1
WB JEE 2023
+1
-0.25

The function $$y = {e^{kx}}$$ satisfies $$\left( {{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}}} \right)\left( {{{dy} \over {dx}} - y} \right) = y{{dy} \over {dx}}$$. It is valid for

A
exactly one value of k.
B
two distinct values of k.
C
three distinct values of k.
D
infinitely many values of k.
2
WB JEE 2023
+1
-0.25

If $$y = {\log ^n}x$$, where $${\log ^n}$$ means $${\log _e}{\log _e}{\log _e}\,...$$ (repeated n times), then $$x\log x{\log ^2}x{\log ^3}x\,.....\,{\log ^{n - 1}}x{\log ^n}x{{dy} \over {dx}}$$ is equal to

A
$$\log x$$
B
$$x$$
C
1
D
$${\log ^n}x$$
3
WB JEE 2023
+2
-0.5

If $$x = \sin \theta$$ and $$y = \sin k\theta$$, then $$(1 - {x^2}){y_2} - x{y_1} - \alpha y = 0$$, for $$\alpha=$$

A
k
B
$$-$$k
C
$$-$$k$$^2$$
D
k$$^2$$
4
WB JEE 2022
+1
-0.25

If $$y = {e^{{{\tan }^{ - 1}}x}}$$, then

A
$$(1 + {x^2}){y_2} + (2x - 1){y_1} = 0$$
B
$$(1 + {x^2}){y_2} + 2xy = 0$$
C
$$(1 - {x^2}){y_2} - {y_1} = 0$$
D
$$(1 + {x^2}){y_2} + 3x{y_1} + 4y = 0$$
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