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Sunil has $80 and Asha has $75.
(a) Write the ratio $80 : 75$ in its simplest form.
Answer(a).............................. : .............................. [1]
(b) (i) Sunil spends $24.
Work out $24 as a percentage of $80.
Answer(b)(i) ..........................................................% [1]
(ii) Sunil invests $50 at a rate of 2\% per year compound interest.
Calculate the extbf{interest} Sunil has after 20 years.
Answer(b)(ii) $ ........................................................... [4]
(c) During each month, Asha spends $\frac{1}{5}$ of the money that she had at the beginning of the month.
(i) Work out how much of the $75 Asha has at the end of the 2nd month.
Answer(c)(i) $ ............................................................. [2]
(ii) Calculate the number of extbf{whole} months it takes for Asha to have less than $5.
Answer(c)(ii) ............................................................ [3]
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(a) \[ \mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}, \quad \mathbf{q} = \begin{pmatrix} 14 \\ 8 \end{pmatrix} \] (i) Find $2 \mathbf{p} + 3 \mathbf{q}$. \hspace{10cm} \text{Answer (a)(i)} \begin{pmatrix} \end{pmatrix} \hspace{10cm} [2]
(ii) Find $|\mathbf{q} - \mathbf{p}|$. \hspace{10cm} \text{Answer (a)(ii)} \text{..........................................................} [3]
(b) The graph of $y = f(x)$ is mapped onto the graph of $y = f(x + 2)$ by a translation with vector $\begin{pmatrix} u \\ v \end{pmatrix}$.
Find the value of $u$ and the value of $v$.
\hspace{10cm} \text{Answer (b)} \quad u = \text{.............................................}
\hspace{10cm} \qquad \quad v = \text{.............................................} [2]
(c)
(i) Draw the image of triangle $T$ under a rotation of $90^\circ$ clockwise about the point $(-1, -1)$. [3]
(ii) Describe fully the \textbf{single} transformation that maps triangle $T$ onto triangle $D$. \\\\ ................................................................. \text{.................................................................} [2]
(iii) Describe fully the \textbf{single} transformation that maps triangle $T$ onto triangle $E$. \\\\ ................................................................. \text{.................................................................} [3]
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(a) On the diagram, sketch the graph of $y = f(x)$, for values between $x = -5$ and $x = 5$. [2]
(b) Solve the inequality $f(x) < 0$. [2]
(c) Find $f^{-1}(x)$. [3]
(d) On the diagram, sketch the graph of $y = f^{-1}(x)$, for values between $x = -5$ and $x = 5$. [2]
(e) Describe fully the single transformation that maps the graph of $y = f(x)$ onto the graph of $y = f^{-1}(x)$. .................................................................................................................................................................. .................................................................................................................................................................. [2]
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(a)
AB and CD are parallel.
AD and CB intersect at X.
CD = 9 cm, AB = 4 cm, AX = 4.5 cm and BX = 3 cm.
Calculate the length of CX.
Answer(a) ................................................... cm
(b)
P, Q, R and S lie on a circle.
PR and QS intersect at Y.
QR = 6 cm, PS = 8 cm, PY = 7 cm and YS = 4 cm.
Calculate the length of RY.
Answer(b) ................................................... cm
(c)
The two shapes are mathematically similar.
The area of E is 90 cm² and the area of F is 45 cm².
Find the value of w.
Answer(c) w = .................................................
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Calculate
(a) $BC$,
Answer(a) ............................................................. cm [2]
(b) angle $CAD$,
Answer(b) ........................................................... [3]
(c) the area of the quadrilateral $ABCD$.
Answer(c) ............................................................. cm$^2$ [3]
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120 students estimate the mass, $m$ kg, of a bag of oranges. The frequency table shows the results.
[Table_1]
(a) Calculate an estimate of the mean.
Answer(a) $\text{..................................................}$ kg [2]
(b) Complete the histogram to show the information in the table.
[3]
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(a) A solid metal cuboid measures 20 cm by 8 cm by 2 cm. 1 cm$^3$ of the metal has a mass of 7.85 g.
(i) Calculate the mass of the cuboid. Answer(a)(i) ............................................................ g [2]
(ii) The surface of the cuboid is painted at a cost of 8 cents per cm$^2$. Calculate the cost of painting the cuboid. Give your answer in dollars. Answer(a)(ii) $ .......................................................... [3]
(b) Another cuboid measures 16 cm by 6 cm by 4 cm. It is cut into cubes, each of side 2 cm. Calculate the number of cubes. Answer(b) ............................................................ [2]
(c) Another solid metal cuboid measures 20 cm by 12 cm by 4 cm. It is melted down and made into spheres of radius 1.5 cm. Calculate
(i) the largest number of spheres of radius 1.5 cm that can be made, Answer(c)(i) ............................................................ [3]
(ii) the volume of metal remaining after the spheres have been made, Answer(c)(ii) .......................................................... cm$^3$ [2]
(iii) the radius of the sphere that can be made using all the remaining metal. Answer(c)(iii) .......................................................... cm [2]
(d) A plastic cone has radius $r$ cm and perpendicular height $3r$ cm. 1 cm$^3$ of the plastic has a mass of 0.9 g. A wooden hemisphere has a radius of $2r$ cm. 1 cm$^3$ of the wood has a mass of 0.45 g. Find the mass of the cone as a fraction of the mass of the hemisphere. Give your answer in its lowest terms. Answer(d) ............................................................ [4]
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(a) On the diagram, sketch the graphs of $y = f(x)$ and $y = g(x)$ for values between $x = 0$ and $x = 2$. [4]
(b) Solve the equation $3 - x^2 = x^x$ for $0 \leq x \leq 2$.
Answer(b) $x = \text{..............................}$ [1]
(c) Solve the equation $3 - x^2 = 0$ for $0 \leq x \leq 2$.
Answer(c) $x = \text{.......................................................}$ [1]
(d) (i) Find the co-ordinates of the local minimum point on the graph of $y = g(x)$.
Answer(d)(i) $(\text{........................} , \text{..........................} )$ [2]
(d) (ii) Find the range of $g(x)$ for the domain $0 < x \leq 2$.
Answer(d)(ii) $\text{...........................................................}$ [2]
(e) (i) Find the values of the following.
$g(0.1) = \text{.............................}$ $g(0.01) = \text{.............................}$ $g(0.001) = \text{.............................}$ [3]
(e) (ii) Complete the statement.
Starting from $x = 0.1$, as $x$ gets closer and closer to 0,
$g(x)$ gets closer and closer to the value $\text{..................................}$ [1]
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The diagram shows two unbiased dice, $A$ and $B$.
The numbers on die $A$ are 0, 1, 1, 1, 2, 3.
The numbers on die $B$ are 1, 2, 2, 3, 3, 3.
When a die is rolled, the number shown on the top face is recorded.
(a) Both dice are rolled.
Find the probability that
(i) both dice show 3,
\text{Answer}(a)(i) \text{ ............................................................ [2]}
(ii) the numbers showing on the two dice add up to 2.
\text{Answer}(a)(ii) \text{ ............................................................ [3]}
(b) Die $B$ is rolled until it shows 2.
Find the probability that this occurs when the die is rolled for the 4th time.
\text{Answer}(b) \text{ ............................................................ [2]}
(c) Die $A$ is rolled until it shows 3.
The probability that this occurs when the die is rolled for the $n$th time is $\frac{3125}{46656}$.
Find the value of $n$.
\text{Answer}(c) \text{ ............................................................ [2]}
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Given f(x) = 2x + 3, g(x) = x - 1, h(x) = \log{(x + 1)}
(a) Find f(h(9)).
Answer(a) ............................................................. [2]
Given f(x) = 2x + 3, g(x) = x - 1, h(x) = \log{(x + 1)}
(b) Find g(f(x)) in its simplest form.
Answer(b) ............................................................. [2]
Given f(x) = 2x + 3, g(x) = x - 1, h(x) = \log{(x + 1)}
(c) Find \frac{1}{f(x)} + \frac{1}{g(x)} in terms of x.
Give your answer as a single fraction.
Answer(c) ............................................................. [3]
(d) Solve the equation.
h(x) = -1
Answer(d) x = ............................................................. [2]
(e) Solve the equation.
(g(x))^2 = 5
Give exact answers.
Answer(e) x = .......................... or x = .......................... [3]
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(a) Cakes cost $x$ cents each and drinks cost $y$ cents each.
2 cakes and 1 drink cost $1.57$.
1 cake and 3 drinks cost $2.96$.
Find the total cost of 3 cakes and 2 drinks.
Give your answer in dollars.
Answer(a) $ \text{...............................................................} $ [6]
(b) A child’s train ticket costs $ \$x$.
An adult’s train ticket costs $ \$(x + 5)$.
Claudia buys 11 tickets.
She spends $ \$24$ on children’s tickets and $ \$24$ on adults’ tickets.
Write down an equation in $x$ and solve it to find the cost of a child’s ticket.
Answer(b) $ \text{...............................................................} $ [4]
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