A particle is moving in a circle of radius R in such a way that at any instant the total acceleration makes an angle of \[45{}^\circ \] with radius. Initial speed of particle is\[{{v}_{0}}\]. The time taken to complete the first revolution is
A lift of total mass M is raised by cables from rest through a height h. The greatest tension which the cables can safely bear is\[nMg\]. The maximum speed of lift during its journey if the ascent is to be made in shortest time is
A rod of length \[l\] leans by its upper end against a smooth vertical wall, while its other end leans against the floor. The end that leans against the wall moves uniformly downward. Then
A)
The other end also moves uniformly
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B)
The speed of other end goes on decreasing
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C)
The speed of other end goes on increasing
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D)
The speed of other end first decreases and then increases
A bar of mass M and length L is in pure translatory motion with its centre of mass velocity V. It collides with and sticks to a second identical bar which is initially at rest. (Assume that it becomes one composite bar of length \[2L\]). The angular velocity of the composite bar will be
There are three point objects A, B and C of mass M, m and m respectively placed on a smooth surface on an x-y plane. Among them A and C are fixed while B can move. There is a massless target F that is placed at distance r from the object B, and the line joining B and T makes an angle of \[30{}^\circ \]withx-axis. Then the value of M (in terms of m) so that the point object B can hit the target \[T(a>>r)\] is (assuming only mutual gravitation force of attraction between the point objects)
A container has two immiscible liquids of densities \[{{\rho }_{1}}\] and \[{{\rho }_{2}}(>{{\rho }_{1}})\]. A capillary tube of radius r is inserted in the liquid so that its bottom reaches up to the denser liquid. The denser liquid rises in the capillary and attains a height h from the interface of the liquids, which is equal to the column lengths of the lighter liquid. Assuming angle of contact to be zero, the surface tension of heavier liquid is
A steel drill making 180 rpm is used to drill a hole in a block of steel. The mass of the steel block and the drill is 180 g each. Consider all the entire mechanical work is used up in providing heat and the rate of rise of temperature of the block is \[0.5{}^\circ C\,{{s}^{-1}}\]. Specific heat of steel is \[0.10\,cal-{{g}^{-1}}{{K}^{-1}}.\]The couple required to drive the drill is
A solid conducting sphere having a charge Q is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the conducting sphere is now given a charge of \[-3Q,\] the new potential difference between the same two surfaces is
A tower used for power transmission leaks a current I into the ground. The tower is assumed to be a rod having lower end to be hemispherical with radius b. Resistivity of ground is \[\rho \]. The potential difference between the lower end of the rod and a point inside the ground at a distance r is
In the circuit shown in the figure, the emf of battery is \[50V,\] the resistance is \[250\Omega \]. and the capacitance is\[0.5\mu F\]. The switch S is closed for a long time and no voltage is measured across the capacitor. After the switch is opened, the potential difference across the capacitor reaches a maximum value of\[150\text{ }V\]. Calculate the inductance, in \[mH,\]to the nearest three digit integer.
The value of \[\theta \] so that the ray retraces the path after it strikes the mirror in water as shown in the diagram is \[({{\mu }_{of\,water}}=4/3)\]
Light from a monochromatic slit passes a special cylindrical planoconvex lens whose plane surface is made of two planes as shown. The radius of curvature of the curved face is 20 cm. Find the fringe width.
An electron of stationary hydrogen atom passes from the fifth energy level to the ground level. The velocity that the atom of mass m acquired a result of photon emission will be (R, Rydberg constant and h, Planck's constant)
In a YDSE, if the incident light consists of two wavelengths \[{{\lambda }_{1}}\] and \[{{\lambda }_{2}},\] the slit separation is d and the distance between the slit and the screen is D, the maxima due to each wavelength will coincide at a distance from the central maxima, is given by
A)
\[\frac{{{\lambda }_{1}}+{{\lambda }_{2}}}{2Dd}\]
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B)
LCM of \[\frac{{{\lambda }_{1}}D}{d}\]and \[\frac{{{\lambda }_{2}}D}{d}\]
The primary and secondary coils of a transformer have 50 turns and 1500 turns respectively. If the magnetic flux/linked with the primary coil is given by \[\phi ={{\phi }_{0}}+4t,\] where \[\phi \] is in Webers, t is time in seconds and \[{{\phi }_{0}}\] is a constant, the output voltage across the secondary coil is
In the given circuit, ammeters \[{{A}_{1}}\] and \[{{A}_{2}}\] are ideal and the voltmeter (V) is having very large resistance. In the steady state reading of ammeters \[{{A}_{1}},{{A}_{2}}\] and voltmeter (V) will be respectively
A)
\[0,\frac{\varepsilon }{2R}\] and \[\frac{\varepsilon }{2}\]
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B)
\[0,\frac{\varepsilon }{R}\] and \[\frac{\varepsilon }{2}\]
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C)
\[0,\frac{\varepsilon }{2R}\] and \[\frac{5\varepsilon }{2}\]
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D)
\[0,\frac{\varepsilon }{2R}\] and \[\frac{3\varepsilon }{2}\]
Three \[60\text{ }W,\] \[120V\]light bulbs are connected across a \[120V\] power source. If resistance of each bulb does not change with current, then find out the total power delivered to the three bulbs.
An upright object is placed at a distance in front of a converging lens equal to twice the focal length \[20cm\]of the lens. On the other side there is a concave mirror of focal length \[15cm\] separated from the lens by a distance of \[70cm\]. Then select the correct statements from the following.
(i) Magnification for the system is \[+1\]
(ii) Magnification for the system is \[-1\]
(iii) Final image by the system will be real and at distance of \[110\text{ }cm\]from centre of curvature of spherical mirror
(iv) Final image by the system will be real and at distance of \[60\text{ }cm\]from centre of curvature of spherical mirror
Figure shows the path of an electron in a region of uniform magnetic field. The path consists of two straight sections, each between a pair of uniformly charged plates, and two half-circles. The plates are named 1, 2, 3 and 4. Then
A)
1 and 3 at higher (positive) potential and 2 and 4 at lower (negative) potential
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B)
1 and 3 at lower potential and 2 and 4 at higher potential
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C)
1 and 4 at higher potential and 2 and 3 at lower potential
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D)
1 and 4 at lower potential and 2 and 3 at higher potential
Ram pushes eight identical blocks on the horizontal frictionless surface with horizontal force F. The force that block-1 exerts on the block-2 has magnitude \[{{F}_{21}}\] and the force that block-7 exerts on the block-8 is \[{{F}_{87}}\] Find \[\frac{{{F}_{21}}}{{{F}_{87}}}\] .
Two blocks of masses 2 kg and 1 kg respectively are tied to the ends of a string which passes over a light frictionless pulley. The masses are held at rest at the same horizontal level and then released. What is the distance (in m) traversed by centre of mass in 2 seconds? \[(g=10m/{{s}^{2}})\]
A copper sphere is suspended in an evacuated chamber maintained at\[300K\]. The sphere is maintained at constant temperature of \[900K\] by heating electrically. A total of \[300W\]electric power is needed to do this. When half of the surface of the copper sphere is completely blackened, \[600W\] is needed to maintain the same temperature of sphere. What is the emissivity of copper?
The frequency of first overtone of a closed organ pipe of length L is f. A hole is made at a distance \[\frac{L}{6}\] from the closed end so that it becomes an open pipe. Now the frequency of first overtone of open pipe is \[{{f}_{2}}\]. What is the value of \[\frac{{{f}_{1}}}{{{f}_{2}}}\] ?
It is given that \[{{m}_{H}}=1.007825\,amu,\]\[{{m}_{n}}=1.008665\text{ }amu,\] mass of iron \[{{(}_{26}}F{{e}^{56}})=55.934939\,amu,\]mass of bismuth \[{{(}_{83}}B{{i}^{209}})=208.980388\,amu.\]What is the ratio of binding energy per nucleon (in\[MeV\]) of \[_{26}F{{e}^{56}}\] to \[_{83}B{{i}^{209}}\]? \[(1\,\,amu=931.5\,MeV)\]
A sample of Ammonium Phosphate \[{{(N{{H}_{4}})}_{3}}P{{O}_{4}}\] contains 3.18 moles of hydrogen atoms. The number of moles of oxygen atoms in the sample is -
The first ionization energies (in\[KJ\text{ }mo{{l}^{-1}}\]) of carbon, silicon, germanium, tin, and lead are 1086, 786, 761, 708 and 715 respectively.
doneclear
B)
Down the group, ionization energy decreases regular from B to Tl in boron family
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C)
Among oxides of the elements of carbon family, \[CO\]is neutral, \[GeO\] is acidic and \[SnO\] is amphoteric
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D)
The 4f- and 5f-inner transition elements are placed separately at the bottom of the periodic table
Assume that the decomposition of \[HN{{O}_{3}}\] can be represented by the following equation \[4NH{{O}_{3}}(g)\rightleftharpoons 4N{{O}_{2}}(g)+2{{H}_{2}}O(g)+{{O}_{2}}(g)\] and the reaction approaches equilibrium at 400 K temperature and 30 atm pressure. At equilibrium partial pressure of \[HN{{O}_{3}}\] is 2 atm- Calculate \[{{K}_{c}}\]in \[{{\left( mol/L \right)}^{3}}\] at 400 K: \[(Use:R=0.08\text{ }atm-L/mol-K)\]
A gaseous system undergoes a change of state from (1) to (2) by any of the given path : path-I or path- II as shown in figure As per path -I, \[\Delta q=-400\,cal\] and \[\Delta W=14cal\] As per path-II, \[\Delta q=-48cal\]. Therefore work done, \[\Delta W\] in path -(II) is-
What will be the temperature at which a solution containing 6 g of glucose per 1000 g water will boil, if molal elevation constant for water is 0.5 2/1000 g.
In a Cu-voltameter, mass deposited in 30s is m gm. If the time-current graph is shown in the following figure What is the electrochemical equivalent of Cu?
The rate of disappearance of \[S{{O}_{2}}\] in the reaction \[2S{{O}_{2}}+{{O}_{2}}\to 2S{{O}_{3}},\] is\[1.28\times {{10}^{-3}}g\,litr{{e}^{-1}}\text{ }se{{c}^{-1}}\]. Then the rate of formation of \[S{{O}_{3}}\] in g \[litr{{e}^{-1}}Se{{c}^{-1}}\] is-
How many number of Dehydrating agent effectively £2 reaction in give dehydrating agent (mainly) \[A{{l}_{2}}{{O}_{3}},{{H}_{2}}S{{O}_{4}},{{H}_{3}}P{{O}_{4}},Th{{O}_{2}},KHS{{O}_{4}},HCl{{O}_{4}},\] \[POC{{l}_{3}},{{P}_{2}}{{O}_{5}},HI.\]
When a graph is plotted between log x/m and log p, it is straight line with an angle \[45{}^\circ \]and intercept 0.3010 on y-axis. If initial pressure is 0.3 aim, what will be the amount of gas adsorbed per gm of adsorbent?
The specific rate constant of the decomposition of \[{{N}_{2}}{{O}_{5}}\] is\[0.008\text{ }mi{{n}^{-1}}\]. The volume of \[{{O}_{2}}\] collected after 20 minutes is 16 ml. Find the volume that would be collected at the end of reaction. \[N{{O}_{2}}\] formed is dissolved in \[CC{{l}_{4}}\].
The standard free energy change for the reaction: \[{{H}_{2}}\left( g \right)+2AgC{{l}_{(s)}}\xrightarrow{{}}2A{{g}_{\left( s \right)}}+2{{H}^{+}}_{\left( aq \right)}+2C{{l}^{-}}_{(aq)}\] is \[-10.26\] kcal \[mo{{l}^{-1}}\] at\[25{}^\circ C\]. A cell using above reaction is operated at \[25{}^\circ C\] under \[{{P}_{{{H}_{2}}}}=1\] atm, \[[{{H}^{+}}]\] and \[\left[ C{{l}^{-}} \right]=0.1.\] Calculate e.m.f. of cell.
In the expansion of \[{{\left( 1 + x \right)}^{18}}\] , if the coefficients of \[{{(2r+4)}^{th}}\] and \[{{(r-2)}^{th}}\] terms are equal, then the value of r is:
If \[f(x)=\left\{ x\frac{{{e}^{(1/x)}}-{{e}^{(-1/x)}}}{{{e}^{(1/x)}}+{{e}^{(-1/x)}}} \right.,\,\,x\ne 0\] 0, x\[0,\,\,x=0\,\,\] then which of the following is true
A)
f is continuous and differentiable at every point
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B)
f is continuous at every point but not differentiable
Let P be a variable point on the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] with foci \[{{\operatorname{F}}_{1}}\,\,and\,\,{{F}_{2}}\]. If A is the area of the triangle \[P{{F}_{1}}{{F}_{2}}\], then maximum value of A is
Tangents drawn from the point P(1, 8) to the circle \[{{\operatorname{x}}^{2}}+{{y}^{2}}- 6x -4y -11 = 0\] touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is
If \[{{\left| \vec{a}\,\,\times \,\,\vec{b} \right|}^{2}}{{\left| \vec{a}\,.\,\vec{b} \right|}^{2}}=144\] and \[\left| {\vec{a}} \right| =4\], then \[\left| {\vec{b}} \right|\] is equal to:
If for a variable line \[\frac{x}{a}+\frac{y}{b}=1\], the condition \[\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}=\frac{1}{{{c}^{2}}}\] (c is a constant) is satisfied, then locus of foot of perpendicular drawn from origin to the line is
If \[{{e}^{\left[ {{\sin }^{2}}a\,+{{\sin }^{4}}\alpha \,\,{{\sin }^{6}}\alpha +\,...\,\infty \right]lo{{g}_{e}}\,2}}\] of equation \[{{\operatorname{x}}^{2}}- 9x+ 8 = 0,\] where \[0 < \alpha < \frac{\pi }{2}\], then the principle value of \[{{\sin }^{-1}}\sin \,\left( \frac{2\pi }{3} \right)\] is
\[\cos \left( \alpha - \beta \right) = 1 and cos \left( \alpha + \beta \right) =1/e\], where \[\alpha ,\text{ }\beta \text{ }\in (-\pi ,\,\,\pi )\]. Pairs of a, P which satisfy both the equations is/are
Let \[{{\operatorname{P}}_{1}}:2x+y-z=3\,\,and\,\,{{P}_{2}}:x+2y+z=2\] be two planes. Then, which of the following statements(s) is (are) TRUE
A)
The line of intersection of \[{{P}_{1}}\,\,and\,\,{{P}_{2}}\] has direction ratios 1, 2, -1.
doneclear
B)
The line \[\frac{3x-4}{9}=\frac{1-3y}{9}=\frac{z}{3}\] is perpendicular to the line of intersection of \[{{\operatorname{P}}_{1}}\,\,and\,\,{{P}_{2}}\].
doneclear
C)
The acute angle between \[{{\operatorname{P}}_{1}}\,\,and\,\,{{P}_{2}}\] is \[45{}^\circ \].
doneclear
D)
If \[{{P}_{3}}\] is the plane passing through the point (4, 2, -2) and perpendicular to the line of intersection of \[{{\operatorname{P}}_{1}}\,\,and\,\,{{P}_{2}}\], then the distance of the point (2, 1, 1) from the plane \[{{P}_{3}}\,\,is\,\,\frac{2}{\sqrt{3}}\].
The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs Rs. 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to Rs. 300 per hour is
Let \[{{S}_{n}}\] denote the sum of first n terms of an A.P. If \[{{\operatorname{S}}_{2n}}=3{{S}_{n}}\], then the ratio \[{{\operatorname{S}}_{3n}}/{{S}_{n}}\] is equal to: