Three vectors \(\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}\) and \(\overrightarrow{\mathrm{OR}}\) each of magnitude \(A\) are acting as shown in figure. The resultant of the three vectors is \(A \sqrt{x}\). The value of \(x\) is _______.

An observer looks at a distant tree of height \(10\ m\) with a telescope of magnifying power of \(20\). To the observer the tree appears :

Young's modulus is determined by the equation given by \(\mathrm{Y}=49000 \frac{\mathrm{m}}{\ell} \frac{\text { dyne }}{\mathrm{cm}^2}\) where \(\mathrm{M}\) is the mass and \(\ell\) is the extension of wre used in the experiment. Now error in Young modules \((\mathrm{Y})\) is estimated by taking data from \(M-\ell\) plot in graph paper. The smallest scale divisions are \(5 \mathrm{~g}\) and \(0.02\) \(\mathrm{cm}\) along load axis and extension axis respectively. If the value of \(M\) and \(\ell\) are \(500 \mathrm{~g}\) and \(2 \mathrm{~cm}\) respectively then percentage error of \(\mathrm{Y}\) is _______.

In a microscope the objective is having focal length \(f_0=2 cm\) and eye-piece is having focal length \(f_{ e }=4 cm\). The tube length is 32 cm . The magnification produced by this microscope for normal adjustment is _________ .

Thermodynamic process is shown below on a \(P-V\) diagram for one mole of an ideal gas. If \(V _{2}=2 V _{1}\) then the ratio of temperature \(T _{2} / T _{1}\) is ...... .

Planet \(A\) has mass \(\mathrm{M}\) and radius \(\mathrm{R}\). Planet \(\mathrm{B}\) has half the mass and half the radius of Planet \(A\). If the escape velocities from the Planets \(A\) and \(\mathrm{B}\) are \(v_{\mathrm{A}}\) and \(v_{\mathrm{B}},\) respectively, then \(\frac{v_{\mathrm{A}}}{v_{\mathrm{B}}}=\frac{\mathrm{n}}{4}\) The value of \(\mathrm{n}\) is

A capacitor has air as dielectric medium and two conducting plates of area \(12 \mathrm{~cm}^2\) and they are \(0.6 \mathrm{~cm}\) apart. When a slab of dielectric having area \(12 \mathrm{~cm}^2\) and \(0.6 \mathrm{~cm}\) thickness is inserted between the plates, one of the conducting plates has to be moved by \(0.2 \mathrm{~cm}\) to keep the capacitance same as in previous case. The dielectric constant of the slab is _______. (Given \(\left.\epsilon_0=8.834 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right)\)

Four closed surfaces and corresponding charge distributions are shown below Let the respective electric fluxes through the surfaces be \({\phi _1},{\phi _2},{\phi _3}\) and \({\phi _4}\) . Then

Let \(L_1, L_2\) be the lines passing through the point \(\mathrm{P}(0,1)\) and touching the parabola \(9 x^2+12 x+18 y-14=0\). Let \(Q\) and \(R\) be the points on the lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\) such that the \(\triangle \mathrm{PQR}\) is an isosceles triangle with base \(\mathrm{QR}\). If the slopes of the lines \(Q R\) are \(m_1\) and \(m_2\). then \(16\left(m_1^2+m_2^2\right)\) is equal to ..............

Let \(A=\{-4,-3,-2,0,1,3,4\}\) and \(R =\{( a , b ) \in A\) \(\times A : b =| a |\) or \(\left.b ^2= a +1\right\}\) be a relation on \(A\). Then the minimum number of elements, that must be added to the relation \(R\) so that it becomes reflexive and symmetric, is \(........\).

Consider the function : \(f\left( x \right) = \left[ x \right] + \left| {1 - x} \right|,\, - 1 \le x \le 3\) where \([x]\) is the greatest integer function Statement \(1\) :\(f\) is not continuous at \(x = 0, 1, 2\) and \(3\) Statement \(2\) :\(f\left( x \right) = \left( \begin{array}{l}
 - x,\,\,\,\,\,\,\,\,\, - 1 \le x < 0\\
1 - x,\,\,\,\,\,\,\,0 \le x < 1\\
1 + x,\,\,\,\,\,\,\,1 \le x < 2\,\\
2 + x,\,\,\,\,\,\,2 \le x \le 3
\end{array} \right.\)

A mass \(m\) is attached to two springs as shown in figure. The spring constants of two springs are \(K _1\) and \(K _2\). For the frictionless surface, the time period of oscillation of mass \(m\) is

Let \(\mathrm{M}\) and \(\mathrm{m}\) respectively be the maximum and minimum values of the function \(f(x)=\tan ^{-1}(\sin x+\cos x)\) in \(\left[0, \frac{\pi}{2}\right]\), Then the value of \(\tan (\mathrm{M}-\mathrm{m})\) is equal to: