Let N be the set of all natural numbers and f: N→N be such that 1990< f(1990)<2100and satisfies the equation x−f(x)=19[x19]−90[f(x)90], where [y] denotes the greatest integer less than or equal to y. Then the number of possible values of f(1990) is

If 1990 < f(1990) < 2100⇒ 199090 < f(199090)< 210090⇒ 22.1 < f(199090) < 23.3⇒ [f(1990)90]=22, 23Given x−f(x)=19[x19]−90 [f(x)90]put x= 1990⇒ 1990−f(1990)=19[199019]−90[f(1990)90]⇒ 1990−f(1990) =19[104.7]−90[f(1990)90]⇒ 1990−f(1990)=19(104)−90[f(1990)90]∴ f(1990)= 1990−19(104)+90[f(1990)90]=1990−1976+90[f(1990)90]If [f(1990)90]=22 thenf(1990)=14+90(22)=1994If [f(1990)90]=23 thenf(1990)=14+90(23)=2084∴ number of possible values are '2'

Let N be the set of all natural numbers and f: N→N be such that 1990< f(1990)<2100and satisfies the equation x−f(x)=19[x19]−90[f(x)90], where [y] denotes the greatest integer less than or equal to y. Then the number of possible values of f(1990) is

If 1990 < f(1990) < 2100⇒ 199090 < f(199090)< 210090⇒ 22.1 < f(199090) < 23.3⇒ [f(1990)90]=22, 23Given x−f(x)=19[x19]−90 [f(x)90]put x= 1990⇒ 1990−f(1990)=19[199019]−90[f(1990)90]⇒ 1990−f(1990) =19[104.7]−90[f(1990)90]⇒ 1990−f(1990)=19(104)−90[f(1990)90]∴ f(1990)= 1990−19(104)+90[f(1990)90]=1990−1976+90[f(1990)90]If [f(1990)90]=22 thenf(1990)=14+90(22)=1994If [f(1990)90]=23 thenf(1990)=14+90(23)=2084∴ number of possible values are '2'

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$\begin{array}{l} L e t N b e t h e s e t o f a l l n a t u r a l n u m b e r s a n d f : N \to N b e s u c h t h a t 1990 < f (1990) < 2100 \\ a n d s a t i s f i e s t h e e q u a t i o n x - f (x) = 19 [\frac{x}{19}] - 90 [\frac{f (x)}{90}], w h e r e [y] d e n o t e s t h e g r e a t e s t \\ int e g e r l e s s t h a n o r e q u a l t o y . T h e n t h e n u m b e r o f p o s s i b l e v a l u e s o f f (1990) i s \end{array}$

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Detailed Solution

$\begin{array}{l} I f 1990 < f (1990) < 2100 \\ \Rightarrow \frac{1990}{90} < f (\frac{1990}{90}) < \frac{2100}{90} \\ \Rightarrow 22.1 < f (\frac{1990}{90}) < 23.3 \\ \Rightarrow [\frac{f (1990)}{90}] = 22, 23 \\ G i v e n x - f (x) = 19 [\frac{x}{19}] - 90 [\frac{f (x)}{90}] \\ p u t x = 1990 \\ \Rightarrow 1990 - f (1990) = 19 [\frac{1990}{19}] - 90 [\frac{f (1990)}{90}] \\ \Rightarrow 1990 - f (1990) = 19 [104.7] - 90 [\frac{f (1990)}{90}] \end{array}$

$\begin{array}{l} \Rightarrow 1990 - f (1990) = 19 (104) - 90 [\frac{f (1990)}{90}] \\ ∴ f (1990) = 1990 - 19 (104) + 90 [\frac{f (1990)}{90}] \\ = 1990 - 1976 + 90 [\frac{f (1990)}{90}] \\ I f [\frac{f (1990)}{90}] = 22 t h e n \\ f (1990) = 14 + 90 (22) \\ = 1994 \\ I f [\frac{f (1990)}{90}] = 23 t h e n \\ f (1990) = 14 + 90 (23) \\ = 2084 \\ ∴ n u m b e r o f p o s s i b l e v a l u e s a r e' 2' \end{array}$