Expansions — Assertion-Reason Type Questions

Assertion (A): (26)³ + (−15)³ + (−11)³ = 3 × 26 × 15 × 11.

Reason (R): If x + y + z = 0, then x³ + y³ + z³ = 3xyz

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

We know that,

⇒ x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

If x + y + z = 0, then :

⇒ x³ + y³ + z³ - 3xyz = 0

⇒ x³ + y³ + z³ = 3xyz.

So, reason (R) is true.

⇒ 26 + (-15) + (-11)

⇒ 26 - 26

⇒ 0

Since, 26 + (-15) + (-11) = 0,

∴ (26)³ + (−15)³ + (−11)³ = 3 × 26 × -15 × -11

Assertion (A) is false.

Thus, A is false and R is true.

Hence, Option 2 is the correct option.

Assertion (A): If $x^2 - 2x - 1 = 0$, then $x^2 + \dfrac{1}{x^2} = 6$.
Reason (R): x² - 2x - 1 can be written as (x - 1)².

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Given,

$\Rightarrow x^2 - 2x - 1 = 0 \\[1em] \Rightarrow x^2 - 1 = 2x \\[1em] \Rightarrow \dfrac{x^2 - 1}{x} = 2 \\[1em] \Rightarrow \dfrac{x^2}{x} - \dfrac{1}{x} = 2 \\[1em] \Rightarrow x - \dfrac{1}{x} = 2 \\[1em]$

Using identity,

$\Big(x - \dfrac{1}{x}\Big)^2 = x^2 + \dfrac{1}{x^2} - 2$

Substituting,

$\Rightarrow (2)^2 = x^2 + \dfrac{1}{x^2} - 2 \\[1em] \Rightarrow 4 = x^2 + \dfrac{1}{x^2} - 2 \\[1em] \Rightarrow x^2 + \dfrac{1}{x^2} = 4 + 2 \\[1em] \Rightarrow x^2 + \dfrac{1}{x^2} = 6.$

Assertion (A) is true.

⇒ (x - 1)² = x² + 1² - 2(x)(1)

⇒ (x - 1)² = x² - 2x + 1

Reason (R) is false.

A is true, R is false

Hence, Option 1 is the correct option.

Assertion (A): (1 - 3x)³ can be expanded as 1 - 27x³ - 9x - 27x².
Reason (R): (a - b)³ = a³ - b³ - 3ab(a - b)

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Using identity,

(a - b)³ = a³ - b³ - 3ab(a - b)

So, reason (R) is true.

⇒ (1 - 3x)³ = 1³ - (3x)³ - 3 × 1 × 3x (1 - 3x)

⇒ (1 - 3x)³ = 1 - 27x³ - 9x(1 - 3x)

⇒ (1 - 3x)³ = 1 - 27x³ - 9x + 27x²

So, assertion (A) is false.

Hence, Option 2 is the correct option.