Questions - OCR - A-Level Maths

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OCR C1 2011 January Q8

13 marks Moderate -0.8

Find the equation of the tangent to the curve $y = 7 + 6 x - x ^ { 2 }$ at the point $P$ where $x = 5$, giving your answer in the form $a x + b y + c = 0$.
This tangent meets the $x$-axis at $Q$. Find the coordinates of the mid-point of $P Q$.
Find the equation of the line of symmetry of the curve $y = 7 + 6 x - x ^ { 2 }$.
State the set of values of $x$ for which $7 + 6 x - x ^ { 2 }$ is an increasing function.

OCR C1 2011 January Q9

15 marks Standard +0.3

9 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0$.

Find the coordinates of $C$ and the radius of the circle.
Find the values of $k$ for which the line $y = k$ is a tangent to the circle, giving your answers in simplified surd form.
The points $S$ and $T$ lie on the circumference of the circle. $M$ is the mid-point of the chord $S T$. Given that the length of $C M$ is 2 , calculate the length of the chord $S T$.
Find the coordinates of the point where the circle meets the line $x - 2 y - 12 = 0$.

OCR C1 2012 January Q1

4 marks Easy -1.2

1 Express $\frac { 15 + \sqrt { 3 } } { 3 - \sqrt { 3 } }$ in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.

OCR C1 2012 January Q2

4 marks Easy -1.2

2
\includegraphics[max width=\textwidth, alt={}, center]{559e9f1a-340e-4478-adaa-a7361dd70fe8-2_325_479_468_794} The graph of $y = \mathrm { f } ( x )$ for $- 2 \leqslant x \leqslant 2$ is shown above.

Sketch the graph of $y = \mathrm { f } ( - x )$ for $- 2 \leqslant x \leqslant 2$.
Sketch the graph of $y = \mathrm { f } ( x ) + 2$ for $- 2 \leqslant x \leqslant 2$.

OCR C1 2012 January Q3

4 marks Moderate -0.8

3 Given that $$5 x ^ { 2 } + p x - 8 = q ( x - 1 ) ^ { 2 } + r$$ for all values of $x$, find the values of the constants $p , q$ and $r$.

OCR C1 2012 January Q4

5 marks Easy -1.8

4 Evaluate

$3 ^ { - 2 }$,
$16 ^ { \frac { 3 } { 4 } }$,
$\frac { \sqrt { 200 } } { \sqrt { 8 } }$.

OCR C1 2012 January Q5

5 marks Moderate -0.3

5 Find the real roots of the equation $\frac { 3 } { y ^ { 4 } } - \frac { 10 } { y ^ { 2 } } - 8 = 0$.

OCR C1 2012 January Q6

7 marks Moderate -0.8

6 Given that $\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2$,

find $\mathrm { f } ^ { \prime } ( x )$,
find $\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)$.

OCR C1 2012 January Q7

12 marks Moderate -0.3

7 A curve has equation $y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)$.

Find the coordinates of the minimum point, justifying that it is a minimum.
Calculate the discriminant of $x ^ { 2 } - 3 x + 5$.
Explain why $( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)$ is always positive for $x > - 2$.

OCR C1 2012 January Q8

6 marks Standard +0.3

8 The line $l$ has gradient - 2 and passes through the point $A ( 3,5 ) . B$ is a point on the line $l$ such that the distance $A B$ is $6 \sqrt { 5 }$. Find the coordinates of each of the possible points $B$.

OCR C1 2012 January Q9

12 marks Moderate -0.3

Sketch the curve $y = 12 - x - x ^ { 2 }$, giving the coordinates of all intercepts with the axes.
Solve the inequality $12 - x - x ^ { 2 } > 0$.
Find the coordinates of the points of intersection of the curve $y = 12 - x - x ^ { 2 }$ and the line $3 x + y = 4$.

OCR C1 2012 January Q10

13 marks Moderate -0.3

10 A circle has centre $C ( - 2,4 )$ and radius 5 .

Find the equation of the circle, giving your answer in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$.
Show that the tangent to the circle at the point $P ( - 5,8 )$ has equation $3 x - 4 y + 47 = 0$.
Verify that the point $T ( 3,14 )$ lies on this tangent.
Find the area of the triangle $C P T$. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

OCR C1 2013 January Q1

5 marks Moderate -0.8

Solve the equation $x ^ { 2 } - 6 x - 2 = 0$, giving your answers in simplified surd form.
Find the gradient of the curve $y = x ^ { 2 } - 6 x - 2$ at the point where $x = - 5$.

OCR C1 2013 January Q2

6 marks Easy -1.2

2 Solve the equations

$3 ^ { n } = 1$,
$t ^ { - 3 } = 64$,
$\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8$.

OCR C1 2013 January Q3

5 marks Moderate -0.3

Sketch the curve $y = ( 1 + x ) ( 2 - x ) ( 3 + x )$, giving the coordinates of all points of intersection with the axes.
Describe the transformation that transforms the curve $y = ( 1 + x ) ( 2 - x ) ( 3 + x )$ to the curve $y = ( 1 - x ) ( 2 + x ) ( 3 - x )$.

OCR C1 2013 January Q4

6 marks Moderate -0.8

Solve the simultaneous equations $$y = 2 x ^ { 2 } - 3 x - 5 , \quad 10 x + 2 y + 11 = 0$$
What can you deduce from the answer to part (i) about the curve $y = 2 x ^ { 2 } - 3 x - 5$ and the line $10 x + 2 y + 11 = 0$ ?
Simplify $( x + 4 ) ( 5 x - 3 ) - 3 ( x - 2 ) ^ { 2 }$.
The coefficient of $x ^ { 2 }$ in the expansion of $$( x + 3 ) ( x + k ) ( 2 x - 5 )$$ is - 3 . Find the value of the constant $k$.

OCR C1 2013 January Q6

10 marks Moderate -0.8

The line joining the points ( $- 2,7$ ) and ( $- 4 , p$ ) has gradient 4 . Find the value of $p$.
The line segment joining the points $( - 2,7 )$ and $( 6 , q )$ has mid-point $( m , 5 )$. Find $m$ and $q$.
The line segment joining the points $( - 2,7 )$ and $( d , 3 )$ has length $2 \sqrt { 13 }$. Find the two possible values of $d$.

OCR C1 2013 January Q7

8 marks Easy -1.2

7 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases:

$y = \frac { ( 3 x ) ^ { 2 } \times x ^ { 4 } } { x }$,
$y = \sqrt [ 3 ] { x }$,
$y = \frac { 1 } { 2 x ^ { 3 } }$.

OCR C1 2013 January Q8

7 marks Standard +0.3

8 The quadratic equation $k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0$ has no real roots. Find the set of possible values of $k$.

OCR C1 2013 January Q9

9 marks Moderate -0.3

9 A circle with centre $C$ has equation $x ^ { 2 } + y ^ { 2 } - 2 x + 10 y - 19 = 0$.

Find the coordinates of $C$ and the radius of the circle.
Verify that the point $( 7 , - 2 )$ lies on the circumference of the circle.
Find the equation of the tangent to the circle at the point $( 7 , - 2 )$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.