Question

If $x{\log }_{e}x=a,\text{and}aϵR,$ then match the following

Column I	Column II
(A) Equation has one real root	(p) $a<-\frac{1}{e}$
(B) Equation has 2 different real roots	(q) $a\ge 1$
(C) Equation has 2 equal roots	(r) $a=-\frac{1}{e}$
(D) Equation has no real roots	(s) $-\frac{1}{e}<a<0$
	(t) $a\ge 0$

• A. A(q, t), B(s), C(r), D(p)
• B.

Answer 1

The correct option is A A(q, t), B(s), C(r), D(p)


Graph of $x{\log }_{e}x$
When $a<-\frac{1}{e}\to x{\log }_{e}x$ will not cut at any point $y=a$ and no real root
$\Rightarrow a=-\frac{1}{e}\to x{\log }_{e}x$ will touch the line $y=a,$ So two equal root
When $0<a<\frac{-1}{e}\to x{\log }_{e}x$ will cut at two point to line $y=a$, So two unequal real root
When $a\ge 0\to x{\log }_{e}x$ will cut at one point to line $y=a$, So one real root.