A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height \(y\) (in m ) of the elevator, from the ground, with time \(t\) (in s ) is given by \(y=8\left[1+\sin \left(\frac{2 \pi t}{T}\right)\right]\), where \(T=40 \pi \mathrm{~s}\). Taking acceleration due to gravity, \(g=10 \mathrm{~m} / \mathrm{s}^2\), the maximum variation of the object's weight (in N ) as observed in the experiment is _______ .

Put a uniform meter scale horizontally on your extended index fingers with the left one at $0.00 \mathrm{cm}$ and the right one at $90.00 \mathrm{cm}$. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_R$ from the center $(50.00 \mathrm{cm})$ of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are $0.40$ and $0.32$, respectively, the value of $x_R$ (in cm) is___________.

Two inductors $L_i$ (inductance $1 \mathrm{mH}$, internal resistance $3 \mathrm{\Omega }$ ) and $L_2$ (inductance $2 \mathrm{mH}$, internal resistance $4 \mathrm{\Omega }$ ), and a resistor $R$(resistance $12 \mathrm{\Omega }$ ) are all connected in parallel across a $5 \mathrm{V}$ battery. The circuit is switched on at time $t=0$. The ratio of the maximum current $\left(\frac{I_{max}}{I_{min}}\right)$ drawn from the battery is:

A spherical metal shell \(A\) of radius \(R_A\) and a solid metal sphere \(B\) of radius \(R_B < \left(R_A\right)\) are kept far apart and each is given charge \(+Q\). Now, they are connected by a thin metal wire. Then

A table tennis ball has radius \((3 / 2) \times 10^{-2} \mathrm{~m}\) and mass \((22 / 7) \times 10^{-3} \mathrm{~kg}\). It is slowly pushed down into a swimming pool to a depth of \(d=0.7 \mathrm{~m}\) below the water surface and then released from rest. It emerges from the water surface at speed \(v\), without getting wet, and rises up to a height \(H\). Which of the following option(s) is(are) correct?
[Given: \(\pi=22 / 7, g=10 \mathrm{~m} \mathrm{~s}^{-2}\), density of water \(=1 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), viscosity of water \(=1 \times 10^{-3} \mathrm{~Pa}\)-s.]

A circular disc with a groove along its diameter is placed horizontally. A block of mass \(1 \mathrm{~kg}\) is placed as shown. The coefficient of friction between the block and all surfaces of groove in contact is \(\mu=2 / 5\). The disc has an acceleration of \(25 \mathrm{~m} / \mathrm{s}^2\). Find the acceleration of the block with respect to disc.

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(R_{100}, R_{60}\) and \(R_{40}\), respectively, the relation between these resistances is

Consider the set of eight vectors $\text{V}=\left\{\text{a}\widehat{i}+\text{b}\widehat{\text{j}}+\text{c}\widehat{\text{k}}:\; a,\; b,\; c \in \left\{-1,\; 1\right\}\right\}$. Three non-coplanar vectors can be chosen from $\text{V}$ in $2^{\text{p}}$ ways. Then $\text{p}$ is

Let $\mathrm{F}\left(\mathrm{x}\right)=\mathrm{ }\underset{\mathrm{x}}{\overset{{\mathrm{x}}^2+\frac{\mathrm{\pi }}{6}}{\int }}2{\cos }^2\mathrm{t}\mathrm{ }\mathrm{d}\mathrm{t}$ for all $\mathrm{x}\mathrm{ }\in \mathbb{R}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{f}\mathrm{ }:\mathrm{ }\left[0,\frac{1}{2}\right]\to \left[0,\mathrm{ }\mathrm{\infty }\right)$ be a continuous function. For $\mathrm{a}\in \left[0,\frac{1}{2}\right],$ if ${\mathrm{F}}^{\mathrm{\prime }}\left(\mathrm{a}\right)+2$ is the area of the region bounded by $\mathrm{x}=0,\mathrm{ }\mathrm{y}=0,\mathrm{ }\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right)\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{x}=\mathrm{a},$ then $\mathrm{f}\left(0\right)$ is

Let $a_1, a_2, a_3, .........$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, ........$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality $2 \left(a_1+a_2+........+a_n\right)=b_1+b_2+.....+b_n$ holds for some positive integer $n$, is _______

For a non-zero complex number \(z\), let \(\arg (z)\) denote the principal argument of \(z\), with \(-\pi <\arg (z) \leq \pi\). Let \(\omega\) be the cube root of unity for which \(0 <\arg (\omega) <\pi\). Let \(\alpha=\arg \left(\sum_{n=1}^{2025}(-\omega)^n\right)\).
Then the value of \(\frac{3 \alpha}{\pi}\) is ___________

Let $R=\left\{\left(\begin{array}{lll}a& 3& b\\ c& 2& d\\ 0& 5& 0\end{array}\right):a,b,c,d\in \left\{0,3,5,7,11,13,17,19\right\}\right\}$. Then the number of invertible matrices in $R$ is

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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?