Test Index

Functions Part 2

Section: Mathematics

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Question : 47 of 73

Marks: +1, -0

If the domain of the function

f(x)=\;\frac{\sqrt{x^2-25}}{4-x^2}

+\log_{10}(x^2+2x-15)

(-\infty, \alpha) \cup [\beta, \infty)

\alpha^2 + \beta^3

[1-Feb-2024 Shift 2]

140
175
150
125

Solution: 👈: Video Solution

f(x) = \frac{\sqrt{x^2-25}}{4-x^2} + \log_{10}(x^2+2x-15)

Conditions for the domain:

(i)

x^2-25 \ge 0 \Rightarrow x \in (-\infty,-5] \cup [5,\infty)

(ii)

4-x^2 \ne 0 \Rightarrow x \ne -2,\,2

(iii)

x^2+2x-15 > 0 \Rightarrow (x+5)(x-3) > 0 \Rightarrow x \in (-\infty,-5) \cup (3,\infty)

Taking the intersection:

x \in (-\infty,-5) \cup [5,\infty)

(At

x=-5

the argument of the logarithm becomes

0

, so

x=-5

is excluded; at

x=5

all conditions are satisfied.)

Hence

\alpha = -5

and

\beta = 5

.

\alpha^2 + \beta^3 = (-5)^2 + 5^3 = 25 + 125 = 150

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