The binomial expansion of $(2 + 5x)^4$ is given by $(2 + 5x)^4 = 4 + 160x + Bx^{2} + 1000x^{3} + 625x^{4}$. 5 (a) Find the value of A and the value of B. 5 (b) Show that $(2 + 5x)^{4} - (2 - 5x)^{4} = Cx + D x^{3}$ where C and D are constants to be found. 5 (c) Hence, or otherwise, find $ \int ((2 + 5x)^{4} - (2 - 5x)^{4}) \, dx$.

A-Level AQA Maths Pure Binomial Expansion: The binomial expansion of $(2 +

The binomial expansion of $(2 + 5x)^4$ is given by $(2 + 5x)^4 = 4 + 160x + Bx^{2} + 1000x^{3} + 625x^{4}$ - AQA - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Question 5

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The binomial expansion of $(2 + 5x)^4$ is given by $(2 + 5x)^4 = 4 + 160x + Bx^{2} + 1000x^{3} + 625x^{4}$. 5 (a) Find the value of A and the value of B. 5 (b) Sho... show full transcript

Worked Solution & Example Answer:The binomial expansion of $(2 + 5x)^4$ is given by $(2 + 5x)^4 = 4 + 160x + Bx^{2} + 1000x^{3} + 625x^{4}$ - AQA - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Step 1

Find the value of A and the value of B.

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Answer

To find the values of A and B, we need to compare the coefficients in the binomial expansion.

Using the binomial theorem, we expand $(2 + 5x)^{4}$ :

$A = 4$ (this is the constant term when $x = 0$ ) and the other non-constant terms give:

For $x^{2}$ term: From expansion, we have:

$B = \binom{4}{2} \cdot (2)^{2} \cdot (5x)^{2} = 6 \cdot 4 \cdot 25 = 600.$

Thus, we find: $A = 16$ $B = 600$ .

Step 2

Show that (2 + 5x)^{4} - (2 - 5x)^{4} = Cx + D x^{3}$ where C and D are constants to be found.

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Answer

To demonstrate the equality, we need to calculate $(2 + 5x)^{4}$ and $(2 - 5x)^{4}$ . Using the binomial theorem:

$(2 + 5x)^{4} = 16 + 160x + 600x^{2} + 1000x^{3} + 625x^{4}$ $(2 - 5x)^{4} = 16 - 160x + 600x^{2} - 1000x^{3} + 625x^{4}$

When we subtract these expressions:

$(2 + 5x)^{4} - (2 - 5x)^{4} = (16 + 160x + 600x^{2} + 1000x^{3} + 625x^{4}) - (16 - 160x + 600x^{2} - 1000x^{3} + 625x^{4})$

This simplifies to: $320x + 2000x^{3}$ Hence, we identify C and D as $C = 320$ and $D = 2000$ .

Step 3

Hence, or otherwise, find ∫((2 + 5x)^{4} - (2 - 5x)^{4}) dx.

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Answer

To find the integral, we will use the result from part (b):

We can express the integral as: $\int (320x + 2000x^{3}) dx$

Applying the integration rules gives us: $= \int 320x \: dx + \int 2000x^{3} \: dx$ $= 320 \cdot \frac{x^{2}}{2} + 2000 \cdot \frac{x^{4}}{4} + c$ $= 160x^{2} + 500x^{4} + c$ Hence, the final result is: $160x^{2} + 500x^{4} + c$ .

The binomial expansion of $(2 + 5x)^4$ is given by $(2 + 5x)^4 = 4 + 160x + Bx^{2} + 1000x^{3} + 625x^{4}$ - AQA - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Worked Solution & Example Answer:The binomial expansion of $(2 + 5x)^4$ is given by $(2 + 5x)^4 = 4 + 160x + Bx^{2} + 1000x^{3} + 625x^{4}$ - AQA - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Find the value of A and the value of B.

Show that (2 + 5x)^{4} - (2 - 5x)^{4} = Cx + D x^{3}$ where C and D are constants to be found.

Hence, or otherwise, find ∫((2 + 5x)^{4} - (2 - 5x)^{4}) dx.

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