If | , * 20 c a&b&c &c&a &a&b , | = k(a + b + c)( a^2 + b^2 + c^2… - Vidyadip

MCQ

If $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = k(a + b + c)({a^2} + {b^2} + {c^2}$ $ - bc - ca - ab)$, then $k =$

A
$1$
B
$2$
✓
$-1$
D
$-2$

✓

Answer

Correct option: C.

$-1$

c
(c) $\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = a(bc - {a^2}) - b({b^2} - ca) + c(ab - {c^2})$

= $ - {a^3} - {b^3} - {c^3} + 3abc$ = $ - 1\,[{a^3} + {b^3} + {c^3} - 3abc]$

= $ - [(a + b + c)\,({a^2} + {b^2} + {c^2} - ab - bc - ca)]$

==> $k = - 1$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $S=\left\{\theta \in[-\pi, \pi]-\left\{\pm \frac{\pi}{2}\right\}: \sin \theta \tan \theta+\tan \theta=\sin 2 \theta\right\} \text {. }$ If $T =\sum_{\theta \in S } \cos 2 \theta$, then $T + n ( S )$ is equal

If the sum of all the roots of the equation $e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0$ is $\log _{ e } P$, then $p$ is equal to

The number of points of intersection of $| z -(4+3 i )|=2$ and $| z |+| z -4|=6, z \in C$ is

Given that $\pi < \alpha < \frac{{3\pi }}{2},$ then the expression $\sqrt {(4{{\sin }^4}\alpha + {{\sin }^2}2\alpha )} + 4{\cos ^2}\left( {\frac{\pi }{4} - \frac{\alpha }{2}} \right)$ is equal to

Let $f: R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{ccc}\frac{a-b \cos 2 x}{x^2} & ; & x<0 \\ x^2+c x+2 & ; & 0 \leq x \leq 1 \\ 2 x+1 & ; & x>1\end{array}\right.$ If $f$ is continuous everywhere in $R$ and $\mathrm{m}$ is the number of points where $f$ is $NOT$ differential then $\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}$ equals:

The value of $\sqrt {[12 - \sqrt {(68 + 48\sqrt 2 )} ]} = $

If $u = {e^{ - {x^2} - {y^2}}}$, then

Let $X=R \times R$. Define a relation $R$ on $X$ as:
$
\left(a_1, b_1\right) R\left(a_2, b_2\right) \Leftrightarrow b_1=b_2 .
$
Statement - I : R is an equivalence relation.
Statement - II : For some $(a, b) \in X$, the set $S=\{(x, y) \in X:(x, y) R(a, b)\}$ represents a line parallel to $y = x$.In the light of the above statements, choose the correct answer from the options given below:

If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10},$ then the sum of its first $200$ terms is

Let $A$ be a matrix of order $3 \times 3$ and $|A|=5$. If
$|2 \operatorname{adj}(3 A \operatorname{adj}(2 A))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in N$ then $\alpha+\beta+\gamma$ is equal to