Number System Question 4 - CAT 2024

Step-by-Step Explanation

First, convert the base numbers to a uniform base. Notice that 32768 = 2¹⁵ = ( 2³ )⁵ = 8⁵.

Rewrite the equation using the base (1)/(8):

( 1/8 )ᵏ × ( 1/8⁵) ¹⁄³ = 1/8 × ( (1)/(8⁵) )¹⁄ᵏ

( (1)/(8) )ᵏ × ( (1)/(8) )⁵⁄³ = (1)/(8) ×  (1/8)⁵⁄ ᵏ

Since the bases match on both sides, equate the exponents:

k + (5)/(3) = 1 + (5)/(k)

k - 1 + (5)/(3) - (5)/(k) = 0 ⇒ k + (2)/(3) - (5)/(k) = 0

Multiply the entire equation by 3k to form a standard quadratic equation:

3k² + 2k - 15 = 0

Check the discriminant (D = b² - 4ac) to ensure the roots are real numbers:

D = 2² - 4(3)( - 15) = 4 + 180 = 184 > 0 ⇒ Roots are real

The sum of the roots of a quadratic equation ax² + bx + c = 0 is given by - (b)/(a):

Sum of real values of k = - (2)/(3)

Correct Option: C

💡 FundaMakers Tip:

When asked for the "sum of all real values" from a quadratic equation, always check the discriminant first. If D ≥ 0, you can use Viet's formula (- (b)/(a)) to find the sum of the roots instantly without factoring the equation.