A brass wire of length 2 m and radius 1 mm at \(27^{\circ} C\) is held taut between two rigid supports. Initially it was cooled to a temperature of \(-43^{\circ} C\) creating a tension T in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to 1.4 T , is _________ \({ }^{\circ} C\)

A thin bar of length \(L\) has a mass per unit length \(\lambda \), that increases linearly with distance from one end. If its total mass is \(M\) and its mass per unit length at the lighter end is \(\lambda_0\), then the distance of the centre of mass from the lighter end is

A uniform metal chain of mass \(m\) and length ' \(L\) ' passes over a massless and frictionless pulley. It is released from rest with a part of its length ' \(l\) ' is hanging on one side and rest of its length ' \(L -l\) ' is hanging on the other side of the pulley. At a certain point of time, when \(l=\frac{L}{x}\), the acceleration of the chain is \(\frac{g}{2}\). The value of \(x\) is ........

Ship \(A\) is sailing towards north -east with velocity \(\vec v = 30\,\hat i + 50\hat j\,km/hr\) where \(\hat i\) points east and \(\hat j\) , north. Ship \(B\) is at a distance of \(80\, km\) east and \(150\, km\) north of Ship \(A\) and is sailing towards west at \(10\, km/hr\). \(A\) will be at minimum distance from \(B\) is.........\(hrs\)

In a linear simple harmonic motion \((SHM)\) \((A)\) Restoring force is directly proportional to the displacement. \((B)\) The acceleration and displacement are opposite in direction. \((C)\) The velocity is maximum at mean position. \((D)\) The acceleration is minimum at extreme points. Choose the correct answer from the options given below :

\(LED\) is constructed from \(Ga-As-P\) semiconducting material. The energy gap of this \(LED\) is \(1.9\, eV\). Calculate the wavelength of light emitted and its colour. \(\left[ h =6.63 \times 10^{-34} \;Js \right.\) and \(\left. c =3 \times 10^{8}\; ms ^{-1}\right]\)

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Consider a ferromagnetic material :
Assertion (A) : The individual atoms in a ferromagnetic material possess a magnetic dipole moment and interact with one another in such a way that they spontaneously align themselves forming domains.
Reason (R) : At high enough temperature, the domain structure of ferromagnetic material disintegrates. Thus, magnetization will disappear at high enough temperature known as Curie temperature.
In the light of the above statements, choose the correct answer from the options given below :

Let the mirror image of a circle \(c_{1}: x^{2}+y^{2}-2 x-\) \(6 y+\alpha=0\) in line \(y=x+1\) be \(c_{2}: 5 x^{2}+5 y^{2}+10 g x\) \(+10 f y +38=0\). If \(r\) is the radius of circle \(c _{2}\), then \(\alpha+6 r^{2}\) is equal to\(.....\)

For a given transistor amplifier circuit in \(CE\) configuration \(V_{C C}=1 V , R_c=1 k \Omega, R_b=100 k \Omega\) and \(\beta=100\). Value of base current \(I_b\) is

If the variable line \(3 x+4 y=\alpha\) lies between the two circles \((x-1)^{2}+(y-1)^{2}=1\) and \((x-9)^{2}+(y-1)^{2}=4\) without intercepting a chord on either circle, then the sum of all the integral values of \(\alpha\) is .... .

A multiple choice examination has \(5\) questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get \(4\) or more correct answers just by guessing is :

The equation for real gas is given by \(\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}\), where \(\mathrm{P}, \mathrm{V}, \mathrm{T}\) and R are the pressure, volume, temperature and gas constant, respectively. The dimension of \(\mathrm{ab}^{-2}\) is equivalent to that of :

Let \(|\cos \theta \cos (60-\theta) \cos (60-\theta)| \leq \frac{1}{8}, \theta \in[0,2 \pi]\) Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is :