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Mat1501 Assignment 1 2021

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This document contains MAT1501 Assignment 1 Solutions . All working are shown clearly and explanations are provided when necessary

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MAT1501

ASSIGNMENT 1 2021

QUESTION 1


2 𝑏−𝑎 2 𝑏−𝑎
𝐴: ( )( )= ×
𝑎−𝑏 2 𝑎−𝑏 2
2 −1(𝑎 − 𝑏)
= ×
𝑎−𝑏 2

= −1 (𝑇𝑅𝑈𝐸)

1
𝐵: 𝑥 = − 𝑎𝑛𝑑 𝑦 = −2
2
1
4𝑥 1 2 4 (− 2)
−𝑥 2 − + 𝑦 2 = − (− ) − + (−2)2
𝑦 2 (−2)

1 (−2)
= −( ) − +4
4 (−2)

1
=− −1+4
4
1
=− +3
4
−1 + 3(4)
=
4
−1 + 12
=
4
11
=
4
3
=2 (𝑇𝑅𝑈𝐸)
4

√1 − 𝑥
𝐶: 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑓 √𝑥 − 1 ≠ 0
√𝑥 − 1

√𝑥 − 1 ≠ 0

𝑥−1≠0

, 𝑥≠1

𝑥 ∈ ℝ, 𝑥 ≠ 1 (𝑇𝑅𝑈𝐸) ANSWER:[5]



QUESTION 2

𝑥−𝑦+𝑧 𝑥−𝑦+𝑧
𝐴: =
𝑦 − 𝑥 − 𝑧 −(−𝑦 + 𝑥 + 𝑧)

𝑥−𝑦+𝑧
=−
(−𝑦 + 𝑥 + 𝑧)

𝑥−𝑦+𝑧
=−
(𝑥 − 𝑦 + 𝑧)

= −1 (𝐹𝐴𝐿𝑆𝐸)



1
𝐵: 3𝑎 − 𝑏 <
3𝑎 + 𝑏
1
(3𝑎 − 𝑏) × (3𝑎 + 𝑏) < × (3𝑎 + 𝑏)
3𝑎 + 𝑏

(3𝑎 − 𝑏)(3𝑎 + 𝑏) < 1

9𝑎2 + 3𝑎𝑏 − 3𝑎𝑏 − 𝑏 2 < 1

9𝑎2 − 𝑏 2 < 1 (𝑇𝑅𝑈𝐸)



𝐶: 𝑝2 − 𝑝𝑞 + 𝑞 2 = (𝑝 − 𝑞)2

= (𝑝 − 𝑞)(𝑝 − 𝑞)

= 𝑝2 − 𝑝𝑞 − 𝑝𝑞 + 𝑞 2

= 𝑝2 − 2𝑝𝑞 + 𝑞 2 (𝐹𝐴𝐿𝑆𝐸)

ANSWER:[2]



QUESTION 3


1 1
3 − 𝑥 − 𝑥2
2 4

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