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1.     A discrete random variable $X$ has the probability distribution given by the following table.

Mathematics HL Paper 1 - N20/5/MATHL/HP1/ENG/TZ0/XX No 1

Given that $ \ \textrm{E}(X) \ = \ \frac{19}{12}$, determine the value of $p$ and the value of $q$.

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2.     Find the equation of tangent to the curve $ \ y \ = \ \textrm{e}^{2x} \ – \ 3x \ $ at the point where $ \ x \ = \ 0$.
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3.     At Nusaybah’s Breakfast Diner, three types of omelette are available to order: chicken, vegetarian and steak. Each omelette is served with either a portion of fries or hash browns. It is known that 20% of customers choose a chicken omelette, 70% choose a vegetarian omelette and 10% choose a steak omelette.

It is also known that 65% of those ordering the chicken omelette, 70% of those ordering the vegetarian omelette and 60% of those ordering the steak omelette, order fries.

The following tree diagram represents the orders made by each customer.

Mathematics HL Paper 1 - N20/5/MATHL/HP1/ENG/TZ0/XX No 3

    (a)     Complete the tree diagram by adding the respective probabilities to each branch.
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    (b)     Find the probability that a randomly selected customer orders fries.
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    (c)     Find the probability that a randomly selected customer orders fries, given that they do not order a chicken omelette.
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4.     Consider the equation
$$ \frac{2z}{3 \ – \ {z}^{*}} \ = \ \textrm{i} \ ,$$

where $ \ z \ = \ x + \textrm{i} y \ $ and $ \ x, y \in {\Bbb R}$.

Find the value of $x$ and the value of $y$.
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5.     The first term in an arithmetic sequence is $4$ and the fifth term is $ \log_{2} 625$.

Find the common difference of the sequence, expressing your answer in the form $\log_{2} p$, where $ \ p \in {\Bbb Q}$.
$$\tag*{[5]} $$

6.     Consider the equation $a{x}^{2} + bx + c \ = \ 0$, where $ \ a \neq 0$. Given that the roots of this equation are $ \ x \ = \ \sin \theta \ $  and $ \ x \ = \ \cos \theta$, show that $ \ b^2 \ = \ a^2 + 2ac$.
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7.     Consider the complex numbers $ \ z_{1}= \cos \frac{11 \pi}{12} + \textrm{i} \sin \frac{11 \pi}{12} \ $ and $ \ z_{2}= \cos \frac{\pi}{6} + \textrm{i} \sin \frac{\pi}{6} \ $.

    (a)   (i)   Find $ \ \frac{z_{1}}{z_{2}}$

          (ii)   Find $ \ \frac{z_{2}}{z_{1}}$
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    (b)     $ 0$, $\ \frac{z_{1}}{z_{2}} \ $ and $ \ \frac{z_{2}}{z_{1}} \ $ are represented by three points $O$, $A$ and $B$ respectively on an Argand diagram. Determine the area of the triangle $OAB$.
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8.     (a)     Show that $\frac{\sin x \tan x}{1 \ – \ \cos x} \equiv 1 + \frac{1}{\cos x}, x \neq 2n\pi, n \in {\Bbb Z}$.
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        (b)     Hence determine the range of values of $k$ for which $ \ \frac{\sin x \tan x}{1 \ – \ \cos x} \ = \ k \ $ has no real solutions.
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9.     By using the substitution $ \ x \ = \ \tan u$, find the value of $ \displaystyle \int_{0}^{1} \frac{x^2}{(1+{x}^{2})^{3}} \textrm{d}x$.
$$\tag*{[8]} $$

10.     Consider the function $ \ f(x) = a{x}^{3}+b{x}^{2}+cx+d$, where $ \ x \in {\Bbb R} \ $ and $ \ a, b, c, d \in {\Bbb R} $.

    (a)   (i)   Write down an expression for $f'(x)$.

            (ii)   Hence, given that $f'(x)$ does not exist, $ \ b^2 \ – \ 3ac > 0$.
$$\tag*{[4]} $$
    (b)     Consider the function $g(x) = \frac{1}{2}x^{3} \ – \ 3x^{2} + 6x \ – \ 8$, where $\ x \in {\Bbb R} $.

          (i)   Show that $g^{-1}$ exists.

          (ii)   $g(x) \ $ can be written in the form $ \ p(x \ – \ 2)^{3} + q$, where $\ p, q \in {\Bbb R} $.
                  Find the value of $p$ and the value of $q$.

          (iii)   Hence find $g^{-1}(x)$.
$$\tag*{[8]} $$

The graph of $ \ y \ = \ g(x) \ $ may be obtained by transforming the graph of $ \ y \ = \ x^{3} \ $ using a sequence of three transformations.

    (c)   State each of the transformations in the order in which they are applied.

    (d)   Sketch the graphs of $ \ y \ = \ g(x) \ $ and $ \ y \ = \ g^{-1}(x) \ $ on the same set of axes, indicating the points where each graph crosses the coordinate axes.
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11.     Consider the curve $C$ defined by $ \ y^{2} = \sin (xy), y \neq 0$.

    (a)     Show that $ \ \frac{ \textrm{d}y }{ \textrm{d}x } = \frac{y \cos (xy) }{ 2y \ – \ x \cos (xy) }$.
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    (b)     Prove that, when $ \ \frac{ \textrm{d}y }{ \textrm{d}x } = 0, \ y = \pm 1 $.
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    (c)     Hence find the coordinates of all points on $C$, for $ \ 0 < x < 4\pi $, where $ \ \frac{ \textrm{d}y }{ \textrm{d}x } = 0 $. $$\tag*{[5]} $$ 12.     Consider the function defined by $ \ f(x) = \frac{kx \ - \ 5}{x \ - \ k}$, where $ \ x, k \in {\Bbb R} \ $ and $ \ k^{2} \neq 5$.     (a)     State the equation of the vertical asymptote on the graph of $ \ y \ = \ f(x)$. $$\tag*{[1]} $$     (b)     State the equation of the horizontal asymptote on the graph of $ \ y \ = \ f(x)$. $$\tag*{[1]} $$     (c)     Use an algebraic method to determine whether $f$ is a self-inverse function. $$\tag*{[4]} $$           Consider the case where $ \ k \ = \ 3$.     (d)     Sketch the graph of $ \ y \ = \ f(x)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes. $$\tag*{[3]} $$     (e)     The region bounded by the $x$-axis, the curve $ \ y \ = \ f(x)$, and the lines $ \ x = 5 \ $ and $ \ x = 7 \ $ is rotated through $2\pi$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\pi ( a + b \ln 2 )$, where $ \ a, b \in {\Bbb Z}$. $$\tag*{[6]} $$

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