Let \[g\left( x \right)=\int_{0}^{{{\left| x \right|}^{3/4}}}{{{t}^{2/3}}}\sin \frac{1}{t}dt,\] for all real x, then \[_{x\to 0}^{\lim }\frac{g\left( x \right)}{x}\] is equal to
A circle is inscribed into a rhombus ABCD with one angle \[60{}^\circ \]. The distance from the centre of the circle to the nearest vertex is equal to 1. If P is any point of the circle, then \[{{\left| PA \right|}^{2}}+{{\left| PB \right|}^{2}}+{{\left| PC \right|}^{2}}+{{\left| PD \right|}^{2}}\] is equal to
A parabola \[y=a{{x}^{2}}+bx+c\] crosses the X-axis at \[\left( \alpha +0 \right)\], \[\left( \beta +0 \right)\] both to the right of the origin. A circle also passes through those two points. The length of a tangent from the origin to the circle is
Let [n] denotes the greatest integer less than or equal to \[\lambda \]. How many real numbers x satisfy the equation \[{{x}^{2}}+10000\left[ x \right]=10000x\]
In a \[\Delta XYZ\], let a, b and c be the lengths of the sides opposite to the angles X, Y and Z, respectively If \[1+\cos 2X-2\cos 2Y=2\sin X\sin Y\], then \[\frac{a}{b}\] is equal to
Let the harmonic mean of two positive real numbers a and b be 4. If q is positive real number such that a,5,q,b is an arithmetic progression, then the values of q is equal to
Let a, b and c be three non-coplaner unit vectors such that the angle between every pair of them is \[\frac{\pi }{3}\]. If \[a\times b\times b\times c=pa+qb+rc\] where p, q and r are scalars, then the value of \[\frac{{{P}^{2}}+2{{q}^{2}}+{{r}^{2}}}{{{q}^{2}}}\]
The ellipse \[{{E}_{1}}=\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1\] is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse \[{{E}_{2}}\] passing through the point \[(0,4)\] circumscribes the rectangle R. The eccentricity of ellipse \[{{E}_{2}}\] is
Let f be a non-negative function defined on the interval \[\left[ 0,1 \right]\]. If \[\int_{0}^{x}{\sqrt{1-{{\left( f'\left( t \right) \right)}^{2}}}dt=\int_{0}^{x}{f(t)dt}}\] \[0\le x\le 1\] and \[f\left( 0 \right)=0\], then
The particle of mass m and charge q will touch the infinitely large plate of uniform charge density \[\sigma \] if its velocity v is more than: (Given that \[\sigma \,q>0\])
A negative charge is given to a loop and the loop is rotated in the plane of paper about its centre as shown. The magnetic field produced by the ring affects a small magnet placed above the ring in the same plane of paper.
A)
The magnet does not rotate
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B)
The magnet rotates clockwise as seen by observer from below
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C)
The magnet rotates anti-clockwise as seen from below
An inelastic ball of mass m has been thrown vertically upwards (positive z-direction) from the ground at \[z=0.\]Linear momentum of ball is \[{{P}_{z}}.\] The phase trajectory (graph between \[{{P}_{z}}\] versus z) of the ball after successive bouncing on the ground is-
In a cylindrical region uniform magnetic field which is perpendicular to the plane of the figure is increasing with time and a conducting rod PQ is placed in the region. Then-
A)
P will be at higher potential than Q.
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B)
Q will be at higher potential than P.
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C)
Both P and Q will be equipotential
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D)
No emf will be developed across rod as it is not crossing / cutting any line of force
In the plane mirror, the co-ordinates of image after two and half time periods are (initial velocity \[{{V}_{0}}\] is in the xy-plane and the plane mirror is perpendicular to the x-axis. A uniform magnetic field \[B\hat{i}\] exists in the whole space. \[{{P}_{0}}\] is pitch of helix, \[{{R}_{0}}\] is radius of helix).
In a YDSE both slits produce equal intensities on the screen. A 100 % transparent thin film is placed in front of one of the slits. Now the intensity of the geometrical centre of system on the screen becomes 76 % of the previous intensity. The wavelength of the light is 6000\[\overset{\text{o}}{\mathop{\text{A}}}\,\] and \[{{\mu }_{fjlm}}=1.5.\] The thickness of the film cannot be -
A beam of electrons striking a copper target produces x-rays. Its spectrum is as shown. Keeping the voltage same if the copper target is replaced with a different metal, the cut-off wavelength and characteristic lines of the new spectrum will change in comparison with old as -
A)
Cut-off wavelength will remain unchanged while characteristic lines will be different.
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B)
Both cut-off wavelength and characteristic lines will remain unchanged.
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C)
Both cut-off wavelength and characteristic lines will be different.
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D)
Cut-off wavelength will different while characteristic lines will remain unchanged.
Two point charges \[+\,q\] and \[-\,4q\] are placed at \[(-\,a,0)\] and \[(+\,a,\,\,0).\] Take electric field intensity to be positive if it is along positive x-direction. The variation of the electric field intensity as one moves along the x-axis is -
If at t = 0 the switch \[{{S}_{W}}\]is closed, then the charge on capacitor in the given circuit (initially uncharged) when the current through battery becomes 50 % of its maximum value is (assume battery is ideal).
Two rods of same length and areas of cross section \[{{A}_{1}}\] and \[{{A}_{2}}\] have their ends at same temperature. If \[{{K}_{1}}\] and \[{{K}_{2}}\] are their thermal conductivities, \[{{C}_{1}}\] and \[{{C}_{2}}\] their specific heats and \[{{\rho }_{1}}\] and \[{{\rho }_{2}}\] are their densities, then the condition that rate of flow of heat is same in both the rods is -
A chord attached about an end to a vibrating fork 'divides it into 6 loops, when its tension is 36 N. The tension at which it will vibrate in 4 loops is -
Water flows through a frictionless horizontal duct with a cross-section varying as shown in figure. Pressure P at points along the axis is represented by -
Figure shows three vectors \[\vec{a},\] \[\vec{b}\] and \[\overrightarrow{c}\]. If \[\vec{c}\]\[RQ\text{ }=\text{ }2PR,\] which of the following relations is correct?
A pebble is thrown horizontally from the top of a 20 m high tower with an initial velocity of 10 m/s. The air drag is negligible. The speed of the pebble when it is at the same distance from top as well as base of the tower \[(g=10\,m/{{s}^{2}})\]
Two bodies A and B have emissivities 0.5 and 0.8 respectively. At some temperatures the two bodies have maximum spectral emissive powers at wave length 8000\[\overset{\text{o}}{\mathop{\text{A}}}\,\] and 4000\[\overset{\text{o}}{\mathop{\text{A}}}\,\] respectively. The ratio of their emissive powers at these temperatures are -
\[{{\mathbf{S}}_{\mathbf{1}}}:\] An uncharged conductor kept near a charged body is attracted by that body whether the charge on the body is positive or negative.
\[{{\mathbf{S}}_{\mathbf{2}}}:\] A solid conductor is placed in a uniform electric field. The electric field due to conductor will be same at every point inside the conductor.
\[{{\mathbf{S}}_{\mathbf{3}}}\mathbf{:}\] The electric field produced by an infinitely large sheet is same on both sides.
\[{{\mathbf{S}}_{\mathbf{4}}}\mathbf{:}\] Intensity of electric field decreases as you go away from the centre of a uniformly charged solid sphere (having uniform volume charge density). State, in order, whether \[{{S}_{1}},\,\,{{S}_{2}},\,\,{{S}_{3}},\,\,{{S}_{4}}\]are true or false.
The increasing order of the boiling points for the following compounds is: \[\begin{align} & {{\operatorname{C}}_{2}}{{H}_{5}}OH{{C}_{2}}{{H}_{5}}Cl{{C}_{2}}{{H}_{5}}C{{H}_{3}}{{C}_{2}}{{H}_{5}}OC{{H}_{3}} \\ & \,\,\,\,(I)\,\,\,\,(II)\,\,\,(III)\,\,\,\,(IV) \\ \end{align}\]
Ice and water are placed in a closed container at a pressure of 1 atm and temperature 273.15 K. If pressure of the system is increased to 2 atm while keeping temperature constant, which of the following would be the correct observation?
Gradual addition of potassium iodide solution to \[\operatorname{Bi}{{\left( N{{O}_{3}} \right)}_{3}}\]solution initially produces a dark brown precipitate which dissolves in excess of KI to give a clear yellow solution. Identify the yellow precipitate.
The average life of an excited state of hydrogen atom is of the order of \[{{10}^{-8}}s\]. The number of revolutions made by an electron when it is in state n = 2 and before it suffers a transition to state n= 1, are
Pure water freezes at 273 K and 1 bar. The addition of 34.5 g of ethanol to 500 g of water changes the freezing point of the solution. Use the freezing point depression constant of water as 2 K kg \[{{\operatorname{mol}}^{-1}}\]. The figures shown below represent plots of vapour pressure (V.P.) versus temperature (T). molecular weight of ethanol is 46 g \[{{\operatorname{mol}}^{-1}}\]] Among the following, the option representing change in the freezing point is
The major products obtained from the following sequence of reactions are: \[\left( C{{H}_{3}} \right) CHC{{H}_{2}}N {{(C{{H}_{2}}C{{H}_{3}})}_{2}}\xrightarrow{C{{H}_{3}}I}\] \[\xrightarrow[{{H}_{2}}O]{A{{g}_{2}}O}\xrightarrow{heat}products\]
During the process of digestion, the proteins present in food materials are hydrolysed to amino acids. The two enzymes involved in the process\[\Pr oteins\xrightarrow{Enzyme(A)}Polypeptides\]\[\xrightarrow{Enzyme(B)}A\min o\,acids\] are:
To an aqueous solution containing anions a few drops of acidified \[KMn{{O}_{4}}\] are added. Which one of the following anions, if present will not decolourise the \[KMn{{O}_{4}}\] solution?
The rate constant, the activation energy and the Arrhenius parameter of a chemical reaction at \[25{}^\circ C\] are \[3.0\times 1{{0}^{-4}}{{s}^{-1}},\]\[104.4{{\operatorname{kJmol}}^{-1}}\] 'and \[6.0\times 1{{0}^{14}}{{s}^{-1}}\]respectively. The value of the rate constant as\[\operatorname{T}\to \infty \] is
Let A, B, C be the angles of \[\Delta ABC\] with vertex \[A\left( 4,-1 \right)\] and \[x-1=0\] and \[x-y=1\]are internal angle bisectors through B and C respectively. Let D, E, F be points of contact of sides BC. CA and AB with incircle of \[\Delta ABC\]. If D?, E?, F' are images of D, E and F in internal angle bisector of A, B, C, then equation of circumcircle of \[\Delta D'\,E'\,F\] is
Initially there is 50 gm of salt in tank with 100 L of water present. A liquid at rate of 5 L/min with 2 gm/L of salt is coming into tank. After proper mixing in tank it is running out with 4l/min. The amount of salt present in tank after time t=100 min is
Let f(x) be a twice differentiable function all real values of x and satisfies \[f\left( 1 \right)=1,f\left( 2 \right)=4,f\left( 3 \right)=9\] then which of the following is definitely true?
A)
\[f''\left( x \right)=2,x\in \left( 1,3 \right)\]
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B)
\[f''\left( x \right)=f'\left( x \right)=5,\,\] for some \[x\in \left( 2,3 \right)\]
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C)
\[f''\left( x \right)=3,\]\[x\in \left( 2,3 \right)\]
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D)
\[f''\left( x \right)=2\] for some \[x\in \left( 1,3 \right)\]
Two blocks of masses \[{{m}_{1}}=1\,\,kg\] and \[{{m}_{2}}=2\,\,kg\] are connected by a non-deformed light spring. They are lying on a rough horizontal surface. The coefficient of friction between the blocks and the surface is 0.4. What minimum constant force F has to be applied in horizontal direction to the block of mass mi in order to shift the other block? \[(g=10m/{{s}^{2}})\]
A hemisphere of radius R and of mass \[4\,\,m\]is free to slide with its base on a smooth horizontal table. A particle of mass \[m\] is placed on the top of the hemisphere. The angular velocity of the particle relative to centre of hemisphere at an angular displacement \[\theta \] when velocity of hemisphere has become \[v\] is -
Two particles A and B are situated at a distance d = 2m apart. Particle A has a velocity of 10 m/s at an angle of \[60{}^\circ \] and particle B has a velocity \[v\] at an angle \[30{}^\circ \] as shown in figure. The distance d between A and B the instant shown in figure is constant. The angular velocity of B with respect to A is -
The mirror of length \[2\ell \] makes 10 revolutions per minute about the axis crossing its midpoint \[O\] and perpendicular to the plane of the figure There is a light source in point A and an observer at point B of the circle of radius R drawn around centre \[O\,\,(\angle AOB=90{}^\circ )\] What is the proportion \[\frac{R}{\ell }\] if the observer B sees the light source first time when the angle of mirror \[\psi =15{}^\circ ?\]
The mass per unit length of a non-uniform rod \[OP\] of length L varies as \[m=k\frac{x}{L}\] where \[k\] is a constant and \[x\] is the distance of any point on the rod from end \[O\] The distance of the centre of mass of the rod from end \[O\] is -
A point charge q is placed inside a conducting spherical shell of inner radius 2R and outer radius 3R at a distance of R from centre of the shell. The electric potential at the centre of shell will be.
A positive point charge is placed at P in front of an earthed metal sheet S. Q and R are two points between P and S as shown in figure. If the electric field strength at Q and R are respectively \[{{E}_{Q}}\] and \[{{E}_{R}}\] and potential at Q and R are respectively \[{{V}_{Q}}\]and \[{{V}_{R}},\] then -
A circular current loop is shown in the adjacent figure. The magnetic field in the region is along x-axis and its magnitude in the space is increasing with increasing y-coordinate. The net magnetic force or the loop is -
Molar conductivity of aqueous solution of sodium stearate, which behaves as a strong electrolyte is recorded at varying concentration [C] of sodium stearate. Which one of the following plots provides the correct representation of micelle formation in the solution? (Critical micelle concentration (CMC) is marked with an arrow in the figures)
3 g of activated charcoal was added to 50 mL of acetic acid solution (0.06 N) in a flask. After an hour it was filtered and the strength of the filtrate was found to be 0.042 N. The amount of acetic acid adsorbed (per gram of charcoal) is :
The standard state Gibbs free energies of formation of C(graphite) and C(diamond) at T=298 K are \[{{\Delta }_{f}}{{G}^{0}}\] [C(graphite)] \[=0\text{ }kJmo{{l}^{-1}}\]\[{{\Delta }_{f}}{{G}^{0}}\] [C (diamond)] \[=2.9kJmo{{l}^{-1}}\] The standard state means that the pressure should be 1 bar, and substance should be pure at a given temperature. The conversion of graphite [C (graphite)] to diamond [C (diamond)] reduces its volume by \[2\times {{10}^{-6}}{{m}^{3}}mo{{l}^{-1}}\]. If C (graphite) is converted to C (diamond) isothermally at T = 298 K, the pressure at which C (graphite) is in equilibrium with C (diamond), is [Useful information: 1 J=1 kg \[{{\operatorname{m}}^{2}}{{s}^{-2}};\] Pa = 1 kg \[{{\operatorname{m}}^{-2}}{{s}^{-2}};\]1 bar \[{{10}^{5}}\]pa]
Lithium forms a \[\operatorname{b}.c.c.\] lattice. If the lattice constant is \[3.50\times {{10}^{-10}}m\] and the experimental density is \[5.30\times {{10}^{2}} kg {{m}^{-3}}\], the percentage occupancy of Li metal is (Li = 7).
In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are \[{{\operatorname{S}}_{2}}{{O}_{3}}^{2-}\xrightarrow{A{{g}^{+}}}\underset{\begin{smallmatrix} Clear \\ solution \end{smallmatrix}}{\mathop{X}}\,\xrightarrow{A{{g}^{+}}}\underset{\begin{smallmatrix} white\, \\ preeoitate \end{smallmatrix}}{\mathop{Y}}\,\]\[\xrightarrow{with\,\,time}\underset{black\,preecipitate}{\mathop{Z}}\,\]
If the \[{{\operatorname{IE}}_{1}}\], and \[{{\operatorname{IE}}_{2}}\] values are \[27\text{ }kJmo{{l}^{-1}}\]and \[51\text{ }kJ\,mo{{l}^{-1}}\] respectively, then the value of is \[I{{E}_{2}}\]_____\[kJ\,mo{{l}^{-1}}\]
Boric acid cannot be titrated with \[NaOH\]satisfactorily because as a strong acid. However, on addition by it is a weak acid. But on addition of diols it behaves as a strong acid. However, on addition of which of the following diols, the titration with \[NaOH\] is still not satisfactory on using phenolphthalein as indicator?
The gas phase decomposition of dimethyl ether follows first order kinetics. \[C{{H}_{3}}-O-C{{H}_{3}}(g)\to C{{H}_{4}}(g)+{{H}_{2}}(g)+CO(g)\]The reaction is carried out in a constant volume container at \[500 {}^\circ C\] and has a half-life of 14.5 minutes. Initially, only dimethyl ether is present at a pressure of 0.40 atmosphere. What is the total pressure of the system after 12 minutes? Assume ideal gas behaviour.
Rice has a diploid genome with \[2\,n=24.\] If crossing over is stopped in a rice plant and then selfed seeds are collected, will all the off springs be genetically identical to the parent plant?
A)
yes, because crossing-over is the only source of genetic variation
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B)
no, because stopping of crossing over automatically increases rate of point mutation
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C)
yes, only if the parent plant was a completely inbred line
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D)
yes, only if the parent plant was a hybrid between two pure-bred lines