A bob of mass M is suspended by a massless string of length L. The horizontal velocity v at position A is just sufficient to make it reach the point B. The angle \[\theta \]at which the speed of the bob is half of that at A, satisfies
A diatomic molecule is made of two masses \[{{m}_{1}}\]and \[{{m}_{2}}\] which are separated by a distance r. If we calculate its rotational energy by applying Bohrs rule of angular momentum quantization, its energy will be given by (n is an integer)
The concentration of hole-electron pairs in pure Silicon at T = 300 K is \[7\times {{10}^{15}}{{m}^{-}}^{3}\]. Antimony is doped into Silicon in a proportion of 1 atom in \[{{10}^{7}}Si\]atoms. Assuming that half of the impurity atoms contribute electron in the conduction band, calculate the factor by which the number of charge carriers increases due to doping of \[5\times {{10}^{28}}{{m}^{-3}}\]of Si atoms.
A uniform rod of length L and mass M is held vertical, with its bottom end pivoted to the floor. The rod falls under gravity, freely turning about the pivot. If acceleration due to gravity is g, what is the instantaneous angular speed of the rod when it makes an angle \[60{}^\circ \]with the vertical?
Two long parallel wires carry currents of equal magnitude but in opposite directions. These wires are suspended from rod PQ by four chords of same length L as shown in the figure. The mass per unit length of the wire is \[\lambda \]. Determine the value of\[\theta \]assuming it to be small.
A circular coil of radius 8.0 cm and of 20 turns is rotated about its vertical diameter with an angular speed of 50 rad \[{{s}^{-1}}\]in a uniform horizontal magnetic field of magnitude \[3.0\times {{10}^{-2}}T\]. If the coil forms a closed loop of resistance \[10\Omega \], calculate the average power loss due to Joule heating.
A train moves from rest with acceleration\[\alpha \]and in time \[{{t}_{1}}\]covers a distance x. It then decelerates to rest at constant retardation\[\beta \]for distance y in time \[{{t}_{2}}\] Then
A bottle has an opening of radius a and length b. A cork of length b and radius \[(a+\Delta a)\]where\[(\Delta a<<a)\] is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is then the force needed to push the cork into the bottle is
A particle is projected from the ground with an initial speed of v at angle \[\theta \]with horizontal. The average velocity of the particle between its point of projection and the highest point of trajectory is
In the circuit shown, switch \[{{S}_{2}}\]is closed first and is kept closed for a long time. Now \[{{S}_{1}}\] is closed. Just after that instant the current through \[{{S}_{1}}\]is
A particle is executing simple harmonic motion with a time period T. At time t = 0, it is at its position of equilibrium. The kinetic energy-time graph of the particle will look like
From a solid sphere of mass M and radius R, a spherical portion of radius \[\frac{R}{2}\] is removed, as shown in the figure. Taking gravitational potential V = 0 at \[r=\infty \], the potential at the centre of the cavity thus formed is (G = gravitational constant)
A monochromatic beam of light has a frequency \[\upsilon =\frac{3}{2\pi }\times {{10}^{12}}\]Hz and is propagating along the direction\[\frac{\hat{i}+\hat{j}}{\sqrt{2}}\] It is polarized along the \[\hat{k}\]direction. The acceptable form for the magnetic field is
A body of mass m rests on a horizontal floor with which it has a coefficient of static friction \[\mu \]. It is desired to make the body move by applying a minimum possible force \[\vec{F}\] as shown in the diagram. The values of \[\theta \] and \[{{F}_{\min }}\]shall be respectively equal to
An electron of mass m with an initial velocity \[\vec{v}={{v}_{0}}\hat{i}({{v}_{0}}>0)\]enters an electric field \[\vec{E}=-{{\vec{E}}_{0}}\hat{i}\](\[{{E}_{0}}=\]constant > 0) at t = 0. If \[{{\lambda }_{0}}\] is its de-Broglie wavelength initially, then its de- Broglie wavelength at time t is
Consider an ideal gas confined in an isolated closed chamber. As the gas undergoes an adiabatic expansion, the average time of collision between molecules increases as \[{{V}^{q}}\], where V is the volume of the gas. The value of q is \[\left( \gamma =\frac{{{C}_{P}}}{{{C}_{V}}} \right)\]
Two identical thin rings, each of radius a are placed coaxially at a distance a apart. Let charges \[{{Q}_{1}}\]and \[{{Q}_{2}}\]be placed uniformly on the two rings. The work done in moving a charge q from the centre of one ring to that of the other is
A cylindrical tube open at both the ends has a fundamental frequency of 390 Hz in air. If 1/4th of the tube is immersed vertically in water the fundamental frequency of air column is
A square ABCD of side 1 mm is kept at distance 15 cm in front of the concave mirror as shown in the figure. If focal length of the mirror is 10 cm, then the length of the perimeter of its image will be ___mm.
In the given situation, radius of disc is\[\sqrt{3}b\]and distance of point charge from the disc is b. The ratio of electric flux not going through the disc and electric flux of charge through the disc is x: 1. The value of x is ___.
In Youngs double slit experiment, slits are separated by 2 mm and the screen is placed at a distance of 1.2 m from the slits. Light consisting of two wavelengths \[\text{6500}\overset{\text{o}}{\mathop{\text{A}}}\,\] and \[\text{5200}\overset{\text{o}}{\mathop{\text{A}}}\,\] are used to obtain interference fringes. The separation between the fourth bright fringes of two different patterns produced by the two wavelengths is ___ mm.
When a light of photons of energy 4.2 eV is incident on a metallic sphere of radius 10 cm and work function 2.4 eV, photoelectrons are emitted. The number of photoelectrons liberated before the emission is stopped, is \[1.25\times {{10}^{x}}\]. The value of x is ___. \[(e=1.6\times {{10}^{-19}}C\,\text{and}\frac{1}{4\pi {{\varepsilon }_{0}}}=9\times {{10}^{9}}N{{m}^{2}}{{C}^{-2}}\])
Water is in streamline flow along a horizontal pipe with non-uniform cross-section. At a point in the pipe where the area of cross- section is \[10c{{m}^{2}}\], the velocity of water is \[1m\,{{s}^{-1}}\]and the pressure is 2000 Pa. The pressure at another point where the cross- sectional area is \[5c{{m}^{2}}\]is ____ Pa.
A system is said to possess \[3N\] (where N particles are bound together to form a molecule) independent degree of freedom. Then the \[3N\] coordinates and components of motion are conveniently chosen as, translational (trans), rotational (rot) and vibrational (vib) motions. The total average energy contribution due to all the three modes for a linear miecule is
I. \[Xe{{F}_{6}}+{{H}_{2}}O\xrightarrow[hydrolysis]{Partial}(A)+2HF\]
II. \[Xe{{F}_{6}}+2{{H}_{2}}O\xrightarrow[hydrolysis]{Partial}(B)+4HF\]
III. \[Xe{{F}_{6}}+3{{H}_{2}}O\xrightarrow[hydrolysis]{Complete}(C)+6HF\]
IV. \[Xe{{F}_{4}}+12{{H}_{2}}O\xrightarrow[hydrolysis]{Complete}(D)+4Xe+24HF+3{{O}_{2}}\]
Which of the following statements is INCORRECT?
A)
Product has \[s{{p}^{3}}{{d}^{2}}\]hybridisation and is square pyramid in shape.
doneclear
B)
Product has \[s{{p}^{3}}d\] hybridisation and is folded square in shape.
doneclear
C)
Products and are same, having \[s{{p}^{3}}\]hybridisation and is pyramidal in shape.
doneclear
D)
Products and are different, having \[s{{p}^{3}}\] and \[s{{p}^{3}}d\] hybridisation with tetrahedral and (distorted /irregular) tetrahedral shapes respectively.
A compound made of particles A, B, and C forms ccp lattice. Ions A are at lattice points, B occupy TVs and C occupy OVs. If all the ions along one of the edge axis are removed, then the formula of the compound is
If 80% of a radioactive element undergoing decay is left over after a certain period of time t from the start, how many such periods should elapse from the start for just over 50% of the element to be left over?
Optically pure \[(+)-2\]-chlorooctane, \[[\alpha ]=+40{}^\circ ,\] reacts with aq. \[NaOH\] in acetone to give optically pure \[(-)-2\]-octanol, \[[\alpha ]=-12.0{}^\circ \]. With partially racemised chloro compound whose \[[\alpha ]=+30{}^\circ ,\]the \[[\alpha ]\] of alcohol product is\[-6.0{}^\circ \]. The percentage of front side and back side attacks are, respectively,
For 109% labelled oleum, if the number of moles of \[{{H}_{2}}S{{O}_{4}}\]and free \[S{{O}_{3}}\] be p and q respectively, then what will be the value of \[\frac{p-q}{p+q}\]?
Consider the electrode: \[Ag|AgCl(s),\] \[C{{l}^{\bigcirc -}}(0.1m),\] i.e., silver electrode in contact with \[0.1\text{ }M\text{ }KCl\]solution saturated with\[AgCl\]. If it is combined with the electrode \[Ag|A{{g}^{\oplus }}(0.1M)\] to form a complete cell, the EMF would be (\[{{K}_{sp}}\] of \[AgCl={{10}^{-10}}\] at\[25{}^\circ C\])
In hydrogen atom an orbit has a diameter of about\[16.92\text{ }\overset{o}{\mathop{A}}\,\]. The maximum number of electron that can accommodated is __________.
Two liquids A and B have vapour pressures in the ratio of \[{{P}_{A}}{}^\circ :{{P}_{B}}{}^\circ =1:2\] at a certain temperature. Suppose we have an ideal solution of A and B in the mole fraction ratio \[A:B=1:2\]. The mole fraction of A in the vapour phase in equilibrium with the solution at a given temperature is __________.
If the coefficient of 4th term in the expansion of \[{{\left( x+\frac{\alpha }{2x} \right)}^{n}}\]is 20, then the respective values of \[\alpha \]and n are
If the roots of the quadratic equation \[{{x}^{2}}+px+q=0\] are \[\tan 30{}^\circ \] and \[\tan 15{}^\circ \] respectively, then the value of \[2+q-p\]is
Let C be the circle with centre \[(0,0)\] and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of \[\frac{2\pi }{3}\] at its center is
The length of the perpendicular from the origin to a line is 7 and line makes an angle of \[150{}^\circ \] with the positive direction of y-axis, then the equation of the line is
ABC is triangular park with \[AB=AC=100m\] A clock tower is situated at the mid-point of BC. The angles of elevation of the top of the tower at A and B are \[{{\cot }^{-1}}3.2\] and \[\cos e{{c}^{-}}^{1}\,2.6\] respectively. The height of the tower is
If the vectors \[\overrightarrow{AB}=-3\hat{i}+4\hat{k}\] and \[\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}\] are the sides of a triangle ABC, then the length of the median through A is
Let \[f(x)=\left\{ \begin{align} & {{x}^{p}}\sin \frac{1}{x},x\ne 0 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,x=0 \\ \end{align} \right.\] then f(x) is continuous but not differentiable at \[x=0\] if
If \[\Delta (x)=\left| \begin{matrix} {{e}^{x}} & \sin x \\ \cos x & In\,(1+{{x}^{2}}) \\ \end{matrix} \right|,\]then the value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\Delta (x)}{x}\] is
A box contains two white balls, three black balls and four red balls. The number of ways such that three balls can be drawn from the box if at least one black ball is to be included in the draw is