AQA A-level MATHEMATICS Paper 1, 7357/1.PB/KL/Jun23/E7,Solution.

Solution

AQA A-level MATHEMATICS Paper 1

7357/1

PB/KL/Jun23/E7

shahbaz ahmed

September 2024



Relevant formula and skills

0! = 1
n n!
Cr = r!(n−r)!
d n d n
dx cx = c dx x = nxn−1

limn−→∞ (1 + n1 )n = e

Where e is an irrational number such that 2 < e < 3

If ax = b ⇐⇒ loga b = x

loge x = ln x

................................................................

1

, To expand cos(θ) using Taylor’s theorem up to three terms around

x = 0, we can use the formula:


f ′′ (0) 2 f ′′′ (0) 3
f (x) = f (0) + f ′ (0)x + x + x + R3 (x)
2! 3!

where R3 (x) is the remainder term. For cos(θ):

1. **Function and its derivatives**: - f (θ) = cos(θ) - f ′ (x) =

− sin(x) - f ′′ (x) = − cos(x) - f ′′′ (x) = sin(x) - f (4) (x) = cos(x) (and it

repeats)

2. **Evaluating at x = 0**: - f (0) = cos(0) = 1 - f ′ (0) = − sin(0) =

0 - f ′′ (0) = − cos(0) = −1 - f ′′′ (0) = sin(0) = 0

3. **Constructing the expansion**: Using the derivatives:

−1 2 θ3
cos(θ) ≈ 1 + 0 · θ + θ +0·
2! 3!

Simplifying this gives:

θ2
cos(θ) ≈ 1 −
2

Thus, the Taylor series expansion of cos(θ) up to three terms is:




2

, θ2
cos(θ) ≈ 1 − + 0 · θ3
2

For practical purposes, we can stop at the quadratic term, resulting

in:


θ2
cos(θ) ≈ 1 −
2

............................................................................

cos(α + β) = cos α cos β − sin α sin β

Put α = β = x

cos 2x = cos2 x − sin2 x

From equation

cos2 x + sin2 x = 1 ⇐⇒ cos2 x = 1 − sin2 x

=⇒ cos 2x = cos2 x − sin2 x = 1 − sin2 x − sin2 x = 1 − 2 sin2 x

Similarly

From equation

cos2 x + sin2 x = 1 ⇐⇒ sin2 x = 1 − cos2 x

=⇒

cos 2x = cos2 x − sin2 x = cos2 x − (1 − cos2 x) = 2 cos2 x − 1


3

, .....................................................................

log m + log n = log mn

log m
n = log m − log n

log mn = n log m

...........................................................................

Integration by parts

Z Z
u dv = uv − v du




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Q1. Find the coefficient of x7 in the expansion of

(2x − 3)7

Circle your answer.
128
−2187 − 128 2

Solution

Using the formula


4