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6 (a) Find the first two terms, in ascending powers of x, of the binomial expansion of \( \left( 1 - \frac{x}{2} \right)^{\frac{1}{2}} \) 6 (b) Hence, for small values of x, show that \( sin(4x) + \sqrt{\cos x} \approx A + Bx + Cx^2 \) where A, B and C are constants to be found. - AQA - A-Level Maths Pure - Question 6 - 2022 - Paper 1
Question 6
6 (a) Find the first two terms, in ascending powers of x, of the binomial expansion of \( \left( 1 - \frac{x}{2} \right)^{\frac{1}{2}} \) 6 (b) Hence, for small val... show full transcript
Worked Solution & Example Answer:6 (a) Find the first two terms, in ascending powers of x, of the binomial expansion of \( \left( 1 - \frac{x}{2} \right)^{\frac{1}{2}} \) 6 (b) Hence, for small values of x, show that \( sin(4x) + \sqrt{\cos x} \approx A + Bx + Cx^2 \) where A, B and C are constants to be found. - AQA - A-Level Maths Pure - Question 6 - 2022 - Paper 1
Step 1
Find the first two terms, in ascending powers of x, of the binomial expansion of \( \left( 1 - \frac{x}{2} \right)^{\frac{1}{2}} \)
Answer
To find the first two terms of the binomial expansion of ( \left( 1 - \frac{x}{2} \right)^{\frac{1}{2}} ), we use the binomial theorem:
[ (1 + u)^n = 1 + nu + \frac{n(n-1)}{2!}u^2 + \cdots ] where ( u = -\frac{x}{2} ) and ( n = \frac{1}{2} ).
First term: [ 1 ]
Second term: [ \frac{1}{2} \left(-\frac{x}{2}\right) = -\frac{x}{4} ]
Thus, the first two terms are: [ 1 - \frac{x}{4} ]
Step 2
Hence, for small values of x, show that \( sin(4x) + \sqrt{\cos x} \approx A + Bx + Cx^2 \)
Answer
For small values of x, we can use the approximations:
[ sin(kx) \approx kx \quad \text{and} \quad \cos x \approx 1 - \frac{x^2}{2} ]
Applying these:
-
For (sin(4x)): [ sin(4x) \approx 4x ]
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For (\sqrt{\cos x}): [ \sqrt{\cos x} \approx \sqrt{1 - \frac{x^2}{2}} \approx 1 - \frac{x^2}{4} \quad (using ; \sqrt{1-u}\approx 1 - \frac{u}{2} ; for ; small ; u) ]
Combining these: [ sin(4x) + \sqrt{\cos x} \approx 4x + \left(1 - \frac{x^2}{4}\right) ]
Thus: [ sin(4x) + \sqrt{\cos x} \approx 1 + 4x - \frac{x^2}{4} ]
From this, we identify constants:
- A = 1
- B = 4
- C = -(\frac{1}{4})
Hence, the required form is: [ sin(4x) + \sqrt{\cos x} \approx A + Bx + Cx^2 ]