For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 8& 13& 18\end{array}\right], P=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \& F=\left[\begin{array}{ccc}1& 3& 2\\ 8& 18& 13\\ 2& 4& 3\end{array}\right]$ If $Q$ is a nonsingular matrix of order $3\times 3$, then which of the following statements is (are) TRUE ?

If \(|z|=1\) and \(z \neq \pm 1\), then all the values of \(\frac{z}{1-z^2}\) lie on

Let \(f_{1}:(0, \infty) \rightarrow \mathbb{R}\) and \(f_{2}:(0, \infty) \rightarrow \mathbb{R}\) be defined by

\[

f_{1}(x)=\int_{0}^{x} \prod_{j=1}^{21}(t-j)^{j} d t, \quad x>0

\]

and

\(

f_{2}(x)=98(x-1)^{50}-600(x-1)^{49}+2450, \quad x>0

\)

where, for any positive integer \(n\) and real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \prod_{i=1}^{n} a_{i}\) denotes the product of \(a_{1}, a_{2}, \ldots, a_{n} .\) Let \(m_{i}\) and \(n_{i}\), respectively, denote the number of points of local minima and the number of points of local maxima of function \(f_{i}, i=1,2\), in the interval \((0, \infty)\).

The value of $2m_1+3n_1+m_1{n}_1$ is _____.

Let $\alpha$ and $\beta$ be real numbers such that $-\frac{\pi }{4}<\beta <0<\alpha <\frac{\pi }{4}$. If $\sin \left(\alpha +\beta \right)=\frac{1}{3}$ and $\cos \left(\alpha -\beta \right)=\frac{2}{3}$, then the greatest integer less than or equal to ${\left(\frac{\sin \alpha }{\cos \beta }+\frac{\cos \beta }{\sin \alpha }+\frac{\cos \alpha }{\sin \beta }+\frac{\sin \beta }{\cos \alpha }\right)}^2$ is

Let $\left|M\right|$ denote the determinant of a square matrix $M$. Let $g:\left[0,\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be the function defined by $g\left(\theta \right)=\sqrt{f\left(\theta \right)-1}+\sqrt{f\left(\frac{\pi }{2}-\theta \right)-1}$ where $f\left(\theta \right)=\frac{1}{2}\left|\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta +\frac{\pi }{4}\right)& \tan \left(\theta -\frac{\pi }{4}\right)\\ \sin \left(\theta -\frac{\pi }{4}\right)& -\cos \frac{\pi }{2}& {\log }_e\left(\frac{4}{\pi }\right)\\ \cot \left(\theta +\frac{\pi }{4}\right)& {\log }_e\left(\frac{\pi }{4}\right)& \tan \pi \end{array}\right|$
Let $p\left(x\right)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g\left(\theta \right)$, and $p\left(2\right)=2-\sqrt{2}$. Then, which of the following is/are TRUE ?

Paragraph:

Let \(f:[0,1] \rightarrow \mathbb{R}\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0)=f(1)=0\) and satisfies \(f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .\)


Question:

Which of the following is true for \(f(x)\) ?

The option(s) with the value of $a$ and $L$ that satisfy the following equation is(are) $\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t\left({\sin }^{6}at+{\cos }^{4}at\right) dt}{\underset{0}{\overset{\pi }{\int }}\left({\sin }^{6}at+{\cos }^{4}at\right) dt}=L ?$

Atom $\mathrm{X}$ occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in $\%$) of the resultant solid is closest to

A ball is thrown from ground at an angle $\mathrm{\theta }$ with horizontal and with an initial speed ${\mathrm{u}}_0.$ For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is ${\mathrm{V}}_1.$ After hitting the ground, ball rebounds at the same angle $\mathrm{\theta }$ but with a reduced speed of $\frac{{\mathrm{u}}_0}{\mathrm{\alpha }}.$ Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8{\mathrm{V}}_1,$ the value of $\mathrm{\alpha }$ is______

If $2x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$ , then which of the following CANNOT be sides of a right angled triangle?

The boiling point of water in a \(0.1\) molal silver nitrate solution (solution \(\mathbf{A}\) ) is \(\mathbf{x}^{\circ} \mathrm{C}\). To this solution \(\mathbf{A}\), an equal volume of \(0.1\) molal aqueous barium chloride solution is added to make a new solution B. The difference in the boiling points of water in the two solutions \(\mathbf{A}\) and \(\mathbf{B}\) is \(\mathbf{y} \times 10^{-2}{ }^{\circ} \mathrm{C}\).
(Assume: Densities of the solutions \(\mathbf{A}\) and \(\mathbf{B}\) are the same as that of water and the soluble salts dissociate completely.
Use: Molal elevation constant (Ebullioscopic Constant), \(K_{b}=0.5 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\); Boiling point of pure water as \(100^{\circ} \mathrm{C}\).)
The value of $\left|\mathrm{y}\right|$ is ___.

The filament of a light bulb has surface area $64 {\mathrm{mm}}^2.$ The filament can be considered as a black body at temperature $2500 \mathrm{K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \mathrm{m}$. Assume the pupil of the eyes of the observer to be circular with radius $3 \mathrm{mm}$. Then: (Take Stefan-Boltzmann constant $=5.67\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-4},$ Wiens' displacement constant $=2.90\times {10}^{-3} \mathrm{m} \mathrm{K}$, Planck's constant $=6.60\times {10}^{-34} \mathrm{J} \mathrm{s}$, speed of light in vacuum $=3.00\times {10}^8 \mathrm{m}{\mathrm{s}}^{-1}$)

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV - diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

	LIST -I		LIST -II
A)	In process $I$	P)	Work done by the gas is zero
B)	In process $II$	Q)	Temperature of the gas remains unchanged
C)	In process $III$	R)	No heat is exchanged between the gas and its surroundings
D)	In process $IV$	S)	Work done by the gas is $6P_0{V}_0$