Given \[\left| {{{\vec{A}}}_{1}} \right|=2,\left| {{{\vec{A}}}_{2}} \right|=3\] and \[\left| {{{\vec{A}}}_{1}}+{{{\vec{A}}}_{2}} \right|=3\]. The magnitude of \[\left( {{{\vec{A}}}_{1}}+2{{{\vec{A}}}_{2}} \right)\cdot \left( 3{{{\vec{A}}}_{1}}-4{{{\vec{A}}}_{2}} \right)\] is
A very broad elevator is going up vertically with a constant acceleration of\[2\text{ }m/{{s}^{2}}\]. At the instant when its velocity is \[4\text{ }m/s,\]a ball is projected from the floor of the lift with a speed of 4 m/s relative to the floor at an elevation of \[30{}^\circ \]. The time taken by the ball to return to the floor is \[(g=10\text{ }m/{{s}^{2}})\]
A circus acrobat of mass M leaps straight up with initial velocity \[{{V}_{0}}\] from a trampoline. As he rises up, he takes a trained monkey of mass m hanging from a branch at a height h above the trampoline. What is the maximum height attained by the pair (from the branch)?
Moment of inertia of a uniform hexagonal plate about an axis LL' is I as shown in the figure (a). The moment of inertia (about axis XX') of an equilateral uniform triangular plate of thickness which is double that of the hexagonal plate is (Ratio of specific gravity \[\frac{{{\rho }_{t}}}{{{\rho }_{h}}}=3\])
A calorimeter of negligible heat capacity contains \[100\text{ }c{{m}^{3}}\] of water at\[40{}^\circ C\]. The water cools to \[35{}^\circ C\] in 5 minutes. The water is now replaced by oil of equal volume at\[40{}^\circ C\]. Find the time taken for the temperature to become \[35{}^\circ C\] under similar conditions Specific heat capacities of water and oil are \[4200\frac{J}{kg-K}\] and \[2100\frac{J}{kg-K}\] respectively. Density of oil is \[800\,\,kg/{{m}^{3}}.\]
Two identical carts A and B each with mass m are connected via a spring with spring constant k. Two additional springs, identical to the first, connect the carts to two fixed points. The carts are free to oscillate under the effect of the springs in one dimensional frictionless motion.
Under suitable initial conditions, the two carts will oscillate in phase according to \[{{x}_{A}}(t){{x}_{0}}\sin {{\omega }_{1}}t={{x}_{B}}(t)\] Where \[{{x}_{A}}\] and \[{{x}_{B}}\] are the locations of carts A and B relative to their respective equilibrium positions. Under other suitable initial conditions, the two carts will oscillate exactly out of phase according to \[{{x}_{A}}(t)={{x}_{0}}\sin {{\omega }_{2}}t=-{{x}_{B}}(t)\] Determine the ratio\[{{\omega }_{2}}/{{\omega }_{1}}\].
A solid non-conducting hemisphere of radius R has a uniformly distributed positive charge of volume density\[\rho \]. A negatively charged particle having charge q is transferred from centre of its base to infinity. Work performed in the process is
A conducting sphere S of radius 1 cm is connected to a parallel plate air capacitor C of plate area \[9\times {{10}^{-2}}\text{ }{{m}^{2}}\] and plate separation 8.85 mm. The capacitor is earthed with the help of switch S and inductor \[L=20\text{ }mH\] as shown in the figure. The sphere is imparted a charge \[{{q}_{0}}=36\,\mu C\] and switch is closed at t = 0. The maximum current in the circuit after closing the switch is
In a Vernier calipers, n divisions of its main scale match with (n + 5) divisions on its Vernier scale. Each division of the main scale is of \['\ell '\] units. Using the Vernier principle, find the least count.
Figure shows a circular region of radius \[R=\sqrt{3}m\]which has a uniform magnetic field \[B=0.2\text{ }T\]directed into the plane of the figure. A particle having mass\[m=2g\], speed \[v=0.3\text{ }m/s\]and charge \[q=1\text{ }mC\]is projected along the radius of the circular region as shown in figure. Calculate the angular deviation produced in the path of the particle as it comes out of the magnetic field. Neglect any other force apart from the magnetic force.
In the figure shown ABC is a circle of radius a. Arc AB and AC each have resistance R. Arc BC has resistance 2R. A current I enters at point A and leaves the circle at B and C. All straight wires are radial. Calculate the magnetic field at the centre of the circle. Each arc AB, BC and AC subtends \[120{}^\circ \] at the centre of the circle.
A glass block of refractive index \[\mu =1.5\] has an L cross-section with both arms identical. A light ray enters the block from left at an angle of incidence of\[45{}^\circ \], as shown in the figure. If the block was absent, the ray would pass through the point P. Calculate the angle at which the ray will emerge from the bottom face after refraction through the block.
Electric potential in a 3-dimensional space is given by \[V=\left( \frac{1}{x}+\frac{1}{y}+\frac{2}{z} \right)\] volt where x, y and z are in metre. A particle has charge \[q={{10}^{-12}}\text{ }C\]and mass \[m={{10}^{-9}}\text{ }g\] and is constrained to move in xy-plane. Find the initial acceleration of the particle if it is released at (1, 1, 1) m.
A neutral conducting ball of radius R is connected to one plate of a capacitor (Capacitance = C, the other plate of which is grounded. The capacitor is at a large distance from the ball. Two point charges, q each, begin to approach the ball from infinite distance. The two point charges move in mutually perpendicular directions. Calculate the charge on the capacitor when the two point charges are at distance x and y from the centre of the sphere.
In Young's double slit experiment, when the slit plane is illuminated with light of wavelength\[{{\lambda }_{1}}\], it was observed that point P is closest point from central maximum O, where intensity was 75% the intensity at O. When the light of wavelength \[{{\lambda }_{2}}\] is used, point P happens to be the nearest point from O where intensity is 50% of that at O. Find the ratio \[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}\] .
The self-inductance of a choke coil is\[10\text{ }mH\]. When it is connected with a 10 V dc source, then the loss of power is 20 watt. When it is connected with 10 volt ac source, loss of power is 10 watt. The frequency of ac source is\[8\times {{10}^{M}}Hz\]. Find the value of M.
A block is kept on a smooth inclined plane of angle of inclination \[30{}^\circ \] that moves with a constant acceleration so that the block does not slide relative to the inclined plane. Let \[{{F}_{1}}\] be the contact force between the block and the plane. Now the inclined plane stops and let \[{{F}_{2}}\] be the contact force between the two in this case. Find the ratio of\[{{F}_{1}}/{{F}_{2}}\].
A block of mass m is kept on a rough horizontal surface. If minimum force needed to pull the block is F and minimum horizontal force to pull the block is 2F. The value of coefficient of friction is\[\sqrt{N}\]. Find the value of N.
A plane longitudinal wave having angular frequency \[\omega =500\text{ }rad/sec\] is travelling in positive x-direction in a medium of density \[\rho =1\text{ }g/{{m}^{3}}\]and bulk modulus\[4\times {{10}^{4}}N/{{m}^{2}}\]. The loudness at a point in the medium is observed to be 20 dB. Assuming at t = 0 initial phase of the medium particles to be zero. The maximum pressure change in the medium is \[K\times {{10}^{-4}}N/{{m}^{2}}\]. Find the value of K.
If \[0.01\text{ }M\]solution of an electrolyte has a resistance of 40 ohms in a cell having a cell constant of \[0.4\,c{{m}^{-1}},\] then its molar conductance in \[oh{{m}^{-1}}\,\,c{{m}^{2}}\,\,mo{{l}^{-1}}\].
A process has \[\Delta H=200\text{ }J\text{ }mo{{l}^{-1}}\]and \[\Delta \text{S}=40\text{ }J{{K}^{-1}}\text{ }mo{{l}^{-1}}\]Out of the values given below, the minimum temperature above which the process will be spontaneous:
Consider the reaction: \[{{N}_{2}}(g)+3{{H}_{2}}(g)\to 2N{{H}_{3}}(g)\]The equality relationship between \[\frac{d[N{{H}_{3}}]}{dt}\] and \[-\frac{d[{{H}_{2}}]}{dt}\]is
One part of an element A combines with two parts of another B. Six parts of the element C combine with four parts of the element B. If A and C combine together the ratio of their weights will be governed by
The dissociation constant of two acids \[H{{A}_{1}}\]and \[H{{A}_{2}}\] are \[3.14\times {{10}^{-4}}\] and \[1.96\times {{10}^{-5}}\] respectively The relative strength of the acids will be approximately
The pure crystalline substance on being heated gradually first forms a turbid liquid at constant temperature and still at higher temperature turbidity completely disappears. The behaviour is a characteristic of substance forming
The dipole moment of \[HBr\] is \[1.6\times {{10}^{-30}}\] coulomb meter and interatomic spacing is \[1\overset{o}{\mathop{A}}\,\]. What is the % ionic character of\[HBr\]?
16 g of oxygen and \[3\text{ }g\] of hydrogen are mixed and kept at \[760\text{ }mm\] of \[Hg\] pressure and \[0{}^\circ C\]. Calculate the total volume occupied by the mixture in mL.
Naturally occurring boron consists of two isotopes whose atomic weights are \[10.01\] and\[11.01\]. The atomic weight of natural boron is\[10.81\]. Calculate the percentage of isotope with atomic weight \[11.01\] in natural boron.
The value of \[P{}^\circ \] for benzene is \[640\text{ }mm\] of \[Hg\]. The vapour pressure of solution containing \[2.5g\] substance in \[39g\] benzene is 600mm of Hg. What is the molecular mass of X?
The value of a for which the system of equations \[{{a}^{3}}x+{{(a+1)}^{3}}y+{{(a+2)}^{3}}z=0,\]\[ax+(a+1)y+\]\[(a+2)z=0,x+y+z=0\]has a non zero solution is
The value of p for which the function \[f(x)=\left\{ \begin{align} & \frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{x}{p}\log \left[ 1+\frac{{{x}^{2}}}{3} \right]},x\ne 0 \\ & 12{{(log\,4)}^{3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\] may be continuous at x = 0, is
A purse contains 4 copper and 3 silver coins, and a second purse contains 6 copper and 2 silver coins. A coin is taken out from any purse, the probability that it is a copper coin is
If the ad joint of a \[3\times 3\] matrix P is \[\left[ \begin{matrix} 1 & 2 & 1 \\ 4 & 1 & 1 \\ 4 & 7 & 3 \\ \end{matrix} \right],\] then the sum of squares of possible values of determinant of P is _______.
The number of polynomials of the form \[{{x}^{3}}+a{{x}^{2}}+bx+c\] which are divisible by \[{{x}^{2}}+1\]where \[a,b,c\in \]{1, 2, 3,.... 10} is K, then 10 K is _______.
If the sum to infinity of a decreasing G.P. with the common ratio x is k such that \[|x|<1;x\ne 0\]. The ratio of the fourth term to the second term is 1/16 and the ratio of third term to the square of the second term is 1/9. Find the value of k.
The plane \[2x-2y+z=3\] is rotated about the line where it cuts the xy plane by an acute angle \[\alpha \]. If the new position of plane contains the point (3, 1, 1), then \[\cos \alpha \] equal to _______.
If the points with position vectors \[10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\] and \[\lambda \hat{i}+11\hat{j}\] are collinear, then \[\frac{3}{2}\lambda \] is equal to __________.