In the expression for torque \[\tau =a\times L+b\times I/\omega \],L represents angular momentum, I is moment of inertia and \[\omega \] is angular velocity. The dimensions of a x b are
The electric field in a region is radially outward with magnitude E = Ar. What will be the charge contained in a sphere of radius a centred at the origin? Take \[A=100V{{m}^{-2}}\] and a = 20.0 cm.
Water drops fall at regular intervals from a tap which is 5.0 m above the ground. The third drop is leaving the tap at the instant the first drop reaches the ground. How far above the ground is the second drop at that instant?
A spring balance has a scale that reads from 0 to 60 kg. The length of the scale is 30 cm. A body suspended from this balance and when displaced and released, oscillates with a period of 0.8 s, what is the weight of the body when oscillating?
A block of mass m = 1 kg moving on a horizontal surface with speed \[{{v}_{i}}=2\,\text{m}\,{{\text{s}}^{-1}}\]enters a rough patch ranging from x = 0.10 m to x = 2.01 m. The retarding force \[{{F}_{r}}\]on the block in this range is inversely proportional to x over this range, \[{{F}_{r}}=-\frac{k}{x}\]for\[0.1<x<2.01m\]= 0 for x < 0.1 m and x > 2.01 m where k = 0.5 J. What is the final kinetic energy of the block as it crosses this patch?
A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches \[30{}^\circ \], the box starts to slip and slides 4.0 m down the plank in 4.0 s. The coefficients of static and kinetic friction between the box and the plank will be, respectively (Take\[g=10\,\text{m}\,{{\text{s}}^{-2}}\])
A vibration magnetometer placed in magnetic meridian has a small bar magnet. The magnet executes oscillations with a time period of 2s in earths horizontal magnetic field of \[24\mu T\]. When a horizontal field of \[18\mu T\]is produced opposite to the earths field by placing a current carrying wire, the new time period of the magnet will be
A semicircular lamina of mass m, radius r and centre at C is shown in the figure. Its centre of mass is at a distance x from C. Its moment of inertia about an axis through its centre of mass and perpendicular to its plane is
An inductor of inductance L = 400 mH and resistors of resistances \[{{R}_{1}}=2\Omega \]. and \[{{R}_{2}}=2\,\Omega \] are connected to a battery of emf 12V as shown in figure. The internal resistance of the battery is negligible. The switch S is closed at t = 0. The potential drop across L as a function of time is
The instantaneous values of alternating current and voltage in a circuit are given as \[I=\frac{1}{\sqrt{2}}\sin (100\pi t)\]A and \[\varepsilon =\frac{1}{\sqrt{2}}\sin (100\pi t+\pi /3)V\] The average power in watts consumed in the circuit is
A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The gravitational potential at a point situated at distance \[\frac{a}{2}\] from the centre, will be
In Youngs double slit experiment, first slit has width four times the width of the second slit. The ratio of the maximum intensity to the minimum intensity in the interference fringe system is
In a compound microscope, the focal lengths of two lenses are 1.5 cm and 6.25 cm. If an object is placed at 2 cm from objective and the final image is formed at 25 cm from eye lens, the distance between the two lenses is
A carrier frequency of 1 MHz and peak value of 10 V is amplitude modulated with a signal frequency of 10 kHz with peak value of 0.5 V. Then the modulation index and the side band frequencies respectively are
A thin copper wire of length L increases its length by 1% when heated from temperature \[{{T}_{1}}\]to \[{{T}_{2}}\]. What is the percentage change in area when a thin copper plate having dimensions \[2L\times L\] is heated from\[{{T}_{1}}\] to\[{{T}_{2}}\]?
A boat of 90 kg is floating in still water. A boy of mass 30 kg walks from the stern to the bow. The length of the boat is 3 m. The distance moved by the boat is _____ m.
Two soap bubbles A and B are kept in a closed chamber where the air is maintained at pressure a of \[8\text{N}\,{{\text{m}}^{-2}}\]. The radii of bubbles A and B are 2 cm and 4 cm respectively. Surface tension of soap water used to make bubbles is \[\text{0}\text{.04}\,\text{N}\,{{\text{m}}^{-1}}\]. If \[{{n}_{A}}\]and \[{{n}_{B}}\]are the number of moles of air in bubbles A and B respectively. Then the value of \[\frac{{{n}_{B}}}{{{n}_{A}}}\]is ____. (Neglect effect of gravity).
An electric drill of output 0.2 hp is used to drill a hole in 100 g of iron. It takes 20 s to drill the hole. Assuming that all the energy spent is absorbed by the iron, its rise in temperature is _____\[{}^\circ C\]. (Given specific heat of iron \[=450J\,k{{g}^{-1}}{{\,}^{o}}{{C}^{-1}}\]).
Monochromatic radiation of wavelength 640.2 nm from a neon lamp irradiates photosensitive material made of cesium. The stopping voltage is measured to be 0.54 V. The source is replaced by another source of wavelength 427.2 nm which irradiates the same photocell. The new stopping voltage is_____V.
\[S{{m}^{+2}},E{{u}^{2+}}\] and \[Y{{b}^{2+}}\] ions in solution are good oxidising agents but an aqueous solution of \[C{{e}^{4+}}\] is a good reducing agent.
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B)
\[M{{n}^{+2}}\]shows maximum paramagnetic character amongst the \[+2\] ions of \[{{I}^{st}}\] transition series.
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C)
\[La{{(OH)}_{3}}\] is more basic than \[Lu{{(OH)}_{3}}\]
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D)
\[F{{e}^{2+}},M{{n}^{2+}}\] and \[C{{r}^{3+}}\] ions are coloured.
With very dil. \[HN{{O}_{3}}\]and cold dil. \[HN{{O}_{3}},\] which metal will not give ammonium nitrate (i.e., \[\overset{\oplus }{\mathop{N}}\,{{H}_{4}}\]ion) and hydroxylamine \[(N{{H}_{2}}OH)\] respectively?
Equimolal solution of \[KCl\] and compound \[X\] in water show depression in freezing point in the ratio of\[4:1\]. Assuming \[KCl\]to be completely ionized, the compound X in solution must
For the reaction, \[X2Y\] and \[ZP+Q\] occurring at two different pressure \[{{P}_{1}}\] and \[{{P}_{2}},\] respectively. The ratio of the two pressure is\[1:3\]. What will be the ratio of equilibrium constant, if the degree of dissociation of X and Z are equal?
A second order reaction requires 70 min to change the concentration of reactants from \[0.08\text{ }M\]to\[0.01\text{ }M\]. The time required to become\[0.04\text{ }M=2x\text{ }min\]. Find the value of x.
For \[{{H}_{2}}\] gas molecule, the most probable speed at temperature TK. is equal to \[{{10}^{2}}m{{s}^{-1}}\]. The temperature \[T(K)\] is _____. (Take \[R\approx 8.0\,J{{K}^{-1}}mo{{l}^{-1}}\])
\[{{F}_{2}}\] can be prepared by reacting hexafluoro magnate (IV) with antimony pentafluoride as: \[{{K}_{2}}Mn{{F}_{6}}+Sb{{F}_{5}}\xrightarrow{{}}KSb{{F}_{6}}+Mn{{F}_{3}}+{{F}_{2}}\] The number of equivalent \[{{K}_{2}}Mn{{F}_{6}}\] required to react completely with one mole of \[Sb{{F}_{5}}\] in the given reaction is _____.
The EMF of the following concentration cells at \[30{}^\circ C\] in V is _____. [Assume Kw does not change at 3 \[0{}^\circ C\]] (Take \[2.303\frac{RT}{F}=0.06\])
If the Potential energy of an electron in the first Bohr orbit of H atom is zero, the total energy of the electron in second Bohr orbit in eV is _____.
Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is
If the arithmetic mean of the numbers \[{{x}_{1}},\,\,{{x}_{2}},\,\,{{x}_{3}}......{{x}_{n}}\] is \[\overline{x}\] then the arithmetic mean of numbers \[a{{x}_{1}}+b,\,\,a{{x}_{2}}+b\]+ b, \[a{{x}_{3}}+b,\,\,.......a{{x}_{n}}+b\], where a, b are two constants would be
Perpendicular are drawn from points on the line \[\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}\] to the plane \[x+y+z=13\]. The feet of 2-13 perpendiculars lie on the line
If \[\overline{a},\,\,\bar{b},\,\,\bar{c}\,\,and\,\,\bar{d}\] are unit vector such that \[(\vec{a}\,\times \vec{b}).(\vec{c}\,\times \vec{d})=1\] and \[\vec{a}.\vec{c}=\frac{1}{2}\], then
A)
\[\vec{a},\,\,\vec{b},\,\,\vec{c}\] are non-coplanar
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B)
\[\vec{b},\,\,\vec{c},\,\,\vec{d}\] are non-coplanar
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C)
\[\vec{b},\,\,\vec{d}\] are non-parallel
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D)
\[\vec{a},\,\,\vec{d}\] are parallel and \[\vec{b},\,\,\vec{c}\] are parallel
Suppose the cubic \[{{\operatorname{x}}^{3}}- px + q\] has three distinct real roots where \[\operatorname{p} > 9 and q > 9\]. Then which one of the following holds?
A)
The cubic has minima at \[-\sqrt{\frac{p}{3}}\] and maxima at \[\sqrt{\frac{p}{3}}\]
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B)
The cubic has minima at \[\sqrt{\frac{p}{3}}\] and maxima at \[-\sqrt{\frac{p}{3}}\].
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C)
The cubic has maxima at both \[\sqrt{\frac{p}{3}}\,\,and\,\,-\sqrt{\frac{p}{3}}\]
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D)
The cubic has minima at both \[\sqrt{\frac{p}{3}}\,\,and\,\,-\sqrt{\frac{p}{3}}\]
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?
A line is drawn through the point \[\operatorname{P}(3,\,\,11)\] to cut the circle \[{{x}^{2}}+{{y}^{2}}=9\] at A and B. Then \[\operatorname{PA}\cdot PB\] is equal to