1
JEE Main 2021 (Online) 17th March Evening Shift
+4
-1
Let y = y(x) be the solution of the differential equation

$$\cos x(3\sin x + \cos x + 3)dy = (1 + y\sin x(3\sin x + \cos x + 3))dx,0 \le x \le {\pi \over 2},y(0) = 0$$. Then, $$y\left( {{\pi \over 3}} \right)$$ is equal to :
A
$$2{\log _e}\left( {{{\sqrt 3 + 7} \over 2}} \right)$$
B
$$2{\log _e}\left( {{{3\sqrt 3 - 8} \over 4}} \right)$$
C
$$2{\log _e}\left( {{{2\sqrt 3 + 10} \over {11}}} \right)$$
D
$$2{\log _e}\left( {{{2\sqrt 3 + 9} \over 6}} \right)$$
2
JEE Main 2021 (Online) 17th March Morning Shift
+4
-1
Which of the following is true for y(x) that satisfies the differential equation

$${{dy} \over {dx}}$$ = xy $$-$$ 1 + x $$-$$ y; y(0) = 0 :
A
y(1) = 1
B
y(1) = e$$-$$$${1 \over 2}$$ $$-$$ 1
C
y(1) = e$${1 \over 2}$$ $$-$$ e$$-$$$${1 \over 2}$$
D
y(1) = e$${1 \over 2}$$ $$-$$ 1
3
JEE Main 2021 (Online) 16th March Evening Shift
+4
-1
If y = y(x) is the solution of the differential equation

$${{dy} \over {dx}}$$ + (tan x) y = sin x, $$0 \le x \le {\pi \over 3}$$, with y(0) = 0, then $$y\left( {{\pi \over 4}} \right)$$ equal to :
A
$${1 \over 2}$$loge 2
B
$$\left( {{1 \over {2\sqrt 2 }}} \right)$$ loge 2
C
loge 2
D
$${1 \over 4}$$ loge 2
4
JEE Main 2021 (Online) 16th March Evening Shift
+4
-1
Let C1 be the curve obtained by the solution of differential equation

$$2xy{{dy} \over {dx}} = {y^2} - {x^2},x > 0$$. Let the curve C2 be the

solution of $${{2xy} \over {{x^2} - {y^2}}} = {{dy} \over {dx}}$$. If both the curves pass through (1, 1), then the area enclosed by the curves C1 and C2 is equal to :
A
$${\pi \over 4}$$ + 1
B
$$\pi$$ + 1
C
$$\pi$$ $$-$$ 1
D
$${\pi \over 2}$$ $$-$$ 1
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