Question

Statement-1: The equation $3^{x-1}+x=105$, where $x$ is an integer, has exactly one solution.
Statement-2: If $f ( x )$ is a monotonic continuous function in $[ a , b ]$ such that $f ( a ) f ( b )< 0$, then the equation $f ( x )=0$ has exactly one solution in $( a , b )$.

• A. Statement-1 is true, statement-2 is true and statement- 2 is correct explanation for statement- 1 .
• B. Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
• C. Statement-1 is true, statement- 2 is false.
• D. Statement-1 is false, statement-2 is true.

Answer 1

Answer: (D)

$ f ( x )=3^{ x -1}+ x -105$ is strictly increasing and so $f ( x )=0$ has at most one solution and $f(5)=-19<0$ and $f(6)=144>0$, so $f(x)$ has exactly one solution in $(5,6)$ but $x$ is an integer and hence no solution in $(5,6) \Rightarrow$ (D)