AP TET Paper 2 Math and Science Model Paper 3

Concept:Use the algebraic identity $(a-b)^2 = a^2 + b^2 - 2ab$ after scaling the given equation to match the required terms.Explanation:Given equation: $3x - \frac{1}{2x} = 4$.We need $4x^2 + \frac{1}{9x^2}$, which can be written as $(2x)^2 + \left(\frac{1}{3x}\right)^2$.Multiply the given equation by $\frac{2}{3}$ on both sides.$\frac{2}{3}\left(3x - \frac{1}{2x}\right) = 4 \times \frac{2}{3}$This simplifies to: $2x - \frac{1}{3x} = \frac{8}{3}$.Square both sides of this equation.$\left(2x - \frac{1}{3x}\right)^2 = \left(\frac{8}{3}\right)^2$Expanding using the identity: $(2x)^2 + \left(\frac{1}{3x}\right)^2 - 2(2x)\left(\frac{1}{3x}\right) = \frac{64}{9}$$4x^2 + \frac{1}{9x^2} - \frac{4}{3} = \frac{64}{9}$Rearrange to isolate the required expression.$4x^2 + \frac{1}{9x^2} = \frac{64}{9} + \frac{4}{3}$Convert $\frac{4}{3}$ to $\frac{12}{9}$ and add.$4x^2 + \frac{1}{9x^2} = \frac{64 + 12}{9} = \frac{76}{9}$Answer:The value of $4x^2 + \frac{1}{9x^2}$ is $\frac{76}{9}$, which is option D.