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HL Paper 2

A function $f$ is defined by $f (x) = arcsin (\frac{x^{2} - 1}{x^{2} + 1}), x \in ℝ$ .

A function $g$ is defined by $g (x) = arcsin (\frac{x^{2} - 1}{x^{2} + 1}), x \in ℝ, x \geq 0$ .

Show that $f$ is an even function.

[1]

By considering limits, show that the graph of $y = f (x)$ has a horizontal asymptote and state its equation.

[2]

Show that $f' (x) = \frac{2 x}{\sqrt{x^{2}} (x^{2} + 1)}$ for $x \in ℝ, x \neq 0$ .

[6]

c.i.

By using the expression for $f' (x)$ and the result $\sqrt{x^{2}} = |x|$ , show that $f$ is decreasing for $x < 0$ .

[3]

c.ii.

Find an expression for $g^{- 1} (x)$ , justifying your answer.

[5]

State the domain of $g^{- 1}$ .

[1]

Sketch the graph of $y = g^{- 1} (x)$ , clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.

[3]

Show that ${\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}$ .

[1]

Verify that $x = {\text{tan}}\,\theta$ and $x = - \,{\text{cot}}\,\theta$ satisfy the equation ${x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0$ .

[7]

Hence, or otherwise, show that the exact value of ${\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3$ .

[5]

Using the results from parts (b) and (c) find the exact value of ${\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}$ .

Give your answer in the form $a + b\sqrt 3$ where $a$ , $b \in \mathbb{Z}$ .

[6]

A continuous random variable $X$ has a probability density function given by

$f (x) = \{\begin{matrix} arccos x & 0 \leq x \leq 1 \\ 0 & otherwise \end{matrix}$

The median of this distribution is $m$ .

Determine the value of $m$ .

[2]

Given that $P (|X - m| \leq a) = 0.3$ , determine the value of $a$ .

[4]

The height of water, in metres, in Dungeness harbour is modelled by the function $H (t) = a \sin (b (t - c)) + d$ , where $t$ is the number of hours after midnight, and $a, b, c$ and $d$ are constants, where $a > 0, b > 0$ and $c > 0$ .

The following graph shows the height of the water for $13$ hours, starting at midnight.

The first high tide occurs at $04 : 30$ and the next high tide occurs $12$ hours later. Throughout the day, the height of the water fluctuates between $2.2 m$ and $6.8 m$ .

All heights are given correct to one decimal place.

Show that $b = \frac{π}{6}$ .

[1]

Find the value of $a$ .

[2]

Find the value of $d$ .

[2]

Find the smallest possible value of $c$ .

[3]

Find the height of the water at $12 : 00$ .

[2]

Determine the number of hours, over a 24-hour period, for which the tide is higher than $5$ metres.

[3]

A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur $50$ minutes earlier than at Dungeness.

Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.

[2]

The voltage $v$ in a circuit is given by the equation

$v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)$ , $t \geqslant 0$ where $t$ is measured in seconds.

The current $i$ in this circuit is given by the equation

$i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)$ .

The power $p$ in this circuit is given by $p\left( t \right) = v\left( t \right) \times i\left( t \right)$ .

The average power ${p_{av}}$ in this circuit from $t = 0$ to $t = T$ is given by the equation

${p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t}$ , where $T > 0$ .

Write down the maximum and minimum value of $v$ .

[2]

Write down two transformations that will transform the graph of $y = v\left( t \right)$ onto the graph of $y = i\left( t \right)$ .

[2]

Sketch the graph of $y = p\left( t \right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

[3]

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left( t \right)$ ≥ 3.

[3]

Find ${p_{av}}$ (0.007).

[2]

With reference to your graph of $y = p\left( t \right)$ explain why ${p_{av}}\left( T \right)$ > 0 for all $T$ > 0.

[2]

Given that $p\left( t \right)$ can be written as $p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d$ where $a$ , $b$ , $c$ , $d$ > 0, use your graph to find the values of $a$ , $b$ , $c$ and $d$ .

[6]

Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = $p$ , OQ = $q$ and OR = $r$ , such that 0 < $p$ < $q$ < $r$ < 10.

Let ${\theta _p}$ be the angle APO, ${\theta _q}$ be the angle AQO and ${\theta _r}$ be the angle ARO.

Consider the case when ${\theta _p} = {\theta _q} + {\theta _r}$ and QR = 1.

Find an expression for ${\theta _p}$ in terms of $p$ .

[3]

Show that $p = \frac{{{q^2} + q - 1}}{{2q + 1}}$ .

[6]

By sketching the graph of $p$ as a function of $q$ , determine the range of values of $p$ for which there are possible values of $q$ .

[4]

The plane $Π_{1}$ has equation $3 x - y + z = - 13$ and the line $L$ has vector equation

$r = (\begin{matrix} 1 \\ 2 \\ - 2 \end{matrix}) + λ (\begin{matrix} - 3 \\ - 1 \\ 4 \end{matrix}), λ \in ℝ$ .

The plane $Π_{2}$ contains the point $O$ and the line $L$ .

Given that $L$ meets $Π_{1}$ at the point $P$ , find the coordinates of $P$ .

[4]

Find the shortest distance from the point $O (0, 0, 0)$ to $Π_{1}$ .

[4]

Find the equation of $Π_{2}$ , giving your answer in the form $r . n = d$ .

[3]

Determine the acute angle between $Π_{1}$ and $Π_{2}$ .

[5]

A particle $P$ moves in a straight line such that after time $t$ seconds, its velocity, $v$ in ${m s}^{- 1}$ , is given by $v = e^{- 3 t} \sin 6 t$ , where $0 < t < \frac{π}{2}$ .

At time $t$ , $P$ has displacement $s (t)$ ; at time $t = 0$ , $s (0) = 0$ .

At successive times when the acceleration of $P$ is $0 {m s}^{- 2}$ , the velocities of $P$ form a geometric sequence. The acceleration of $P$ is zero at times $t_{1}, t_{2}, t_{3}$ where $t_{1} < t_{2} < t_{3}$ and the respective velocities are $v_{1}, v_{2}, v_{3}$ .

Find the times when $P$ comes to instantaneous rest.

[2]

Find an expression for $s$ in terms of $t$ .

[7]

Find the maximum displacement of $P$ , in metres, from its initial position.

[2]

Find the total distance travelled by $P$ in the first $1.5$ seconds of its motion.

[2]

Show that, at these times, $\tan 6 t = 2$ .

[2]

e.i.

Hence show that $\frac{v_{2}}{v_{1}} = \frac{v_{3}}{v_{2}} = - e^{- \frac{π}{2}}$ .

[5]

e.ii.

Find the set of values of $k$ that satisfy the inequality ${k^2} - k - 12 < 0$ .

[2]

The triangle ABC is shown in the following diagram. Given that $\cos B < \frac{1}{4}$ , find the range of possible values for AB.

M17/5/MATHL/HP2/ENG/TZ2/04.b

[4]

Two airplanes, $A$ and $B$ , have position vectors with respect to an origin $O$ given respectively by

$r_{A} = (\begin{matrix} 19 \\ - 1 \\ 1 \end{matrix}) + t (\begin{matrix} - 6 \\ 2 \\ 4 \end{matrix})$

$r_{B} = (\begin{matrix} 1 \\ 0 \\ 12 \end{matrix}) + t (\begin{matrix} 4 \\ 2 \\ - 2 \end{matrix})$

where $t$ represents the time in minutes and $0 \leq t \leq 2.5$ .

Entries in each column vector give the displacement east of $O$ , the displacement north of $O$ and the distance above sea level, all measured in kilometres.

The two airplanes’ lines of flight cross at point $P$ .

Find the three-figure bearing on which airplane $B$ is travelling.

[2]

Show that airplane $A$ travels at a greater speed than airplane $B$ .

[2]

Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.

[4]

Find the coordinates of $P$ .

[5]

d.i.

Determine the length of time between the first airplane arriving at $P$ and the second airplane arriving at $P$ .

[2]

d.ii.

Let $D (t)$ represent the distance between airplane $A$ and airplane $B$ for $0 \leq t \leq 2.5$ .

Find the minimum value of $D (t)$ .

[5]

A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is $\theta$ radians.

M17/5/MATHL/HP2/ENG/TZ1/08

The volume of water is increasing at a constant rate of $0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}$ .

Find an expression for the volume of water $V{\text{ }}({{\text{m}}^3})$ in the trough in terms of $\theta$ .

[3]

Calculate $\frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ when $\theta = \frac{\pi }{3}$ .

[4]

Consider the function $f$ defined by $f(x) = 3x\arccos (x)$ where $- 1 \leqslant x \leqslant 1$ .

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

[3]

State the range of $f$ .

[2]

Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .

[4]

Consider the function $f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}$ .

Let $u = \tan x$ .

Determine an expression for $f’(x)$ in terms of $x$ .

[2]

a.i.

Sketch a graph of $y = f’(x)$ for $0 \leqslant x < \frac{\pi }{2}$ .

[4]

a.ii.

Find the $x$ -coordinate(s) of the point(s) of inflexion of the graph of $y = f(x)$ , labelling these clearly on the graph of $y = f’(x)$ .

[2]

a.iii.

Express $\sin x$ in terms of $\mu$ .

[2]

b.i.

Express $\sin 2x$ in terms of $u$ .

[3]

b.ii.

Hence show that $f(x) = 0$ can be expressed as ${u^3} - 7{u^2} + 15u - 9 = 0$ .

[2]

b.iii.

Solve the equation $f(x) = 0$ , giving your answers in the form $\arctan k$ where $k \in \mathbb{Z}$ .

[3]

Two submarines A and B have their routes planned so that their positions at time t hours, 0 ≤ t < 20 , would be defined by the position vectors r_A $= \left( \begin{gathered} \,2 \hfill \\ \,4 \hfill \\ - 1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 1 \hfill \\ \,1 \hfill \\ - 0.15 \hfill \\ \end{gathered} \right)$ and r_B $= \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right)$ relative to a fixed point on the surface of the ocean (all lengths are in kilometres).

To avoid the collision submarine B adjusts its velocity so that its position vector is now given by

r_B $= \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)$ .

Show that the two submarines would collide at a point P and write down the coordinates of P.

[4]

Show that submarine B travels in the same direction as originally planned.

[1]

b.i.

Find the value of t when submarine B passes through P.

[2]

b.ii.

Find an expression for the distance between the two submarines in terms of t.

[5]

c.i.

Find the value of t when the two submarines are closest together.

[2]

c.ii.

Find the distance between the two submarines at this time.

[1]

c.iii.

The points A, B and C have the following position vectors with respect to an origin O.

$\overrightarrow {{\rm{OA}}} = 2$ i + j – 2k

$\overrightarrow {{\rm{OB}}} = 2$ i – j + 2k

$\overrightarrow {{\rm{OC}}} =$ i + 3j + 3k

The plane Π $_2$ contains the points O, A and B and the plane Π $_3$ contains the points O, A and C.

Find the vector equation of the line (BC).

[3]

Determine whether or not the lines (OA) and (BC) intersect.

[6]

Find the Cartesian equation of the plane Π $_1$ , which passes through C and is perpendicular to $\overrightarrow {{\rm{OA}}}$ .

[3]

Show that the line (BC) lies in the plane Π $_1$ .

[2]

Verify that 2j + k is perpendicular to the plane Π $_2$ .

[3]

Find a vector perpendicular to the plane Π $_3$ .

[1]

Find the acute angle between the planes Π $_2$ and Π $_3$ .

[4]

Consider the planes $Π_{1}$ and $Π_{2}$ with the following equations.

$Π_{1} : 3 x + 2 y + z = 6$

$Π_{2} : x - 2 y + z = 4$

Find a Cartesian equation of the plane $Π_{3}$ which is perpendicular to $Π_{1}$ and $Π_{2}$ and passes through the origin $(0, 0, 0)$ .

[3]

Find the coordinates of the point where $Π_{1}$ , $Π_{2}$ and $Π_{3}$ intersect.

[2]

Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.

Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.

[3]

Calculate the probability that Iqbal passes at least two of the papers he attempts.

[2]

Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.

[3]

The diagram shows two circles with centres at the points A and B and radii $2r$ and $r$ , respectively. The point B lies on the circle with centre A. The circles intersect at the points C and D.

N16/5/MATHL/HP2/ENG/TZ0/09

Let $\alpha$ be the measure of the angle CAD and $\theta$ be the measure of the angle CBD in radians.

Find an expression for the shaded area in terms of $\alpha$ , $\theta$ and $r$ .

[3]

Show that $\alpha = 4\arcsin \frac{1}{4}$ .

[2]

Hence find the value of $r$ given that the shaded area is equal to 4.

[3]

The plane П₁ contains the points P(1, 6, −7) , Q(0, 1, 1) and R(2, 0, −4).

The Cartesian equation of the plane П₂ is given by $x - 3y - z = 3$ .

The Cartesian equation of the plane П₃ is given by $ax + by + cz = 1$ .

Consider the case that П₃ contains $L$ .

Find the Cartesian equation of the plane containing P, Q and R.

[6]

Given that П₁ and П₂ meet in a line $L$ , verify that the vector equation of $L$ can be given by r $= \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)$ .

[3]

Given that П₃ is parallel to the line $L$ , show that $a + 2b - 5c = 0$ .

[1]

Show that $5a - 7c = 4$ .

[2]

d.i.

Given that П₃ is equally inclined to both П₁ and П₂, determine two distinct possible Cartesian equations for П₃.

[7]

d.ii.

A scientist conducted a nine-week experiment on two plants, $A$ and $B$ , of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant $A$ was given fertilizer regularly, while Plant $B$ was not.

The scientist found that the height of Plant $A, h_{A} cm$ , at time $t$ weeks can be modelled by the function $h_{A} (t) = \sin (2 t + 6) + 9 t + 27$ , where $0 \leq t \leq 9$ .

The scientist found that the height of Plant $B, h_{B} cm$ , at time $t$ weeks can be modelled by the function $h_{B} (t) = 8 t + 32$ , where $0 \leq t \leq 9$ .

Use the scientist’s models to find the initial height of

Plant $B$ .

[1]

a.i.

Plant $A$ correct to three significant figures.

[2]

a.ii.

Find the values of $t$ when $h_{A} (t) = h_{B} (t)$ .

[3]

For $t > 6$ , prove that Plant $A$ was always taller than Plant $B$ .

[3]

For $0 \leq t \leq 9$ , find the total amount of time when the rate of growth of Plant $B$ was greater than the rate of growth of Plant $A$ .

[6]

Consider the following diagram.

The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].

Find AM.

[3]

a.i.

Find ${\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}$ in radians.

[2]

a.ii.

Find the area of the shaded region.

[3]

Let $f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)$ where $0 < x < \frac{\pi }{2}$ .

Express $f\left( x \right)$ in terms of sin $x$ and cos $x$ .

Consider the vectors $a$ and $b$ such that $a = (\begin{matrix} 12 \\ - 5 \end{matrix})$ and $|b| = 15$ .

Consider the vector $p$ such that $p = a + b$ .

Consider the vector $q$ such that $q = (\begin{matrix} x \\ y \end{matrix})$ , where $x, y \in ℝ^{+}$ .

Find the possible range of values for $|a + b|$ .

[2]

Given that $|a + b|$ is a minimum, find $p$ .

[2]

Find $q$ such that $| q | = | b |$ and $q$ is perpendicular to $a$ .

[5]

Three points $A (3, 0, 0), B (0, - 2, 0)$ and $C (1, 1, - 7)$ lie on the plane $Π_{1}$ .

Plane $Π_{2}$ has equation $3 x - y + 2 z = 2$ .

The plane $Π_{3}$ is given by $2 x - 2 z = 3$ . The line $L$ and the plane $Π_{3}$ intersect at the point $P$ .

The point $B (0, - 2, 0)$ lies on $L$ .

Find the vector $\vec{AB}$ and the vector $\vec{AC}$ .

[2]

a.i.

Hence find the equation of $Π_{1}$ , expressing your answer in the form $a x + b y + c z = d$ , where $a, b, c, d \in ℤ$ .

[5]

a.ii.

The line $L$ is the intersection of $Π_{1}$ and $Π_{2}$ . Verify that the vector equation of $L$ can be written as $r = (\begin{matrix} 0 \\ - 2 \\ 0 \end{matrix}) + λ (\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix})$ .

[2]

Show that at the point $P, λ = \frac{3}{4}$ .

[2]

c.i.

Hence find the coordinates of $P$ .

[1]

c.ii.

Find the reflection of the point $B$ in the plane $Π_{3}$ .

[7]

d.i.

Hence find the vector equation of the line formed when $L$ is reflected in the plane $Π_{3}$ .

[2]

d.ii.

The points $A (5, - 2, 5)$ , $B (5, 4, - 1)$ , $C (- 1, - 2, - 1)$ and $D (7, - 4, - 3)$ are the vertices of a right-pyramid.

The line $L$ passes through the point $D$ and is perpendicular to $Π$ .

Find the vectors $\vec{AB}$ and $\vec{AC}$ .

[2]

Use a vector method to show that $B \hat{A} C = 60 °$ .

[3]

Show that the Cartesian equation of the plane $Π$ that contains the triangle $ABC$ is $- x + y + z = - 2$ .

[3]

Find a vector equation of the line $L$ .

[1]

d.i.

Hence determine the minimum distance, $d_{min}$ , from $D$ to $Π$ .

[4]

d.ii.

Find the volume of right-pyramid $ABCD$ .

[4]

In triangle ABC, AB = 5, BC = 14 and AC = 11.

Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.

The following shape consists of three arcs of a circle, each with centre at the opposite vertex of an equilateral triangle as shown in the diagram.

For this shape, calculate

the perimeter.

[2]

the area.

[5]

Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.
“Seaview” $(S)$ is at an angle of depression of 25°.
“Nauti Buoy” $(N)$ is at an angle of depression of 35°.
The following three dimensional diagram shows Barry and the two yachts at S and N.
X lies at the foot of the cliff and angle ${\text{SXN}} =$ 70°.

N17/5/MATHL/HP2/ENG/TZ0/05

Find, to 3 significant figures, the distance between the two yachts.

Consider $\lim_{x \to 0} \frac{arctan (\cos x) - k}{x^{2}}$ , where $k \in ℝ$ .

Show that a finite limit only exists for $k = \frac{π}{4}$ .

[2]

Using l’Hôpital’s rule, show algebraically that the value of the limit is $- \frac{1}{4}$ .

[6]

Given that a $\times$ b $=$ b $\times$ c $\ne$ 0 prove that a $+$ c $=$ sb where s is a scalar.

Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time t hours after midday are given by

r_A = $\left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 5 \\ 8 \end{array}} \right)$

r_B = $\left( {\begin{array}{*{20}{c}} 7 \\ { - 3} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 0 \\ {12} \end{array}} \right)$

where distances are measured in kilometres.

Find the minimum distance between the two ships.

In a triangle ${\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}$ and ${\rm{B\hat AC}} = \frac{\pi }{9}$ .

Use the cosine rule to find the two possible values for AC.

[5]

Find the difference between the areas of the two possible triangles ABC.

[3]

Two boats $A$ and $B$ travel due north.

Initially, boat $B$ is positioned $50$ metres due east of boat $A$ .

The distances travelled by boat $A$ and boat $B$ , after $t$ seconds, are $x$ metres and $y$ metres respectively. The angle $θ$ is the radian measure of the bearing of boat $B$ from boat $A$ . This information is shown on the following diagram.

Show that $y = x + 50 cot θ$ .

[1]

At time $T$ , the following conditions are true.

Boat $B$ has travelled $10$ metres further than boat $A$ .
Boat $B$ is travelling at double the speed of boat $A$ .
The rate of change of the angle $θ$ is $- 0.1$ radians per second.

Find the speed of boat $A$ at time $T$ .

[6]

This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius ${\text{OB}} = 9{\text{ cm}}$ and four equal sectors of a smaller circle of radius ${\text{OA}} = 3{\text{ cm}}$ .
The angle ${\text{BOC}} =$ 20°.

N17/5/MATHL/HP2/ENG/TZ0/03

Find the area of the pendant.

Find the Cartesian equation of plane Π containing the points ${\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)$ and ${\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)$ and perpendicular to the plane $x + 2y - z - 6 = 0$ .

Find the acute angle between the planes with equations $x + y + z = 3$ and $2x - z = 2$ .

Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.