Vector \(\vec{A}\) of magnitude \(5 \sqrt{3}\) units, another vector \(\vec{B}\) of magnitude of 10 units are inclined to each other at an angle of \(30^{\circ}\). The magnitude of vector product of the two vectors is \(\left[\sin 30^{\circ}=\frac{1}{2}\right]\)
Four particles each of mass M are placed at the corners of a square of side L . The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is
In Fraunhofer diffraction pattern, slit width is 0.3 mm and screen is at 1.5 m away from the lens. If wavelength of light used is \(4500 \mathop {\rm{A}}\limits^{\rm{o}}\), then the distance between the first minimum on either side of the central maximum is [ \(\theta\) is small and measured in radian.]
If the electron in hydrogen atom jumps from third Bohr orbit to ground state directly and the difference between energies of the two states is radiated in the form of photons. If the work function of the material is 4.1 eV , then stopping potential is nearly
[Energy of electron in \(n^{\text {th }}\) orbit \(=\frac{-13 \cdot 6}{n^2} \mathrm{eV}\) ]
The magnetic field intensity H at the centre of a long solenoid having \(n\) turns per unit length and carrying a current I, when no material is kept in it is ( \(\mu_0=\) permeability of free space)
Two discs of moment of inertia ' \(\mathrm{I}_1\) ' and ' \(\mathrm{I}_2\) ' and angular speeds ' \(\omega_1\) ' and ' \(\omega_2\) ' are rotating along the collinear axes passing through their centre of mass and perpendicular to their plane. If the two discs are made to rotate together along the same axis. The rotational kinetic energy of the system will be
A wire has a mass \(0.3 \pm 0.003\) gram, radius \(0.5 \pm 0.005 \mathrm{~mm}\) and length \(6 \pm 0.06 \mathrm{~cm}\). The maximum percentage error in the measurement of its density is
Two particles ' \(A\) ' and ' \(B\) ' execute SHMs of periods ' \(T\) ' and \(\frac{3 T}{2}\). If they start from the mean position then the phase difference between them, when the particle ' \(A\) ' completes two oscillations will be
The reactance of a capacitor is \(\mathrm{X}_{\mathrm{C}}\). If the frequency and the capacitance are doubled, then the new reactance will be
The current (I) drawn from the battery in the given circuit is
A rectangular black body of temperature \(127^{\circ} \mathrm{C}\) has surface area \(4 \mathrm{~cm} \times 2 \mathrm{~cm}\) and rate of radiation is E . If its temperature is increased by \(400^{\circ} \mathrm{C}\) and surface area is reduced to half of the initial value then the rate of radiation is
Two planets A and B have densities ' \(\rho_1\) ', ' \(\rho_2\) ' and have radii ' \(r_1\) ', ' \(r_2\) ', respectively. The ratio of acceleration due to gravity on \(A\) to that of \(B\) is
The equation of the plane containing the line \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) and perpendicular to the plane containing the lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{1}\) and \(\frac{x}{3}=\frac{y}{2}=\frac{z}{1}\) is
The equation of the plane containing the line \(\frac{x+1}{2}=\frac{y+2}{1}=\frac{z-2}{3}\) and the point \((1,-1,3)\) is
If \(\mathrm{f}(x)=3 x+10, \mathrm{~g}(x)=x^2-1\), then \((\mathrm{fog})^{-1}(x)=\)
Let \(\bar{a}=2 \hat{i}+\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-\hat{k}\) and vector \(\bar{c}\) be coplanar. If \(\bar{c}\) is perpendicular to \(\bar{a}\), then \(\bar{c}\) is
If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors such that \(|\overrightarrow{\mathrm{a}}|=\sqrt{31}, 4|\overrightarrow{\mathrm{~b}}|=|\overrightarrow{\mathrm{c}}|=2\) and \(2(\overline{\mathrm{a}} \times \overline{\mathrm{b}})=3(\overline{\mathrm{c}} \times \overline{\mathrm{a}})\) and if the angle between \(\overline{\mathrm{b}}\) and \(\overline{\mathrm{c}}\) is \(\frac{2 \pi}{3}\) then \(\left|\frac{\bar{a} \times \bar{c}}{\bar{a} \cdot \bar{b}}\right|^2=\)
If the tangent at the point \((2 \sec \theta, 3 \tan \theta)\) to the hyperbola \(\frac{x^2}{4}-\frac{y^2}{9}=1\) is parallel to \(3 x-y+4=0\), then the value of \(\theta\) is
Bag I contains 3 red and 2 green balls and Bag II contains 5 red and 3 green balls. A ball is drawn from one of the bag at random and it is found to be green. Then the probability that it is drawn from Bag I is
If \(x=-2+\sqrt{-3}\), then the value of \(2 x^4+5 x^3+7 x^2-x+38\) is equal to
If \(x=a \sin 2 t(1+\cos 2 t), y=b \cos 2 t(1-\cos 2 t)\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to
The contrapositive of the statement \(\sim p \vee(q \wedge \sim r)\) is
The distance of the point \(\mathrm{P}(3,4,4)\) from the point of intersection of the line joining the points \(\mathrm{Q}(3,-4,-5), \mathrm{R}(2,-3,1)\) and the plane \(2 x+y+z=7\) is
If the lines \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\) and \(\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}\) intersect, then the value of k is
