If \[\left| z-4+3i \right|\le 1\] and \[\alpha \] and \[\beta \] are the least and greatest value of \[\left| z \right|\] and K be the least value of \[\frac{{{x}^{2}}+{{x}^{2}}+4}{x}\] interval \[(0,\infty ),\] then K is equal to
Let E' denote the complement of an event E. Let E, F, G be pairwise independent events with \[P\,(G)>0\] and \[P\,(E\cap F\cap G)=0.\] Then, \[\frac{P\,(E'\cap F')}{G}\] equals
Let \[f\,(x)=\frac{x}{{{(1+{{x}^{n}})}^{1/n}}}\] for \[n\ge 2\] and \[g(x)=\underbrace{({{f}_{0}}{{f}_{0}}...f)}_{n\,\,\text{times}}f(x).\] Then, \[\int{{{x}^{n-2}}g(x)\,dx}\] is equal to
Suppose two perpendicular tangents can be drawn from the origin to the circle \[{{x}^{2}}+{{y}^{2}}-6x-2py+17=0,\] for some real p. Then,\[|p|\]is equal to
One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated to his wife is
Let the curve C be the mirror image of the parabola \[{{y}^{2}}=4x\] with respect to the line \[x+y+4=0.\] If A and B are the points of intersection of C with the line \[y=-\,5,\] then the distance between A and B is
The value of a for which the points A, B, C with position vector \[2\hat{i}-\hat{j}+\hat{k},\] \[\hat{i}-3\hat{j}-5\hat{k},\] and \[a\hat{i}-3\hat{j}+\hat{k}\] respectively are the vertices of a right angle triangle with \[C=\frac{\pi }{2}\] are
If \[{{\log }_{x}}8=z,\]\[{{\log }_{y}}x=-\,1\]and \[{{\log }_{\frac{1}{4}}}y=-\,1,\] then \[{{\left( \frac{1}{x}+1 \right)}^{\log \sqrt{5}({{y}^{2}}+4{{z}^{2}})}}\] is equal to
One ticket is selected at random from 50 tickets numbered 00, 01, 02,..., 49. Then, the probability that the sum of the digits on the selected ticket is 8 given that the product of these digits is zero, equals
In a binomial distribution \[B\left( n,p=\frac{1}{4} \right).\] If the probability of at least one success is greater than or equal to \[\frac{9}{10},\] then n is greater than
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 60W/200V 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 & 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
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B)
half of the energy finally stored in the capacitor
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C)
twice the energy finally stored in the capacitor
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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 partitions which can move freely without friction inside the ring. The angle \[\alpha \] 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 photo electrons are emitted 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 wavefront of a light beam is given by the equation \[x+2y+3z=c,\] (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 each of same mass are attached by a thin inextensible string through an ideal pulley. Initially block B is held in position as shown in figure. Now the block B is released. Block A will slide to right and 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)
\[{{t}_{A}}>{{t}_{B}}\]
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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 \[2\,\,m{{m}^{2}}\]cross-section is cooled from \[40{}^\circ C\] to \[20{}^\circ C.\] The co-efficient of the linear expansion of the rod is \[12\times {{10}^{-\,6}}\] per degree & it's young modulus of elasticity is \[{{10}^{11}}\,N/{{m}^{2}}.\] The energy stored per unit volume of the rod is:
A long cyclindrical drum is filled with water. Two small holes are made on the side of the drum as shown in the fig. Find the depth of the liquid in the drum if the ranges of water from the holes are equal.
(I) When copper ore is mixed with silica, in a reverberatory furnace copper matte is produced. The copper matte contains sulphides of copper (II) and iron (II).
(II) Zone refining is based on the principle that impurities are more soluble in molten metal than in solid metal.
(III) In the metallurgy of aluminium, graphite anode is oxidized to carbon monoxide and carbon dioxide.
Among the following statements which is INCORRECT -
A)
In the preparation of compounds of Xe, Bartlett had taken \[{{O}_{2}}Pt{{F}_{6}}\] as a base compound because both \[{{O}_{2}}\] and Xe have almost same ionization enthalpy.
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B)
Nitrogen does not show allotropy.
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C)
A brown ring is formed in the ring test for \[N{{O}_{{{3}^{-}}}}\] ion. It is due to the formation of \[{{[Fe{{({{H}_{2}}O)}_{5}}(NO)]}^{2+}}\]
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D)
On heating with concentrated NaOH solution in an inert atmosphere of \[C{{O}_{2}},\] red phosphorus gives \[P{{H}_{3}}\] gas
For a given reaction A\[\to \]Product, rate is \[1\times {{10}^{-\,4}}M{{s}^{-\,1}}\] when [A] = 0.01 M and rate is\[1.41\times {{10}^{-\,4}}M{{s}^{-\,1}}\] when [A] = 0.02 M. Hence, rate law is -
The electrode potentials for \[C{{u}^{2+}}_{(aq)}+{{e}^{-}}\to C{{u}^{+}}_{(aq)}+\] and \[C{{u}^{+}}_{(aq)}{{e}^{-}}\to C{{u}_{(s)}}\]respectively. The value of \[E{{{}^\circ }_{C{{u}^{2+}}/Cu}}\] will be -
\[{{H}_{2}}S\] reacts with lead acetate forming a black compound which reacts with \[{{H}_{2}}{{O}_{2}}\] to form another compound. The colour of the compound is -
In FCC lattice A, B, C, D atoms are arranged at comers, face centres, octahedral voids and tetrahedral voids respectively, then the body diagonal contains -
Decreasing order of relative nucleophilicity of the following nucleophiles in protic solvent is - \[\overset{\Theta }{\mathop{S}}\,H,\,\,Ac\overset{\Theta }{\mathop{O}}\,,\,\,Ph\overset{\Theta }{\mathop{O}}\,,\overset{\Theta }{\mathop{\,\,O}}\,\,H,\,\,{{H}_{2}}O\]
The enantiomeric excess and observed specific rotation of a mixture containing \[6\,\,gm\left( + \right)-2-\]butanol and 4 (gm) of \[\left( - \right)-2-\] butanol are respectively (If the specific rotation of enentiomerically pure \[\left( + \right)-2-\] butanol is + 13.5 unit).
The resting membrane potential for neuron A is -70m V, while the resting potential for neuron B is -50mV. The threshold voltage for the production of an action potential is -35m V for both neurons. Which of the following statements is false?
A)
Neuron A must depolarise by 35m V to reach the threshold voltage.
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B)
Neuron B must hyperpolarise by 15mV to Reach the threshold voltage.
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C)
The inside of both neurons is negatively charged with respect to the outside.
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D)
A single EPSP received by neuron A would cause it to depolarise slightly.
A person with unknown blood group under ABO system, has suffered much blood loss in an accident and needs immediate blood ansfusion. His one friend who has a valid certificate of his own blood type offers blood donation without delay. What would have been the type of blood group of the donor friend?
Given below is the diagrammatic representation of one of the categories of small molecular weight organic compounds in the living tissues. Identify the category shown and the one blank Component "X" in it.
A small segment of DNA contains the base Sequence CGT. If an mRNA transcript is made That includes this DNA sequence, what will be The anticodon on the tRNA that will bind to the Corresponding mRNA codon for this DNA triplet?
If \[a\in \left[ -20,0 \right],\]then probability that the graph of the function\[y=16{{x}^{2}}+8\left( a+5 \right)x-7a-5\] is strictly above the \[\operatorname{x}-axis\,is\]
For all complex number \[\left| {{z}_{1}} \right|=12\] and \[\left| {{z}_{2}}-3-4i \right|=5\] the minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|\] is
Let be a circle passing through \[\left( 2,0 \right)\] and \[\left( 0,2 \right)\] and having the small possible area. \[{{S}_{2}}\] is the circle touching \[{{S}_{1}}\] externally and passing through origin. \[{{S}_{3}}\] is a circle that touches and \[{{S}_{2}}\] both externally. If the area of each of \[{{S}_{1}}\] \[{{S}_{2}}\] and \[{{S}_{3}}\] is same, then
A)
Equation of \[{{S}_{2}}\] is \[{{x}^{2}}+{{y}^{2}}+2x-2y=0\]
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B)
Equation of the smallest circle containing \[{{S}_{1}}\operatorname{and}\,{{S}_{2}}\,\operatorname{is}{{x}^{2}}+{{y}^{2}}=8\]
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C)
Coordinated of the Centre of \[{{S}_{3}}\] and \[\left( 2,-2 \right)\] or \[\left( -2,2 \right)\]
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D)
The circle intersecting \[S{{ }_{1}},{{S}_{2}}\,\,and\,\,{{S}_{3}}\] orthogonally has radius \[\sqrt{\frac{3}{2}}\]
Let \[\left( x+y \right)=f\left( x \right)+f\left( y \right)+2xy-1,y\in R\] if \[f\left( x \right)\] is differentiable and \[f'\left( 0 \right)=\sin \phi \], then
The focal length of a convex lens of refractive index 1.5 is 2 cm. The focal length of the lens when immerged in liquid spirit of refractive index 1.25 is -
In the figure shown \[{{V}_{1}},\]\[{{V}_{2}}\] are AC voltmeters and A is AC ammeter. The readings of \[{{V}_{1}},\]\[{{V}_{2}},\]\[{{V}_{3}}\] and A are 10V, 20V, 20V, 2A respectively. Find the values of R, C, L and the source voltage \[{{V}_{s}}.\] If the inductor is short circuit then what will be the reading of and \[{{V}_{1}},\]\[{{V}_{2}}\] A.
A segment of angle \[\theta \] is cut from a half disc with symmetry of symmetrically as shown. If the centre of mass of the remaining part is at a distance 'a' from O and the centre of mass of the original disc was at distance d then it can be definitely said that
A)
a = d
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B)
a > d
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C)
a < d
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D)
A, B, C depends on the angle of segment cut from disc.
A large open tank has two small holes in its vertical wall as shown in figure. One is a square hole of side 'L' at a depth '4y' from the top and the other is a circular hole of radius 'R' at a depth 'y' from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then, 'R' is equal to -
If a charged particle of charge to mass ratio \[\frac{q}{m}=\alpha \] is entering in a magnetic field of strength B at a speed \[v=\text{(2}\alpha d\text{)}\,(B),\] then which of the following is correct -
A)
Angle subtended by charged particle at the centre of circular path is \[2\pi \]
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B)
The charge will move on a circular path and will come out from magnetic field at a distance 4d from the point of insertion.
doneclear
C)
The time for which particle will he the magnetic field is \[\frac{2\pi }{\alpha B}.\]
doneclear
D)
The charged particle will subtend an angle of \[90{}^\circ \]at the centre of circular path.
Two identical rectangular rods of metal are welded end to end in series between temperature \[0{}^\circ C\] and \[100{}^\circ C\] and 10J of heat is conducted (in steady state process) through the rod in 2.00 min. If 5 such rods are taken and joined as shown in figure maintaining the same temperature difference between A and B, then the time in which 20 J heat will flow through the rods is -
Two points A and B on a disc have velocities \[{{v}_{1}}\]& \[{{v}_{2}}\] at some moment. Their directing make angles \[60{}^\circ \] and \[30{}^\circ \] respectively with the lnie of separation as shown in figure. The angular velocity of disc is -
A chain of length L is placed on a horizontal surface as shown in figure. At any instant x is the length of chain on rough surface and the remaining portion lies on smooth surface. Initially x = 0. A horizontal force P is applied to the chain (as shown in figure). In the duration x changes from x = 0 to x = L, for chain to move with constant speed.
A)
The magnitude of P should increase with time
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B)
The magnitude of P should decrease with time
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C)
The magnitude of P should increase first and then decrease with time
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D)
The magnitude of P should decrease first and then increase with time
An object approaches a fixed diverging lens with a constant velocity from infinity along the principal axis. The relative velocity between object and its image will be-
A ring of radius R is placed in the plane with its center at origin and its axis along the x-axis and having uniformly distributed positive charge. A ring of radius r (<<R) and coaxial with the larger ring is moving along the axis with constant velocity then the variation of electrical flux \[(\phi )\] passing through the smaller ring with position will be best represented by -
An organic compound [A] with molecular formula \[{{C}_{9}}{{H}_{10}}O\] forms an orange-red precipitate with 2, 4-DNP reagent and gives yellow precipitate on heating with iodine and \[NaOH.\] It does not reduces Tollen's reagent or Fehling's solution nor it decolourises bromine water as Baeyer's reagent. On drastic oxidation with chromic acid, it gives a carboxylic acid having molecular formula \[{{C}_{7}}{{H}_{6}}{{O}_{2}}.\] Identify the compound [A].
The emf of the cell, \[Pb|PbC{{l}_{2}}||AgCl||Ag\] at 300 K is 0.50 V. If temperature coefficient of emf is \[-\,2\times {{10}^{-\,4}}\]volt \[{{\deg }^{-\,1}}.\] Then calculate the enthalpy change for the cell reaction.
If dominant C and P genes are essential for the development of purple colour in sweet pea flowers, what would be the ratio of white and purple colour in a cross between \[CcPp\times Ccpp\]
Given below sequence of the processed m-RNA ready for translation: 5'AUG CUA UACCUCCUUUAUCUGUGA - 3' How many different t - RNA molecule require to translate this m - RNA -
In a cross between individuals homozygous for (a, b) and wild type (++) 700 out of 1000 individuals were of parental type. Then the distance between a and b is
Given below is an incomplete table about certain hormones, their source glands and one major effect of each on the body in humans. Identify the correct option for the three blanks A, B and C.
GLAND
SECRETION
EFFECT ON BODY
Ovary
A
Maintenance of secondary sexual characters in female