A photograph of a landscape is captured by a drone camera at a height of 18 km . The size of the camera film is \(2 \mathrm{~cm} \times 2 \mathrm{~cm}\) and the area of the landscape photographed is \(400 \mathrm{~km}^2\). The focal length of the lens in the drone camera is :

From \(18\) m height above the ground a ball is dropped from rest. The height above the ground at which the magnitude of velocity equal to the magnitude of acceleration (in the same set of units) due to gravity is _____ m.
(Take \(g = 10\) m/s\(^2\) and neglect the air resistance)

Consider a Young’s double slit experiment as shown in figure. What should be the slit separation \(d\) in terms of wavelength \(\lambda \) such that the first minima occurs directly in front of the slit \((S_1)\)?

Given below are two statements: Statement \(I :\) A second's pendulum has a time period of \(1\) second. Statement \(II :\) It takes precisely one second to move between the two extreme positions. In the light of the above statements, choose the correct answer from the options given below:

Consider two containers \(A\) and \(B\) containing monoatomic gases at the same Pressure \((P)\), Volume \((V)\) and Temperature \((T)\). The gas in \(A\) is compressed isothermally to \(\frac{1}{8}\) of its original volume while the gas \(B\) is compressed adiabatically to \(\frac{1}{8}\) of its original volume. The ratio of final pressure of gas in \(B\) to that of gas in \(A\) is ...........

A ring is hung on a nail. It can oscillate, without slipping or sliding \((i)\) in its plane with a time period \(T_{1}\) and, \((ii)\) back and forth in a direction perpendicular to its plane, with a period \(T _{2}\). the ratio \(\frac{ T _{1}}{ T _{2}}\) will be 

As shown in the figure, two parallel plate capacitors having equal plate area of \(200\,cm ^2\) are joined in such a way that \(a \neq b\). The equivalent capacitance of the combination is \(x \varepsilon_0 F\). The value of \(x\) is \(..........\).

The position vectors of two 1 kg particles, (A) and (B), are given by
\(\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}\) and \(\overrightarrow{\mathrm{r}}_{\mathrm{B}}=(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}\) \(+\beta_3 \mathrm{t} \hat{k}) \mathrm{m}\), respectively; \((\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s},\) \(\alpha_3=2 \mathrm{~m} / \mathrm{s},\) \(\beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s})\), where t is time, n and p are constants. At \(t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|\) and velocities \(\vec{V}_A\) and \(\vec{V}_B\) of the particles are orthogonal to each other. At \(t=1 \mathrm{~s}\), the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is \(\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}\). The value of L is _______ .

The probability that a randomly chosen one-one function from the set \(\{a, b, c, d\}\) to the set \(\{1,2,3,4,5\}\) satisfies \(f(a)+2 f(b)-f(c)=f(d)\) is

Consider the circle \(\mathrm{C}: \mathrm{x}^2+\mathrm{y}^2=4\) and the parabola \(P: y^2=8 x\). If the set of all values of \(\alpha\), for which three chords of the circle \(\mathrm{C}\) on three distinct lines passing through the point \((\alpha, 0)\) are bisected by the parabola \(P\) is the interval \((p, q)\), then \((2 q-p)^2\) is equal to .............

The distance of the point  \((1, -2, 4)\)  from the plane passing through the point \((1, 2, 2 )\)  and perpendicular to the planes  \(x - y + 2 z = 3\)  and \(2x - 2y+ z+ 12=0,\)  is

If \(\mathrm{S}=\left\{\mathrm{z} \in \mathbb{C}: \frac{\mathrm{z}-i}{\mathrm{z}+2 i} \in \mathbb{R}\right\}\), then :

Choose the incorrect statement : \((a)\) The electric lines of force entering into a Gaussian surface provide negative flux. \((b)\) A charge ' \(q\) ' is placed at the centre of a cube. The flux through all the faces will be the same. \((c)\) In a uniform electric field net flux through a closed Gaussian surface containing no net charge, is zero. \((d)\) When electric field is parallel to a Gaussian surface, it provides a finite non-zero flux. Choose the most appropriate answer from the options given below