A point object O is placed in front of a transparent slab at a distance \[x\]from its closer surface. It is seen from the other side of the slab by light incident nearly normally to the slab. The thickness of the slab is t and its refractive index is \[\mu \]. The apparent shift in the position of the object is independent of \[x\] then its value, is
A telescope has an objective lens of 10 diameter and is situated at a distance of 1 km from two objects. The minimum distance between these two objects, which can be resolved by the telescope, when the mean wavelength of light is \[5000\,\overset{o}{\mathop{\text{A}}}\,\], is of the order of
A point performs simple harmonic oscillation of period T and the equation of motion is given by \[x=a\,\sin \,\left( \omega t+\frac{\pi }{6} \right)\].After the elapse of what fraction of the time period, the velocity of the point will be equal to half of its maximum velocity?
Two points are located at a distance of 10 m and 15 m from the source of oscillation. The period of oscillation is 0.05 s and the velocity of the wave is \[300\,\,m{{s}^{-1}}\]. What is the phase difference between the oscillations of two points?
A fireman of mass 60 kg slides down a pole. He is pressing the pole with a force of 600 N. The coefficient of friction between the hands and the pole is 0.5, with what acceleration will the fireman slide down? (\[g=10\,\,m{{s}^{-2}}\])
A tuning fork of frequency 340 Hz is vibrated just above the tube of 120 cm height. Water is poured slowly in the tube. What is the minimum height of water necessary for the resonance? (Speed of sound in air \[=340\,\,m{{s}^{-1}}\])
A train moving with \[20\,\,m{{s}^{-1}}\] towards a stationary observer produces frequency of 440 Hz. The apparent frequency heard will be (\[v=330\,\,m{{s}^{-1}}\])
In Young's experiment, the distance between two slits is halved and the distance between the screen and slit is made three times. Then width of the fringe
A Light Emitting Diode (LED) has a voltage drop of 2 V across it and passes a current of10 mA. When it operates with a 6 V batery through a limiting resistor R, the value of R is
The gravitational field in a region is given by\[E=(10\,N/kg)\,\,(i+j)\]. The work done by an external agent to slowly shift a particle of mass 2 kg from the point (0, 0) to a point(5m, 4m)
At \[t=0\], three particles A, Band Care located at the origin of the coordinate system. Then they start moving simultaneously, A moves with a constant velocity of 3i (m/s) and B moves under a constant acceleration of 2k (m/s2) with an initial velocity of 8j (m/s). Particle C moves with constant velocity v0 in such a way that B and C collide at t =4s. Then,
A)
v0 is 8j + 4k
doneclear
B)
position vector of location where two particles collide is 16i + 32k
doneclear
C)
Both [a] and [b] are correct
doneclear
D)
it is not possible that B and C collide with each other for any value of v0
A particle of mass 1 kg is released from rest at origin to move ina conservative force field. The variation of potential energy corresponding to this conservative force with X-coordinate is as shown in the figure. Mark out the correct statement(s) for this situation.
A)
Particle is moving along +ve X-axis
doneclear
B)
Particle may move along+ve X-axis
doneclear
C)
Particle may move along -ve X-axis
doneclear
D)
When particle crosses x =-2 m, the speed of particle is \[\sqrt{60}\,m/s\]
A time varying magnetic field is present in a cylindrical region of radius R as shown in the figure. A positive charge q is taken slowly from P to Q \[via\]two paths-first through POQ second through PAQ. Work done by external agent in 1stpath is W1 and through 2nd path is W2, then \[\frac{{{W}_{1}}}{{{W}_{2}}}\] is
(To compute \[\frac{{{W}_{1}}}{{{W}_{2}}}\], various distances and time variation of B (magnetic field) must be known.)
Direction: For the following questions choose the correct answers from the codes [a], [b], [c] and [d] defined as follows.
Statement I Xenon-139 has a half-life of 41 s, and is produced at a constant rate during the fission of a particular sample of U-235. (Assume that number xenon-139 atom escape d from the sample). Then in this situation, number of nuclei of xenon-139 atom becomes constant after a certain time.
Statement II Half-life of U-235 to decay to xenon-139 is very near to 41 s.
A)
Statement I is true. Statement II is also true and Statement II is the correct explanation of Statement I.
doneclear
B)
Statement I is true. Statement n is also true and Statement II is not the correct explanation of Statement I.
Direction: For the following questions choose the correct answers from the codes [a], [b], [c] and [d] defined as follows.
Statement I Systematic errors have same size and sign for all measurements.
Statement II While computing systematic error in some physical quantity, say X, which AB depends on other physical quantities, say A, B and C, by the expression \[X=\frac{AB}{{{C}^{2}}}\], we won't consider the worst case.
A)
Statement I is true. Statement II is also true and Statement II is the correct explanation of Statement I.
doneclear
B)
Statement I is true. Statement n is also true and Statement II is not the correct explanation of Statement I.
Direction: Light having photons energy \[hv\] is incident on a metallic plate having work function \[\phi \] to eject the electrons.
The most energetic electrons are then allowed to enter in a region of uniform magnetic field B as shown in the figure.
The electrons are projected in XZ-plane making an angle of\[\theta \] with X-axis and magnetic field is \[\mathbf{B}={{B}_{0}}\,i\] along X-axis. Maximum pitch of the helix described by electron is found to be p. Take mass of electron as m and charge as q. Based on above information, answer the following questions
The correct relation between P and \[{{B}_{0}}\] is
Direction: Light having photons energy \[hv\] is incident on a metallic plate having work function \[\phi \] to eject the electrons.
The most energetic electrons are then allowed to enter in a region of uniform magnetic field B as shown in the figure.
The electrons are projected in XZ-plane making an angle of\[\theta \] with X-axis and magnetic field is \[\mathbf{B}={{B}_{0}}\,i\] along X-axis. Maximum pitch of the helix described by electron is found to be p. Take mass of electron as m and charge as q. Based on above information, answer the following questions
Considering the instant of crossing origin at \[t=0,\] the Z-coordinate of location of electron as a function of time is
Direction: Light having photons energy \[hv\] is incident on a metallic plate having work function \[\phi \] to eject the electrons.
The most energetic electrons are then allowed to enter in a region of uniform magnetic field B as shown in the figure.
The electrons are projected in XZ-plane making an angle of\[\theta \] with X-axis and magnetic field is \[\mathbf{B}={{B}_{0}}\,i\] along X-axis. Maximum pitch of the helix described by electron is found to be p. Take mass of electron as m and charge as q. Based on above information, answer the following questions
The plot between X-coordinates of location of electron as a function of time for different frequencies v of the incident light, is
The nucleus of an atom is located at\[x=y=z\]. If the probability of finding an electron of s-orbital in a tiny volume around \[x=a,\,\,y=z=0\], then what is the probability of finding the electron in the same sized volume around \[x=z=0,\,\,y=a\]?
Direction: The dissociation of complex may be expressed as \[{{[M{{L}_{x}}]}^{4+}}\,+xL\] and equilibrium constant of this is known as instability constant which is a measure of stability. The stability of complex depends on EAN of central atom, charge on metalion, basic nature of ligand and chelation.
The EAN of \[Co\] in \[Co{{(CO)}_{4}}\] is 35 and it is less stable. It attains stability by
Direction: The dissociation of complex may be expressed as \[{{[M{{L}_{x}}]}^{4+}}\,+xL\] and equilibrium constant of this is known as instability constant which is a measure of stability. The stability of complex depends on EAN of central atom, charge on metalion, basic nature of ligand and chelation.
Which one of the following does not follow EAN rule?
Direction: The dissociation of complex may be expressed as \[{{[M{{L}_{x}}]}^{4+}}\,+xL\] and equilibrium constant of this is known as instability constant which is a measure of stability. The stability of complex depends on EAN of central atom, charge on metalion, basic nature of ligand and chelation.
Which complex is most stable where \[{{K}_{d}}\] is un stability constant?
\[\frac{{{N}_{0}}}{2}\] atoms of \[X(g)\] are converted into \[{{X}^{+}}(g)\] by energy \[{{E}_{1}}\cdot \frac{{{N}_{0}}}{2}\] atoms of \[X(g)\] are converted into \[{{X}^{-}}(g)\]by energy \[{{E}_{2}}\]. Hence, ionization potential and electron affinity of \[X(g)\] are
If \[\Delta G=\Delta H-T\,\,\Delta S\]and \[\Delta G=\Delta H+T\,\,{{\left[ \frac{d(\Delta G)}{dT} \right]}_{P}},\] then variation of EMF of a cell E, with temperature T, is given by
Using the following Latimer diagram for bromine,\[pH=0;\,\,Br{{O}_{2}}\xrightarrow{1.82\,V}\,BrO_{3}^{-}\xrightarrow{1.50\,V}\,\] \[HBr{{O}_{2}}\xrightarrow{1.595\,V}\,B{{r}_{2}}\xrightarrow{1.652\,V}B{{r}^{-}}\] The species undergoing dispropotionation is
An organic compound of molecular formula \[{{C}_{4}}{{H}_{6}},\] , forms precipitates with ammonia cal silver nitrate and ammonia cal cuprous chloride, 'A' has an isomer 'B' one mole of which reacts with one mole of \[B{{r}_{2}}\] to form 1, 4-dibromo-2-butene. 'A and 'B are
A)
\[C{{H}_{3}}-C{{H}_{2}}-C\equiv CH\] and \[C{{H}_{2}}=CH-CH=C{{H}_{2}}\]
doneclear
B)
\[C{{H}_{3}}-C\equiv C-C{{H}_{3}}\] and \[C{{H}_{3}}-CH=C=C{{H}_{2}}\]
The surface tension of several alcohols at \[20{}^\circ C\] is \[\gamma (C{{H}_{3}}OH)=22.61dyne\,\,c{{m}^{-1}},\]\[\gamma ({{C}_{2}}{{H}_{5}}OH)=2.275\times {{10}^{-2}}N{{m}^{-1}}\] and \[\gamma ({{C}_{3}}{{H}_{7}}OH)=23.78\,\,J{{m}^{-2}}\]. The alcohol having the highest surface tension is
Based on the given diagram, which of the following statements regarding the solutions of two miscible volatile liquids are correct?
1. Plots AD and BC show that Raoult's law is obeyed for the solution in which B is a solvent and A is the solute and as well as for that in which A is solvent and B is solute.
2. Plot CD shows that Dalton's law of partial pressure is observed by the binary solution of components A and B.
3.\[EF+EG+EH\]; and AC and BD corresponds to the vapour pressure of the pure solvents A and B respectively.
Select the correct answer using the codes given below.
Calculate the standard reduction potential for the reaction, \[{{H}_{2}}O+{{e}^{-}}\to \,\frac{1}{2}\,{{H}_{2}}+O{{H}^{-}}\], using Nernst equation and the fact that the standard reduction potential for the reaction; \[{{H}^{+}}+{{e}^{-}}\to \frac{1}{2}\,{{H}_{2}}\], is by definition equal to 0.00 V at \[25{}^\circ C\].
For the reaction,\[{{[Cu{{\left( N{{H}_{3}} \right)}_{4}}]}^{2+}}+{{H}_{2}}O\xrightarrow{\,}\,{{[Cu{{(N{{H}_{3}})}_{3}}{{H}_{2}}O]}^{2+}}\]\[+N{{H}_{3}}\]the net rate is \[\frac{dx}{dt}=2.0\times {{10}^{-4}}{{s}^{-1}}\]\[\,{{[Cu{{(N{{H}_{3}})}_{4}}]}^{2+}}-3.0\times {{10}^{5}}\,L\,mo{{l}^{-1}}\,{{s}^{-1}}\] \[{{[Cu{{(N{{H}_{3}})}_{3}}{{H}_{2}}O]}^{2+}}\,[N{{H}_{3}}]\] Then the ratio of rate constants of the forward and backward reactions is
In an adsorption experiment, a graph between \[\log \frac{x}{m}\] versus log p was found to be linear with a slope of \[{{45}^{o}}\]. The intercept on the \[\log \frac{x}{m}\] axis was found to be 0.30103. Calculate the amount of the gas adsorbed per gram of charcoal under a pressure 1 atm. \[[{{\log }_{10}}2=0.301032]\]
The hair dyes available in the market generally contain two bottles, one containing the dye and the other \[{{H}_{2}}{{O}_{2}}\]. Before applying the dye, the two solutions are mixed. The \[{{H}_{2}}{{O}_{2}}\]
Let \[{{x}^{2006}}{{y}^{2007}}{{z}^{2008}},\]\[{{x}^{2007}}{{y}^{2008}}{{z}^{2009}},\]\[{{x}^{2008}}{{y}^{2009}}{{z}^{2010}}\] be in AP, where, \[x,\,\,y,\,\,z>0\]. Te least value of \[x+y+z\] is
Ramya has 4 different toys and Saumya has 7 different toys. The number of ways in which they can exchange their toys, so that each keep her initial number of toys are
A bag contains 10 black and 3 white balls. Balls are drawn one-by-one with out replacement till all the white balls are drawn. The probability that the procedure of drawing balls will come to an end at the seventh draw is
The maximum number of solutions of the equation\[5\,\,\cos \,x+\frac{5}{2\,\cos \,x}-5\,=2\,{{\cos }^{2}}\,x+\frac{1}{2\,{{\cos }^{2}}x},\]\[x\,\in [0,\,2\pi ]\]
A circle concentric to the ellipse \[\frac{4{{x}^{2}}}{289}+\frac{{{y}^{2}}}{{{k}^{2}}}=1,\] where \[k<\frac{17}{2},\] passes through foci \[{{S}_{1}}\] and\[{{S}_{2}}\] of the ellipse and cuts ellipse at point P. If area of \[\Delta P{{S}_{1}}{{S}_{2}}\] is 35 sq units, then \[{{S}_{1}}{{S}_{2}}\] is equal to
Let a variable line has its intercepts on the coordinate axes, respectively as e,e' where \[\frac{e}{2}\]and \[\frac{e'}{2}\] are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle \[{{x}^{2}}+{{y}^{2}}={{a}^{2}},\] where a is equal to
Let the length of the tangent drawn from a variable point to one. given circle is\[\mu \,(\mu \ne 1)\]times the length of the tangent from it to another circle. The locus of the variable point is
Let \[0\le \theta <\frac{\pi }{2},\]\[\le \phi <\frac{\pi }{2}\] and \[0\le \psi <\frac{\pi }{2}\] and \[\theta +\psi =\frac{\pi }{2}\]. The product \[\tan \,\theta \,\,\tan \,\phi \,\,\tan \,\psi \] attains the greatest value when
A)
\[\theta =\phi =\psi \]
doneclear
B)
\[\theta =\frac{\pi }{2}\]
doneclear
C)
\[\phi =\psi =\pi \]
doneclear
D)
any one of \[\theta ,\,\,\phi \,\,\psi \] is \[\frac{\pi }{3}\]
Two straight lines \[{{L}_{1}}=0\] and \[{{L}_{2}}=0\] pass through the origin forming an angle of \[{{\tan }^{-1}}\left( \frac{7}{9} \right)\]with each other. If the ratio of the slopes of \[{{L}_{2}}=0\] and \[{{L}_{1}}=0\] is \[\frac{9}{2}\],then their equations are
The minimum and maximum distances of a point (3, 12) from the ellipse\[25{{x}^{2}}+9{{y}^{2}}-150x-90y+225=0\] are \[a\] and b, then a and b satisfy the relation.
The number of solution of the equation is greater than or equal to\[{{2}^{\cos \,x}}=\,|\,\sin \,x|\] in \[[-2\pi ,\,\,2\pi ]\] is greater than or equal to
In a class of 30 pupils, 12 take needle work 16 take Physics and 18 take History. If all the30 students take at least one subject and no one take all three, then the number of pupils taking 2 subjects is
The lines \[\frac{x-3}{1}=\frac{y-}{2}=\frac{z-3}{-\lambda }\] and \[\frac{x-1}{\lambda }=\frac{y-3}{3}=\frac{z-1}{4}\] are coplanar. Then, the number of possible value(s) of \[\lambda \] is
Direction: A straight line will touch a given conic if there is only one point of intersection of the line and the given conic. If the conic is specified by quadratic equation in\[x\] and \[y,\] then the straight line will touch if the discriminant of the equation obtained by the elimination of one of the variable is zero. Let us consider parabola \[{{y}^{2}}=8x\] and an ellipse\[15{{x}^{2}}+4{{y}^{2}}=60\].
The equation of a tangent common to both the parabola and the ellipse is
Direction: A straight line will touch a given conic if there is only one point of intersection of the line and the given conic. If the conic is specified by quadratic equation in\[x\] and \[y,\] then the straight line will touch if the discriminant of the equation obtained by the elimination of one of the variable is zero. Let us consider parabola \[{{y}^{2}}=8x\] and an ellipse\[15{{x}^{2}}+4{{y}^{2}}=60\].
The equation of the normal at the point of contact of the common tangent which makes an acute angle with the positive direction of \[x-\]axis to the parabola is