1
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

The value of $\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}$ is equal to

A
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)+\frac{x}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
B
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)+\frac{\mathrm{e}^x}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
C
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)+\frac{1}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
D
$\log \left(\frac{x \mathrm{e}^x}{1+x \mathrm{e}^x}\right)-\frac{x}{1+x \mathrm{e}^x}+\mathrm{c}$, where c is a constant of integration
2
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

If $\mathrm{p} \rightarrow(\sim \mathrm{p} \vee \sim \mathrm{q})$ is false, then the truth values of p and q are respectively

A
$\mathrm{F, F}$
B
$\mathrm{F}, \mathrm{T}$
C
$\mathrm{T, T}$
D
$\mathrm{T}, \mathrm{F}$
3
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

Let $\alpha, \beta$ be the roots of the equation $x^2-\mathrm{p} x+\mathrm{r}=0$ and $\frac{\alpha}{2}, 2 \beta$ be the roots of the equation $x^2-q x+r=0$. Then the value of r is

A
$\frac{2}{9}(\mathrm{p}-\mathrm{q})(2 \mathrm{q}-\mathrm{p})$
B
$\frac{2}{9}(\mathrm{q}-\mathrm{p})(2 \mathrm{p}-\mathrm{q})$
C
$\frac{2}{9}(q-2 p)(2 q-p)$
D
$\frac{2}{9}(2 p-q)(2 q-p)$
4
MHT CET 2024 9th May Evening Shift
MCQ (Single Correct Answer)
+2
-0

Let $\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \quad$ and $\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$, where $\alpha, \beta \in \mathbb{R}$, be three vectors. If the projection at $\overline{\mathrm{a}}$ on $\overline{\mathrm{c}}$ is $\frac{10}{3}$ and $\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$, then the value of $\alpha^2+\beta^2-\alpha \beta$ is equal to

A
1
B
2
C
3
D
4
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