The equation of the stationary wave is\[y=2A\,\sin \,\left( \frac{2\pi ct}{\lambda } \right)\cos \,\left( \frac{2\pi x}{\lambda } \right)\]. Which of the following statement(s) is wrong?
A)
The unit of \[ct\] is same as that of\[\lambda \].
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B)
The unit of \[x\] is same as that of\[\lambda \].
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C)
The unit of \[2\pi c/\lambda \] is same as that of \[2\pi \times /\lambda t\]
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D)
The unit of \[c/\lambda \] is same as that of\[x/\lambda \].
A particle is projected from ground making an angle \[\theta \] with the horizontal. The value of \[\theta \] for which, at the highest point of its trajectory, kinetic energy of particle will be equal to its potential energy, is
If B is the bulk modulus of a metal and a pressure p is applied uniformly on all sides of the metal with density D, then the fractional increase in density is given by
A water barrel having water up to a depth d is placed on a table of height h. A small hole is made on the wall of the barrel at its bottom. If the stream of water coming out of the hole falls on the ground at a horizontal distance R from the barrel, then the value of d is
If two like charges of magnitude \[1\times {{10}^{-9}}C\] and \[9\times {{10}^{-9}}C\] are separated by a distance of 1 m, then the point on the line joining the charges, where the force experienced by a charge placed at that point is zero, is
Flux \[\phi \] (in weber) in a closed circuit of resistance \[10\,\Omega \] varies with time t (in seconds) according to the equation\[\phi =6{{t}^{2}}-5t+1\] . The magnitude of the induced current in the circuit at \[t=0.25\,s\] is
A 1 m long mirror is placed at a distance 4 m from a tall building as shown in figure. What height of the building can be seen in the mirror from a point \[O\] at a distance 2 m from the mirror?
A block of pure silicon at 300 K has a length of 10 cm and an area of\[1.0\times {{10}^{-4}}\,{{m}^{2}}\]. If a battery of \[emf\] 2 V is connected across it, what is the electron-current? The mobility of electrons is \[0.14{{m}^{2}}{{V}^{-1}}{{s}^{-1}}\] and their number density is\[1.5\times {{10}^{16}}{{m}^{-3}}\].
A ball collides elastically with another ball of the same mass. The collision is oblique and initially one of the balls was at rest. After the collision, the two balls move with same speeds. What will be the angle between the velocity of the balls after the collision?
A cubical block of mass m and edge a slides down a rough inclined plane of inclination \[\theta \] with a uniform velocity. The torque of the normal force on the block about its centre has a magnitude
We have two (narrow) capillary tubes \[{{T}_{1}}\] and \[{{T}_{2}}\]. Then lengths are \[{{l}_{1}}\] and \[{{l}_{2}}\] and radii of cross-sections are \[{{r}_{1}}\] and \[{{r}_{2}}\] respectively. The rate of flow of water under a pressure difference \[p\] through the tube \[{{T}_{1}}\] is\[8\,\,c{{m}^{3}}/s\]. If \[{{l}_{1}}=2{{l}_{2}}\] and \[{{r}_{1}}={{r}_{2}}\]. What will be the rate of flow when the two tubes are connected in series and pressure difference across the combination is same as before (=p)?
The elastic flux \[\phi \] through a hemisphere surface of radius R, placed in a uniform electric field of intensity E parallel to the axis of its plane is
If the velocity v of a particle moving along a straight line decreases linearly with its displacement S from \[20\,\,m{{s}^{-1}}\] to a value approaching zero at \[S=30\,m\], then acceleration of the particle at \[S=15\,m\], is
A disc of mass 10 kg is kept floating horizontally by throwing 10 marbles per second against it from below. The marbles strike the disc normally and rebound downwards with the same speed. If the mass of each marble is 5 g, the velocity with which the marbles are strike the disc is \[(g=9.8\,m\,{{s}^{-2}})\]
The displacement \[x\] of a particle of mass m kg moving in one dimension, under the action of a force is related to the time t by the equation \[t=\sqrt{x}+3,\] where x is in metre and t is in second. The work done by the force in the first six seconds in joule is
Two cones A and B are made of two different materials, the density of A being greater than that of B. The height of B is greater than that of A but their base areas and masses are the same. The correct statement about the moment of inertia of the two cones about their axes, is
A)
A will have larger moment of inertia than B
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B)
B will have larger moment of inertia than A
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C)
in such a situation, it is dependent upon the height of the cone, the mass of the cone and radius of the base
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D)
the moment of inertia of the two will be the same as it is not dependent upon height of the cone but depends only upon the mass and the base area
There are some passengers inside a stationary railway compartment. The centre of mass of the compartment itself is\[{{c}_{1}}\], while the centre of mass of the 'compartment plus passengers' system is \[{{c}_{2}}\]. If the passengers move about inside the compartment
A)
both \[{{c}_{1}}\] and \[{{c}_{2}}\] will move w.r.t. ground
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B)
neither \[{{c}_{1}}\] and \[{{c}_{2}}\] will move w.r.t. ground
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C)
\[{{c}_{1}}\] will move but \[{{c}_{2}}\] will be stationary w.r.t. ground
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D)
\[{{c}_{2}}\] will move but \[{{c}_{1}}\], will be stationary w.r.t. ground
If g is the acceleration due to gravity at the surface of the earth, then the energy required to launch a satellite of mass m from the surface of the earth into a circular orbit at an altitude of 2 R where R being the radius of the earth, is
A cubical wooden block of side 10 cm floats at the interface between oil and water with its lower face 2 cm below the interface. If the density of oil is \[0.6\,g\,c{{m}^{-3}}\] the mass of the block is
Springs of constants k, 2k, 4k, 8 k ...2048 k are connected in series. A mass \[m\] is attached to one end and the system is allowed to oscillate. The time period is approximately
A source of sound emitting a note of constant frequency is moving towards a stationary listener, and then recedes from the listener with constant velocity maintained throughout the motion. The frequency head by the listener (f) when plotted against time (f) will give one of the following curve(s)
A system is taken from state A to state B along two different paths 1 and 2. The work done on the system along these two paths are \[{{W}_{1}}\] and \[{{W}_{2}}\], respectively. The heat absorbed by the system along these two paths are \[{{Q}_{1}}\] and \[{{Q}_{2}}\], respectively. The internal energies at A and B are \[{{U}_{A}}\] and \[{{U}_{B}}\] respectively. Then,
A solid rubber ball of density \[\rho \] and radius R falls vertically through air. Assume the air resistance on the ball is \[F=kRv,\] where k is a constant and v is the velocity. Because of this air resistance, the ball attains a constant velocity called terminal velocity \[{{v}_{T}}\] after some time, then \[{{v}_{T}}\] is
\[C{{H}_{3}}C\equiv CH\xrightarrow{NaN{{H}_{2}}}\,A\xrightarrow{C{{H}_{3}}Br}B\xrightarrow[{{H}_{2}}S{{O}_{4}}]{HgS{{O}_{4}}}C\]The number of hydrogen atoms present in one molecule of C
Pressure versus temperature graph of an ideal gas of equal number of moles of different volumes are plotted as shown in figure. Choose the correct alternative.
A)
\[{{V}_{1}}={{V}_{2}}={{V}_{3}}={{V}_{4}}\]
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B)
\[{{V}_{4}}>{{V}_{3}}>{{V}_{2}}>{{V}_{1}}\]
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C)
\[{{V}_{1}}={{V}_{2}},\,\,{{V}_{3}}={{V}_{4}}\] and \[{{V}_{2}}>{{V}_{3}}\]
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D)
\[{{V}_{1}}={{V}_{2}},\,\,{{V}_{3}}={{V}_{4}}\] and \[{{V}_{2}}<{{V}_{3}}\]
For the given reaction\[,\] \[2A(s)+B(g)\,C(g)+2D(s)+E(s)\]. The degree of dissociation of B was found to be 20% at 300 K and 24% at 500 K. The rate of backward reaction
A)
increases with increase in pressure and temperature
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B)
increases with increase in pressure and decrease in temperature
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C)
depends on temperature only and decreases with increase in temperature
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D)
increases with increasing-the concentration of 6 and increasing the temperature
The concentration of \[O{{H}^{-}}\] ion in a solution left after mixing 100 mL of \[0.1\,\,M\,MgC{{l}_{2}}\] and 100 mL of \[0.2\,M\,NaOH\] \[({{K}_{sp}}\,[Mg{{(OH)}_{2}}]=1.2\times {{10}^{-11}})\] is
Two reactions one of first order and other of second order have same values of rate constants (\[{{k}_{1}}\] and \[{{k}_{2}}\]) when concentrations are expressed in \[mol/d{{m}^{3}}\]. If the concentrations are expressed in \[mol/mL,\] the relationship between their rate constants \[{{k}_{1}}\] and \[{{k}_{2}}\] will be
A 0.1 M solution of a certain cation will form a precipitate with 0.1 M solution of all these anions \[O{{H}^{\odot -}},\]\[CO_{3}^{2-},\,C{{l}^{-}},\,SO_{4}^{2-}\]. Which cation fits in this description?
The stop cock connecting the two bulbs of volume 5 L and 10 L containing an ideal gas at 9 atm and 6 atm respectively, is opened. What is the final pressure, if the temperature remains same?
Direction (Q. Nos. 53): A solution of sucrose [M (mass) = 342] has been prepared In dissolving 68.4 g of sucrose in 1 kg of water. \[{{K}_{f}}\] for water is \[1.\,86\,kg\,\,mo{{l}^{-1}}\] and vapour pressure of water at 298 K is 0.024 atm. Assume density of the solution is 1 g/mL.
The vapour pressure of the solution at 298 K will be
Direction (Q. Nos. 54): A solution of sucrose [M (mass) = 342] has been prepared In dissolving 68.4 g of sucrose in 1 kg of water. \[{{K}_{f}}\] for water is \[1.\,86\,kg\,\,mo{{l}^{-1}}\] and vapour pressure of water at 298 K is 0.024 atm. Assume density of the solution is 1 g/mL.
Direction (Q. Nos. 55): A solution of sucrose [M (mass) = 342] has been prepared In dissolving 68.4 g of sucrose in 1 kg of water. \[{{K}_{f}}\] for water is \[1.\,86\,kg\,\,mo{{l}^{-1}}\] and vapour pressure of water at 298 K is 0.024 atm. Assume density of the solution is 1 g/mL.
Direction (Q. Nos. 56): Electrolysis involves electronation and de-electronation al the respective electrodes. Anode of electrolytic cell is the electrode at winch de-electronation takes place whereas at cathode electronation is noticed. If two or more ions of same charge are to be electronated or de-electronated, the ion having lesser discharge potential is discharged. Discharge potential of an ion refers for \[{{E}^{o}}{{\,}_{OP}}\] or \[{{E}^{o}}_{RP}\] as the case may be. The products formed at either electrode is given in terms of Faraday?s law of electrolysis, i.e., \[W=\frac{Eit}{96500}\]
During electrolysis of \[C{{H}_{3}}COONa(aq),\] the mole ratio of gases formed at cathode and anode is
Direction (Q. Nos. 57): Electrolysis involves electronation and de-electronation al the respective electrodes. Anode of electrolytic cell is the electrode at winch de-electronation takes place whereas at cathode electronation is noticed. If two or more ions of same charge are to be electronated or de-electronated, the ion having lesser discharge potential is discharged. Discharge potential of an ion refers for \[{{E}^{o}}{{\,}_{OP}}\] or \[{{E}^{o}}_{RP}\] as the case may be. The products formed at either electrode is given in terms of Faraday?s law of electrolysis, i.e., \[W=\frac{Eit}{96500}\]
During electrolysis of \[C{{H}_{3}}COONa(aq),\] the gas liberated at anode and cathode are respectively
Direction (Q. Nos. 58): Electrolysis involves electronation and de-electronation al the respective electrodes. Anode of electrolytic cell is the electrode at winch de-electronation takes place whereas at cathode electronation is noticed. If two or more ions of same charge are to be electronated or de-electronated, the ion having lesser discharge potential is discharged. Discharge potential of an ion refers for \[{{E}^{o}}{{\,}_{OP}}\] or \[{{E}^{o}}_{RP}\] as the case may be. The products formed at either electrode is given in terms of Faraday?s law of electrolysis, i.e., \[W=\frac{Eit}{96500}\]
During electrolysis of \[CuS{{O}_{4}}(aq)\] using platinum electrodes, the pH of solution (electrolyte)
If a circle of radius 'r' is concentric with ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\], then the common tangent is inclined to major axis at an angle
If the inclination of the diameter \[PP'\] of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] to the major axis is \[\theta \] and \[PP{{'}^{2}}\] is the AM of squares of major and minor axis, then tan 9 is equal to
A ray of light travels along the line \[2x-3y+5=0\] and strikes a plane mirror lying along the line\[x+y=2\]. The equation of the straight line containing the refracted ray is
Given, two points \[A\equiv (-2,\,0)\] and\[B\equiv (0,\,\,4)\]. The coordinates of a point M lying on the line \[y=x\], so that the perimeter of the \[\Delta \,AMB\] is least, is
Consider any set of observations \[{{x}_{1}},\,{{x}_{2}},\,....,\,\,{{x}_{10}},\] it being given that \[{{x}_{1}}<{{x}_{2}}<{{x}_{3}}<...<{{x}_{100}}<{{x}_{101}},\] then the mean deviation of this set of observations about a point k is minimum when k is equal to
The locus of the mid-points of the chords of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] which are tangent to the ellipse \[\frac{{{x}^{2}}}{{{p}^{2}}}+\frac{{{y}^{2}}}{{{q}^{2}}}=1\] is
Read the following mathematical statements carefully.
I. If \[x,\,\,y\] and z are all different real numbers, then \[\frac{1}{{{(x-y)}^{2}}}+\frac{1}{{{(y-z)}^{2}}}+\frac{1}{{{(z-x)}^{2}}}=\,{{\left( \frac{1}{x-y}+\frac{1}{y-z}+\frac{1}{z-x} \right)}^{2}}\]
II. \[{{\log }_{3}}\,x\cdot \,{{\log }_{4}}\,x\,\cdot \,{{\log }_{5}}\,\,x\] \[=\,\,({{\log }_{3}}\,x\cdot \,{{\log }_{4}}\,x)\] \[+\,({{\log }_{4}}\,x\cdot \,{{\log }_{5}}\,x)+\,({{\log }_{5}}\,x\cdot {{\log }_{3}}\,x)\] is true for exactly for one real value of x.
III. A matrix has 12 elements. Number of possible orders it can have a six.
If nth root of unity be \[1,\,\,{{a}_{1}},\,\,{{a}_{2}},\,....,\,{{a}_{n-1,}}\] then \[\sum\limits_{r\,=\,\,1}^{n\,-\,1}{\,\,\frac{1}{2+{{a}_{r}}}}\] is equal to
Let \[{{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}},...\] and \[{{b}_{1}},\,\,{{b}_{2}},\,{{b}_{3}},\,...\] be two distinct infinite GP?s. The sum of each one is 1. If \[{{a}_{2}}={{b}_{2}}\] and \[{{a}_{3}}=\frac{1}{8},\] then\[{{b}_{3}}\] is equal to
In \[\Delta ABC,\] the slope of the median through A is \[-2,\,\,B=(-1,\,3)\] and \[C=(3,5)\]. If its area is 5, then the distance of the vertex A from the origin is
From the point (6, 2) three normal?s are drawn to the parabola \[{{y}^{2}}=8x\]. If a circle is drawn through the feet of normal?s, then the length of its intercept on the Y-axis is
Direction: (Q. Nos. 88) For the following questions, the correct answers from the codes (a), (b), (c) and (d) defined as follows.
We have, \[a\cdot \,(b\times c)=[\,a\,b\,c]\] Statement I If a, b and c are unit coplanar vectors, then \[[2a-b\,\,\,2b-c\,\,\,2c-a]=0\] Statement II [a b c] = 0
A)
Statement I is true, Statement II is also true and Statement II is the correct explanation of Statement I.
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B)
Statement I is true, Statement II is also true and Statement II is not the correct explanation of Statement I.
Direction: (Q. Nos. 89) For the following questions, the correct answers from the codes (a), (b), (c) and (d) defined as follows.
We have, \[\sum{n=\frac{n(n+1)}{2},\,\,\sum{{{n}^{2}}}=\frac{n(n+1)\,(2n+1)}{6}}\] and \[\sum{{{n}^{3}}={{\left[ \frac{n(n+1)}{2} \right]}^{2}},\,\,n\in N}\] Statement I The sum of the series \[1+\,\,(1+2+4)\,+\,(4+6+9)\,+\,(9+12+16)\] \[+...+\,\,(361+380\,+400)\] is 8000. Statement II \[\sum\limits_{k=1}^{n}{[{{k}^{3}}-{{(k-1)}^{3}}]={{n}^{3}}}\] for any natural number\[n\].
A)
Statement I is true, Statement II is also true and Statement II is the correct explanation of Statement I.
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B)
Statement I is true, Statement II is also true and Statement II is not the correct explanation of Statement I.
Direction: (Q. Nos. 90) For the following questions, the correct answers from the codes (a), (b), (c) and (d) defined as follows.
In onto functions, each image must be assigned at least one preimage. Statement I Let \[E=\,\{1,\,\,2,\,\,3,\,\,4\}\] and \[F=\,\{\,a,\,\,b\},\] then the number of onto function from E to F is 14. Statement II Number of ways in which 4 distinct object can be distribution in two different boxes is 14, if no box remain empty.
A)
Statement I is true, Statement II is also true and Statement II is the correct explanation of Statement I.
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B)
Statement I is true, Statement II is also true and Statement II is not the correct explanation of Statement I.