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For each of these sequences, find the next term and an expression for the $n$th term.
(a) 17 \hspace{2mm} 14 \hspace{2mm} 11 \hspace{2mm} 8 \hspace{2mm} 5 \ldots
next term \hspace{10mm} \text{...............................................}
$n$th term \hspace{10mm} \text{............................................... [3]}
(b) \( \frac{1}{2} \hspace{3mm} \frac{2}{3} \hspace{3mm} \frac{3}{4} \hspace{3mm} \frac{4}{5} \hspace{3mm} \frac{5}{6} \ldots \)
next term \hspace{10mm} \text{...............................................}
$n$th term \hspace{10mm} \text{............................................... [2]}
(c) \hspace{2mm} 4 \hspace{4mm} 8 \hspace{4mm} 16 \hspace{4mm} 32 \hspace{4mm} 64 \ldots
next term \hspace{10mm} \text{...............................................}
$n$th term \hspace{10mm} \text{............................................... [3]}
(d) \hspace{2mm} -2 \hspace{4mm} 5 \hspace{4mm} 24 \hspace{4mm} 61 \hspace{4mm} 122 \ldots
next term \hspace{10mm} \text{...............................................}
$n$th term \hspace{10mm} \text{............................................... [3]}
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The population of a species of bird is estimated to be decreasing by 4\% per year. At the end of 2020 the population was 4.32 million.
(a) Find the population at the end of 2019. ...................................... million [2]
(b) Calculate an estimate for the population at the end of 2025. ...................................... million [2]
(c) Find the year in which the population is first expected to be below 2 million. .............................................. [4]
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(a) Reflect triangle $A$ in the line $y = -1$. [2]
(b) Translate triangle $A$ by the vector $\begin{pmatrix} -5 \\ 3 \end{pmatrix}$. [2]
(c) Describe fully the \textbf{single} transformation that maps triangle $A$ onto triangle $B$.
..................................................................................................................................................
.................................................................................................................................................. [3]
(d) Describe fully the \textbf{single} transformation that maps triangle $A$ onto triangle $C$.
..................................................................................................................................................
.................................................................................................................................................. [3]
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The masses, $m$ kg, of 160 students are recorded in the table.
[Table_1]
(a) Draw a cumulative frequency curve for these results.
(b) Use your cumulative frequency curve to estimate
(i) the median .........................................................kg [1]
(ii) the interquartile range. .........................................................kg [2]
(c) The masses of 60% of the students lie in the range $p$ kg $< m \leq 80$ kg.
Use your cumulative frequency curve to estimate the value of $p$.
$p = .........................................................$ [3]
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536
(a) The diagram shows a regular pentagon with sides of 10 cm and centre $O$.
(i) Find angle $AOB$.
Angle $AOB = \text{................................................}$ [1]
(ii) Show that $OA = 8.51$ cm correct to 3 significant figures. [3]
(iii) Find the area of the pentagon.
$\text{...................................... cm}^2$ [2]
(b) The regular pentagon in part (a) is the base of a pyramid. The sloping edges, $VA, VB, VC, VD$, and $VE$, are each of length 18 cm.
(i) Calculate the perpendicular height, $VO$, of the pyramid.
$VO = \text{................................................ cm}$ [3]
(ii) Calculate the volume of the pyramid.
$\text{...................................... cm}^3$ [2]
(iii) A geometrically similar pyramid has volume $1500 \text{ cm}^3$. Calculate the length of a side of the base of this pyramid.
$\text{................................................ cm}$ [3]
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535
(a) On the diagram sketch the graph of $y = f(x)$ for values of $x$ between $-6$ and $6$.
[3]
(b) Write down the equations of the asymptotes parallel to the $y$-axis.
...............................................................
[2]
(c) Find the zeros of the graph of $y = f(x)$.
...............................................................
[2]
(d) $g(x) = x - 3$
(i) On the diagram sketch the graph of $y = g(x)$ for $-6 \le x \le 6$. [1]
(ii) Use your graphs to solve $f(x) = g(x)$.
...............................................................
[3]
(iii) Solve $g(x) > f(x)$.
...............................................................
[3]
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A is the point \((-8, 2)\) and C is the point \((8, 10)\).
(a) Find the equation of the line AC. ............................................................ [3]
(b) N is the point \((4, 8)\).
Show that N lies on AC. .......................................................... [1]
(c) Find the equation of the line that is perpendicular to AC and passes through N. ............................................................ [3]
(d) A and C are two vertices of a quadrilateral ABCD.
B is the point \((2, 12)\).
D is the reflection of B in the line AC.
(i) Find the coordinates of D. \((....................., .....................)\) [2]
(ii) Write down the name of the special quadrilateral ABCD. ............................................................ [1]
(iii) Find the length AC. ............................................................ [2]
(iv) Find the area of the quadrilateral ABCD. ............................................................ [3]
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A ship sails from port A at a constant speed of 18 km/h on a bearing of 040°. A motorboat sails in a straight line at a constant speed from port B to intercept the ship.
Port B is 30 km due south of port A. The ship leaves port A at 08 20 and the motorboat leaves port B at 08 30.
The motorboat intercepts the ship at point C at 09 50.
(b) Find the bearing on which the motorboat sails.
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Asa and Bernice have these 10 letter cards.
A, E, I, O and U are vowels. All other letters are consonants.
(a) Asa picks a card at random.
Write down the probability that Asa’s card shows the letter T.
[1]
(b) Asa replaces his card.
Bernice picks two cards at random without replacement.
Calculate the probability that both of Bernice’s cards are vowels.
[2]
(c) Bernice replaces her cards.
Asa picks 3 cards at random without replacement.
Calculate the probability that Asa’s cards can be arranged to spell the word PEN.
[3]
(d) Asa replaces his cards.
Bernice picks cards at random with replacement until she first gets a consonant.
The probability that she first gets a consonant on her n-th pick is $\frac{48}{3125}$.
Find the value of $n$.
[3]
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(a) Simplify.
$3x - 5y + 4x - 6y$ .................................................. [2]
(b) Expand.
$x(x + 2)$ .......................................................... [1]
(c) Factorise.
$10ab + 8ac - 15b^2 - 12bc$ ............................................. [2]
(d) $\frac{2}{2x+1} - \frac{5}{x-3} = 3$ .................................................... [2]
(i) Show that $6x^2 - 7x + 2 = 0$.
(ii) Solve $6x^2 - 7x + 2 = 0$. You must show all your working.
$x = \text{.................} \text{or } x = \text{.................}$ [3]
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f(x) = 2x + 5 \hspace{10pt} g(x) = 1 - 3x
(a) Find f(-2). ............................................................ [1]
(b) Solve \hspace{5pt} f(g(x)) = 19. ............................................................ [3]
(c) Find \hspace{5pt} g^{-1}(x).
\hspace{25pt} g^{-1}(x) = ............................................................ [2]
(d) \hspace{5pt} y = \frac{g(x)}{f(x)}
\hspace{5pt} Find \hspace{5pt} x \hspace{5pt} in \hspace{5pt} terms \hspace{5pt} of \hspace{5pt} y.
\hspace{25pt} x = ............................................................ [3]
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