Questions - OCR C1 - A-Level Maths

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OCR C1 Q5

6 marks Moderate -0.3

Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
What can you deduce from the solution to part (i) about the graphs of $y = x ^ { 2 } - 3 x + 2$ and $y = 3 x - 7$ ?
Hence, or otherwise, find the equation of the normal to the curve $y = x ^ { 2 } - 3 x + 2$ at the point ( 3,2 ), giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers.

OCR C1 Q7

13 marks Moderate -0.8

7 \includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points $A ( 2,9 )$ and $B ( 10,3 ) . A B$ is a diameter of the circle.

Calculate the radius of the circle and the coordinates of the centre.
Show that the equation of the circle may be written in the form $x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0$.
The tangent to the circle at the point $B$ cuts the $x$-axis at $C$. Find the coordinates of $C$.

OCR C1 2005 January Q1

6 marks Easy -1.8

Express $11 ^ { - 2 }$ as a fraction.
Evaluate $100 ^ { \frac { 3 } { 2 } }$.
Express $\sqrt { 50 } + \frac { 6 } { \sqrt { 3 } }$ in the form $a \sqrt { } 2 + b \sqrt { } 3$, where $a$ and $b$ are integers.

OCR C1 2005 January Q2

4 marks Moderate -0.8

2 Given that $2 x ^ { 2 } - 12 x + p = q ( x - r ) ^ { 2 } + 10$ for all values of $x$, find the constants $p , q$ and $r$.

OCR C1 2005 January Q3

4 marks Easy -1.2

The curve $y = 5 \sqrt { } x$ is transformed by a stretch, scale factor $\frac { 1 } { 2 }$, parallel to the $x$-axis. Find the equation of the curve after it has been transformed.
Describe the single transformation which transforms the curve $y = 5 \sqrt { } x$ to the curve $y = ( 5 \sqrt { } x ) - 3$.

OCR C1 2005 January Q4

5 marks Moderate -0.5

4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$

OCR C1 2005 January Q5

7 marks Easy -1.2

5 On separate diagrams,

sketch the curve $y = \frac { 1 } { x }$,
sketch the curve $y = x \left( x ^ { 2 } - 1 \right)$, stating the coordinates of the points where it crosses the $x$-axis,
sketch the curve $y = - \sqrt { } x$.

OCR C1 2005 January Q6

7 marks Moderate -0.8

Calculate the discriminant of $- 2 x ^ { 2 } + 7 x + 3$ and hence state the number of real roots of the equation $- 2 x ^ { 2 } + 7 x + 3 = 0$.
The quadratic equation $2 x ^ { 2 } + ( p + 1 ) x + 8 = 0$ has equal roots. Find the possible values of $p$.

OCR C1 2005 January Q7

9 marks Easy -1.3

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

$y = \frac { 1 } { 2 } x ^ { 4 } - 3 x$,
$y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )$,
$y = \sqrt [ 5 ] { x }$.

OCR C1 2005 January Q8

8 marks Moderate -0.8

8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is $x$ metres.

The perimeter of the playground is greater than 64 m . Write down a linear inequality in $x$.
The area of the playground is less than $299 \mathrm {~m} ^ { 2 }$. Show that $( x - 13 ) ( x + 23 ) < 0$.
By solving the inequalities in parts (i) and (ii), determine the set of possible values of $x$.

OCR C1 2005 January Q9

9 marks Moderate -0.3

Find the gradient of the curve $y = 2 x ^ { 2 }$ at the point where $x = 3$.
At a point $A$ on the curve $y = 2 x ^ { 2 }$, the gradient of the normal is $\frac { 1 } { 8 }$. Find the coordinates of $A$. Points $P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)$ and $P _ { 3 } \left( 1.1 , y _ { 3 } \right)$ lie on the curve $y = k x ^ { 2 }$. The gradient of the chord $P _ { 1 } P _ { 3 }$ is 6.3 and the gradient of the chord $P _ { 1 } P _ { 2 }$ is 6.03.
What do these results suggest about the gradient of the tangent to the curve $y = k x ^ { 2 }$ at $P _ { 1 }$ ?
Deduce the value of $k$.

OCR C1 2005 January Q10

13 marks Moderate -0.8

10 The points $D , E$ and $F$ have coordinates $( - 2,0 ) , ( 0 , - 1 )$ and $( 2,3 )$ respectively.

Calculate the gradient of $D E$.
Find the equation of the line through $F$, parallel to $D E$, giving your answer in the form $a x + b y + c = 0$.
By calculating the gradient of $E F$, show that $D E F$ is a right-angled triangle.
Calculate the length of $D F$.
Use the results of parts (iii) and (iv) to show that the circle which passes through $D , E$ and $F$ has equation $x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0$.

OCR C1 2006 January Q1

4 marks Easy -1.8

1 Solve the equations

$x ^ { \frac { 1 } { 3 } } = 2$,
$10 ^ { \prime } = 1$,
$\left( y ^ { - 2 } \right) ^ { 2 } = \frac { 1 } { 81 }$.

OCR C1 2006 January Q2

5 marks Easy -1.2

Simplify $( 3 x + 1 ) ^ { 2 } - 2 ( 2 x - 3 ) ^ { 2 }$.
Find the coefficient of $x ^ { 3 }$ in the expansion of $$\left( 2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 3 \right) \left( x ^ { 2 } - 2 x + 1 \right)$$

OCR C1 2006 January Q3

5 marks Easy -1.2

3 Given that $y = 3 x ^ { 5 } - \sqrt { x } + 15$, find

$\frac { \mathrm { d } y } { \mathrm {~d} x }$,
$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.

OCR C1 2006 January Q4

7 marks Easy -1.2

Sketch the curve $y = \frac { 1 } { x ^ { 2 } }$.
Hence sketch the curve $y = \frac { 1 } { ( x - 3 ) ^ { 2 } }$.
Describe fully a transformation that transforms the curve $y = \frac { 1 } { x ^ { 2 } }$ to the curve $y = \frac { 2 } { x ^ { 2 } }$.

OCR C1 2006 January Q5

7 marks Moderate -0.8

Express $x ^ { 2 } + 3 x$ in the form $( x + a ) ^ { 2 } + b$.
Express $y ^ { 2 } - 4 y - \frac { 11 } { 4 }$ in the form $( y + p ) ^ { 2 } + q$. A circle has equation $x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0$.
Write down the coordinates of the centre of the circle.
Find the radius of the circle.

OCR C1 2006 January Q6

11 marks Moderate -0.8

Find the coordinates of the stationary points on the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 4$.
Determine whether each stationary point is a maximum point or a minimum point.
For what values of $x$ does $x ^ { 3 } - 3 x ^ { 2 } + 4$ increase as $x$ increases?

OCR C1 2006 January Q7

11 marks Standard +0.3

Solve the equation $x ^ { 2 } - 8 x + 11 = 0$, giving your answers in simplified surd form.
Hence sketch the curve $y = x ^ { 2 } - 8 x + 11$, labelling the points where the curve crosses the axes.
Solve the equation $y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0$, giving your answers in the form $p \pm q \sqrt { 5 }$.

OCR C1 2006 January Q8

11 marks Moderate -0.3

Given that $y = x ^ { 2 } - 5 x + 15$ and $5 x - y = 10$, show that $x ^ { 2 } - 10 x + 25 = 0$.
Find the discriminant of $x ^ { 2 } - 10 x + 25$.
What can you deduce from the answer to part (ii) about the line $5 x - y = 10$ and the curve $y = x ^ { 2 } - 5 x + 15$ ?
Solve the simultaneous equations $$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
Hence, or otherwise, find the equation of the normal to the curve $y = x ^ { 2 } - 5 x + 15$ at the point $( 5,15 )$, giving your answer in the form $a x + b y = c$, where $a , b$ and $c$ are integers.

OCR C1 2006 January Q9

11 marks Moderate -0.3

9 The points $A , B$ and $C$ have coordinates $( 5,1 ) , ( p , 7 )$ and $( 8,2 )$ respectively.

Given that the distance between points $A$ and $B$ is twice the distance between points $A$ and $C$, calculate the possible values of $p$.
Given also that the line passing through $A$ and $B$ has equation $y = 3 x - 14$, find the coordinates of the mid-point of $A B$.