1
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

If $$f(\alpha)=\int\limits_{1}^{\alpha} \frac{\log _{10} \mathrm{t}}{1+\mathrm{t}} \mathrm{dt}, \alpha>0$$, then $$f\left(\mathrm{e}^{3}\right)+f\left(\mathrm{e}^{-3}\right)$$ is equal to :

A
9
B
$$\frac{9}{2}$$
C
$$\frac{9}{\log _{e}(10)}$$
D
$$\frac{9}{2 \log _{e}(10)}$$
2
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

Let $$I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$$ Then :

A
$$50 I_{6}-9 I_{5}=x I_{5}^{\prime}$$
B
$$50 I_{6}-11 I_{5}=x I_{5}^{\prime}$$
C
$$50 I_{6}-9 I_{5}=I_{5}^{\prime}$$
D
$$50 I_{6}-11 I_{5}=I_{5}^{\prime}$$
3
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

The minimum value of the twice differentiable function $$f(x)=\int\limits_{0}^{x} \mathrm{e}^{x-\mathrm{t}} f^{\prime}(\mathrm{t}) \mathrm{dt}-\left(x^{2}-x+1\right) \mathrm{e}^{x}$$, $$x \in \mathbf{R}$$, is :

A
$$-\frac{2}{\sqrt{\mathrm{e}}}$$
B
$$-2 \sqrt{\mathrm{e}}$$
C
$$-\sqrt{\mathrm{e}}$$
D
$$\frac{2}{\sqrt{\mathrm{e}}}$$
4
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1

Let $$f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}$$.

Consider

$$(\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2$$

$$(\mathrm{S} 2): \int\limits_{-2}^{2} f(x) \mathrm{d} x=12$$

Then,

A
both (S1) and (S2) are correct
B
both (S1) and (S2) are wrong
C
only (S1) is correct
D
only (S2) is correct
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