If the domain of the function f(x) = frac1sqrt10 + 3x - x^2 + frac1sqrtx + |x|$$f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$$ is (a, b)$(a, b)$, then (1 + a)^2 + b^2$(1 + a)^2 + b^2$ is equal to:

### Related Formula textFor frac1sqrtg(x) text to be defined, we require: g(x) > 0$$\text{For } \frac{1}{\sqrt{g(x)}} \text{ to be defined, we require: } g(x) > 0$$ ### Core Logic We find the domains of the two constituent terms separately and then find their intersection. ### Step 1: Find the domain of the first term For the first term to be defined: 10 + 3x - x^2 > 0 implies x^2 - 3x - 10 < 0$$10 + 3x - x^2 > 0 \implies x^2 - 3x - 10 < 0$$ (x - 5)(x + 2) < 0 implies x in (-2, 5) quad text--- (1)$$(x - 5)(x + 2) < 0 \implies x \in (-2, 5) \quad \text{--- (1)}$$ ### Step 2: Find the domain of the second term For the second term to be defined: x + |x| > 0$x + |x| > 0$ - If x ge 0$x \ge 0$: x + x = 2x > 0 implies x > 0$x + x = 2x > 0 \implies x > 0$. - If x < 0$x < 0$: x - x = 0 ngtr 0$x - x = 0 \ngtr 0$. Thus, the domain of the second term is: x in (0, infty) quad text--- (2)$$x \in (0, \infty) \quad \text{--- (2)}$$ ### Step 3: Find intersection and calculate the final expression Intersecting domains (1) and (2): x in (-2, 5) cap (0, infty) implies x in (0, 5)$$x \in (-2, 5) \cap (0, \infty) \implies x \in (0, 5)$$ Comparing this with (a, b)$(a, b)$ gives a = 0$a = 0$ and b = 5$b = 5$. Now calculate the value: (1 + a)^2 + b^2 = (1 + 0)^2 + 5^2 = 1 + 25 = 26$$(1 + a)^2 + b^2 = (1 + 0)^2 + 5^2 = 1 + 25 = 26$$ ### Pattern Recognition Modulus domain constraint: The function x + |x|$x + |x|$ is non-zero only for positive values of x$x$. This is a standard math trick that collapses complex domains down to x > 0$x > 0$ instantly. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Relations and Functions