Order of Magnitude

Order of Magnitude

1. If x = √3 + √2, then the value of (x + 1)/(x - 1) - (x - 1)/(x + 1) is ?

1. (x + 1)/(x - 1) - (x - 1)/(x + 1) = {(x + 1)2 - (x - 1)2} / {(x - 1) (x + 1)}
= (x2 + 1 + 2x) - (x2 + 1 - 2x) / (x2 - 1)
= 4x / (x2 - 1) = 4(√3 + √2) / {(√3 + √2)2 - 1}

Correct Option: B

(x + 1)/(x - 1) - (x - 1)/(x + 1) = {(x + 1)2 - (x - 1)2} / {(x - 1) (x + 1)}
= (x2 + 1 + 2x) - (x2 + 1 - 2x) / (x2 - 1)
= 4x / (x2 - 1) = 4(√3 + √2) / {(√3 + √2)2 - 1}
= 4(√3 + √2) / (3 + 2 + 2√6 - 1) = 4(√3 + √2) / (4 + 2 √6)
= 4(√3 + √2) / 2 √2 (√3 + √2)
= √2

1. Which among ∛8 and ∛1000 is greater ?

1. ∛8 and ∛1000 = 81/3 and 10001/3

Correct Option: B

∛8 and ∛1000 = 81/3 and 10001/3
Since 1000 > 8
∴ ∛1000 > ∛8

1. Which among √81 and √1 is smaller ?

1. √81 and √1 = 811/2 and 11/2

Correct Option: B

√81 and √1 = 811/2 and 11/2
Since, 1 < 81
∴ √1 < √81

1. IF (14)3 is added to the square of a number, the answer to obtained is 4425. What is the number ?

1. Let the number be x.
According to the question,
x2 + (14)3 = 4425

Correct Option: C

Let the number be x.
According to the question,
x2 + (14)3 = 4425
⇒ x2 = 4425 - (14)3
⇒ x2 = 4425 - 2744
⇒ x2 = 1681
∴ x = √1681 = 41

1. Which among ∜10 and ∛8 is greater ?

1. ∜10 and ∛8 = 101/4 and 81/3
Since, LCM of 4 and 3 is 12.
∴ 101/4 = 103/12 = (103)1/12 = (1000)1/12 = 121000
and 81/3 = 84/12 = (84)1/12 (4096)1/2 = 124096
∴ 4096 > 1000

Correct Option: B

∜10 and ∛8 = 101/4 and 81/3
Since, LCM of 4 and 3 is 12.
∴ 101/4 = 103/12 = (103)1/12 = (1000)1/12 = 121000
and 81/3 = 84/12 = (84)1/12 (4096)1/2 = 124096
∴ 4096 > 1000
Hence, 81/3 >101/4
∴ ∛8 > ∜10