If a^2 + b^2 + c^2 + ab + bc + ca 0 , a, ,b, ,c , ,R , then the v… - Vidyadip

MCQ

If ${a^2} + {b^2} + {c^2} + ab + bc + ca \leq 0\,\forall a,\,b,\,c\, \in \,R$ , then the value of determinant $\left| {\begin{array}{*{20}{c}}
{{{(a + b + c)}^2}}&{{a^2} + {b^2}}&1 \\
1&{{{(b + c + 2)}^2}}&{{b^2} + {c^2}} \\
{{c^2} + {a^2}}&1&{{{(c + a + 2)}^2}}
\end{array}} \right|$

✓
$65$
B
$a^2+b^2+c^2+31$
C
$4(a^2+b^2+c^2)$
D
$0$

✓

Answer

Correct option: A.

$65$

a
$\therefore \sum \mathrm{a}^{2}+\sum \mathrm{ab} \leq 0 $

$\Rightarrow(\mathrm{a}+\mathrm{b})^{2}+(\mathrm{b}+\mathrm{c})^{2}+(\mathrm{c}+\mathrm{a})^{2} \leq 0$

$\Rightarrow \mathrm{a}+\mathrm{b}=0, \mathrm{b}+\mathrm{c}=0, \mathrm{c}+\mathrm{a}=0$

$\Rightarrow \mathrm{a}=\mathrm{b}=\mathrm{c}=0$

$\therefore \left|\begin{array}{lll}{4} & {0} & {1} \\ {1} & {4} & {0} \\ {0} & {1} & {4}\end{array}\right|=65$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 3 and 4 . determinant and metrices→3 and 4 . determinant and metrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $|a\,.\,b|\, = 3$ and $|a \times b|\, = 4,$ then the angle between $a $ and $ b$ is

If $y=y(x)$ satisfies the differential equation

$8 \sqrt{x}(\sqrt{9+\sqrt{x}}) d y=(\sqrt{4+\sqrt{9+\sqrt{x}}})^{-1} d x, \quad x>0$

and $y(0)=\sqrt{7}$, then $y(256)=$

If $X$ follows a binomial distribution with parameters $n = 6$ and $p$. If $9P\,(X = 4) = P\,(X = 2),$ then $p = $

If A is any square matrix, then which of the following is skew-symmetric?

A + A^T
A - A^T
AA^T
A^TA

The point which lies in the half-plane $2 x+y-4 \leq 0$ is:

The rate of growth of bacteria in a culture is proportional to the number of bacteris present and the bacteria count is $1000$ at initial time $t =0 .$ The number of bacteria is increased by $20 \%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{ k }{\log _{ e }\left(\frac{6}{5}\right)}$ hours, then $\left(\frac{ k }{\log _{ c } 2}\right)^{2}$ is equal to

If the value of the integral $\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,d x=\alpha e^{-1}+\beta$

where $\alpha, \beta \in R, 5 \alpha+6 \beta=0$, and $[\mathrm{x}]$ denotes the greatest integer less than or equal to $x$; then the value of $(\alpha+\beta)^{2}$ is equal to :

If $y = {x^2}\log x + {2 \over {\sqrt x }},$ then ${{dy} \over {dx}} = $

If $f\left( x \right) = \left| \begin{array}{*{20}{c}}
{\cos x}&x&1\\
{2\sin x}&{{x^2}}&{2x}\\
{\tan x}&x&1
\end{array}\right|$ , then $\mathop {\lim }\limits_{x \to 0} \frac{{f'\left( x \right)}}{x}$

The value of $\tan ^{-1}(1)+\tan ^{-1}(0)+\tan ^{-1}(-1)$ is equal to