1
WB JEE 2022
+1
-0.25

Let $$\mathop {\lim }\limits_{ \in \to 0 + } \int\limits_ \in ^x {{{bt\cos 4t - a\sin 4t} \over {{t^2}}}dt = {{a\sin 4x} \over x} - 1,\left( {0 < x < {\pi \over 4}} \right)}$$. Then a and b are given by

A
$$a = 2,b = 2$$
B
$$a = {1 \over 4},b = 1$$
C
$$a = - 1,b = 4$$
D
$$a = 2,b = 4$$
2
WB JEE 2022
+1
-0.25

Let $$f(x) = \int\limits_{\sin x}^{\cos x} {{e^{ - {t^2}}}dt}$$. Then $$f'\left( {{\pi \over 4}} \right)$$ equals

A
$$\sqrt {{1 \over e}}$$
B
$$- \sqrt {{2 \over e}}$$
C
$$\sqrt {{2 \over e}}$$
D
$$- \sqrt {{1 \over e}}$$
3
WB JEE 2022
+2
-0.5

If I is the greatest of $${I_1} = \int\limits_0^1 {{e^{ - x}}{{\cos }^2}x\,dx}$$, $${I_2} = \int\limits_0^1 {{e^{ - {x^2}}}{{\cos }^2}x\,dx}$$, $${I_3} = \int\limits_0^1 {{e^{ - {x^2}}}dx}$$, $${I_4} = \int\limits_0^1 {{e^{ - {x^2}/2}}dx}$$, then

A
I = I1
B
I = I2
C
I = I3
D
I = I4
4
WB JEE 2021
+1
-0.25
$$\int\limits_1^3 {{{\left| {x - 1} \right|} \over {\left| {x - 2} \right| + \left| {x - 3} \right|}}dx}$$ is equal to
A
$$1 + {4 \over 3}{\log _e}3$$
B
$$1 + {3 \over 4}{\log _e}3$$
C
$$1 - {4 \over 3}{\log _e}3$$
D
$$1 - {3 \over 4}{\log _e}3$$
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