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Date	May 2018	Marks available	2	Reference code	18M.1.SL.TZ1.S_10
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	S_10	Adapted from	N/A

Question

The first two terms of an infinite geometric sequence are u₁ = 18 and u₂ = 12sin² θ , where 0 < θ < 2 $π$ $\pi$ , and θ ≠ $π$ $\pi$ .

Find an expression for r in terms of θ.

[2]

a.i.

Find the values of θ which give the greatest value of the sum.

[6]

c.

Markscheme

valid approach (M1)

eg $\frac{u_{2}}{u_{1}}, \frac{u_{1}}{u_{2}}$ $\frac{{{u_2}}}{{{u_1}}},\,\,\frac{{{u_1}}}{{{u_2}}}$

$r = \frac{12 {sin}^{2} θ}{18} (= \frac{2 {sin}^{2} θ}{3})$ $r = \frac{{12\,{{\sin }^2}\,\theta }}{{18}}\left( { = \frac{{2\,{{\sin }^2}\,\theta }}{3}} \right)$ A1 N2

[2 marks]

a.i.

METHOD 1 (using differentiation)

recognizing $\frac{d S_{\infty}}{d θ} = 0$ $\frac{{{\text{d}}{S_\infty }}}{{{\text{d}}\theta }} = 0$ (seen anywhere) (M1)

finding any correct expression for $\frac{d S_{\infty}}{d θ}$ $\frac{{{\text{d}}{S_\infty }}}{{{\text{d}}\theta }}$ (A1)

eg $\frac{0 - 54 \times (- 2 sin 2 θ)}{{(2 + cos 2 θ)}^{2}}, - 54 {(2 + cos 2 θ)}^{- 2} (- 2 sin 2 θ)$ $\frac{{0 - 54 \times \left( { - 2\,{\text{sin}}\,2\,\theta } \right)}}{{{{\left( {2 + {\text{cos}}\,2\,\theta } \right)}^2}}},\,\, - 54{\left( {2 + {\text{cos}}\,2\,\theta } \right)^{ - 2}}\,\left( { - 2\,{\text{sin}}\,2\,\theta } \right)$

correct working (A1)

eg sin 2θ = 0

any correct value for sin⁻¹(0) (seen anywhere) (A1)

eg 0, $π$ $\pi$ , … , sketch of sine curve with x-intercept(s) marked both correct values for 2θ (ignore additional values) (A1)

2θ = $π$ $\pi$ , 3 $π$ $\pi$ (accept values in degrees)

both correct answers $θ = \frac{π}{2}, \frac{3 π}{2}$ $\theta = \frac{\pi }{2},\,\frac{{3\pi }}{2}$ A1 N4

Note: Award A0 if either or both correct answers are given in degrees.
Award A0 if additional values are given.

METHOD 2 (using denominator)

recognizing when S_∞ is greatest (M1)

eg 2 + cos 2θ is a minimum, 1−r is smallest
correct working (A1)

eg minimum value of 2 + cos 2θ is 1, minimum r = $\frac{2}{3}$ $\frac{2}{3}$

correct working (A1)

eg $cos 2 θ = - 1, \frac{2}{3} si n^{2} θ = \frac{2}{3}, si n^{2} θ = 1$ ${\text{cos}}\,2\,\theta = - 1,\,\,\frac{2}{3}\,{\text{si}}{{\text{n}}^2}\,\theta = \frac{2}{3},\,\,{\text{si}}{{\text{n}}^2}\theta = 1$

EITHER (using cos 2θ)

any correct value for cos⁻¹(−1) (seen anywhere) (A1)

eg $π$ $\pi$ , 3 $π$ $\pi$ , … (accept values in degrees), sketch of cosine curve with x-intercept(s) marked

both correct values for 2θ (ignore additional values) (A1)

2θ = $π$ $\pi$ , 3 $π$ $\pi$ (accept values in degrees)

OR (using sinθ)

sinθ = ±1 (A1)

sin⁻¹(1) = $\frac{π}{2}$ $\frac{\pi }{2}$ (accept values in degrees) (seen anywhere) A1

THEN

both correct answers $θ = \frac{π}{2}, \frac{3 π}{2}$ $\theta = \frac{\pi }{2},\,\frac{{3\pi }}{2}$ A1 N4

Note: Award A0 if either or both correct answers are given in degrees.
Award A0 if additional values are given.

[6 marks]

c.

Examiners report

[N/A]

a.i.

[N/A]

c.

Syllabus sections

Topic 1—Number and algebra » SL 1.3—Geometric sequences and series

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