If equations \[{{x}^{n}}-n{{x}^{n\,-\,1}}+{{a}_{2}}{{x}^{n\,-\,2}}+{{a}_{3}}{{x}^{n\,-\,3}}+...\] has \[+{{a}_{n-1}}x+{{(-1)}^{n}}=0\] positive roots, then the least value of n for which \[{{a}_{2}}+{{a}_{3}}\]is negative is
If in an isosceles triangle with base 'a' vertical angle \[20{}^\circ \]and lateral side each of length 'b' is given, then the value of \[{{a}^{3}}+{{b}^{3}}\]equals
Given \[\int_{0}^{\frac{\pi }{2}}{\frac{\sin x}{1+\sin x+\cos x}dx=K,}\]then the value of the definite integral \[\int_{0}^{\frac{\pi }{2}}{\frac{dx}{1+\sin x+\cos x}}\]is equal to
If the line \[y=\sqrt{3}x\] cuts the curve \[{{x}^{4}}+a{{x}^{2}}y+bxy+cx+dy+6=0\] at A, B, C and D, then the value of \[\frac{1}{12}OA\cdot OB\cdot OC\cdot OD\](where O is origin) is equal to
If \[f(x)\] is a function satisfying the relation \[{{x}^{2}}f(x)-2f\left( \frac{1}{x} \right)=g(x),\] where \[g\,(x)\] is odd function, then the value of \[f(2)\] is
If the chords of contact of tangents from two points\[(-\,4,2)\] and (2, 1) to the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are at right angle, then the eccentricity of the hyperbola is
If Z is a complex number satisfying \[\left| {{Z}^{3}}+{{Z}^{-3}} \right|\le 2,\] then the maximum possible value of \[\left| Z+{{Z}^{-\,1}} \right|\]is
If \[f\,(x)={{x}^{2}}+\alpha {{x}^{2}}+\beta x+\gamma ,\] where \[\alpha ,\beta ,\gamma ,\] are rational numbers and two roots of \[f(x)=0\] are eccentricities of a parabola and a rectangular hyperbola, then \[\alpha +\beta +\gamma \] is equal to
A ten-digit number is formed without repeating any digit the probability that the difference of digits at equal distances from the beginning and the end is always 1, is
A hollow conducting sphere is placed in an electric field produced by a point charge as shown. Let \[{{V}_{A}},\]\[{{V}_{B}},\]\[{{V}_{C}}\] be the electric potentials at points A, B, C respectively and \[{{V}_{0}}\] is the potential at centre O due to induced charge on shell -
Five bulbs \[{{B}_{1}},\] \[{{B}_{2}},\] \[{{B}_{3}}\]and \[{{B}_{4}}\] each of rating 60 W / 200 V and \[{{B}_{5}}\] of rating 120W / 400V are connected as shown in circuit. Total power consumption by all the bulbs is -
The mirror of length L moves horizontally as shown in the figure with a velocity v. The mirror is illuminated by a point source of light 'P' placed on the ground. The rate at which the length of the light spot on the ground increases is -
In the circuit shown the cells are ideal & of equal e.m.f., the capacitance of the capacitor is C and the resistance of the resistor is R. X is first joined to Y and then to Z. After a long time the total heat produced in the resistor will be -
A)
Equal to the energy finally stored in the capacitor
doneclear
B)
Half of the energy finally stored in take capacitor
doneclear
C)
Twice the energy finally stored in the capacitor
doneclear
D)
4 times the energy finally stored in the capacitor
A ring shaped tube contains two ideal gases with equal masses and atomic mass numbers \[{{M}_{1}}=32\] and \[{{M}_{2}}=28.\] The gases are separated by one fixed partition P and another movable conducting partition S which can move freely without friction inside the ring. The angle a as shown in the figure in equilibrium is -
The wave-function for a certain standing wave on a string fixed at both ends is \[y(x,\,\,t)=0.5\,\,\sin \,\,(0.025\,\pi x)\,\,\cos \,\,500t\] where x and y are in centimeters and t is in seconds. The shortest possible length of the string is -
1. In photo electric effect, even for monochromatic incident radiation, the photoelectrons are emitter with a spread of velocities.
2. Photoelectrons are emitted without delay once the incident light reaches the surface of the emitter.
3. Frequency of monochromatic light (well above the cutoff frequency), that is incident on a emitter in a photoelectric effect, is increased while keeping the intensity constant. It results in decrease in magnitude of stopping potential.
Correct order of the true/false for the above statements is -
An element X decays, first by positron emission and then two \[\alpha \]-particles are emitted in successive radioactive decay. If the product nuclei has a mass number 229 and atomic number 89, the mass number and atomic number of element X are -
The diagram shows the arrangement of three small uniformly charged spheres A, B and C. The arrows indicate the direction of the electrostatic forces acting between the spheres (for example, the left arrow on sphere A indicates the electrostatic force on sphere A due to sphere -B). At least two of the spheres are positively charged. Which sphere, if any, could be negatively charged
The path difference between two interfering waves at a point on the screen is \[\lambda /6.\] The ratio of intensity at this point and that at the central bright fringe will be \[-\] (Assume that intensity due to each slit in same)
The wave front of a light beam is given by the equation \[x+2y+8z=c,\] (where c is arbitrary constant) then the angle made by the direction of light with the y-axis is-
A charged particle moves in a uniform magnetic field but constant with time such that initial velocity is perpendicular to the magnetic field. If no other force acts on the particle, then -
Two blocks A and B of same mass are attached by a thin inextensible string through an ideal pulley. Initially block B is held in position as shown m figure. Now the block B is released. Block A will slide to right angle hit the pulley in time \[~{{t}_{A}}.\] Block B will swing and hit the surface in time \[{{t}_{B}}.\] Assume the surface as frictionless.
A)
\[{{t}_{A}}={{t}_{B}}\]
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B)
\[{{t}_{A}}<{{t}_{B}}\]
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C)
\[{{t}_{A}}>{{t}_{B}}\]
doneclear
D)
Data are not sufficient to get relationship between \[{{t}_{A}}\]and \[{{t}_{B}}\]
A sample contains number of stable nuclei equal to\[{{N}_{s}}\]and number of unstable nuclei equal to\[{{N}_{u}}.\] After a time T the activity of the sample decreased to one third of the initial activity, while the total number of nuclei (excluding decayed nuclei) became half. The ratio\[{{N}_{s}}/{{N}_{u}}\] initially is -
Suppose in gravity free space a disc of mass \[{{m}_{0}}\]rotates freely about a fixed horizontal axis through its centre. A thin cotton pad is fixed to its rim, which can absorb water. The mass of water dripping onto the pad is \[\mu \] kg per second. After what time will the angular velocity of the disc get reduced to half of its initial value?
The surface tension and bulk modulus of elasticity of water are S and B respectively. Then the ratio \[\frac{B}{S}\] is dimensionally equivalent to the dimension of-
A uniform metal rod (fixed at both ends) of \[2\text{ }m{{m}^{2}}\]cross-section is cooled from \[40{}^\circ C\] to \[20{}^\circ C.\] The co-efficient of the linear expansion of the rod is \[12\times {{10}^{-\,4}}\] per degree & It's young modulus of elasticity is \[{{10}^{11}}\text{ }N/{{m}^{2}}.\]The energy stored per unit volume of the rod is -
A long cylindrical drum is filled with water. Two small holes are made on the side of the drum as shown in the figure. Find the depth of the liquid in the drum if the ranges of water from the holes are equal-
Diameter of a planoconvex lens of focal length 37 cm is 6 cm. It's thickness at the centre is 5 mm. The speed of light in the material of the lens is -
Two complexes \[[Cr({{H}_{2}}{{O}_{6}})C{{l}_{3}}]\][A] and \[[Cr{{(N{{H}_{3}})}_{6}}]C{{l}_{3}}\][B] are violet and yellow coloured, respectively. The incorrect statement regarding them is:
A)
\[{{\Delta }_{0}}\]value of [a] is less than that of [b]
doneclear
B)
\[{{\Delta }_{0}}\]value of [a] and [b] are calculated from the energies of violet and yellow light, respectively.
doneclear
C)
Both absorb energies corresponding to their complementary colours.
doneclear
D)
Both are paramagnetic with three unpaired electrons
Adsorption of a gas follows Freundlich adsorption isotherm. In the given plot, x is the mass of the gas y adsorbed on mass m of the adsorbent at pressure p.\[\frac{x}{m}\]
For emission line of atomic hydrogen from \[{{n}_{i}}=8\] to \[{{n}_{i}}=\]the plot of wave number \[(\overline{v})\] against \[\left( \frac{1}{{{n}^{2}}} \right)\]will be (The Rydberg constant, \[{{\text{R}}_{H}}\] is in wave number unit).
A)
Linear with slope \[-{{\text{R}}_{\text{H}}}\]
doneclear
B)
Linear with intercept \[-{{\text{R}}_{\text{H}}}\]
20 mL of \[\text{0}\text{.1M}\,{{\text{H}}_{\text{2}}}\text{S}{{\text{O}}_{\text{4}}}\] solution is added to 30 mL of \[\text{0}\text{.2}\,\text{M}\,\text{N}{{\text{H}}_{\text{4}}}\text{OH}\] solution. The pH of the resultant mixture is: \[\text{ }\!\![\!\!\text{ p}{{\text{K}}_{\text{b}}}\,\text{of}\,\text{N}{{\text{H}}_{\text{4}}}\text{OH=4}\text{.7 }\!\!]\!\!\text{ }\]
0.5 moles of gas A and x moles of gas B exert a pressure of 200 Pa in a container of volume \[10\,{{\text{m}}^{3}}\] at 1000 K, given R is the gas constant in \[\text{J}{{\text{K}}^{-1}}\]\[\text{mo}{{\text{l}}^{-1}}\]m, x is:
Consider the reversible isothermal expansion of an ideal gas in a closed system at two different \[{{\text{T}}_{1}}\] temperatures and \[{{\text{T}}_{2}}\,({{\text{T}}_{1}}<{{\text{T}}_{2}}).\] The correct graphical depiction of the dependence of work done (w) on the final volume (V) is:
The major product of following reaction is \[\text{R}\,-\,\text{C}\,=\,\text{N}\xrightarrow[(2){{\text{H}}_{2}}O]{(1)\text{AlH}(i-\text{B}{{\text{u}}_{2}})}?\]
A solution of sodium sulphate contains 92 g \[\text{N}{{\text{a}}^{+}}\]of ions per kilogram of water. The molality of \[\text{N}{{\text{a}}^{+}}\]ions in that solution in \[\text{K}{{\text{g}}^{-1}}\] is:
A water sample has ppm level concentration of the following metals: \[Fe=0.2;Mn=5.0;\]\[Cu=3.0;Zn=5.0\]. The metal that makes the water sample unsuitable drinking is:
A triangle has base 10 cm long and the base angles of \[50{}^\circ \]and \[70{}^\circ .\] If the perimeter of the triangle is\[x+y\,\cos z{}^\circ ,\]where \[z\in (0,90{}^\circ ),\]then the value of \[x+y+z\]equals to
If the expression \[{{z}^{5}}-32\] can be factorized into linear and quadratic factors over real coefficient as \[({{z}^{5}}-32)=(z-2)({{z}^{2}}-pz+4)({{z}^{2}}-qz+4),\] then\[({{p}^{2}}+2q)\] is equal to
If the line \[x+y=1\] is a tangent to a parabola with focus (1, 2) at A and intersects the directrix at B and tangent at vertex at C respectively, then \[AC\cdot BC\] is equal to
There are 6 boxes labelled \[{{B}_{1}},{{B}_{2}},{{B}_{3}},...,{{B}_{6}}.\] By In each trial two fair dice \[{{D}_{1}},{{D}_{2}}\] are thrown. If \[{{D}_{1}}\] shows j and \[{{D}_{2}}\]shows k, then j balls are put into the box \[{{B}_{K}},\] After n trials, what is the probability that \[{{B}_{1}}\]contains at most one ball?
Let \[[x]\] and \[\{x\}\] be the integral part and fractional part of a real number x respectively. Then the value of the integral \[\int_{0}^{5}{[x]\{x\}dx}\] is
A particle moves constantly in a circle centered at the origin with a period T. If its position at time \[t=0\] seconds is (A, 0) meters, which graph represents \[{{v}_{x}},\] the x-component of the particle's velocity, as a function of time?
Two plane mirrors are placed as shown in figure. A point object O is approaching the intersection point A of mirrors with a speed of 100 cm/s. The velocity of image of the object formed by \[{{M}_{2}}\], with respect to velocity of image of object formed by \[{{M}_{1}}\] is:
A nut is screwed onto a bolt with 12 turns per cm and diameter more than 1 cm. The bolt is lying in a horizontal position. The nut spins at 216 rpm. Time taken by the nut to cover 3 cm along the bolt is:
A motor bike accelerates from rest at a constant rate of \[3\text{ }m{{s}^{-2}}\] for some time and then moves with uniform velocity for the same duration. Then it retards at a constant rate of \[6\text{ }m{{s}^{-2}}\] and comes to rest. The bike was in motion for a period of 5 seconds. The total distance travelled by the bike is:
One conducting U-tube can slide inside another as shown in figure, maintaining electrical contacts between the tubes. The magnetic field B is perpendicular to the plane of the figure. If each tube moves towards the other at a constant speed \[v,\]then the emf induced in the circuit in terms of B, \[l\] and \[v,\] where is the width of each tube, will be:
The current I in an inductance coil varies with time t according to the graph shown in figure. Which one of the following plots shows the variation of voltage in the coil with time?
A non-planar loop of conducting wire carrying a current I is placed as shown in the figure. Each of the straight sections of the loop of length 2a. The magnetic field due to this loop at the point P(a, 0, a) points in the direction
The ground state energy of hydrogen atom is \[-\,13.6\] Consider an electronic state \[\Psi \]of \[\text{H}{{\text{e}}^{+}}\] whose energy, quantum number and magnetic quantum number are \[-\,3.4\] eV, 2 and 0, respectively Which of statement(s) is (are) true for the state\[\Psi \] ?
A)
The nuclear charge experienced by the electron in this state is less than 2e, where e is the magnitude of the electronic charge
The cyanide process of gold extraction involves leaching out gold from its ore with \[\text{C}{{\text{N}}^{-}}\] in the presence of Q in water to form R. Subsequently R is treated with T to obtain Au and Z. Choose the correct option(s):
A)
Z is \[{{[\text{Zn}{{(\text{CN})}_{4}}]}^{2\,-}}\]
In Antirrhinum majus, the Red (RR) flowerd plant crossed with white flowered (rr) plant & in\[{{F}_{1}}\]generation pink (Rr) flowered plants obtained on selfing of \[{{F}_{1}}\] generation, \[{{F}_{2}}\] generation obtained. In which the ratio of Red & white flowerd plants will.