Class 12th

$f\left(x\right)=\stackrel{?}{a}\cdot (\stackrel{?}{b}*\stackrel{?}{c})=\left|\begin{array}{ccc}x& -2& 3\\ -2& x& -1\\ 7& -2& x\end{array}\right|$

$=x^3-27x+26$

$f^{\mathrm{\text{'}}}\left(x\right)=3x^2-27=0\Rightarrow x=\pm 3$ and $f^{\mathrm{\text{'}}\mathrm{\text{'}}}(-3)<0$

$\Rightarrow$ local maxima at $\mathrm{x}={\mathrm{x}}_0=-3$

Thus, $\stackrel{?}{a}=-3\hat{i}-2\hat{j}+3\hat{k},\stackrel{?}{b}=2\hat{i}-3\hat{j}-\hat{k}$ , and $\stackrel{?}{c}=7\hat{i}-2\hat{j}-3\hat{k}$

$\Rightarrow \stackrel{?}{a}\cdot \stackrel{?}{b}+\stackrel{?}{b}\cdot \stackrel{?}{c}+\stackrel{?}{c}\cdot \stackrel{?}{a}=9-5-26=-22$

Yes, MIMTS accepts Class 12 scores for UG course admission. The college offers three courses, i.e., BSc, BCom and BBA. The selection criteria are merit-based. For BSc and BCom courses, candidates are selected for admission according to Directorate of Higher Education (DHE) Government of Odisha through Student Academic Management System (SAMS). 

Candidates seeking admission to the GNM, BSc and MSc programme can enroll for admission with Class 12 and graduation marks. Candidates must complete Class 12 to enroll for UG course. Similarly, for PG course, candidates are selected based on BSc Nursing. Apollo School and College of Nursing, Hyderabad admissions are merit + entrance-based.

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This is Other Questions as classified in NCERT Exemplar

$\begin{array}{l}Giventhat:\frac{{\left(a^2+1\right)}^2}{2a-i}=x+yi\dots \left(i\right)\\ Takingconjugateonbothsides\\ \frac{{\left(a^2+1\right)}^2}{2a+i}=x-yi\dots \left(ii\right)\\ Multiplyingeqn.\left(i\right)and\left(ii\right)wehave\\ \frac{{\left(a^2+1\right)}^2{\left(a^2+1\right)}^2}{\left(2a-i\right)\left(2a+i\right)}=x^2+y^2\\ \Rightarrow \frac{{\left(a^2+1\right)}^4}{4a^2-i^2}=x^2+y^2\\ \Rightarrow \frac{{\left(a^2+1\right)}^4}{4a^2+1}=x^2+y^2\\ Hence,thevalueof{x}^2+y^2=\frac{{\left(a^2+1\right)}^4}{4a^2+1}.\end{array}$