Questions C1 - A-Level Maths

Find the coordinates of the points where $C$ crosses the $x$-axis.
Find the exact coordinates of the stationary point of $C$.
Determine the nature of the stationary point.
Sketch the curve $C$.

OCR C1 Q1

Easy -1.2

Express $\sqrt { 50 } + 3 \sqrt { 8 }$ in the form $k \sqrt { 2 }$.
Find the coordinates of the stationary point of the curve with equation

$$y = x + \frac { 4 } { x ^ { 2 } } .$$

OCR C1 Q3

Standard +0.3

\includegraphics[max width=\textwidth, alt={}]{4a5e8809-b4f6-4d24-b3f9-741eea5cc450-1_522_919_705_411}

The diagram shows the curve with equation $y = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b$ and $c$ are constants. The curve crosses the $x$-axis at the point $( - 1,0 )$ and touches the $x$-axis at the point $( 3,0 )$. Show that $a = - 5$ and find the values of $b$ and $c$.

OCR C1 Q4

Moderate -0.8

4. The curve $C$ has the equation $y = ( x - a ) ^ { 2 }$ where $a$ is a constant. Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$

find the value of $a$,
describe fully a single transformation that would map $C$ onto the graph of $y = x ^ { 2 }$.

OCR C1 Q5

Moderate -0.8

5. The straight line $l _ { 1 }$ has the equation $3 x - y = 0$. The straight line $l _ { 2 }$ has the equation $x + 2 y - 4 = 0$.

Sketch $l _ { 1 }$ and $l _ { 2 }$ on the same diagram, showing the coordinates of any points where each line meets the coordinate axes.
Find, as exact fractions, the coordinates of the point where $l _ { 1 }$ and $l _ { 2 }$ intersect.

OCR C1 Q6

Moderate -0.8

6. (a) Given that $y = 2 ^ { x }$, find expressions in terms of $y$ for

$2 ^ { x + 2 }$,
$2 ^ { 3 - x }$.
(b) Show that using the substitution $y = 2 ^ { x }$, the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$ can be rewritten as $$4 y ^ { 2 } - 33 y + 8 = 0$$ (c) Hence solve the equation $$2 ^ { x + 2 } + 2 ^ { 3 - x } = 33$$

OCR C1 Q7

Moderate -0.8

The point $A$ has coordinates ( 4,6 ).

Given that $O A$, where $O$ is the origin, is a diameter of circle $C$,

find an equation for $C$. Circle $C$ crosses the $x$-axis at $O$ and at the point $B$.
Find the coordinates of $B$.
Find an equation for the tangent to $C$ at $B$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.

OCR C1 Q8

Moderate -0.5

8. (i) Express $3 x ^ { 2 } - 12 x + 11$ in the form $a ( x + b ) ^ { 2 } + c$.
(ii) Sketch the curve with equation $y = 3 x ^ { 2 } - 12 x + 11$, showing the coordinates of the minimum point of the curve. Given that the curve $y = 3 x ^ { 2 } - 12 x + 11$ crosses the $x$-axis at the points $A$ and $B$,
(iii) find the length $A B$ in the form $k \sqrt { 3 }$.

OCR C1 Q9

Standard +0.3

9. A curve has the equation $y = x ^ { 3 } - 5 x ^ { 2 } + 7 x$.

Show that the curve only crosses the $x$-axis at one point. The point $P$ on the curve has coordinates $( 3,3 )$.
Find an equation for the normal to the curve at $P$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers. The normal to the curve at $P$ meets the coordinate axes at $Q$ and $R$.
Show that triangle $O Q R$, where $O$ is the origin, has area $28 \frac { 1 } { 8 }$.

OCR C1 Q1

Easy -1.8

Solve the inequality

$$4 ( x - 2 ) < 2 x + 5$$

OCR C1 Q2

Moderate -0.8

2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that $\mathrm { f } ( x )$ is decreasing for all values of $x$.

OCR C1 Q3

Moderate -0.3

3. (i) Solve the equation $$y ^ { 2 } + 8 = 9 y .$$ (ii) Hence solve the equation $$x ^ { 3 } + 8 = 9 x ^ { \frac { 3 } { 2 } } .$$

OCR C1 Q4

Moderate -0.8

Given that

$$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } } ,$$

find $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }$.

OCR C1 Q5

Moderate -0.5

5. Find the pairs of values $( x , y )$ which satisfy the simultaneous equations $$\begin{aligned} & 3 x ^ { 2 } + y ^ { 2 } = 21 \\ & 5 x + y = 7 \end{aligned}$$

OCR C1 Q6

Moderate -0.8

(i) Evaluate $\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }$.
(ii) Find the value of $x$ such that

$$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form $a + b \sqrt { 3 }$ where $a$ and $b$ are rational.

OCR C1 Q7

Moderate -0.8

7. The straight line $l$ passes through the point $P ( - 3,6 )$ and the point $Q ( 1 , - 4 )$.

Find an equation for $l$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The straight line $m$ has the equation $2 x + k y + 7 = 0$, where $k$ is a constant.
Given that $l$ and $m$ are perpendicular,
find the value of $k$.

OCR C1 Q8

Standard +0.3

8. (i) Describe fully a single transformation that maps the graph of $y = \frac { 1 } { x }$ onto the graph of $y = \frac { 3 } { x }$.
(ii) Sketch the graph of $y = \frac { 3 } { x }$ and write down the equations of any asymptotes.
(iii) Find the values of the constant $c$ for which the straight line $y = c - 3 x$ is a tangent to the curve $y = \frac { 3 } { x }$.

OCR C1 Q9

Moderate -0.3

9. The circle $C$ has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$

Find the coordinates of the centre of $C$.
Find the radius of $C$.
Sketch $C$. Given that $C$ crosses the $x$-axis at the points $A$ and $B$,
find the length $A B$, giving your answer in the form $k \sqrt { 5 }$.

OCR C1 Q10

Moderate -0.3

10.
\includegraphics[max width=\textwidth, alt={}, center]{76efaf91-a6f3-4493-88d4-3654b023441d-3_646_773_986_477} The diagram shows the curve $y = x ^ { 2 } - 3 x + 5$ and the straight line $y = 2 x + 1$. The curve and line intersect at the points $P$ and $Q$.

Using algebra, show that $P$ has coordinates $( 1,3 )$ and find the coordinates of $Q$.
Find an equation for the tangent to the curve at $P$.
Show that the tangent to the curve at $Q$ has the equation $y = 5 x - 11$.
Find the coordinates of the point where the tangent to the curve at $P$ intersects the tangent to the curve at $Q$.

OCR MEI C1 Q2

Moderate -0.3

2 Fig. 8 shows a right-angled triangle with base $2 x + 1$, height $h$ and hypotenuse $3 x$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a3414ad8-7959-49f8-b43d-3972b2c03642-1_316_590_631_582} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure} Not to scale

Show that $h ^ { 2 } = 5 x ^ { 2 } - 4 x - 1$.
Given that $h = \sqrt { 7 }$, find the value of $x$, giving your answer in surd form.

OCR MEI C1 Q3

Moderate -0.3

Find the set of values of $k$ for which the line $y = 2 x + k$ intersects the curve $y = 3 x ^ { 2 } + 12 x + 13$ at two distinct points.
Express $3 x ^ { 2 } + 12 x + 13$ in the form $a ( x + b ) ^ { 2 } + c$. Hence show that the curve $y = 3 x ^ { 2 } + 12 x + 13$ lies completely above the $x$-axis.
Find the value of $k$ for which the line $y = 2 x + k$ passes through the minimum point of the curve $y = 3 x ^ { 2 } + 12 x + 13$.