Download App

STD 12 - 3 and 4 . determinant and metrices — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 3 and 4 . determinant and metrices4 Marks
MCQ
Let $A=\left[\begin{array}{l}a_{1} \\ a_{2}\end{array}\right]$ and $B=\left[\begin{array}{l}b_{1} \\ b_{2}\end{array}\right]$ be two $2 \times 1$ matrices with real entries such that $A = XB,$where $X=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & -1 \\ 1 & k\end{array}\right],$ and $k \in R$. If $a _{1}^{2}+ a _{2}^{2}=\frac{2}{3}\left( b _{1}^{2}+ b _{2}^{2}\right)$ and $\left( k ^{2}+1\right) b _{2}^{2} \neq-2 b _{1} b _{2}$ then the value of $k$ is $....... .$
  • A
    $2$
  • ✓
    $1$
  • C
    $4$
  • D
    $5$
✓

Answer

Correct option: B.
$1$
$A=X B$$\left[\begin{array}{l} a _{1} \\ a _{2}\end{array}\right]=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & -1 \\ 1 & k \end{array}\right]\left[\begin{array}{l} b _{1} \\ b _{2}\end{array}\right]$
$\left[\begin{array}{c}\sqrt{3} a_{1} \\ \sqrt{3} a_{2}\end{array}\right]=\left[\begin{array}{c}b_{1}-b_{2} \\ b_{1}+k b_{2}\end{array}\right]$
$b_{1}-b_{2}=\sqrt{3} a_{1} ....(1)$
$b _{1}+ kb _{2}=\sqrt{3} a _{2} ....(2)$
Given, $a_{1}^{2}+a_{2}^{2}=\frac{2}{3}\left(b_{1}^{2}+b_{2}^{2}\right)$
$(1)^{2}+(2)^{2}$
$\left(b_{1}+b_{2}\right)^{2}+\left(b_{1}+k b_{2}\right)^{2}=3\left(a_{1}^{2}+a_{2}^{2}\right)$
$a_{1}^{2}+a_{2}^{2}=\frac{2}{3} b_{1}^{2}+\frac{\left(1+k^{2}\right)}{3} b_{2}^{2}+\frac{2}{3} b_{1} b_{2}(k-1)$
Given, $a_{1}^{2}+a_{2}^{2}=\frac{2}{3} b_{1}^{2}+\frac{2}{3} b_{2}^{2}$
On comparing we get
$\frac{ k ^{2}+1}{3}=\frac{2}{3} $
$\Rightarrow k ^{2}+1=2$
$\Rightarrow k =\pm 1 ....(3)$
And $\frac{2}{3}( k -1)=0 $
$\Rightarrow k =1 ....(4)$
From both we get $k =1$

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

Consider the regions for complex number $z$

$A:\frac{1}{{\log_2 \left| z \right|}} - \frac{1}{{\log_2 \left| z \right| - 1}} - 1 < 0$ and $\left( {B:\operatorname{Im} \left( z \right) = 0} \right).$ The range of values of $Re(z)$
lying in the region $A \cap B$  is

→
The value of $\mathop {\lim }\limits_{x \to \infty } \frac{{\log x}}{{{x^n}}},\;n > 0$ is
→
If $f:R \to R$ is a differentiable function and $f\left( 2 \right) = 6$, then $\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {\frac{{2\,tdt}}{{\left( {x - 2} \right)}}} $ is
→
There are $30$ questions in a multiple-choice test.$A$ student gets $1$ mark for each unattempted question, $0$ mark for each wrong answer and $4$ marks for each correct answer. A student answered $x$ questions correctly and scored $60$.Then, the number of possible value of $x$ is
→
The co-ordinates of the extremities of the latus rectum of the parabola $5{y^2} = 4x$ are
→
Let $P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q=\left[q_{i j}\right]$ is a matrix such that $P Q=k I$, where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order $3$ . If $q_{23}=-\frac{k}{8}$ and $\operatorname{det}(Q)=\frac{k^2}{2}$, then

($A$) $\quad \alpha=0, k=8$

($B$) $4 \alpha-k+8=0$

($C$) $\operatorname{det}(P \operatorname{adj}(Q))=2^9$

($D$) $\operatorname{det}(Q \operatorname{adj}(P))=2^{13}$

→
Let $f(x + y) = f(x)f(y)$ and $f(x) = 1 + \sin (3x)g(x)$ where $g(x)$ is continuous then $f'(x)$ is
→
The curve $xy = c, (c > 0)$, and the circle $x^2 + y^2 = 1$ touch at two points. Then the distance between the points of contacts is
→
If $A = \left[ {\begin{array}{*{20}{c}}{1/3}&2\\0&{2x - 3}\end{array}} \right],B = \left[ {\begin{array}{*{20}{c}}3&6\\0&{ - 1}\end{array}} \right]$and $AB = I$, then $x =$
→
For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument with $-\pi<\arg ( z ) \leq \pi$. Then, which of the following statement (s) is (are) $FALSE$ ?

$(A)$ $\arg (-1- i )=\frac{\pi}{4}$, where $i =\sqrt{-1}$

$(B)$ The function $f: R \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-1+i t)$ for all $t \in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$

$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)\right.$ is an integer multiple of $2 \pi$.

$(D)$ For any three given distinct complex numbers, $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{\left( z - z _1\right)\left( z _2- z _3\right)}{\left( z - z _3\right)\left( z _2- z _1\right)}\right)=\pi$, lies on a straight line

→