MHTCET 18 Apr 2026 Shift 2 Paper

Concept:The absolute value function changes its form at the point where its inside expression is zero; split the integral at that point.Explanation:Set $4x - 5 = 0$, which gives $x = \frac{5}{4}$.For $0 \le x < \frac{5}{4}$, the expression $4x-5$ is negative, so $|4x-5| = 5 - 4x$.For $\frac{5}{4} \le x \le 2$, the expression $4x-5$ is positive, so $|4x-5| = 4x - 5$.Therefore, split the given integral as:$$\int_{0}^{2} |4x-5|\,dx = \int_{0}^{5/4} (5-4x)\,dx + \int_{5/4}^{2} (4x-5)\,dx.$$Evaluate the first part:$$\int_{0}^{5/4} (5-4x)\,dx = \left[5x - 2x^{2}\right]_{0}^{5/4} = \frac{25}{8}.$$Evaluate the second part:$$\int_{5/4}^{2} (4x-5)\,dx = \left[2x^{2} - 5x\right]_{5/4}^{2} = \frac{9}{8}.$$Add both results:$$\frac{25}{8} + \frac{9}{8} = \frac{34}{8} = \frac{17}{4}.$$Thus, the value of the integral is $\frac{17}{4}$. The correct option is A.Answer:$\frac{17}{4}$ (Option A)