1
JEE Main 2021 (Online) 1st September Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If y = y(x) is the solution curve of the differential equation $${x^2}dy + \left( {y - {1 \over x}} \right)dx = 0$$ ; x > 0 and y(1) = 1, then $$y\left( {{1 \over 2}} \right)$$ is equal to :
A
$${3 \over 2} - {1 \over {\sqrt e }}$$
B
$$3 + {1 \over {\sqrt e }}$$
C
3 + e
D
3 $$-$$ e
2
JEE Main 2021 (Online) 1st September Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
The function $$f(x) = {x^3} - 6{x^2} + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements :

Statement 1 : there exists x1, x2 $$\in$$(2, 4), x1 < x2, such that f'(x1) = $$-$$1 and f'(x2) = 0.

Statement 2 : there exists x3, x4 $$\in$$ (2, 4), x3 < x4, such that f is decreasing in (2, x4), increasing in (x4, 4) and $$2f'({x_3}) = \sqrt 3 f({x_4})$$.

Then
A
both Statement 1 and Statement 2 are true
B
Statement 1 is false and Statement 2 is true
C
both Statement 1 and Statement 2 are false
D
Statement 1 is true and Statement 2 is false
3
JEE Main 2021 (Online) 1st September Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx} $$, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A = {[{a_{ij}}]_{3 \times 3}}$$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$$. Then $$\left| {adj{A^{ - 1}}} \right|$$ is :
A
(15)2 $$\times$$ 242
B
(15)2 $$\times$$ 234
C
(105)2 $$\times$$ 238
D
(105)2 $$\times$$ 236
4
JEE Main 2021 (Online) 1st September Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
The area, enclosed by the curves $$y = \sin x + \cos x$$ and $$y = \left| {\cos x - \sin x} \right|$$ and the lines $$x = 0,x = {\pi \over 2}$$, is :
A
$$2\sqrt 2 (\sqrt 2 - 1)$$
B
$$2(\sqrt 2 + 1)$$
C
$$4(\sqrt 2 - 1)$$
D
$$2\sqrt 2 (\sqrt 2 + 1)$$
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