MCQ
$\int_{}^{} {\frac{{dx}}{{(x + 1)(x + 2)}} = } $
- A$\log \frac{{x + 2}}{{x + 1}} + c$
- B$\log (x + 1) + \log (x + 2) + c$
- ✓$\log \frac{{x + 1}}{{x + 2}} + c$
- DNone of these
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For any event $H$, if $H ^{ C }$ denotes its complement, then which of the following statements is(are) $TRUE$?
$(A)$ $P \left( E \cap F \cap G ^{ C }\right) \leq \frac{1}{40}$
$(B)$ $P\left(E^C \cap F \cap G\right) \leq \frac{1}{15}$
$(C)$ $P ($E$\cup F \cup G ) \leq \frac{13}{24}$
$(D)$ $P \left( E ^{ C } \cap F ^{ C } \cap G ^{ C }\right) \leq \frac{5}{12}$
$1.$ Which of the following is true?
$(A)$ $(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=0$
$(B)$ $(2-a)^2 f^{\prime}(1)-(2+a)^2 f^{\prime \prime}(-1)=0$
$(C)$ $f^{\prime}(1) f^{\prime}(-1)=(2-a)^2$
$(D)$ $f^{\prime}(1) f^{\prime}(-1)=-(2+a)^2$
$2.$ Which of the following is true?
$(A)$ $f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$
$(B)$ $f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$
$(C)$ $f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
$(D)$ $f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
$3.$ Let $g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t$ which of the following is true?
$(A)$ $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
$(B)$ $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
$(C)$ $\mathrm{g}^{\prime}(\mathrm{x})$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$
$(D)$ $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$
Give the answer question $1,2$ and $3.$