(B) For $f(x)$ to be a decreasing function,the coefficient of $x$ must be less than or equal to $0$. Let $k = [t]+1$. Then $\frac{|k|+a}{|k|+1-a} \leq

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(B) For $f(x)$ to be a decreasing function,the coefficient of $x$ must be less than or equal to $0$. Let $k = [t]+1$. Then $\frac{|k|+a}{|k|+1-a} \leq 0$. Since $a 0$. For the fraction to be $\leq 0$,the numerator must be $\leq 0$ (since the denominator is always positive). $|k| + a \leq 0 \implies |k| \leq -a$. Since $-a > 0$,this implies $a \leq k \leq -a$. Substituting $k = [t]+1$,we get $a \leq [t]+1 \leq -a$. Subtracting $1$ from all sides,$a-1 \leq [t] \leq -a-1$. Since $a \notin I$,the range for $t$ is $[[a], [-a])$. Thus,the correct option is $B$.

Class 11 Mathematics Linear Inequalities Word problem of Linear inequalities

Let $a$ $(a < 0, a \notin I)$ be a constant and $t$ be a parameter. Then the set of values of $t$ for which the function $f(x) = \left( \frac{|[t]+1|+a}{|[t]+1|+1-a} \right)x$ is a decreasing function of $x$ (where $[.]$ denotes the greatest integer function) is:

A
$([a], [-a+1])$
B
$[[a], [-a])$
C
$[[a+1], [-a+1]]$
D
$[[a-1], [-a+1])$

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Class 11

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Linear Inequalities

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Word problem of Linear inequalities

If $x \in R$ and $1 \leq \frac{3x^2-7x+8}{x^2+1} \leq 2$,then the minimum and maximum values of $x$ are respectively.

MediumAP EAMCET 2019

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The value of the greatest integer $k$ satisfying the inequation $2^{n+4} + 12 \geq k(n+4)$ for all $n \in \mathbb{N}$ is

DifficultTS EAMCET 2025

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$A$ manufacturer has $600 \text{ litres}$ of a $12 \%$ solution of acid. How many litres of a $30 \%$ acid solution must be added to it so that the acid content in the resulting mixture will be more than $15 \%$ but less than $18 \%$?

Difficult

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There are $30$ questions in a multiple-choice test. $A$ student gets $1$ mark for each unattempted question,$0$ mark for each wrong answer,and $4$ marks for each correct answer. If a student answered $x$ questions correctly and scored $60$,then the number of possible values of $x$ is:

DifficultKVPY 2011

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In the first four examinations,each of $100$ marks,Ravi scored $94, 73, 72,$ and $84$ marks. If a final average greater than or equal to $80$ and less than $90$ is needed to obtain a final grade $'B'$ in a course,find the minimum marks that Ravi must obtain in the fifth examination to get grade $'B'$.

Medium

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Let $a$ $(a < 0, a \notin I)$ be a constant and $t$ be a parameter. Then the set of values of $t$ for which the function $f(x) = \left( \frac{|[t]+1|+a}{|[t]+1|+1-a} \right)x$ is a decreasing function of $x$ (where $[.]$ denotes the greatest integer function) is:

Similar Questions

If $x \in R$ and $1 \leq \frac{3x^2-7x+8}{x^2+1} \leq 2$,then the minimum and maximum values of $x$ are respectively.

The value of the greatest integer $k$ satisfying the inequation $2^{n+4} + 12 \geq k(n+4)$ for all $n \in \mathbb{N}$ is

$A$ manufacturer has $600 \text{ litres}$ of a $12 \%$ solution of acid. How many litres of a $30 \%$ acid solution must be added to it so that the acid content in the resulting mixture will be more than $15 \%$ but less than $18 \%$?

There are $30$ questions in a multiple-choice test. $A$ student gets $1$ mark for each unattempted question,$0$ mark for each wrong answer,and $4$ marks for each correct answer. If a student answered $x$ questions correctly and scored $60$,then the number of possible values of $x$ is:

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