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1. The following diagram shows a sector $OAB$ of radius 15 cm. The length of [AB] is 11 cm.
Find the area of the shaded region.
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2. Jenna is a keen book reader. The number of books she reads during one week can be modelled by a Poisson distribution with mean 2.6.
Determine the expected number of weeks in one year, of 52 weeks, during which Jenna reads at least four books.
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3. The following diagram shows part of the graph of $ \ y = p + q \sin (rx)$. The graph has a local maximum point at $ \ \Big( – \frac{9 \pi}{4}, 5 \Big) \ $ and a local minimum point at $ \ \Big( – \frac{3 \pi}{4}, -1 \Big)$.
(a) Determine the values of $p$, $q$ and $r$.
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(b) Hence find the area of the shaded region.
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4. Find the term independent of $x$ in the expansion of $ \ \frac{1}{x^3} {\Big( \frac{1}{3x^2} – \frac{x}{2} \Big)}^{9} $.
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5. A survey of British holidaymakers found that 15% of those surveyed took a holiday in the Lake District in 2019.
(a) A random sample of 16 British holidaymakers was taken. The number of people in the sample who took a holiday in the Lake District in 2019 can be modelled by a binomial distribution.
(i) State two assumptions made in order for this model to be valid.
(ii ) Find the probability that at least three people from the sample took a holiday in the Lake District in 2019.
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(b) From a random sample of $n$ holidaymakers, the probability that at least one of them took a holiday in the Lake District in 2019 is greater than 0.999.
Determine the least possible value of $n$.
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6. Use mathematical induction to prove that $ \ \frac{ \textrm{d}^{n} }{ \textrm{d}{x}^{n}} \big( x{\textrm{e}}^{px}\big) \ = \ {p}^{n-1} (px + n) {\textrm{e}}^{px}\ $ for $ \ n \in {{\Bbb Z}}^{+}, p \in {\Bbb Q}$.
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7. At a gathering of 12 teachers, seven are male and five are female. A group of five of these teachers go out for a meal together. Determine the possible number of groups in each of the following situations:
(a) There are more males than females in the group.
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(b) Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.
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8. A small bead is free to move along a smooth wire in the shape of the curve $ \ \displaystyle y = \frac{10}{3 \ – \ 2 {\textrm{e}}^{-0.5x}}(x \geq 0)$.
(a) Find an expression for $ \ \frac{ \textrm{d}y }{ \textrm{d}x}$.
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At the point on the curve where $ x = 4$, it is given that $ \ \frac{ \textrm{d}y }{ \textrm{d}t} = -0.1 \textrm{m}{\textrm{s}}^{-1}$.
(b) Find the value of $ \ \frac{ \textrm{d}x}{ \textrm{d}t} \ $ at this exact same instant.
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9. The weights, in grams, of individual packets of coffee can be modelled by a normal distribution, with mean 102g and standard deviation 8g.
(a) Find the probability that a randomly selected packet has a weight less than 100g.
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(b) The probability that a randomly selected packet has a weight greater than $w$ grams is 0.444. Find the value of $w$. [2]
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(c) A packet is randomly selected. Given that the packet has a weight greater than 105 g, find the probability that it has a weight greater than 110 g.
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(d) From a random sample of 500 packets, determine the number of packets that would be expected to have a weight lying within 1.5 standard deviations of the mean.
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(e) Packets are delivered to supermarkets in batches of 80. Determine the probability that at least 20 packets from a randomly selected batch have a weight less than 95g.
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10. The plane ${\Pi}_{1}$ has equation $3x – y + z = -13$ and the line $L$ has vector equation
$$ \textbf{r} = \pmatrix {1 \cr 2 \cr -2} + \lambda \pmatrix {-3 \cr -1 \cr 4}, \lambda \in {\Bbb R}. $$
(a) Given that $L$ meets ${\Pi}_{1}$ at the point P, find the coordinates of P.
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(b) Find the shortest distance from the point O (0, 0, 0) to ${\Pi}_{1}$.
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The plane ${\Pi}_{2}$ contains the point O and the line $L$.
(c) Find the equation of ${\Pi}_{2}$, giving your answer in the form $\textbf{r.n} = d$.
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(d) Determine the acute angle between ${\Pi}_{1}$ and ${\Pi}_{2}$.
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11. A particle $P$ moves in a straight line such that after time $t$ seconds, its velocity, $v$ in ${\small \mathrm{m} {\mathrm{s}}^{-1} }$, is given by $v = {\textrm{e}}^{−3t} \sin 6t $, where $ 0 \lt t \lt \frac{\pi}{2} $.
(a) Find the times when $P$ comes to instantaneous rest.
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At time $t$, $P$ has displacement $s(t)$; at time $t=0$, $s(0)=0$.
(b) Find an expression for $s$ in terms of $t$.
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(c) Find the maximum displacement of $P$, in metres, from its initial position.
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(d) Find the total distance travelled by $P$ in the first 1.5 seconds of its motion.
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At successive times when the acceleration of $P$ is 0 ${\small \mathrm{m} {\mathrm{s}}^{-2} }$, the velocities of $P$ form a geometric sequence. The acceleration of $P$ is zero at times $t_{1}, t_{2}, t_{3}$ where $t_{1} \lt t_{2} \lt t_{3}$ and the respective velocities are $v_{1}, v_{2}, v_{3}$.
(e) (i) Show that, at these times, $\tan 6t = 2$.
(ii) Hence show that $ \displaystyle \frac{v_{2}}{v_{1}} = \frac{v_{3}}{v_{2}} = -\textrm{e}^{-\frac{\pi}{2}}$.
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