If $$y = y(x),y \in \left[ {0,{\pi \over 2}} \right)$$ is the solution of the differential equation $$\sec y{{dy} \over {dx}} - \sin (x + y) - \sin (x - y) = 0$$, with y(0) = 0, then $$5y'\left( {{\pi \over 2}} \right)$$ is equal to ______________.
Your Input ________
Answer
Correct Answer is 2
Explanation
$$\sec y{{dy} \over {dx}} = 2\sin x\cos y$$
$${\sec ^2}ydy = 2\sin xdx$$
$$\tan y = - 2\cos x + c$$
$$c = 2$$
$$\tan y = - 2\cos x + 2 \Rightarrow $$ at $$x = {\pi \over 2}$$
$$\tan y = 2$$
$${\sec ^2}y{{dy} \over {dx}} = 2\sin x$$
$$ \therefore $$ $$5{{dy} \over {dx}} = 2$$
3
JEE Main 2021 (Online) 27th July Morning Shift
Numerical
Let $$F:[3,5] \to R$$ be a twice differentiable function on (3, 5) such that $$F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt} $$. If $$F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}$$, then $$\alpha$$ + $$\beta$$ is equal to _______________.
Now, put value of x = 4 we will get $$\alpha$$ = 12 & $$\beta$$ = 4
4
JEE Main 2021 (Online) 27th July Evening Shift
Numerical
Let y = y(x) be the solution of the differential equation dy = e$$\alpha$$x + y dx; $$\alpha$$ $$\in$$ N. If y(loge2) = loge2 and y(0) = loge$$\left( {{1 \over 2}} \right)$$, then the value of $$\alpha$$ is equal to _____________.
Your Input ________
Answer
Correct Answer is 2
Explanation
$$\int {{e^{ - y}}} dy = \int {{e^{\alpha x}}} dx$$