Questions - OCR MEI C1 - A-Level Maths

OCR MEI C1 2011 June Q12

12 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fecb40da-cf47-45e0-801a-1d3d8811b5a0-3_840_919_849_612} \captionsetup{labelformat=empty} \caption{Fig. 12}

\end{figure} Fig. 12 shows the graph of $y = \frac { 4 } { x ^ { 2 } }$.

On the copy of Fig. 12, draw accurately the line $y = 2 x + 5$ and hence find graphically the three roots of the equation $\frac { 4 } { x ^ { 2 } } = 2 x + 5$.
Show that the equation you have solved in part (i) may be written as $2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0$. Verify that $x = - 2$ is a root of this equation and hence find, in exact form, the other two roots.
By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation $x ^ { 3 } + 2 x ^ { 2 } - 4 = 0$.

OCR MEI C1 2011 June Q13

13 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fecb40da-cf47-45e0-801a-1d3d8811b5a0-4_783_766_255_687} \captionsetup{labelformat=empty} \caption{Fig. 13}

\end{figure} Fig. 13 shows the circle with equation $( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 16$.

Write down the radius of the circle and the coordinates of its centre.
Find the $x$-coordinates of the points where the circle crosses the $x$-axis. Give your answers in surd form.
Show that the point $\mathrm { A } ( 4 + 2 \sqrt { 2 } , 2 + 2 \sqrt { 2 } )$ lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it.
Find the equations of both these tangents.

OCR MEI C1 2012 June Q1

1 Find the equation of the line with gradient - 2 which passes through the point $( 3,1 )$. Give your answer in the form $y = a x + b$. Find also the points of intersection of this line with the axes.

OCR MEI C1 2012 June Q2

2 Make $b$ the subject of the following formula. $$a = \frac { 2 } { 3 } b ^ { 2 } c$$

OCR MEI C1 2012 June Q3

3

Evaluate $\left( \frac { 1 } { 5 } \right) ^ { - 2 }$.
Evaluate $\left( \frac { 8 } { 27 } \right) ^ { \frac { 2 } { 3 } }$.

OCR MEI C1 2012 June Q4

4 Factorise and hence simplify the following expression. $$\frac { x ^ { 2 } - 9 } { x ^ { 2 } + 5 x + 6 }$$

OCR MEI C1 2012 June Q5

5

Simplify $\frac { 10 ( \sqrt { 6 } ) ^ { 3 } } { \sqrt { 24 } }$.
Simplify $\frac { 1 } { 4 - \sqrt { 5 } } + \frac { 1 } { 4 + \sqrt { 5 } }$.

OCR MEI C1 2012 June Q6

6

Evaluate ${ } ^ { 5 } \mathrm { C } _ { 3 }$.
Find the coefficient of $x ^ { 3 }$ in the expansion of $( 3 - 2 x ) ^ { 5 }$.

OCR MEI C1 2012 June Q8

8 The function $\mathrm { f } ( x ) = x ^ { 4 } + b x + c$ is such that $\mathrm { f } ( 2 ) = 0$. Also, when $\mathrm { f } ( x )$ is divided by $x + 3$, the remainder is 85 . Find the values of $b$ and $c$.

OCR MEI C1 2012 June Q9

9 Simplify $( n + 3 ) ^ { 2 } - n ^ { 2 }$. Hence explain why, when $n$ is an integer, $( n + 3 ) ^ { 2 } - n ^ { 2 }$ is never an even number. Given also that $( n + 3 ) ^ { 2 } - n ^ { 2 }$ is divisible by 9 , what can you say about $n$ ? \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0088b5e7-d587-419a-a13b-87527ac658c4-3_442_762_379_648} \captionsetup{labelformat=empty} \caption{Fig. 10}

\end{figure} Fig. 10 is a sketch of quadrilateral ABCD with vertices $\mathrm { A } ( 1,5 ) , \mathrm { B } ( - 1,1 ) , \mathrm { C } ( 3 , - 1 )$ and $\mathrm { D } ( 11,5 )$.

Show that $\mathrm { AB } = \mathrm { BC }$.
Show that the diagonals AC and BD are perpendicular.
Find the midpoint of AC . Show that BD bisects AC but AC does not bisect BD .

OCR MEI C1 2012 June Q11

11 A cubic curve has equation $y = \mathrm { f } ( x )$. The curve crosses the $x$-axis where $x = - \frac { 1 } { 2 } , - 2$ and 5 .

Write down three linear factors of $\mathrm { f } ( x )$. Hence find the equation of the curve in the form $y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c$.
Sketch the graph of $y = \mathrm { f } ( x )$.
The curve $y = \mathrm { f } ( x )$ is translated by $\binom { 0 } { - 8 }$. State the coordinates of the point where the translated curve intersects the $y$-axis.
The curve $y = \mathrm { f } ( x )$ is translated by $\binom { 3 } { 0 }$ to give the curve $y = \mathrm { g } ( x )$. Find an expression in factorised form for $\mathrm { g } ( x )$ and state the coordinates of the point where the curve $y = \mathrm { g } ( x )$ intersects the $y$-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0088b5e7-d587-419a-a13b-87527ac658c4-4_1287_1410_292_315} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of $y = \frac { 1 } { x - 3 }$.
Draw accurately, on the copy of Fig. 12, the graph of $y = x ^ { 2 } - 4 x + 1$ for $- 1 \leqslant x \leqslant 5$. Use your graph to estimate the coordinates of the intersections of $y = \frac { 1 } { x - 3 }$ and $y = x ^ { 2 } - 4 x + 1$.
Show algebraically that, where the curves intersect, $x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0$.
Use the fact that $x = 4$ is a root of $x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0$ to find a quadratic factor of $x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4$. Hence find the exact values of the other two roots of this equation. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR MEI C1 2013 June Q1

1 Find the equation of the line which is perpendicular to the line $y = 2 x - 5$ and which passes through the point $( 4,1 )$. Give your answer in the form $y = a x + b$.

OCR MEI C1 2013 June Q2

2 Find the coordinates of the point of intersection of the lines $y = 3 x - 2$ and $x + 3 y = 1$.

OCR MEI C1 2013 June Q3

3

Evaluate $( 0.2 ) ^ { - 2 }$.
Simplify $\left( 16 a ^ { 12 } \right) ^ { \frac { 3 } { 4 } }$.

OCR MEI C1 2013 June Q4

4 Rearrange the following formula to make $r$ the subject, where $r > 0$. $$V = \frac { 1 } { 3 } \pi r ^ { 2 } ( a + b )$$

OCR MEI C1 2013 June Q5

5 You are given that $\mathrm { f } ( x ) = x ^ { 5 } + k x - 20$. When $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 18 . Find the value of $k$.

OCR MEI C1 2013 June Q6

6 Find the coefficient of $x ^ { 3 }$ in the binomial expansion of $( 2 - 4 x ) ^ { 5 }$.

OCR MEI C1 2013 June Q7

7

Express $125 \sqrt { 5 }$ in the form $5 ^ { k }$.
Simplify $10 + 7 \sqrt { 5 } + \frac { 38 } { 1 - 2 \sqrt { 5 } }$, giving your answer in the form $a + b \sqrt { 5 }$.

OCR MEI C1 2013 June Q8

8 Express $3 x ^ { 2 } - 12 x + 5$ in the form $a ( x - b ) ^ { 2 } - c$. Hence state the minimum value of $y$ on the curve $y = 3 x ^ { 2 } - 12 x + 5$.
$9 \quad n - 1 , n$ and $n + 1$ are any three consecutive integers.

Show that the sum of these integers is always divisible by 3 .
Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3 .

OCR MEI C1 2013 June Q10

10 The circle $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20$ has centre C.

Write down the radius of the circle and the coordinates of C .
Find the coordinates of the intersections of the circle with the $x$ - and $y$-axes.
Show that the points $\mathrm { A } ( 1,6 )$ and $\mathrm { B } ( 7,4 )$ lie on the circle. Find the coordinates of the midpoint of AB . Find also the distance of the chord AB from the centre of the circle.

OCR MEI C1 2013 June Q11

11 You are given that $\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )$.

Sketch the graph of $y = \mathrm { f } ( x )$.
State the roots of $\mathrm { f } ( x - 2 ) = 0$.
You are also given that $\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15$.
(A) Show that $\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9$.
(B) Show that $\mathrm { g } ( 1 ) = 0$ and hence factorise $\mathrm { g } ( x )$ completely.

OCR MEI C1 2013 June Q12

4 marks

12 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ec0ef896-9290-4cdd-8a6f-11ece1cb141d-4_1255_1202_255_434} \captionsetup{labelformat=empty} \caption{Fig. 12}

\end{figure} Fig. 12 shows the graph of $y = \frac { 1 } { x - 2 }$.

Draw accurately the graph of $y = 2 x + 3$ on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of $y = \frac { 1 } { x - 2 }$ and $y = 2 x + 3$.
Show algebraically that the $x$-coordinates of the points of intersection of $y = \frac { 1 } { x - 2 }$ and $y = 2 x + 3$ satisfy the equation $2 x ^ { 2 } - x - 7 = 0$. Hence find the exact values of the $x$-coordinates of the points of intersection.
Find the quadratic equation satisfied by the $x$-coordinates of the points of intersection of $y = \frac { 1 } { x - 2 }$ and $y = - x + k$. Hence find the exact values of $k$ for which $y = - x + k$ is a tangent to $y = \frac { 1 } { x - 2 }$. [4]

OCR MEI C1 2014 June Q1

1

Evaluate $\left( \frac { 1 } { 27 } \right) ^ { \frac { 2 } { 3 } }$.
Simplify $\frac { \left( 4 a ^ { 2 } c \right) ^ { 3 } } { 32 a ^ { 4 } c ^ { 7 } }$.

OCR MEI C1 2014 June Q2

2 A is the point $( 1,5 )$ and B is the point $( 6 , - 1 )$. M is the midpoint of AB . Determine whether the line with equation $y = 2 x - 5$ passes through M.

OCR MEI C1 2014 June Q3

3 \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-2_798_819_836_623} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Fig. 3 shows the graph of $y = \mathrm { f } ( x )$. Draw the graphs of the following.

$y = \mathrm { f } ( x ) - 2$
$y = \mathrm { f } ( x - 3 )$

Questions — OCR MEI C1 (472 questions)

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