1
JEE Advanced 2022 Paper 2 Online
Numerical
+3
-1
Change Language
The greatest integer less than or equal to

$$ \int_{1}^{2} \log _{2}\left(x^{3}+1\right) d x+\int_{1}^{\log _{2} 9}\left(2^{x}-1\right)^{\frac{1}{3}} d x $$

is ___________.
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2
JEE Advanced 2022 Paper 2 Online
Numerical
+3
-1
Change Language
Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$$ f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases} $$

If $\alpha$ is the area of the region

$$ \left\{(x, y) \in \mathbb{R} \times \mathbb{R}:|x| \leq \frac{3}{4}, 0 \leq y \leq \min \{f(x), g(x)\}\right\}, $$

then the value of $9 \alpha$ is
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3
JEE Advanced 2021 Paper 2 Online
Numerical
+2
-0
Change Language
Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} } $$, x > 0 and $${f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0$$, where, for any positive integer n and real numbers a1, a2, ....., an, $$\prod\nolimits_{i = 1}^n {{a_i}} $$ denotes the product of a1, a2, ....., an. Let mi and ni, respectively, denote the number of points of local minima and the number of points of local maxima of function fi, i = 1, 2 in the interval (0, $$\infty$$).

The value of $$2{m_1} + 3{n_1} + {m_1}{n_1}$$ is ___________.
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4
JEE Advanced 2021 Paper 2 Online
Numerical
+2
-0
Change Language
Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} } $$, x > 0 and $${f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0$$, where, for any positive integer n and real numbers a1, a2, ....., an, $$\prod\nolimits_{i = 1}^n {{a_i}} $$ denotes the product of a1, a2, ....., an. Let mi and ni, respectively, denote the number of points of local minima and the number of points of local maxima of function fi, i = 1, 2 in the interval (0, $$\infty$$).

The value of $$6{m_2} + 4{n_2} + 8{m_2}{n_2}$$ is ___________.
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