Identify the logic operation carried out.

Two metallic wires \(P\) and \(Q\) have same volume and are made up of same material. If their area of cross sections are in the ratio \(4: 1\) and force \(F_1\) is applied to \(\mathrm{P}\), an extension of \(\Delta l\) is produced. The force which is required to produce same extension in \(Q\) is \(\mathrm{F}_2\).The value of \(\frac{\mathrm{F}_1}{\mathrm{~F}_2}\) is _______.

For a transparent prism, if the angle of minimum deviation is equal to its refracting angle, the refractive index n of the prism satisfies.

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.
Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor \(\left(R / \sqrt{R^2+\omega^2 L^2}\right)\), where \(\omega\) is frequency of the supply across resistor \(R\) and inductor L . If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.
In the light of the above statements, choose the most appropriate answer from the options given below :

A hoop and a solid cylinder of same mass and radius are made of a permanent magnetic material with their magnetic moment parallel to their respective axes. But the magnetic moment of hoop is twice of solid cylinder. They are placed in a uniform magnetic field in such a manner that their magnetic moments make a small angle with the field. If the oscillation periods of hoop and cylinder are \(T_h\) and \(T_c\) respectively, then the relation between them is....

Two light beams fall on a transparent material block at point 1 and 2 with angle \(\theta_1\) and \(\theta_{2^{\prime}}\) respectively, as shown in figure. After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block. Given : the distance between 1 and 2, \(\mathrm{d}=4 \sqrt{3} \mathrm{~cm}\) and \(\theta_1=\theta_2=\cos ^{-1}\left(\frac{\mathrm{n}_2}{2 \mathrm{n}_1}\right)\), where refractive index of the block \(\mathrm{n}_2\gt\) refractive index of the outside medium \(\mathrm{n}_1\), then the thickness of the block is ________ cm.

A body of mass \(m\) is moving in a circular orbit of radius \(R\) about a planet of mass \(M\). At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \(\frac{R}{2}\) , and the other mass, in a circular orbit of radius \(\frac{3R}{2}\). The difference between the final and initial total energies is

The value of \(\lambda \) such that sum of the squares of the roots of quadratic equation, \(x^2 + (3 - \lambda )x + 2 = \lambda \)  has the lest value is

Let \(\vec a\, = \,\hat i\, + \,\hat j\, + \,\sqrt 2 \hat k,\,\,\vec b\, = \,{b_1}\hat i\, + \,{b_2}\hat j\, + \sqrt 2 \hat k\) and \(\vec c\, = \,5\hat i\, + \,\hat j + \sqrt 2 \hat k\) be three vectors such that the projection vector of \(\vec b\) on \(\vec a\) is \(\vec a\). If \(\vec a\, + \vec b\) is perpendicular to \(\vec c\) , then \(\left| {\vec b} \right|\) is equal to

Some distant star is to be observed by some telescope of diameter of objective lens \(a\), at an angular resolution of \(3.0\times 10^{-7}\) radian. If the wavelength of light from the star reaching the telescope is \(500\) nm, the minimum diameter of the objective lens of the telescope is ________ cm. (nearest integer)

Let \(\hat{a}\) be a unit vector perpendicular to the vectors \(\overrightarrow{\mathrm{b}}=\hat{i}-2 \hat{j}+3 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=2 \hat{i}+3 \hat{j}-\hat{k}\), and makes an angle of \(\cos ^{-1}\left(-\frac{1}{3}\right)\) with the vector \(\hat{i}+\hat{j}+\hat{k}\). If \(\hat{\mathrm{a}}\) makes an angle of \(\frac{\pi}{3}\) with the vector \(\hat{i}+\alpha \hat{j}+\hat{k}\), then the value of \(\alpha\) is :

Let \(P(3\cos\alpha, 2\sin\alpha)\), \(\alpha \neq 0\), be a point on the ellipse \(\dfrac{x^2}{9}+\dfrac{y^2}{4}=1\), \(Q\) be a point on the circle \(x^2+y^2-14x-14y+82=0\) and \(R\) be a point on the line \(x+y=5\) such that the centroid of the triangle \(PQR\) is \(\left(2+\cos\alpha, 3+\dfrac{2}{3}\sin\alpha\right)\). Then the sum of the ordinates of all possible points \(R\) is:

The area of the region in the first quadrant inside the circle \(x^2+y^2=8\) and outside the parabola \(\mathrm{y}^2=2 \mathrm{x}\) is equal to :