Which of the following statement(s) is/are CORRECT?

$$\begin{gathered} \mathit{A}\mathbf{)} f\left(0\right)=0,f\left(1\right)=1 \\ {(f\text{'}\left(x\right)-1)}^2\ge 0 \\ \Rightarrow {(f\text{'}\left(x\right))}^2-2f\text{'}\left(x\right)+1\ge 0 \\ \Rightarrow \underset{0}{\overset{1}{\int }}{(f\text{'}\left(x\right))}^{2}dx-2\underset{0}{\overset{1}{\int }}f\text{'}\left(x\right)dx+\underset{0}{\overset{1}{\int }}1dx\ge 0 \\ \Rightarrow \underset{0}{\overset{1}{\int }}{(f\text{'}\left(x\right))}^{2}dx\ge 1 \end{gathered}$$

$$\begin{gathered} \mathit{B}\mathbf{)} 3{(1+x)}^2+3{(1-y)}^2+4(1+x)(1-y)=0 \\ \Rightarrow {(1+x)}^2+{(1-y)}^2+2{((1+x)+(1-y))}^2=0 \\ \Rightarrow x=-1,y=1 \end{gathered}$$

$$\begin{gathered} \mathit{C}\mathbf{)}\mathbf{ }\mathbf{ } {\overline{n}}_1=(x_1,x_2,x_3),{\overline{n}}_2=(y_1,y_2,y_3),{\overline{n}}_3=(1,1,1) \\ {\overline{n}}_1.{\overline{n}}_3=0,{\overline{n}}_1.{\overline{n}}_2=0,{\overline{n}}_2.{\overline{n}}_3=0 \\ Consider \overline{p}=(1,0,0) \end{gathered}$$

$$\begin{gathered} \frac{\overline{p}.{\overline{n}}_1}{\left|{\overline{n}}_1\right|}=\frac{x_1}{\sqrt{x_1^2+x_2^2+x_3^2}},\frac{\overline{p}.{\overline{n}}_2}{\left|{\overline{n}}_2\right|}=\frac{y_1}{\sqrt{y_1^2+y_2^2+y_3^2}},\frac{\overline{p}.{\overline{n}}_3}{\left|{\overline{n}}_3\right|}=\frac{1}{\sqrt{3}} \\ {\left(\frac{\overline{p}.{\overline{n}}_1}{\left|{\overline{n}}_1\right|}\right)}^2+{\left(\frac{\overline{p}.{\overline{n}}_2}{\left|{\overline{n}}_2\right|}\right)}^2+\left(\frac{\overline{p}.{\overline{n}}_3}{\left|{\overline{n}}_3\right|}\right)={\left|\overline{p}\right|}^2 \\ \Rightarrow \frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}+\frac{1}{3}=1 \end{gathered}$$

D) 112, 116, 120, 125, 132
Variance is independent of change of scale shift the origin to 120
The observations are –8, –4, 0, 5, 12
Variance  $={\sigma }^2=\sum _{i=1}^n\frac{{(x_i-\overline{x})}^2}{n}=\sum _{i=1}^5\frac{{(x_i-1)}^2}{5}$
$=\frac{244}{5}=48.80$