Mathematics International Advanced Subsidiary/ Advanced Level Pure Mathematics 1 Exam practice (Solutions)

Solutions:

Edexcel IAL / AS Mathematics – Pure Mathematics 1

Complete Step-by-Step Solutions

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Question 1: Simplify

1(a)
(2a√7b)²
= 2² × a² × (√7b)²
= 4a² × 7b
= 28a²b

1(b)
(2a²∛6b)³
= 2³ × a⁶ × (∛6b)³
= 8a⁶ × 6b
= 48a⁶b

1(c)
(1 − √7) / (3 − √7)
Multiply numerator and denominator by (3 + √7):
= [(1 − √7)(3 + √7)] / (9 − 7)
= (−4 − 2√7) / 2
= −2 − √7

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Question 2: Simultaneous Equations

y + 3x + 1 = 0 ⇒ y = −3x − 1

Substitute into y² + 11x² + 3x = 0:
(−3x − 1)² + 11x² + 3x = 0
9x² + 6x + 1 + 11x² + 3x = 0
20x² + 9x + 1 = 0

(5x + 1)(4x + 1) = 0
x = −1/5 or x = −1/4

Solutions:
(−1/5, −2/5) and (−1/4, −1/4)

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Question 3

f′(x) = √x − 1/√x

Integrate:
f(x) = (2/3)x^3/2 − 2x^1/2 + C

Use point (9, 5):
5 = 18 − 6 + C ⇒ C = −7

f(x) = (2/3)x^3/2 − 2x^1/2 − 7

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Question 4

(x² − 10) / (5√x)
= (1/5)x^3/2 − 2x^−1/2

4(a)
A = 1/5, p = 3/2, B = −2, q = −1/2

4(b)
dy/dx = (3/10)x^1/2 + x^−3/2

4(c)
∫y dx = (2/25)x^5/2 − 4x^1/2 + C

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Question 5: Graph Transformations

Original maximum A(−4, 9), minimum B(2, −3)

(a) y = 2f(x): A(−4, 18), B(2, −6)
(b) y = f(x) − 9: A(−4, 0), B(2, −12)
(c) y = f(x + 2): A(−6, 9), B(0, −3)
(d) y = f(2x): A(−2, 9), B(1, −3)

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Question 6

y = 2x² + 9x^1/3 + (x³ − 6)/(4x^1/2)

dy/dx = 4x + 3x^−2/3 + (5/8)x^3/2 + (3/4)x^−3/2

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Question 7

Distinct roots ⇒ discriminant > 0
k² − 4k − 16 > 0

k = 2 ± 2√5
k < 2 − 2√5 or k > 2 + 2√5

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Question 8

Substitute x = 1, y = q:
p = 2/q and q = 7
p = 2/7, q = 7

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Question 9

y = cos(x − π/4), 0 ≤ x ≤ 2π

Intercepts:
y-axis: (0, √2/2)
x-axis: (3π/4, 0), (7π/4, 0)

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Question 10

Line: y = 3x − 5
Intersection with 2x − 3y + 6 = 0
x = 3, y = 4
P(3, 4)