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JEE Main 06-Apr-2024 Paper - Shift 2 — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEEJEE Main 06-Apr-2024 Paper - Shift 24 Marks
Question
Let $(t)$ denote the largest integer less than or equal to $t$. If $\int_0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) d x=a+b \sqrt{2}-\sqrt{3}-\sqrt{5}+c \sqrt{6}-\sqrt{7},$ where $a , b , c \in z$, then $a + b + c$ is equal to $............$
✓

Answer

$\int_0^3\left[x^2\right] d x+\int_0^3\left[\frac{x^2}{2}\right] dx$
$=\int_0^1 0 d x+\int_1^{12} 1 d x+\int_{\sqrt{2}}^{\sqrt{3}} 2 d x$
$+\int_{\sqrt{3}}^2 3 dx +\int_2^{\sqrt{5}} 4 dx +\int_{\sqrt{5}}^{\sqrt{6}} 5 dx$
$+\int_{\sqrt{6}}^{\sqrt{7}} 6 dx +\int_{\sqrt{7}}^{\sqrt{8}} 7 dx +\int_{\sqrt{8}}^3 8 dx$
$+\int_0^{\sqrt{2}} 0 dx +\int_{\sqrt{2}}^2 1 dx$
$+\int_2^{\sqrt{6}} 2 dx +\int_{\sqrt{6}}^{\sqrt{8}} 3 dx +\int_{\sqrt{8}}^3 4 dx $
$=31-6 \sqrt{2}-\sqrt{3}-\sqrt{5}$
$-2 \sqrt{6}-\sqrt{7}$
$a =31 b=-6 c =-2$
$a + b + c $
$=31-6-2=23$

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