Number Systems 1 - Free MCQ Practice Test with solutions, GMAT Quant Reasoning

Factoring √12 and √27 and solving, we get
√12 = √(4 x 3) = √(2 x 2 x 3) = 2√3
√27 = √(9 x 3) = √(3 x 3 x 3) = 3√3

Now the equation we get is 3√12 / 6√27. On substituting the values from above, we get

The given expression is in the form of (a - b)(a + b), which is a standard identity:

(a - b)(a + b) = a² - b²

Here,
a = 3
b = √3

Now apply the identity:

(3 - √3)(3 + √3) = 3² - (√3)² = 9 - 3 = 6

So, the result is 6, which is a rational number.

0.375 = 375/1000 because there are three digits after the decimal point, so the denominator is 1000

Find the highest common factor of 375 and 1000
HCF (375,1000) = 125
Divide both numerator and denominator by 125
So, 375 ÷ 125 = 3 and 1000 ÷ 125 = 8

Therefore the fraction in simplest form is 3/8
So, option A is correct.

We use the identity
(a + b)² = a² + 2ab + b²

Here,
a = √5
b = √7

Now apply the identity:

(√5 + √7)² = (√5)² + 2 × √5 × √7 + (√7)²
= 5 + 2√35 + 7
= 12 + 2√35

√8 = 2√2.

√32 = 4√2.

Substitute these into the expression to get: 2√2 + 2×4√2 - 5√2

Combine like terms: (2 + 8 - 5)√2 = 5√2

Therefore the value of the expression is 5√2,
So, option A is correct.