The question consists of two statements numbered “I” and “II” given below it. You have to decide whether the data provided in the statements are sufficient to answer the question or not and choose the correct option accordingly.

Equation (i) : ⁿ√(qⁿ) × (p + 1) = 27

'p' is the largest even prime number.

Equation (ii) : (q + m - n)² = 64

Find the sum of q, m, and n

(I) : (x - 1) and (x + 1) are factors of the equation mx³ + nx² - 2x - 3 = 0

(II) : q = n + 6 and 2m + q = 13

Equation (i) : n√(qn) × (p + 1) = 27

'p' is the largest even prime number.

Equation (ii) : (q + m - n)2 = 64

Find the sum of q, m, and n

Largest even prime number = 2 so p = 2

⇒ [(qⁿ)]^(1/n)  × (2 +1) = 27

⇒ q = 9

Equation (ii) : (q + m - n)² = 64

Using statement (I) :

If (x - 1) and (x + 1) are factors of the equation mx3 + nx2 - 2x - 3 = 0 then x = 1 and x = -1 are roots of the given equation and it will satisfy the given equation.

Therefore, m × (1)³ + n × (1)² - 2 × (1) - 3 = 0

⇒ m + n = 5 ...(i)

Similarly, m × (-1)³ + n × (-1)² - 2 × (-1) - 3 = 0

⇒ -m + n = 1 ...(ii)

Using (i) and (ii)

⇒ 2n = 6

⇒ n = 3

And, m = 2 

Sum of  q + m + n = 9 + 2  + 3 = 14

Hence, statement (I) alone is sufficient to answer the question.

Using statement (II) :

⇒ q = n + 6

⇒ n = q - 6 = 9 - 6 = 3

And, 2m + q = 13

⇒ 2m = 13 - 9 = 4

⇒ m = 2

Sum of  q + m + n = 9 + 2  + 3 = 14

Hence, statement (II) alone is sufficient to answer the question.