A metal wire of resistance \(3\,\Omega \) is elongated to make a uniform wire of double its previous length. The new wire is now bent and the ends joined to make a circle. If two points on this circle make an angle \(60^o\) at the centre, the equivalent resistance between these two points will be

The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength \(500\,nm\) and \(600\, nm\), each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in \(nm\) is

A radioactive nucleus decays by two different process. The half life of the first process is \(5\) minutes and that of the second process is \(30\,s\). The effective half-life of the nucleus is calculated to be \(\frac{\alpha}{11}\,s\). The value of \(\alpha\) is \(..............\)

In Young's double slit experiment, the distance between slits and the screen is \(1 .0\,\,m\) and monochromatic light of \(600\,\,nm\)  is being used. A person standing near the slits is looking at the fringe pattern. When the separation between the slits is varied, the interference pattern disappears for a particular distance \(d_0\) between the slits . If the angular resolution of the eye is \(\frac {1}{60}^o,\) the value of \(d_0\)  is close to......\(mm\)

The de Broglie wavelength associated with an electron accelerated through a potential difference V is \(\lambda_e\) and the de Broglie wavelength associated with a proton accelerated through the same potential difference is \(\lambda_p\). If their corresponding masses are \(m_e\) and \(m_p\), respectively, then the ratio of their de Broglie wavelengths \(\left(\dfrac{\lambda_e}{\lambda_p}\right)\) is ______.

A telescope with objective diameter \(R\) is used to observe a distant star emitting light of wavelength \(500\) nm, at a resolution of \(5 \times 10^{-7}\) radian. The value of \(R\) is _____ cm.

\(Y = A \sin \left(\omega t +\phi_{0}\right)\) is the time-displacement equation of a SHM. At \(t=0\) the displacement of the particle is \(Y =\frac{ A }{2}\) and it is moving along negative \(x\) -direction. Then the initial phase angle \(\phi_{0}\) will be ...... .

If the distance between the foci of an ellipse is \(6\) and the distance between its directrices is \(12\), then the length of its latus rectum is

If the letters of the word \(MOTHER\) be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word \(MOTHER\) is

Arrange the following in the ascending order of wavelength \((\lambda)\) :
(A) Microwaves \(\left(\lambda_1\right)\)
(B) Ultraviolet rays \(\left(\lambda_2\right)\)
(C) Infrared rays \(\left(\lambda_3\right)\)
(D) X-rays \(\left(\lambda_4\right)\)
Choose the most appropriate answer from the options given below :

The size of the images of an object, formed by a thin lens are equal when the object is placed at two different positions 8 cm and 24 cm from the lens. The focal length of the lens is ___________ cm.

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its \(11^{\text {th }}\) term is :

For a body projected at an angle with the horizontal from the ground, choose the correct statement.