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 $A = \left( {\begin{array}{*{20}{c}}
{\cos \,\alpha }&{ - \sin \,\alpha }\\
{\sin \,\alpha }&{\cos \,\alpha }
\end{array}} \right)$, $\left( {\alpha  \in R} \right)$ such that ${A^{32}} = \left( {\begin{array}{*{20}{c}}
0&{ - 1}\\
1&0
\end{array}} \right)$. Then a value of $\alpha $ is
  • A
    $0$
  • B
    $\frac{\pi }{{16}}$
  • C
    $\frac{\pi }{{32}}$
  • ✓
    $\frac{\pi }{{64}}$
✓

Answer

Correct option: D.
$\frac{\pi }{{64}}$
d
$A = \left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]$

${A^2} = \left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{ - \sin 2\alpha }\\
{\sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]$

${A^3} = \left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{ - \sin 2\alpha }\\
{\sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }\\
{\sin \alpha }&{\cos \alpha }
\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}
{\cos 3\alpha }&{ - \sin 3\alpha }\\
{\sin 3\alpha }&{\cos 3\alpha }
\end{array}} \right]$

Simiarly ${A^{32}} = \left[ {\begin{array}{*{20}{c}}
{\cos 32\alpha }&{ - \sin 32\alpha }\\
{\sin 32\alpha }&{\cos 32\alpha }
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
0&{ - 1}\\
1&0
\end{array}} \right]$

 $ \Rightarrow \cos 32\alpha  = 0$ and $\sin 32\alpha  = 0$

$ \Rightarrow 32\alpha  = \left( {4n + 1} \right)\frac{\pi }{2},n \in 1$

$\alpha  = \left( {4n + 1} \right)\frac{\pi }{{64}},n \in 1$

$\alpha  = \frac{\pi }{{64}}$ for $n = 0$

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

Given $(1 - 2x + 5x^2 - 10x^3) (1 + x)^n = 1 + a_1x + a_2x^2 + ....$ and that $a_1^2\,= 2a_2$ then the value of $n$ is
→
Let $A \left(\frac{3}{\sqrt{ a }}, \sqrt{ a }\right) a >0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. If $D(3 \cos \theta$, a $\sin \theta)$ is a point in the fourth quadrant such that the maximum area of $\triangle ACD$ is $12$ square units, then $a$ is equal to
→
If $a \in R$ and the equation $ - 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0$ (where $[x]$ denotes the greatest integer $\leq\,x$) has no integral solution, then all possible values of $a$ lie in the interval
→
Locus of the vertex $A$ of a triangle $ABC$ whose base $BC$ is fixed and the perimeter of triangle is also constant, is
→
‘$A$’ draws two cards with replacement from a pack of $52$ cards and ‘$B$' throws a pair of dice what is the chance that ‘$A$’ gets both cards of same suit and ‘$B$’ gets total of $6$
→
The co-ordinate of the point on the circle $x^2 + y^2 - 12x - 4y + 30 = 0,$ which is farthest from the origin are :
→
If $\omega$ is one of the imaginary cube roots of unity, then the value of the determinant $\left| {\begin{array}{*{20}{c}}1&{{\omega ^3}}&{{\omega ^2}}\\ {{\omega ^3}}&1&\omega \\{{\omega ^2}}&\omega &1\end{array}} \right|$ $=$
→
If $f(x)=\left\{\begin{array}{cl}x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$, then
→
If $a$ and $b$ are real numbers such that $(2+\alpha)^{4}=a+b \alpha,$ where $\alpha=\frac{-1+i \sqrt{3}}{2},$ then $a+b$ is equal to
→
If $A = \left[ {\begin{array}{*{20}{c}}2&3\\4&6\end{array}} \right]$, then ${A^{ - 1}}$=
→