1
WB JEE 2023
+1
-0.25

Suppose $$f:R \to R$$ be given by $$f(x) = \left\{ \matrix{ 1,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,x = 1 \hfill \cr {e^{({x^{10}} - 1)}} + {(x - 1)^2}\sin {1 \over {x - 1}},\,\mathrm{if}\,x \ne 1 \hfill \cr} \right.$$

then

A
f'(1) does not exist
B
f'(1) exists and is zero
C
f'(1) exist and is 9
D
f'(1) exists and is 10
2
WB JEE 2023
+1
-0.25

Let $${\cos ^{ - 1}}\left( {{y \over b}} \right) = {\log _e}{\left( {{x \over n}} \right)^n}$$, then $$A{y_2} + B{y_1} + Cy = 0$$ is possible for, where $${y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}$$

A
$$A = 2,B = {x^2},C = n$$
B
$$A = {x^2},B = x,C = {n^2}$$
C
$$A = x,B = 2x,C = 3n + 1$$
D
$$A = {x^2},B = 3x,C = 2n$$
3
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.
4
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
$${\log ^n}x$$
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