JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
If \(A\, = \,\left[ {\begin{array}{*{20}{c}}
{{e^t}}&{{e^{ - t}}\,\cos \,t}&{{e^{ - t}}\,\sin \,t}\\
{{e^t}}&{ - {e^{ - t}}\,\cos \, - {e^{ - t}}\,\sin \,t}&{ - {e^{ - t}}\,\sin \,t\, + \,{e^{ - t}}\,\cos \,t}\\
{{e^t}}&{2{e^{ - t}}\,\sin \,t}&{2{e^{ - t}}\,\cos \,t}
\end{array}} \right]\) Then \(A\) is
- A Invertible only if \(t = \frac {\pi }{2}\)
- B not invertible for any \(t \in R\)
- C invertible for all \(t \in R\)
- D invertible only if \(t = \pi \)
Answer & Solution
Correct Answer
(C) invertible for all \(t \in R\)
Step-by-step Solution
Detailed explanation
\(\left| A \right| = {e^{ - t}}\left| {\begin{array}{*{20}{c}} 1&{\cos \,t}&{\sin \,t}\\ 1&{ - \cos \,t - \sin \,t}&{\, - \sin \,t + \cos \,t}\\ 1&{2\sin \,t}&{ - 2\cos \,t} \end{array}} \right|\) \( = {e^{ - t}}\left[ {5{{\cos }^2}t + 5{{\sin }^2}t} \right]\forall t \in R\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(\mathrm{g}: \mathrm{N} \rightarrow \mathrm{N}\) be defined as \(g(3 n+1)=3 n+2\) \(g(3 n+2)=3 n+3\) \(g(3 n+3)=3 n+1, \text { for all } n \geq 0\) Then which of the following statements is true?JEE Mains 2021 Hard
- Let \(z\,\ne -i\) be any complex number such that \(\frac{{z - i}}{{z + i}}\) is a purely imaginary number. Then \(z +\frac {1}{z}\) isJEE Mains 2014 Hard
- If \(z = \frac{{\sqrt 3 }}{2} + \frac{i}{2}\,\,\,\left( {i = \sqrt { - 1} } \right)\), then \({\left( {1 + iz + {z^5} + i{z^8}} \right)^9}\) is equal toJEE Mains 2019 Hard
- A line passing through the point \(\mathrm{P}(\mathrm{a}, 0)\) makes an acute angle \(\alpha\) with the positive x -axis. Let this line be rotated about the point \(P\) through an angle \(\frac{\alpha}{2}\) in the clock-wise direction. If in the new position, the slope of the line is \(2-\sqrt{3}\) and its distance from the origin is \(\frac{1}{\sqrt{2}}\), then the value of \(3 a^2 \tan ^2 \alpha-2 \sqrt{3}\) isJEE Mains 2025 Medium
- The set of all real values of \(\lambda \) for which exactly two common tangents can be drawn to the circles \(x^2 + y^2 - 4x - 4y+ 6\, = 0\) and \(x^2 + y^2 - 10x - 10y + \lambda \, = 0\) is the interval:JEE Mains 2014 Hard
- The sum of an infinite geometric series with positive terms is \(3\) and the sum of the cubes of its terms is \(\frac {27}{19}\). Then the common ratio of this series isJEE Mains 2019 Hard
More PYQs from JEE Mains
- Let \(A =\left\{1, a _{1}, a _{2} \ldots \ldots a _{18}, 77\right\}\) be a set of integers with \(1< a _{1}< a _{2}<\ldots \ldots< a _{18}<77\). Let the set \(A + A =\{ x + y : x , y \in A \} \quad\) contain exactly \(39\) elements. Then, the value of \(a_{1}+a_{2}+\ldots \ldots+a_{18}\) is equal to...........JEE Mains 2022 Hard
- If \(A = \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&{ - 2}\\a&2&b\end{array}} \right]\) is a matrix satisfying the equation \(AA^T=9I \) where\( I\) is \(3×3\) identity matrix, then the ordered pair \((a, b)\) is equal to:JEE Mains 2015 Medium
- The number of elements in the set \(\{x \in R :(|x|-3)|x+4|=6\}\) is equal toJEE Mains 2021 Hard
- Let \(S=\left\{\left(\begin{array}{cc}-1 & a \\ 0 & b\end{array}\right) ; a, b \in\{1,2,3, \ldots 100\}\right\}\) and let \(T_{n}=\left\{A \in S: A^{n(n+1)}=I\right\}\). Then the number of elements in \(\bigcap \limits_{n=1}^{100} T_{n}\) isJEE Mains 2022 Hard
- The area (in sq, units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse \(\frac{{{x^2}}}{9} + \frac{{{y^2}}}{5} = 1\) is :JEE Mains 2015 Hard
- Let \(S=\mathbf{N} \cup\{0\}\). Define a relation \(R\) from \(S\) to \(\mathbf{R}\) by :
\(\mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\}\)
Then, the sum of all the elements in the range of \(R\) is equal to :JEE Mains 2025 Medium