enEnglishguગુજરાતી
JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
If \(\quad A=\left[\begin{array}{cc}\cos \theta & \text { isin } \theta \\ \operatorname{isin} \theta & \cos \theta\end{array}\right], \left(\theta=\frac{\pi}{24}\right)\) and \(A^{5}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) where \(i=\sqrt{-1},\) then which one of the following is not true?
- A \(0 \leq a^{2}+b^{2} \leq 1\)
- B \(a^{2}-d^{2}=0\)
- C \(a^{2}-b^{2}=\frac{1}{2}\)
- D \(a^{2}-c^{2}=1\)
Answer & Solution
Correct Answer
(C) \(a^{2}-b^{2}=\frac{1}{2}\)
Step-by-step Solution
Detailed explanation
\(A ^{2}=\left(\begin{array}{cc}\cos 2 \theta & i \sin 2 \theta \\ i \sin 2 \theta & \cos 2 \theta\end{array}\right)\) Similarly,…
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
- If \(\cos \,\left( {\alpha + \beta } \right) = \frac{3}{5},\,\sin \,\left( {\alpha - \beta } \right) = \frac{5}{{13}}\) and \(0 < \alpha ,\beta < \frac{\pi }{4}\) then \(\tan \,\left( {2\alpha } \right)\) is equal toJEE Mains 2019 Hard
- Let the lines \( L_1: \vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda(2\hat{i}+3\hat{j}+4\hat{k}) \), \( \lambda \in R \) and \( L_{2}:\vec{r}=(4\hat{i}+\hat{j})+\mu(5\hat{i}+2\hat{j}+\hat{k}) \), \( \mu\in\mathbb{R} \), intersect at the point R. Let P and Q be the points lying on lines \( L_{1} \) and \( L_{2} \), respectively, such that \({|\overrightarrow{ PR }|}=\sqrt{29}\) and \({|\overrightarrow{ PQ }|}=\sqrt{\frac{47}{3}}\). If the point P lies in the first octant, then \( 27(QR)^{2} \) is equal toJEE Mains 2026 Medium
- A line is a common tangent to the circle \((x-3)^{2}+y^{2}=9\) and the parabola \(y^{2}=4 x.\) If the two points of contact \(( a , b )\) and \(( c , d )\) are distinct and lie in the first quadrant, then \(2(a+c)\) is equal to ........ .JEE Mains 2021 Hard
- Let a variable line of slope \(m>0\) passing through the point \((4,-9)\) intersect the coordinate axes at the points \(A\) and \(B\). the minimum value of the sum of the distances of \(\mathrm{A}\) and \(\mathrm{B}\) from the origin isJEE Mains 2024 Hard
- The mean and variance of \(20\) observations are found to be \(10\) and \(4,\) respectively. On rechecking, it was found that an observation \(9\) was incorrect and the correct observation was \(11\). Then the correct variance isJEE Mains 2020 Hard
- Let \(\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c}=\vec{a} \times \vec{b}\). If \(\vec{a} \cdot \vec{c}=-12\), \(\vec{c} .(\hat{i}-2 \hat{j}+\hat{k})=5\), then \(\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})\) is equal to \(.............\).JEE Mains 2023 Hard
More PYQs from JEE Mains
- The absolute minimum value, of the function \(f(x)=\left|x^2-x+1\right|+\left[x^2-x+1\right], \quad\) where \([t]\) denotes the greatest integer function, in the interval \([-1,2]\), is :JEE Mains 2023 Hard
- Let \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) meet the line \(L\) through \(\mathrm{C}\) parallel to the \(\mathrm{x}\)-axis at the point \(\mathrm{D}\). Let \(\mathrm{M}\) and \(\mathrm{N}\) respectively be the feet of the perpendiculars from \(\mathrm{A}\) and \(\mathrm{B}\) on \(\mathrm{L}\). Then \(\left(\frac{\mathrm{AM} \cdot \mathrm{BN}}{\mathrm{CD}}\right)^2\) is equal to ...........JEE Mains 2024 Hard
- If \(m\) and \(M\) are the minimum and the maximum values of \(4 + \frac{1}{2}\,{\sin ^2}\,2x - 2\,{\cos ^4}\,x\,,x\, \in R,\) then \(M - m\) is equal toJEE Mains 2016 Hard
- A square \(ABCD\) has all its vertices on the curve \(x ^{2} y ^{2}=1\). The midpoints of its sides also lie on the same curve. Then, the square of area of \(ABCD\) isJEE Mains 2021 Hard
- The random valuable \(X\) follows binomial distribution \(B (n, p)\) for which the difference of the mean and the variance is \(1\). If \(2 P(X=2)=3 P(X=1)\), then \(n^2 P(X > 1)\) is equal to \(......\).JEE Mains 2023 Hard
- Let \(\mathrm{ABCD}\) and \(\mathrm{AEFG}\) be squares of side \(4\) and \(2\) units, respectively. The point \(\mathrm{E}\) is on the line segment \(\mathrm{AB}\) and the point \(\mathrm{F}\) is on the diagonal \(\mathrm{AC}\). Then the radius \(r\) of the circle passing through the point \(\mathrm{F}\) and touching the line segments \(\mathrm{BC}\) and \(\mathrm{CD}\) satisfies :JEE Mains 2024 Hard