MCQ
On which of the following intervals, the function $x^{100} + sin x - 1$ is strictly increasing.
- A$(0, \pi /2)$
- B$(0, 1)$
- C$(\pi /2, \pi )$
- ✓All of the above
for $x\, \in \,(0,\,1)\,$ and $\,\left( {0\,,\,\frac{\pi }{2}} \right)\,, cosx$ and $x$ are both $+ve$ ==>$\uparrow$
for, $x\, \in \,\,\left( {\frac{\pi }{2},\;\pi } \right)\,x > 1$ hence $100 x^{99}$ obviously $> cosx $==>$\uparrow$ ]
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$(A)$ $N ^{\top} M N$ is symmetric or skew symmetric, according as $M$ is symmetric or skew symmetric
$(B)$ $M N-N M$ is skew symmetric for all symmetric matrices $M$ and $N$
$(C)$ $M N$ is symetric for all symmetric matrices $M$ and $N$
$(D)$ $(\operatorname{adj} M)(\operatorname{adj} N)=\operatorname{adj}(M N)$ for all invertible matrices $M$ and $N$
| Column $I$ | Column $II$ |
| $(A)$ The minimum value of $\frac{x^2+2 x+4}{x+2}$ is | $(p)$ $0$ |
| $(B)$ Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skewsymmetric, and $(A+B)(A-B)=(A-B)(A+B)$. If $(A B)^t=(-1)^k A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then the possible values of $k$ are | $(q)$ $1$ |
| $(C)$ Let $\mathrm{a}=\log _3 \log _3 2$. An integer $\mathrm{k}$ satisfying $1<2^{\left(-k+3^{-2}\right)}<2$, must be less than | $(r)$ $2$ |
| $(D)$ If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are | $(s)$ $3$ |