Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be unit vectors such that \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\). Which one of the following is correct?

Internal bisector of \(\angle A\) of \(\triangle A B C\) meets side \(B C\) at \(D\). A line drawn through \(D\) perpendicular to \(A D\) intersects the side \(A C\) at \(E\) and side \(A B\) at \(F\). If \(a, b\) and \(c\) represent sides of \(\triangle A B C\), then

Of the three independent events ${\text{E}}_1$, ${\text{E}}_2$ and ${\text{E}}_3$, the probability that only ${\text{E}}_1$ occurs is $\alpha$, only ${\text{E}}_2$ occurs is $\beta$ and only  ${\text{E}}_3$ occurs is $\gamma$. Let the probability $p$ that none of events ${\text{E}}_1$, ${\text{E}}_2$ or ${\text{E}}_3$ occurs satisfy the equations $\left(\alpha -2\beta \right)p=\alpha \beta$ and $\left(\beta -3\gamma \right)p = 2\beta \gamma$. All the given probabilities are assumed to lie in the interval $\left(\text{0, 1}\right)$.
Then   $\frac{{\text{Probability of occurrence of E}}_1}{{\text{Probability of occurrence of E}}_3}=$ ​

Let $X$ and $Y$ be two events such that $P\left(X\right)=\frac{1}{3}$, $P\left(X|Y\right)=\frac{1}{2}$ and $P\left(Y|X\right)=\frac{2}{5}.$ Then

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 \(f: R \rightarrow R\) be a continuous function, which satisfies \(f(x)=\int_0^x f(t) d t\). Then, the value of \(f(\ln 5)\) is

Three lines are given by $\overrightarrow{\mathrm{r}}=\mathrm{\lambda }\hat{\mathrm{i}}, \lambda \in \mathrm{R}$
$\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right),\mathrm{\mu }\in \mathrm{R}$ and
$\overrightarrow{\mathrm{r}}=\mathrm{\nu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}}\right),\mathrm{\nu }\in \mathrm{R}.$
Let the lines cut the plane $\mathrm{x}+\mathrm{y}+\mathrm{z}=1$ at the points $\mathrm{A},\mathrm{ }\mathrm{B}$ and $\mathrm{C}$ respectively. If the area of the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of ${5.7}^{\mathrm{o}}\mathrm{C}$ was measured for the beaker and its contents (Expt.1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant $(-57.0\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})$ , this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid $({\mathrm{K}}_{\mathrm{a}}=2.0\times {10}^{-5})$ was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of ${5.6}^{\mathrm{o}}\mathrm{C}$ was measured. (Consider heat capacity of all solutions as $4.2\mathrm{ }\mathrm{J}\mathrm{ }{\mathrm{g}}^{-1}\mathrm{ }{\mathrm{K}}^{-1}$ and density of all solutions as $1.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{m}{\mathrm{L}}^{-1}$ )
The pH of the solution after Expt. 2 is

Let $a$ and $b$ be positive real numbers such that $a>1$ and $b<a.$ Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ Suppose the tangent to the hyperbola at $P$ passes through the point $\left(1,0\right),$ and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$ -axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

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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?

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Consider a simple \(R C\) circuit as shown in Figure \(1\).

Process 1: In the circuit the switch \(S\) is closed at \(t=0\) and the capacitor is fully charged to voltage \(V_{0}\) (i.e., charging continues for time \(T>>R C\) ). In the process some dissipation \(\left(E_{D}\right)\) occurs across the resistance \(R\). The amount of energy finally stored in the fully charged capacitor is \(E_{C}\).

Process 2: In a different process the voltage is first set to \(\frac{V_{0}}{3}\) and maintained for a charging time \(T>>R C\). Then the voltage is raised to \(\frac{2 V_{0}}{3}\) without discharging the capacitor and again maintained for a time \(T>>R C\). The process is repeated one more time by raising the voltage to \(V_{0}\) and the capacitor is charged to the same final voltage \(V_{0}\) as in Process 1.

These two processes are depicted in Figure \(2 .\)







Question:

In Process 1 , the energy stored in the capacitor \(E_{C}\) and heat dissipated across resistance \(E_{D}\) are related by:

Consider a spherical gaseous cloud of mass density $\mathrm{\rho }\left(\mathrm{r}\right)$ in free space where $\mathrm{r}$ is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy $\mathrm{K}$ . The force acting on the particles is their mutual gravitational force. If $\mathrm{\rho }\left(\mathrm{r}\right)$ is constant in time, the particle number density $\mathrm{n}\left(\mathrm{r}\right)=\mathrm{\rho }\left(\mathrm{r}\right)/\mathrm{m}$ is: [ $\mathrm{G}$ is universal gravitational constant]

Let \(S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.\) and \(\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}\). If the area of the region \(S\) is \(\alpha \sqrt{2}\), then \(\alpha\) is equal to