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(a) Reflect triangle $T$ in the line $x = -1$. Label the image $A$. [2]
(b) Translate triangle $T$ using the vector $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Label the image $B$. [2]
(c) Describe fully the single transformation that maps triangle $T$ onto triangle $V$.
..............................................................................................................................
.............................................................................................................................. [3]
(d) Stretch triangle $V$ with factor $\frac{1}{2}$ and invariant line $x = 2$. Label the image $C$. [2]
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f(x) = 2x + 4
g(x) = x - 1
h(x) = x^2 - 3x
(a) Find
(i) f(3)
......................................... [1]
(ii) h(7)
......................................... [1]
(b) Find the value of x when g(x) = -6.
x = ......................................... [1]
(c) Find $f^{-1}(x)$.
$f^{-1}(x) = .........................................$ [2]
(d) Simplify $f(x) \times g(x) + 1$.
......................................... [2]
(e) Solve h(g(x)) = 0.
x = .................... or .................... [3]
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y is inversely proportional to the square of (x + 1).
(a) When x = 5, y = 1.
Find y in terms of x.
y = \text{.............................................} \; [2]
(b) Find y when x = 3.
y = \text{.............................................} \; [2]
(c) Find the value of x when y = (x + 1).
x = \text{.............................................} \; [3]
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(a) A group of 10 people were asked the time, correct to the nearest minute, they each spent listening to the news and reading the news on Monday.
The results are shown in the table.
| Minutes listening (x) | 1 | 9 | 3 | 3 | 11 | 8 | 2 | 7 | 1 | 4 |
| Minutes reading (y) | 5 | 10 | 9 | 7 | 2 | 1 | 3 | 6 | 11 | 5 |
(i) Complete the scatter diagram.
The first six points have been plotted for you.
[2]
(ii) Find the median time spent reading the news on Monday.
$\text{..........................} \text{ min }[1]$
(iii) Find the equation of the line of regression.
Give your answer in the form $\text{ } y = mx+c\text{ }$.
$y = \text{..........................} [2]$
(iv) On Tuesday, each person spends the same time listening to the news as they did on Monday.
They each spend 5 minutes longer reading the news than they did on Monday.
Write down the equation of the line of regression for Tuesday.
$y = \text{..........................} [1]$
(b) In February Sancho read the news for a total of 8 hours.
This was a reduction of 36\% from January.
Work out how long Sancho read the news in January.
$\text{.......................... hours }[2]$
(c) The bar chart shows the number of news articles read one day by each of 23 people.
Calculate the mean number of articles read.
$\text{..........................} [2]$
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(a)
(i) On the diagram sketch the lines $y = -\frac{1}{2}x + 3$, $2y = x + 5$ and $y = x$ for $-6 \leq x \leq 7$. [4]
(ii) Show, by shading, the region that satisfies these inequalities.
$y > -\frac{1}{2}x + 3$
$2y < x + 5$
$y > x$ [2]
(b)
$f(x) = (x-2)^3 - 5x + 12$ for $-1 \leq x \leq 4.7$
(i) On the diagram, sketch the graph of $y = f(x)$. [2]
(ii) Write down the coordinates of the local maximum. [ ( .................. , .................. ) ] [2]
(iii) The equation $(x-2)^3 - 5x + 12 = k$ has exactly 2 solutions. Find the values of $k$.
$k = ..................$ or $k = ..................$ [2]
(iv) $g(x) = -(x-1)^2$ for $-1 \leq x \leq 4.7$ On the diagram, sketch the graph of $y = g(x)$. [2]
(v) Solve $f(x) = g(x)$.
$x = .............................................$ [1]
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(a) (i) Kayla walks from $A$ to $B$ on a bearing of $105^\circ$. She then walks back to $A$.
Calculate the bearing Kayla walks from $B$ to $A$.
(ii) The distance from $A$ to $B$ is $1.5$ km.
(b) It takes Kayla 24 minutes to walk from $A$ to $B$.
Calculate her average speed in km/h.
(c) Kayla has a map with a scale of $1 : 25000$ showing $A$ and $B$.
Work out the length of $AB$ on the map. Give your answer in centimetres.
(d) A train is 770 m long. The train takes 2 minutes and 36 seconds to travel completely through a tunnel. Its speed through the tunnel is 120 km/h.
Work out the length of the tunnel. Give your answer in metres.
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In triangle $ACD$, $AB = 5\, \text{cm}$, $AD = 7.63\, \text{cm}$ and $BD = 7\, \text{cm}$. Angle $BDC = 12^\circ$.
(a) Show that angle $ABD = 77.0^\circ$ correct to 1 decimal place. [3]
(b) Calculate the area of triangle $ABD$. ......................................... $\text{cm}^2$ [2]
(c) Calculate $BC$. ......................................... cm [4]
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(a)
$ABCDE$ is a pyramid with a rectangular base. $AB = 10cm$ and $BC = 24cm$. The length of each sloping edge is $20cm$. Vertex $E$ is vertically above the centre of the base, $M$.
(i) Calculate the length $AC$.
............................................... cm [2]
(ii) Calculate $EM$, the height of the pyramid.
............................................... cm [3]
(iii) $F$ is the mid-point of $BC$.
Find the angle between $EF$ and the base of the pyramid.
................................................ [3]
(b)
Cone $A$ is mathematically similar to cone $B$. The height of cone $A$ is $7cm$ and its volume is $66cm^3$. The volume of cone $B$ is $222.75cm^3$.
(i) Find the height of cone $B$.
............................................... cm [3]
(ii) A sphere also has volume $66cm^3$.
Calculate the radius of the sphere.
............................................... cm [2]
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(a) A bag contains 3 black discs and 5 white discs.
Jani takes a disc from the bag at random.
When the disc is black, he does not replace it.
When the disc is white, he replaces it in the bag.
Jani then takes a second disc at random.
(i) Complete the tree diagram for the first and second discs.
[3]
(ii) Jani takes a third disc from the bag at random.
Find the probability that he takes 2 black discs and 1 white disc.
............................................................ [4]
(b) Another bag contains 10 discs.
$x$ are red and the rest are green.
(i) Write down an expression for the number of green discs.
............................................................ [1]
(ii) $y$ blue discs are added to the bag.
A disc is taken from the bag at random.
(a) The probability of taking a red disc from the bag is $\frac{1}{3}$.
Show that $3x = 10 + y$.
[1]
(b) The probability of taking a green disc is $\frac{2}{9}$.
Write another equation in $x$ and $y$ and find the number of red discs and the number of blue discs.
number of red discs, $x =$ ................................................
number of blue discs, $y =$ ................................................ [5]
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(a) P, Q, R \text{ and } S \text{ are points on the circle centre } O.\newline\text{Find angle } PQR.\newline\text{Angle } PQR = \text{..........................} [1]\newline\newline\text{Diagram Reference: } \newline(b) A, B, C, D \text{ and } E \text{ are points on the circle centre } O.\newline AC \text{ is parallel to } ED.\newline\text{Find the obtuse angle } BOD.\newline\text{Angle } BOD = \text{..........................} [2]\newline\newline\text{Diagram Reference: } \newline(c) ABCDE \text{ is a regular pentagon. }\newline A, B, C, D \text{ and } E \text{ are points on the circle centre } O.\newline\text{The length of the perpendicular from } O \text{ to } ED \text{ is } 7.5\text{ cm.}\newline(i) \text{ Show that the length of one side of the pentagon is } 10.9\text{ cm correct to 3 significant figures.}\newline(ii) \text{Calculate the shaded area.}\newline\text{........................................ } \text{cm}^{2} [4]\newline\newline\text{Diagram Reference: }
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(a) Simplify fully $(64x^6y^3)^{\frac{2}{3}}$.
.................................................. [3]
(b) $3^x \times 2^x = 279936$
Find the value of $x$.
$x = $ ........................................... [2]
(c) ! In triangle $ABC$, $AB = BC$.
The perimeter of triangle $ABC$ is 16 cm.
(i) Show that $4x^2 - 1 = 0$.
.................................................. [5]
(ii) Find the length of $AB$.
$AB =$ ...................................... cm [2]
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