If x = 4t^3 + 3, y = 3t^4 + 4 and d^2x dy^2 ( dx dy )^n is constant then the value of n is

Step-by-step solution

Given x = 4t^3 + 3 and y = 3t^4 + 4 . Differentiating with respect to t : dx dt = 12t^2 dy dt = 12t^3 dx dy = dx dt dy dt = 12t^2 12t^3 = 1 t = t⁻¹ Now, finding the second derivative d^2x dy^2 : d^2x dy^2 = d dy ( dx dy ) = d dt (t⁻¹ ) dt dy d^2x dy^2 = -t⁻² 1 12t^3 = - 1 12 t⁻⁵ Substituting these into the given expression: d^2x dy^2 ( dx dy )^n = - 1 12 t⁻⁵ (t⁻¹)^n = - 1 12 t^ n-5 For this expression to be a constant, it must be independent of t . Therefore, the exponent of t must be zero: n - 5 = 0 n = 5 Answer: