Numerical Ability Test - 26

Given:

The expression: (a + b)(a - b)(a² - ab + b²)(a² + ab + b²)

Calculation:

(a + b)(a - b)(a² - ab + b²)(a² + ab + b²)

⇒ [(a + b)(a² - ab + b²)][(a - b)(a² + ab + b²)]

⇒ [a³ - a²b + ab² + a²b - ab² + b³][a³ + a²b + ab² - a²b - ab² - b³]

⇒ (a³ + b³) × (a³ - b³)

⇒ (a³)² - (b³)²

⇒ = a⁶ - b⁶

∴ The simplified expression is a⁶ - b⁶.

Given:

(6a + 2b + 7c)²

Formula used:

(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Calculation:

Here, x = 6a, y = 2b, and z = 7c.

(6a + 2b + 7c)² = (6a)² + (2b)² + (7c)² + 2(6a)(2b) + 2(2b)(7c) + 2(7c)(6a)

⇒ 36a² + 4b² + 49c² + 24ab + 28bc + 84ca

∴ (6a + 2b + 7c)² = 36a² + 4b² + 49c² + 24ab + 28bc + 84ca

Given:

The expression to be multiplied is (x + a)(x + b).

Formula Used:

Product of two binomials: (x + a)(x + b) = x² + (a + b)x + ab

Calculation:

(x + a)(x + b)

Expanding using the distributive property:

⇒ x(x + b) + a(x + b)

⇒ x² + bx + ax + ab

Combining like terms:

⇒ x² + (a + b)x + ab

The product of (x + a) and (x + b) is x² + (a + b)x + ab.

Given:

Values: 18 , 12 , and -30

Calculation:

We need to find the value of (18)3 + (12)3 + (−30)3

First, calculate each cube:

18³ = 18 × 18 × 18

⇒ 18³ = 5832

12^3 = 12 × 12 × 12

⇒ 12³ = 1728

(-30)³ = (-30) × (-30) × (-30)

⇒ (-30)³ = -27000

Now, sum the cubes:

(18)3 + (12)3 + (−30)3 = 5832 + 1728 - 27000

⇒ 5832 + 1728 = 7560

⇒ 7560 - 27000 = -19440

The correct answer is option 2.

Given:

Solve: (0.1 × 0.1 × 0.1+0.02 × 0.02 × 0.02)/(0.2 × 0.2 × 0.2+0.04 × 0.04 × 0.04)

Formula Used:

Basic arithmetic operations and exponentiation.

Calculation:

(0.1 × 0.1 × 0.1 + 0.02 × 0.02 × 0.02) / (0.2 × 0.2 × 0.2 + 0.04 × 0.04 × 0.04)

⇒ (0.1³ + 0.02³) / (0.2³ + 0.04³)

⇒ (0.001 + 0.000008) / (0.008 + 0.000064)

⇒ 0.001008 / 0.008064

⇒ 0.125

The correct answer is option 2.

Given:

x = 3 + 2√2

Formula used:

The expression we need to solve is: √x - (1/√x)

Calculation:

Let √x = y. So, x = y², and we have:

y² = 3 + 2√2

Now, to find y, consider the expression y = √x, and simplify further:

We can express y as (a + b√2) to match the form of x. So let:

y = a + b√2

Then, y² = (a + b√2)² = a² + 2ab√2 + 2b²

Equating this with x = 3 + 2√2, we get:

a² + 2b² = 3 and 2ab = 2

Solving 2ab = 2 gives ab = 1. Hence, a = 1/b.

Substitute into a² + 2b² = 3:

(1/b)² + 2b² = 3

1/b² + 2b² = 3

Multiply through by b²:

1 + 2b⁴ = 3b²

2b⁴ - 3b² + 1 = 0

Let z = b². The equation becomes:

2z² - 3z + 1 = 0

Solving this quadratic equation, we get z = 1, so b = 1, and a = 1.

Therefore, y = 1 + √2.

Now, calculate the original expression:

√x - (1/√x) = y - (1/y) = (1 + √2) - (1 / (1 + √2))

Rationalize (1 / (1 + √2)) by multiplying the numerator and denominator by (1 - √2):

1 / (1 + √2) × (1 - √2) / (1 - √2) = (1 - √2) / ((1 + √2)(1 - √2)) = (1 - √2) / (1 - 2) = (1 - √2) / -1

⇒ 1 / (1 + √2) = √2 - 1

So, √x - (1/√x) = (1 + √2) - (√2 - 1) = 1 + √2 - √2 + 1 = 2