Download App

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

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 3 and 4 . determinant and metrices1 Mark
MCQ
Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$

Given below are two statements:

Statement I: $f(-x)$ is the inverse of the matrix $f(x)$.

Statement II: $f(x) f(y)=f(x+y)$.

In the light of the above statements, choose the correct answer from the options given below

  • A
    Statement $I$ is false but Statement $II$ is true
  • B
    Both Statement $I$ and Statement $II$ are false
  • C
     Statement $I$ is true but Statement $II$ is false
  • ✓
     Both Statement $I$ and Statement $II$ are true
✓

Answer

Correct option: D.
 Both Statement $I$ and Statement $II$ are true
d
$f(-x)=\left[\begin{array}{ccc}\cos x & \sin x & 0 \\-\sin x & \cos x & 0 \\0 & 0 & 1\end{array}\right]$

$f(x) \cdot f(-x)=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]=I$

Hence statement- $I$ is correct

Now, checking statement $II$

$f(y)=\left[\begin{array}{ccc}\cos y & -\sin y & 0 \\\sin y & \cos y & 0 \\0 & 0 & 1\end{array}\right]$

$f(x) \cdot f(y)=\left[\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & 0 \\\sin (x+y) & \cos (x+y) & 0 \\0 & 0 & 1\end{array}\right]$

$ \Rightarrow f(x) \cdot f(y)=f(x+y)$

Hence statement -$II$ is also correct.

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 STD 12 - 3 and 4 . determinant and metrices→STD 12 - 3 and 4 . determinant and metrices — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If $y = {(1 + x)^x},$ then ${{dy} \over {dx}} = $
→
If $y=\tan ^{-1}(\sqrt{x})-\tan ^{-1}(x)$, then $y^{\prime}(1)$ is equal to
→
Assuming that for a husband-wife couple the chances of their child being a boy or a girl are the same, the probability of their two children being a boy and a girl is
→
The vectors $3\,i + j - 5\,k$ and $a\,i + b\,j - 15\,k$ are collinear, if
→
If $y = A\cos nx + B\sin nx,$ then ${{{d^2}y} \over {d{x^2}}} = $
→
The direction cosines of three lines passing through the origin are ${l_1},{m_1},{n_1};\,{l_2},{m_2},{n_2}$ and ${l_3},{m_3},{n_3}$. The lines will be coplanar, if
→
Choose the correct answer from the given four options.If two events are independent, then :
→
$\int_{}^{} {\frac{{2x{{\tan }^{ - 1}}{x^2}}}{{1 + {x^4}}}} \;dx = $
→
A root of the equation $\left| {\,\begin{array}{*{20}{c}}{3 - x}&{ - 6}&3\\{ - 6}&{3 - x}&3\\3&3&{ - 6 - x}\end{array}\,} \right| = 0$ is
→
Choose the correct answer from the given four options.The value of $\cos^{-1}\Big(\cos\frac{3\pi}{2}\Big)$ is equal to:
→