If x denotes the greatest integer function of x and ∫-3/23/22x-3dx=k, then k+1/2=

Correct answer: (C) 10. ∫-3/23/22x-3dx=∫-3/2-1-6dx+∫-1-1/2-5dx+∫-1/20-4dx+∫01/2-3dx+∫1/21-2dx+∫13/2-4dx =-6x-3/2-1+-5x-1-1/2+-4x-1/20+-3x01/2+-2x1/21+-x13/2 =-6+5+4+3+2+1×1/2 i.e. k=-21/2 Now, k+1/2=-21/2+1/2=10

Q117 / 160 · TS EAMCET 2022 (18 Jul Shift 2) · Mathematics

If $[x]$ denotes the greatest integer function of $x$ and $\int_{- \frac{3}{2}}^{\frac{3}{2}} [2 x - 3] d x = k$ , then $|k + \frac{1}{2}| =$

A
$7$
B
$8$
Correct answer
C
$10$
D
$12$

Solution

$\int_{- \frac{3}{2}}^{\frac{3}{2}} [2 x - 3] d x =$ $\int_{- \frac{3}{2}}^{- 1} (- 6) d x + \int_{- 1}^{- \frac{1}{2}} (- 5) d x + \int_{- \frac{1}{2}}^{0} (- 4) d x + \int_{0}^{\frac{1}{2}} (- 3) d x + \int_{\frac{1}{2}}^{1} (- 2) d x + \int_{1}^{\frac{3}{2}} (- 4) d x$
$= {(- 6 x)}_{- \frac{3}{2}}^{- 1} + {(- 5 x)}_{- 1}^{- \frac{1}{2}} + {(- 4 x)}_{- \frac{1}{2}}^{0} + {(- 3 x)}_{0}^{\frac{1}{2}} + {(- 2 x)}_{\frac{1}{2}}^{1} + {(- x)}_{1}^{\frac{3}{2}}$
$= - (6 + 5 + 4 + 3 + 2 + 1) \times \frac{1}{2}$
i.e. $k = - \frac{21}{2}$
Now, $|k + \frac{1}{2}| = |- \frac{21}{2} + \frac{1}{2}| = 10$

Hence, the correct answer is option: 'C'.

If x denotes the greatest integer function of x and ∫-32322x-3dx=k, then k+12=

Solution

Relevant Questions

If $[x]$ denotes the greatest integer function of $x$ and $\int_{- \frac{3}{2}}^{\frac{3}{2}} [2 x - 3] d x = k$ , then $|k + \frac{1}{2}| =$