If the domain of the function f ( x ) = log ( 10 x 2 - 17 x + 7 ) ( 18 x 2 - 11 x + 1 ) is ( , a ) ( b , c ) ( d , ) - { e } , then 90 ( a + b + c + d + e ) equals: [2026]

Question

If the domain of the function

$f (x) = \log_{(10 x^{2} - 17 x + 7)} (18 x^{2} - 11 x + 1)$

is $(- \infty, a) \cup (b, c) \cup (d, \infty) - {e},$ then $90 (a + b + c + d + e)$ equals: [2026]

Smart Achievers · Accepted Answer

(1)

$18 x^{2} - 11 x + 1 > 0$

$(2 x - 1) (9 x - 1) > 0$

$x < \frac{1}{9} or \frac{1}{2} < x$

$Also 10 x^{2} - 17 x + 7 > 0$

$(x - 1) (10 x - 7) > 0$

$x < \frac{7}{10} or 1 < x$

$and 10 x^{2} - 17 x + 7 \neq 1$

$x \in (- \infty, \frac{1}{9}) \cup (\frac{1}{2}, \frac{7}{10}) \cup (1, \infty) - {\frac{6}{5}}$

$90 (a + b + c + d + e) = 90 (\frac{1}{9} + \frac{1}{2} + \frac{7}{10} + 1 + \frac{6}{5})$

$= 10 + 45 + 63 + 90 + 108 = 316$

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