Question

Number of solution for x between 3 and 15 if $\int _ { 0 } ^ { \mathrm { x } } [ \mathrm { t } ] \mathrm { dt } =$ $\int_{0}^{\lbrack x\rbrack}{tdt,}$ where [.] denotes greatest integer function, is -

• A. 12
• B. 13
• C. 11
• D. Infinitely many

Answer 1

Answer: (A)

12

Answer 2

$\int_{0}^{x}{\lbrack t\rbrack dt}$ = $\int_{0}^{\lbrack x\rbrack}{\lbrack t\rbrack dt}$ + $\int_{\lbrack x\rbrack}^{\lbrack x\rbrack + \{ x\}}{\lbrack t\rbrack dt}$ = $\int_{0}^{\lbrack x\rbrack}{tdt}$

\ $\int_{\lbrack x\rbrack}^{\lbrack x\rbrack + \{ x\}}{\lbrack t\rbrack dt}$ = $\int_{0}^{\lbrack x\rbrack}{tdt}$ – $\int_{0}^{\lbrack x\rbrack}{\lbrack t\rbrack dt}$ = $\int_{0}^{\lbrack x\rbrack}{\{ t\} dt}$

\ {x} [x] = [x] . $\frac{1}{2}$i.e. {x} = $\frac{1}{2}$

thus 3 < x = n + $\frac{1}{2}$ < 15 i.e. 3 – $\frac{1}{2}$ < n < 15 – $\frac{1}{2}$

\ n can take 12 values.

No. of solutions is 12.