MCQ
If $A = \left[ {\begin{array}{*{20}{c}}1&1\\1&1\end{array}} \right],$then ${A^{100}} = $
- A${2^{100}}A$
- ✓${2^{99}}A$
- C${2^{101}}A$
- DNone of these
${A^2} = \left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right]\,\left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right]$= $\left[ {\,\begin{array}{*{20}{c}}2&2\\2&2\end{array}\,} \right] = 2\left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right]$
${A^3} = 2\,\left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right]\,\left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right] = {2^2}\left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right]$
${A^n} = {2^{n - 1}}\left[ {\,\begin{array}{*{20}{c}}1&1\\1&1\end{array}\,} \right]$ ==> ${A^{100}} = {2^{99}}\left[ {\begin{array}{*{20}{c}}1&1\\1&1\end{array}} \right]$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| 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 | $(s)$ $3$ |