Exam (elaborations) study

Q1:Fill in the following blanks the correct answer (50M)



1) 2 −5 =2 − 5

= 2 −5 = (5) −5 =0

2) if 2 = 18 then 3

log 2 = log 18
(𝑎 + 𝑏) log 2 = (2𝑎 − 𝑏) log 18
𝑎 + 𝑏 = 2𝑎 log 18 − 𝑏 log 18
2𝑎 log 18 − 𝑎 = 𝑏 + 𝑏 log 18
𝑎(2 log 18 − 1) = 𝑏(1 + log 18 )

= =.

3) if f(t) = √x t − t then x ∈ ⋯ … … … ..


𝑓(𝑡) = 𝑡 (𝑥 − 1)
𝑥 − 1 ≥ 0 → 𝑥 ∈ 𝑅−] − 1,1[
4) if |x + 2x − 3| = |x − 1| then x ∈ ⋯ … … … … … … … ….
|(𝑥 − 1)(𝑥 + 3)| − |𝑥 − 1| = 0
|𝑥 − 1|(|𝑥 + 3| − 1) = 0
|𝑥 − 1| = 0 → 𝑥 = 1
|𝑥 + 3| = 1 → 𝑥 = −4, −2
𝑥 ∈ {−4, −2,1}
5) if α, β are roots of the x − 2x + 9 = 0 the β + α = ⋯ … … ….
(𝛼 + 𝛽 ) = 2 , 𝛼𝛽 = 9
(𝛼 + 𝛽 ) = 4 → 𝛼 + 2𝛼𝛽 + 𝛽 = 4 → 𝛼 + 𝛽 = 4 − 2(𝛽𝛼) = 4 − 18 = −14
(𝛼 + 𝛽 ) = (14) → 𝛼 + 2(𝛼𝛽) + 𝛽 = 196
𝛼 + 𝛽 = 196 − 2(9) = 196 − 162 = 34

, ×( )
6) 𝑖𝑓 𝑦 = log ( )× ( )× ( ) 𝑡ℎ𝑒𝑛 𝑦 = ⋯ … … … … ..

(3 ) × (2 )
(3 ) ×2 ×3 ×2
3 ×2 =3 ×2 =3

81 × ( 8)
log = log 3 = −3
(27( ) × 6() × 4( ))




7) log √ 𝑥 𝑥 𝑥 √𝑥 = log √ (√𝑥) =


8) if log (x − 1) < 1 𝑡ℎ𝑒𝑛 𝑥 ∈ ⋯ … … … … … … ….


1 3 3 3
𝑥 −1> → 𝑥 − >0 𝑥∈𝑅− − ,
2 2 2 2

1 4 1 1 4 1
9) + + = + +
log 3 + 1 log 4 + 2 log 18 + log 3 + log 2 log 4 + 2 log 3 log 18 + log 4

1 4 1
= + +
log 3 + log 2 log 4 + log 9 log 18 + log 2
1 4 1
= + +
log 6 log 36 log 36
1 4 1
= + +
log 6 log 6 log 6
1 4 1
= + +
log 6 2log 6 2log 6
1
= log 2 + 2 log 3 + log 4 == log 2 + log 9 + log 2 = log 36 = 2
2


‫اكتب المعادلة هنا‬.
1
10) if x −2+ = 2 then the set of solution =
√2 − x
𝐷=∅→𝑠=∅