x = a$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1167a0e5-48c8-48e0-b2d1-76a50bad03ad-4_844_1486_447_331} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure}

Write down the value of $a$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ( 3 x - 2 ) } { ( 3 x - 1 ) ^ { 2 } }$.
Find the exact coordinates of the turning point P . Calculate the gradient of the curve when $x = 0.6$ and $x = 0.8$, and hence verify that P is a minimum point.
Using the substitution $u = 3 x - 1$, show that $\int \frac { x ^ { 2 } } { 3 x - 1 } \mathrm {~d} x = \frac { 1 } { 27 } \int \left( u + 2 + \frac { 1 } { u } \right) \mathrm { d } u$. Hence find the exact area of the region enclosed by the curve, the $x$-axis and the lines $x = \frac { 2 } { 3 }$ and $x = 1$.

OCR MEI C3 2011 June Q1

1 Solve the equation $| 2 x - 1 | = | x |$.

OCR MEI C3 2011 June Q2

2 Given that $\mathrm { f } ( x ) = 2 \ln x$ and $\mathrm { g } ( x ) = \mathrm { e } ^ { x }$, find the composite function $\mathrm { gf } ( x )$, expressing your answer as simply as possible.

OCR MEI C3 2011 June Q4

4 The height $h$ metres of a tree after $t$ years is modelled by the equation $$h = a - b \mathrm { e } ^ { - k t }$$ where $a$, $b$ and $k$ are positive constants.

Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of $a$ and $b$.
Given also that the tree grows to a height of 6 metres in 8 years, find the value of $k$, giving your answer correct to 2 decimal places.

OCR MEI C3 2011 June Q5

5 Given that $y = x ^ { 2 } \sqrt { 1 + 4 x }$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ( 5 x + 1 ) } { \sqrt { 1 + 4 x } }$.

OCR MEI C3 2011 June Q6

6 A curve is defined by the equation $\sin 2 x + \cos y = \sqrt { 3 }$.

Verify that the point $\mathrm { P } \left( \frac { 1 } { 6 } \pi , \frac { 1 } { 6 } \pi \right)$ lies on the curve.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. Hence find the gradient of the curve at the point P .
Multiply out $\left( 3 ^ { n } + 1 \right) \left( 3 ^ { n } - 1 \right)$.
Hence prove that if $n$ is a positive integer then $3 ^ { 2 n } - 1$ is divisible by 8 .

OCR MEI C3 2011 June Q8

8 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{82825739-6239-4afd-9621-538d35c09f28-3_479_1061_342_541} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Fig. 8 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } + 2 }$.

Show algebraically that $\mathrm { f } ( x )$ is an even function, and state how this property relates to the curve $y = \mathrm { f } ( x )$.
Find $\mathrm { f } ^ { \prime } ( x )$.
Show that $\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }$.
Hence, using the substitution $u = \mathrm { e } ^ { x } + 1$, or otherwise, find the exact area enclosed by the curve $y = \mathrm { f } ( x )$, the $x$-axis, and the lines $x = 0$ and $x = 1$.
Show that there is only one point of intersection of the curves $y = \mathrm { f } ( x )$ and $y = \frac { 1 } { 4 } \mathrm { e } ^ { x }$, and find its coordinates.

OCR MEI C3 2011 June Q9

9 Fig. 9 shows the curve $y = \mathrm { f } ( x )$. The endpoints of the curve are $\mathrm { P } ( - \pi , 1 )$ and $\mathrm { Q } ( \pi , 3 )$, and $\mathrm { f } ( x ) = a + \sin b x$, where $a$ and $b$ are constants. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{82825739-6239-4afd-9621-538d35c09f28-4_663_1265_386_440} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure}

Using Fig. 9, show that $a = 2$ and $b = \frac { 1 } { 2 }$.
Find the gradient of the curve $y = \mathrm { f } ( x )$ at the point $( 0,2 )$. Show that there is no point on the curve at which the gradient is greater than this.
Find $\mathrm { f } ^ { - 1 } ( x )$, and state its domain and range. Write down the gradient of $y = \mathrm { f } ^ { - 1 } ( x )$ at the point $( 2,0 )$.
Find the area enclosed by the curve $y = \mathrm { f } ( x )$, the $x$-axis, the $y$-axis and the line $x = \pi$.

OCR MEI C3 2012 June Q2

2 Solve the inequality $| 2 x + 1 | > 4$.

OCR MEI C3 2012 June Q3

3 Find the gradient at the point $( 0 , \ln 2 )$ on the curve with equation $\mathrm { e } ^ { 2 y } = 5 - \mathrm { e } ^ { - x }$.

OCR MEI C3 2012 June Q4

4 Fig. 4 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-2_476_572_861_751} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure}

Find the value of $a$.
Write down the range of $\mathrm { f } ( x )$.
Sketch the curve $y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1$.

OCR MEI C3 2012 June Q5

5 A termites' nest has a population of $P$ million. $P$ is modelled by the equation $P = 7 - 2 \mathrm { e } ^ { - k t }$, where $t$ is in years, and $k$ is a positive constant.

Calculate the population when $t = 0$, and the long-term population, given by this model.
Given that the population when $t = 1$ is estimated to be 5.5 million, calculate the value of $k$.

OCR MEI C3 2012 June Q6

6 Fig. 6 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = 2 \arcsin x , - 1 \leqslant x \leqslant 1$.
Fig. 6 also shows the curve $y = \mathrm { g } ( x )$, where $\mathrm { g } ( x )$ is the inverse function of $\mathrm { f } ( x )$.
P is the point on the curve $y = \mathrm { f } ( x )$ with $x$-coordinate $\frac { 1 } { 2 }$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-3_711_693_466_685} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure}

Find the $y$-coordinate of P , giving your answer in terms of $\pi$. The point Q is the reflection of P in $y = x$.
Find $\mathrm { g } ( x )$ and its derivative $\mathrm { g } ^ { \prime } ( x )$. Hence determine the exact gradient of the curve $y = \mathrm { g } ( x )$ at the point Q . Write down the exact gradient of $y = \mathrm { f } ( x )$ at the point P .

OCR MEI C3 2012 June Q7

7 You are given that $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ are odd functions, defined for $x \in \mathbb { R }$.

Given that $\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )$, prove that $\mathrm { s } ( x )$ is an odd function.
Given that $\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )$, determine whether $\mathrm { p } ( x )$ is odd, even or neither.

OCR MEI C3 2012 June Q8

8 Fig. 8 shows a sketch of part of the curve $y = x \sin 2 x$, where $x$ is in radians.
The curve crosses the $x$-axis at the point P . The tangent to the curve at P crosses the $y$-axis at Q . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-4_712_923_463_571} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$. Hence show that the $x$-coordinates of the turning points of the curve satisfy the equation $\tan 2 x + 2 x = 0$.
Find, in terms of $\pi$, the $x$-coordinate of the point P . Show that the tangent PQ has equation $2 \pi x + 2 y = \pi ^ { 2 }$.
Find the exact coordinates of Q .
Show that the exact value of the area shaded in Fig. 8 is $\frac { 1 } { 8 } \pi \left( \pi ^ { 2 } - 2 \right)$.

OCR MEI C3 2012 June Q9

9 Fig. 9 shows the curve $y = \mathrm { f } ( x )$, which has a $y$-intercept at $\mathrm { P } ( 0,3 )$, a minimum point at $\mathrm { Q } ( 1,2 )$, and an asymptote $x = - 1$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-5_906_944_333_566} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure}

Find the coordinates of the images of the points P and Q when the curve $y = \mathrm { f } ( x )$ is transformed to
(A) $y = 2 \mathrm { f } ( x )$,
(B) $y = \mathrm { f } ( x + 1 ) + 2$. You are now given that $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1$.
Find $\mathrm { f } ^ { \prime } ( x )$, and hence find the coordinates of the other turning point on the curve $y = \mathrm { f } ( x )$.
Show that $\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }$.
Find $\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x$ in terms of $a$ and $b$. Hence, by choosing suitable values for $a$ and $b$, find the exact area enclosed by the curve $y = \mathrm { f } ( x )$, the $x$-axis, the $y$-axis and the line $x = 1$.

OCR MEI C3 2013 June Q1

1 Fig. 1 shows the graphs of $y = | x |$ and $y = a | x + b |$, where $a$ and $b$ are constants. The intercepts of $y = a | x + b |$ with the $x$-and $y$-axes are $( - 1,0 )$ and $\left( 0 , \frac { 1 } { 2 } \right)$ respectively. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{28ce1bcc-e9d5-4ae6-98c0-67b5b8c50bc6-2_624_958_468_539} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure}

Find $a$ and $b$.
Find the coordinates of the two points of intersection of the graphs.

Questions — OCR MEI C3 (366 questions)

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