Let \(f:(-1,1) \rightarrow I R\) be such that \(f(\cos 4 \theta)=\frac{2}{2-\sec ^{2} \theta}\) for \(\theta \in\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\). Then the value (s) of \(f\left(\frac{1}{3}\right)\) is (are)

Consider \(L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0\) where \(p\) is a real number and \(C: x^2+y^2+6 x-10 y+30=0\).
Statement 1 If line \(L_1\) is a chord of circle \(C\), then line \(L_2\) is not always a diameter of circle \(C\).
Statement 2 If line \(L_1\) is a diameter of circle \(C\), then line \(L_2\) is not a chord of circle \(C\).

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Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).


Question:

If the triangle \(P Q R\) varies, then the minimum value of

\(\cos (P+Q)+\cos (Q+R)+\cos (R+P)\) is

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Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is

Let $S$ be the of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$ such that $b_1, b_2, b_3\in \mathrm{ℝ}$ and the system of equations (in real variables)
$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]ϵ S$ ?

Let $\mathrm{S}$ be the sample space of all $3\times 3$ matrices with entries from the set $\left\{0,\mathrm{ }1\right\}.$ Let the events ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ be given by
${\mathrm{E}}_1=\left\{\mathrm{A}\in \mathrm{S}:\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{A}=0\right\}$ and
${\mathrm{E}}_2=\{\mathrm{A}\in \mathrm{S}:$ sum of entries of $\mathrm{A}$ is $7\}.$
If a matrix is chosen at random from $\mathrm{S},$ then the conditional probability $\mathrm{P}\left({\mathrm{E}}_1|{\mathrm{E}}_2\right)$ equals____

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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
If \(f(-10 \sqrt{2})=2 \sqrt{2}\), then \(f^{\prime \prime}(-10 \sqrt{2})\) is equal to

Five persons $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C},\mathrm{ }\mathrm{D}$ and $\mathrm{E}$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is

Statement 1 The geometrical isomer of the complex \(\left.\left[\mathrm{M}_{\left(\mathrm{NH}_3\right.}\right)_4 \mathrm{Cl}_2\right]\) are optically inactive.
Statement 2 Both geometrical isomers of the complex \(\left[\mathrm{M}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right]\) possess axis of symmetry.

If \(w=\alpha+i \beta\), where \(\beta \neq 0\) and \(z \neq 1\) satisfies the condition that \(\left(\frac{w-\bar{w} z}{1-z}\right)\) is purely real, then the set of values of \(z\) is

Let $f:\left[0, 2\right]\to ℝ$ be the function defined by $f\left(x\right)=\left(3-\sin \left(2\pi x\right)\right)\sin \left(\pi x-\frac{\pi }{4}\right)-\sin \left(3\pi x+\frac{\pi }{4}\right)$. If $\alpha , \beta \in \left[0, 2\right]$ are such that $\left\{x\in \left[0, 2\right] :f\left(x\right)\ge 0\right\}=\left[\alpha , \beta \right]$, then the value of $\beta -\alpha$ is _____

For the given aqueous reactions, which of the statement (s) is (are) true?

Consider an $\mathrm{LC}$ circuit, with inductance $L=0.1 \mathrm{H}$ and capacitance $C={10}^{-3} \mathrm{F}$, kept on a plane. The area of the circuit is $1 {\mathrm{m}}^2$. It is placed in a constant magnetic field of strength $B_0$ which is perpendicular to the plane of the circuit. At time $t=0$, the magnetic field strength starts increasing linearly as $B=B_0+\beta \mathrm{t}$ with $\beta =0.04 \mathrm{T} {\mathrm{s}}^{-1}$. The maximum magnitude of the current in the circuit is____$\mathrm{m} \mathrm{A}$.