If $\beta =\underset{x\to 0}{\lim }\frac{e^{x^3}-{\left(1-x^3\right)}^{\frac{1}{3}}+\left({\left(1-x^2\right)}^{\frac{1}{2}}-1\right)\sin x}{x{\sin }^{2}x}$, then the value of $6\beta$ is

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Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(A\) is either symmetric or skew-symmetric or both, and det \((A)\) is divisible by \(p\) is

Considering only the principal values of the inverse trigonometric functions, the value of \(\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)\) is

Let \(L=\lim _{x \rightarrow 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}\), \(a>0\). If \(L\) is finite, then

Let \(f\) be a real-valued differentiable function on \(R\) (the set of all real numbers) such that \(f(1)=1\). If the \(y\)-intercept of the tangent at any point \(P(x, y)\) on the curve \(y=f(x)\) is equal to the cube of the abscissa of \(P\), then the value of \(f(-3)\) is equal to

Let $\overline{z}$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of ${\left(\overline{z}\right)}^2+\frac{1}{z^2}$ are integers, then which of the following is/are possible value(s) of $\left|z\right|$ ?

If $f\left(x\right)=\left|\begin{array}{ccc}\cos 2x& \cos 2x& \sin 2x\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|,$ then

Let \(z_1\) and \(z_2\) be two distinct complex numbers and let \(z=(1-t) z_1+t z_2\) for some real number \(t\) with \(0 < t < 1\). If \(\arg (w)\) denotes the principal argument of a non-zero complex number \(w\), then

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Carbon-14 is used to determine the age of organic material. The procedure is based on the formation of \({ }^{14} \mathrm{C}\) by neutron capture in the upper atmosphere.
\(
{ }_7^{14} \mathrm{~N}+{ }_0^1 n \longrightarrow{ }_6^{14} \mathrm{C}+{ }_1 n^1
\)
\({ }^{14} \mathrm{C}\) is absorbed by living organisms during photosynthesis. The \({ }^{14} \mathrm{C}\) content is constant in living organism once the plant or animal dies, the uptake of carbon dioxide by it ceases and the level of \({ }^{14} \mathrm{C}\) in the dead being, falls to the decay which \({ }^{14} \mathrm{C}\) undergoes.
\(
{ }_6^{14} \mathrm{C} \longrightarrow{ }_7^{14} \mathrm{~N}+\beta^{-}
\)
The half life period of \({ }^{14} \mathrm{C}\) is 5770 years. The decay constant \((\lambda)\) can be calculated by using the following formula \(\lambda=\frac{0.693}{t_{1 / 2}}\).
The comparison of the \(\beta^{-}\)activity of the dead matter with that of the carbon still in circulation enables measurement of the period of the isolation of the material from the living cycle. The method however, ceases to be accurate over periods longer than 30,000 years. The proportion of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) in living matter is \(1: 10^{12}\).
Question:
What should be the age of fossil for meaningful determination of its age?

Consider a triangle $\mathrm{\Delta }$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\mathrm{\Delta }$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\mathrm{\Delta }$ is

Let \(\alpha, \beta\) be the roots of the equation \(x^2-p x+r=0\) and \(\frac{\alpha}{2}, 2 \beta\) be the roots of the equation \(x^2-q x+r=0\). Then, the value of \(r\) is

Among \(\mathrm{V}(\mathrm{CO})_6, \mathrm{Cr}(\mathrm{CO})_5, \mathrm{Cu}(\mathrm{CO})_3,\)\(\operatorname{Mn}(\mathrm{CO})_5, \mathrm{Fe}(\mathrm{CO})_5,\left[\mathrm{Co}(\mathrm{CO})_3\right]^{3-},\left[\mathrm{Cr}(\mathrm{CO})_4\right]^{4-}\), and \(\operatorname{Ir}(\mathrm{CO})_3\), the total number of species isoelectronic with \(\mathrm{Ni}(\mathrm{CO})_4\) is ________.
[Given, atomic number:\(\mathrm{V}=23, \mathrm{Cr}=24, \mathrm{Mn}=25,\) \(\mathrm{Fe}=26, \mathrm{Co}=27, \mathrm{Ni}=28, \mathrm{Cu}=29,\) \(\mathrm{Ir}=77\)]

The total number of cyclic isomers possible for a hydrocarbon with the molecular formula \(\mathrm{C}_4 \mathrm{H}_6\) is