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Date	May 2017	Marks available	5	Reference code	17M.1.AHL.TZ1.H_3
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Solve	Question number	H_3	Adapted from	N/A

Question

Solve the equation ${\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi$ .

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1

use of ${\sec ^2}x = {\tan ^2}x + 1$ M1

${\tan ^2}x + 2\tan x + 1 = 0$

${(\tan x + 1)^2} = 0$ (M1)

$\tan x = - 1$ A1

$x = \frac{{3\pi }}{4},{\text{ }}\frac{{7\pi }}{4}$ A1A1

METHOD 2

$\frac{1}{{{{\cos }^2}x}} + \frac{{2\sin x}}{{\cos x}} = 0$ M1

$1 + 2\sin x\cos x = 0$

$\sin 2x = - 1$ M1A1

$2x = \frac{{3\pi }}{2},{\text{ }}\frac{{7\pi }}{2}$

$x = \frac{{3\pi }}{4},{\text{ }}\frac{{7\pi }}{4}$ A1A1

Note: Award A1A0 if extra solutions given or if solutions given in degrees (or both).

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » AHL 3.8—Unit circle, Pythag identity, solving trig equations graphically

16N.1.AHL.TZ0.H_13c:
Use the principle of mathematical induction to prove that

$\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi$ where $k \in \mathbb{Z}$ .
19M.1.SL.TZ2.S_9d:
The following diagram shows the graph of $f$ for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$ , find the $x$ -coordinate of P.
17M.2.AHL.TZ1.H_10c:
Hence, find the area of the smaller triangle.
19M.1.AHL.TZ1.H_4b:
Find the two possible values for the length of the third side.
16N.1.AHL.TZ0.H_13a:
Find the value of $\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}$ .
19M.1.SL.TZ2.S_9b:
Line $L$ passes through the origin and has a gradient of ${\text{tan}}\,\theta$ . Find the equation of $L$ .
19M.1.SL.TZ2.S_9a:
Find the value of ${\text{tan}}\,\theta$ .
19M.1.AHL.TZ1.H_4a:
Show that ${\text{sin}}\,\theta = \frac{{\sqrt {15} }}{4}$ .
19M.1.SL.TZ2.S_9c:
Find the derivative of $f$ .
17M.2.AHL.TZ2.H_4a:
Find the set of values of $k$ that satisfy the inequality ${k^2} - k - 12 < 0$ .
16N.1.AHL.TZ0.H_13d:
Hence or otherwise solve the equation $\sin x + \sin 3x = \cos x$ in the interval $0 < x < \pi$ .
17M.2.AHL.TZ1.H_10a:
Use the cosine rule to show that ${r^2} - 12\sqrt 3 r + 144 - {p^2} = 0$ .
17M.2.AHL.TZ2.H_4b:
The triangle ABC is shown in the following diagram. Given that $\cos B < \frac{1}{4}$ , find the range of possible values for AB.
16N.2.AHL.TZ0.H_7a:
Use the cosine rule to find the two possible values for AC.
16N.2.AHL.TZ0.H_7b:
Find the difference between the areas of the two possible triangles ABC.
16N.1.AHL.TZ0.H_13b:
Show that $\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi$ where $k \in \mathbb{Z}$ .
17M.2.AHL.TZ1.H_10b:
Calculate the two corresponding values of PQ.
17M.2.AHL.TZ1.H_10d:
Determine the range of values of $p$ for which it is possible to form two triangles.
21N.1.AHL.TZ0.11:
The following diagram shows a corner of a field bounded by two walls defined by lines $L_{1}$ and $L_{2}$ . The walls meet at a point $A$ , making an angle of $40 °$ .

Farmer Nate has $7 m$ of fencing to make a triangular enclosure for his sheep. One end of the fence is positioned at a point $B$ on $L_{2}$ , $10 m$ from $A$ . The other end of the fence will be positioned at some point $C$ on $L_{1}$ , as shown on the diagram.

He wants the enclosure to take up as little of the current field as possible.

Find the minimum possible area of the triangular enclosure $ABC$ .

Topic 3—Geometry and trigonometry

Question

Markscheme

Examiners report

Syllabus sections

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