1
JEE Main 2021 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} = 1 + x{e^{y - x}}, - \sqrt 2 < x < \sqrt 2 ,y(0) = 0$$

then, the minimum value of $$y(x),x \in \left( { - \sqrt 2 ,\sqrt 2 } \right)$$ is equal to :
A
$$\left( {2 - \sqrt 3 } \right) - {\log _e}2$$
B
$$\left( {2 + \sqrt 3 } \right) + {\log _e}2$$
C
$$\left( {1 + \sqrt 3 } \right) - {\log _e}\left( {\sqrt 3 - 1} \right)$$
D
$$\left( {1 - \sqrt 3 } \right) - {\log _e}\left( {\sqrt 3 - 1} \right)$$
2
JEE Main 2021 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
The area (in sq. units) of the region, given by the set $$\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $$ is :
A
$${8 \over 3}$$
B
$${{17} \over 3}$$
C
$${{13} \over 3}$$
D
$${7 \over 3}$$
3
JEE Main 2021 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let g : N $$\to$$ N be defined as

g(3n + 1) = 3n + 2,

g(3n + 2) = 3n + 3,

g(3n + 3) = 3n + 1, for all n $$\ge$$ 0.

Then which of the following statements is true?
A
There exists an onto function f : N $$\to$$ N such that fog = f
B
There exists a one-one function f : N $$\to$$ N such that fog = f
C
gogog = g
D
There exists a function : f : N $$\to$$ N such that gof = f
4
JEE Main 2021 (Online) 25th July Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let $$f:[0,\infty ) \to [0,\infty )$$ be defined as $$f(x) = \int_0^x {[y]dy} $$

where [x] is the greatest integer less than or equal to x. Which of the following is true?
A
f is continuous at every point in $$[0,\infty )$$ and differentiable except at the integer points.
B
f is both continuous and differentiable except at the integer points in $$[0,\infty )$$.
C
f is continuous everywhere except at the integer points in $$[0,\infty )$$.
D
f is differentiable at every point in $$[0,\infty )$$.
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