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In this question you must show all stages of your working - Edexcel - A-Level Maths: Pure - Question 15 - 2022 - Paper 2
Question 15
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In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Given that the first three terms of a geom... show full transcript
Worked Solution & Example Answer:In this question you must show all stages of your working - Edexcel - A-Level Maths: Pure - Question 15 - 2022 - Paper 2
Step 1
show that 4 sin² θ - 52 sin θ + 25 = 0
Answer
To show that the three terms form a geometric series, we start by using the property of geometric sequences where the ratio of successive terms remains constant.
Let the common ratio be r. Then, we have:
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From the first term to the second:
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From the second term to the third:
Setting these two expressions for r equal:
Cross-multiplying gives:
Expanding the left side, we get:
Now we know that (\cos \theta = -\sqrt{1 - \sin^2 \theta}) since θ is obtuse. So substituting this into our equation will yield:
To simplify, we reformulate this into a quadratic in terms of (\sin \theta$$:
Step 2
solve the equation in part (a) to find the exact value of θ
Answer
Using the quadratic formula: where a = 4, b = -52, c = 25.
Calculating the discriminant:
Taking the square root,
Now substituting back into the quadratic formula: This gives us two possible values for (\sin \theta):
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(\sin \theta = \frac{100}{8} = 12.5) → not possible as (\sin \theta) must be in [-1, 1].
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(\sin \theta = \frac{4}{8} = 0.5) → thus θ = 30°.
However, since θ is obtuse, we find θ = 150°.
Step 3
show that the sum to infinity of the series can be expressed in the form k(1 - √3)
Answer
The sum to infinity of a geometric series is given by: where a is the first term and r is the common ratio.
Given the first term: When substituting θ = 150°, we find:
Next, we find the common ratio from our earlier calculations. Substituting our values:
Thus, the sum to infinity becomes: Bringing this into the form k(1 - √3). Simplifying will yield a value for k.