Home » Math HL » Mathematics HL Paper 1 – N20/5/MATHL/HP1/ENG/TZ0/XX

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

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.

(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 $y$.
$$\tag*{[5]}$$

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 $\ p \in {\Bbb Q}$.
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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 $OAB$.
$$\tag*{[2]}$$

8.     (a)     Show that $\frac{\sin x \tan x}{1 \ – \ \cos x} \equiv 1 + \frac{1}{\cos x}, x \neq 2n\pi, n \in {\Bbb Z}$.
$$\tag*{[3]}$$

(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 $\ a, b \in {\Bbb Z}$. $$\tag*{[6]}$$