The smallest division on the main scale of a Vernier calipers is $0.1 \mathrm{cm}$. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

An electric dipole is formed by two charges $+q$ and $-q$ located in $xy$-plane at $(0, 2) \mathrm{mm}$ and $(0, -2) \mathrm{mm}$, respectively, as shown in the figure. The electric potential at point $P(100, 100) \mathrm{mm}$ due to the dipole is $V_0$. The charges $+q$ and $–q$ are then moved to the points $(-1, 2) \mathrm{mm}$ and $(1, -2) \mathrm{mm}$, respectively. What is the value of electric potential at $$

A student uses a simple pendulum of exactly \(1 \mathrm{~m}\) length to determine \(g\), the acceleration due to gravity. He uses a stop watch with the least count of \(1 \mathrm{~s}\) for this and records \(40 \mathrm{~s}\) for 20 oscillations. For this observation, which of the following statement(s) is/are true?

A binary star consists of two stars \(A\) (mass \(2.2 M_S\) ) and \(B\) (mass \(11 M_s\) ), where \(M_S\) is the mass of the sun. They are separated by distance \(d\) and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of \(\operatorname{star} B\) about the centre of mass is

A conducting loop in the shape of right angled isosceles triangle of height 10 cm is kept such that the ${90}^o$ vertex is very close to an infinitely long conducting wire (see the figure). The wire is electrically insulated from the loop. The hypotenuse of the triangle is parallel to the wire. The current in the triangular loop is in counterclockwise direction and increased at constant rate of $10A{s}^{-1}.$ Which of the following statement(s) is (are) true?

Consider a system of three charges \(\frac{q}{3}, \frac{q}{3}\) and \(-\frac{2 q}{3}\) placed at points \(A, B\) and \(C\), respectively, as shown in the figure. Take \(O\) to be the centre of the circle of radius \(R\) and angle \(C A B=60^{\circ}\).

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Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
In the core of nuclear fusion reactor, the gas becomes plasma because of

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The coordination number of \(\mathrm{Ni}^{2+}\) is 4 .
\(\mathrm{NiCl}_2+\mathrm{KCN}\) (excess) \(\rightarrow A\) (cyano complex)
\(\mathrm{NiCl}_2+\) conc. \(\mathrm{HCl}\) (excess) \(\rightarrow B\) (chloro complex)
Question:
Predict the magnetic nature of \(A\) and \(B\) are

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If a continuous \(f\) defined on the real line \(R\), assume positive and negative values in \(R\), then the equation \(f(x)=0\) has a root in \(R\). For example, if it is known that a continuous function \(f\) on \(R\) is positive at some point and its minimum values is negative, then the equation \(f(x)=0\) has a root in \(R\).
Consider \(f(x)=k e^x-x\) for all real \(x\), where \(k\) is real constant.Question:
For \(k>0\), the set of all values of \(k\) for which \(k e^x-x=0\) has two distinct roots, is

The species having bond order different from that in \(\mathrm{CO}\) is

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function with $f\left(0\right)=0$. If $y=f\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}=\left(2+5y\right)\left(5y-2\right)$, then the value of  $\underset{x\to -\infty }{\lim }f\left(x\right)$ is 

Which of the given statement(s) about \(\mathbf{N}, \mathbf{O}, \mathbf{P}\) and \(\mathbf{Q}\) with respect to \(\mathbf{M}\) is (are) correct?

A solid glass sphere of refractive index \(n=\sqrt{3}\) and radius \(R\) contains a spherical air cavity of radius \(\frac{\mathrm{R}}{2}\), as shown in the figure. A very thin glass layer is present at the point O so that the air cavity (refractive index \(n=1\)) remains inside the glass sphere. An unpolarized, unidirectional and monochromatic light source \(S\) emits a light ray from a point inside the glass sphere towards the periphery of the glass sphere. If the light is reflected from the point O and is fully polarized, then the angle of incidence at the inner surface of the glass sphere is \(\theta\). The value of \(\sin \theta\) is ______ [given : θ > 30°]