JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
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
- D Both Statement \(I\) and Statement \(II\) are true
Answer & Solution
Correct Answer
(D) Both Statement \(I\) and Statement \(II\) are true
Step-by-step Solution
Detailed explanation
\(f(-x)=\left[\begin{array}{ccc}\cos x & \sin x & 0 \\-\sin x & \cos x & 0 \\0 & 0 & 1\end{array}\right]\) \(f(x) \cdot f(-x)=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]=I\) Hence statement- \(I\) is correct Now, checking statement \(II\)…
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
- The line, that is coplanar to the line \(\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}\), isJEE Mains 2023 Medium
- Let \(a, b\) and \(c\) be distinct positive numbers. If the vectors \(a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}\) and \(c \hat{i}+c \hat{j}+b \hat{k}\) are co-planar, then \(\mathrm{c}\) is equal to:JEE Mains 2021 Easy
- Let \(N\) be the set of natural numbers and a relation \(R\) on \(N\) be defined by \(R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .\) Then the relation \(R\) is:JEE Mains 2021 Hard
- The number of elements in the set \(S=\left\{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}\) is\(.....\)JEE Mains 2022 Medium
- Let \(f : R -\{0,1\} \rightarrow R\) be a function such that \(f(x)+f\left(\frac{1}{1-x}\right)=1+x\). Then \(f(2)\) is equal to :JEE Mains 2023 Hard
- All the letters of the word "\(GTWENTY\)" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "\(GTWENTY\)" ISJEE Mains 2024 Hard
More PYQs from JEE Mains
- Let \(k\) and \(m\) be positive real numbers such that the function \(\quad f ( x )=\left\{\begin{array}{cc}3 x ^2+ k \sqrt{ x +1}, & 0< x <1 \\ mx ^2+ k ^2, & x \geq 1\end{array}\right.\) is differentiable for all \(x > 0\). Then \(\frac{8 f^{\prime}(8)}{f^{\prime}\left(\frac{1}{8}\right)}\) is equal to \(.............\).JEE Mains 2023 Hard
- Let \(\vec{a}\) be a vector which is perpendicular to the vector \(3 \hat{ i }+\frac{1}{2} \hat{ j }+2 \hat{ k }\). If \(\overrightarrow{ a } \times(2 \hat{ i }+\hat{ k })=2 \hat{ i }-13 \hat{ j }-4 \hat{ k }\), then the projection of the vector \(\vec{a}\) on the vector \(2 \hat{ i }+2 \hat{ j }+\hat{ k }\) isJEE Mains 2022 Medium
- Let \(S_1=\{z \in C:|z| \leq 5\}\), \(S_2=\left\{z \in C: I m\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}\) and \(\mathrm{S}_3=\{\mathrm{z} \in \mathrm{C}: \operatorname{Re}(\mathrm{z}) \geq 0\}\). Then the area of the region \(S_1 \cap S_2 \cap S_3\) is:JEE Mains 2024 Hard
- If the sum of the coefficients of all even powers of \(x\) in the product \(\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)\) is \(61,\) then \(\mathrm{n}\) is equal toJEE Mains 2020 Hard
- If the \(1011^{\text {th }}\) term from the end in the binomial expansion of \(\left(\frac{4 x}{5}-\frac{5}{2 x }\right)^{2022}\) is \(1024\) times \(1011^{\text {th }}\) term from the beginning, then \(|x|\) is equal toJEE Mains 2023 Hard
- Let \(A=\left[\begin{array}{cc}i & -i \\ -i & i\end{array}\right], i=\sqrt{-1}\).Then, the system of linear equations \(A^{8}\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]\) has :JEE Mains 2021 Hard