Question

Consider the equation $||x-1|-2|=\lambda$ . Which of the following statement(s) is/are true?

• A. If the given equation has two solutions, then $\lambda$ belongs to $(2,\infty )\cup \left\{0\right\}$ .
• B. If the given equation has three solutions, then $\lambda$ belongs to $\left\{2\right\}$ .
• C. The number of integral values of $\lambda$ so that the given equation has four solutions, is $1.$
• D. If the given equation has two solutions, then $\lambda$ belongs to $(-\infty ,2)$ .

Answer 1

Answer: (A), (B), (C)

The correct options are
A If the given equation has two solutions, then $\lambda$ belongs to $(2,\infty )\cup \left\{0\right\}$ .
B The number of integral values of $\lambda$ so that the given equation has four solutions, is $1.$
C If the given equation has three solutions, then $\lambda$ belongs to $\left\{2\right\}$ .

Given equation is $||x-1|-2|=\lambda$
$\Rightarrow \lambda \ge 0$
when $\lambda >2$
$||x-1|-2|=2+k$ where $k$ is positive.
$\Rightarrow |x-1|=4+k$ gives two solutions for x. or $|x-1|=-k$ which is not possible.
when $\lambda =0$
$|x-1|=2$ gives two solutions for x.
Hence, If the given equation has two solutions, then $\lambda$ belongs to $(2,\infty )\cup \left\{0\right\}$
when $\lambda =2$
$||x-1|-2|=2$
$\Rightarrow |x-1|=4$ or $|x-1|=0$ gives three solutions for x.
Hence,If the given equation has three solutions, then $\lambda$ belongs to {2}
when $\lambda \in (0,2)$
$|x-1|=2\pm \lambda$
$\Rightarrow x=1\pm (2\pm \lambda )$ gives four solutions for x.
only integer in $(0,2)$ is $1$.
Hence, Number of integral values of $\lambda$ so that the given equation has four solutions, is 1
Hence, options A,B and C.