question_answer2) A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to
AIEEE Solved Paper-2004
question_answer3) A ball is released from the top of a tower of height h metre. It takes T second to reach the ground. What is the position of the ball in \[T/3s\]?
AIEEE Solved Paper-2004
question_answer5) A projectile can have the same range R for two angles of projection. If\[{{T}_{1}}\]and\[{{T}_{2}}\]are the times of flights in the two cases, then the product of the two times of flights is directly proportional to
AIEEE Solved Paper-2004
question_answer7) An automobile travelling with a speed of 60 km/h, can brake to stop within a distance of 20 m. If the car is going twice as fast, i.e., 120 km/h, the stopping distance will be
AIEEE Solved Paper-2004
question_answer8) A machine gun fires a bullet of mass 40 g with a velocity\[1200\text{ }m{{s}^{-1}}\]. The man holding it, can exert a maximum force of 144 N on the gun. How many bullets can he fire per second at the most?
AIEEE Solved Paper-2004
question_answer9) Two masses\[{{m}_{1}}=5\,kg\]and\[{{m}_{2}}=4.8\,kg\]lied to a string are hanging over a light frictionless pulley. What is the acceleration of the masses when lift is free to move? \[(g=9.8\text{ }m/{{s}^{2}})\]
AIEEE Solved Paper-2004
question_answer10) A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table?
AIEEE Solved Paper-2003
question_answer11) A block rests on a rough inclined plane making an angle of\[30{}^\circ \]with the horizontal. The coefficient of static friction between the block and the plane is 0.8. If the frictional force on the block is 10 N, the mass of the block (in kg) is (take\[g=10\text{ }m/{{s}^{2}}\])
question_answer12) A force\[F=(5\hat{i}+3\hat{j}+2\hat{k})\] is applied over a particle which displaces it from its origin to the point\[r=(2\hat{i}-\hat{j})m\]. The work done on the particle in joules is
question_answer13) A body of mass m accelerates uniformly from rest to\[{{v}_{1}}\]in time\[{{t}_{1}}\]. The instantaneous power delivered to the body as a function of time t is
question_answer14) A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of 'the particle. The motion of the particle takes place in a plane, it follows that
question_answer15) A solid sphere is rotating in free space. If the radius of the sphere is increased keeping mass same which one of the following will not be affected?
question_answer16) A ball is thrown from a point with a speed\[{{v}_{0}}\]at an angle of projection\[\theta \]. From the same point and at the same instant, a person starts running with a constant speed \[\frac{{{v}_{0}}}{2}\]to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection?
question_answer17) One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively \[{{l}_{A}}\] and \[{{l}_{B}}\] such that
A)
\[{{l}_{A}}={{l}_{B}}\]
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B)
\[{{l}_{A}}>{{l}_{B}}\]
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C)
\[{{l}_{A}}<{{l}_{B}}\]
doneclear
D)
\[\frac{{{l}_{A}}}{{{l}_{B}}}\,=\frac{{{d}_{A}}}{{{d}_{B}}}\] where\[{{d}_{A}}\]and\[{{d}_{B}}\]are their densities.
question_answer18) A satellite of mass\[m\]revolve around the earth of radius R at a bright\[x\]from its surface. If\[g\]is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellites is
question_answer20) If g is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass m raised from the surface of the earth to a height equal to the radius R of the earth, is
question_answer21) Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius R around the sun will be proportional to
question_answer23) Spherical balls of radius R are falling g in a viscous fluid of viscosity\[\eta \]. The retarding viscous force acting on the spherical ball is
A)
directly proportional to R but inversely proportional to v
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B)
directly proportional to both radius R and velocity v
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C)
Inversely proportional to both radius R and velocity v.
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D)
Inversely proportional to R but directly proportional to velocity v.
question_answer25) The bob of a simple pendulum executes simple harmonic motion in water with a period t, while the period of oscillation of the bob is to in air. Neglecting frictional force of water and given that the density of the bob is\[(4/3)\times 1000\]\[kg/{{m}^{3}}\]. What relationship between t and\[{{t}_{0}}\]is true?
question_answer26) A particle at the end of a spring executes simple harmonic motion with a period\[{{t}_{1}},\]while the corresponding period for another spring is\[{{t}_{2}}\]. If the period of oscillation with the two springs in series is T, then
question_answer28) The displacement y of a particle in a medium can be expressed as\[y={{10}^{-6}}\sin \left( 100t+20x+\frac{\pi }{4} \right)m\], where t is in second and \[x\]in metre. The speed of the wave is
question_answer29) A particle of mass m is attached to a spring (of spring constant\[k\] and has a natural angular frequency\[{{\omega }_{0}}\]. An external force F(t) proportional to \[\cos \omega t(\omega \ne {{\omega }_{0}})\]is applied to the oscillator. The time displacement of the oscillator will be proportional to
question_answer30) In forced oscillation of a particle, the amplitude is maximum for a frequency\[{{\omega }_{1}}\]of the force, while the energy is maximum for a frequency \[{{\omega }_{2}}\]of the force, then
A)
\[{{\omega }_{1}}={{\omega }_{2}}\]
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B)
\[{{\omega }_{1}}>{{\omega }_{2}}\]
doneclear
C)
\[{{\omega }_{1}}<{{\omega }_{2}}\]when damping is small and\[{{\omega }_{1}}>{{\omega }_{2}}\] when damping is large
question_answer31) One mole of ideal monatomic gas\[(\gamma =5/3)\]is mixed with one mole of diatomic gas\[(\gamma =7/5)\]What is\[\gamma \]for the mixture? y denotes the ratio of specific heat at constant pressure, to that at constant volume
question_answer32) If the temperature of the sun were to increase from T to 2T and its radius from R to 2R, then the ratio of the radiant energy received on the earth to what it was previously, will be
question_answer34) Two thermally insulated vessels 1 and 2 are filled with air at temperatures\[({{T}_{1}},{{T}_{2}}),\]volumes \[({{V}_{1}},{{V}_{2}}),\]and pressures\[({{p}_{1}},{{p}_{2}}),\]respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be
question_answer36) The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and\[2K\]and thickness\[x\]and\[4x,\]respectively are\[{{T}_{2}},\]and\[{{T}_{1}}({{T}_{2}}>{{T}_{1}})\]. The rate of heat transfer through the slab, in a steady state is\[\left( \frac{A({{T}_{2}}-{{T}_{1}})K}{x} \right)f,\]with f equals
question_answer37) A light ray is incident perpendicular to one face of a\[90{}^\circ \]prism and is totally internally reflected at the glass-air interface. If the angle of reflection is\[45{}^\circ ,\] we conclude that the refractive index\[n\]is
question_answer38) A plano-convex lens of refractive index 1.5 and radius of curvature 30 cm is silvered at the curved surface. Now, this lens has been used to from the image of an object. At what distance from this lens, an object be placed in order to have a real image of the size of the object?
question_answer40) The maximum number of possible interference maxima for slit-separation equal to twice the wavelength in Young's double-slit experiment, is
question_answer41) An electromagnetic wave of frequency \[v=3.0MHz\]passes from vacuum into a dielectric medium with permittivity\[\varepsilon =4.0.\] Then,
A)
wavelength is doubled and the frequency remains unchanged
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B)
wavelength is' doubled and frequency becomes half
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C)
wavelength is halved and frequency remains unchanged
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D)
(d) wavelength and frequency both remain unchanged.
question_answer42) Two spherical conductors B and C having equal radii and carrying equal charges in them repel each other with a force F when kept apart at some distance. A third spherical conductor having same radius as that of B but uncharged, is brought in contact with B, then brought in contact with C and finally removed away from both. The new force of repulsion between B and C is
question_answer43) A charged particle q is shot towards another charged particle\[Q\]which is fixed, with a speed v. It approaches\[Q\]upto a closest distance r and then returns. If q was given a speed 2 v, the closest distance of approach would be
question_answer44) Four charges equal to\[-Q\]are placed at the four comers of a square and a charge q is at its centre. If the system is in equilibrium, the value of q is
question_answer47) The resistance of the series combination of two resistances is S. When they are joined in parallel, the total resistance is P. If\[S=nP,\] then the minimum possible value of n is
question_answer48) An electric current is passed through a circuit containing two wires of the same material, connected in parallel. If the lengths and radii of the wires are in the ratio of 4/3 and 2/3, then the ratio of the currents passing through the wire will be
question_answer49) In a metre bridge experiment, null point is obtained at 20 cm from one end of the wire when resistance\[X\]is balanced against another resistance\[Y\].If\[X<Y,\]then where will be the new position of the null point from the same end, if one decides to balance a resistance of \[4X\]against\[Y\]?
question_answer52) The thermo-emf of a thermocouple varies with the temperature\[\theta \]of the hot junction as\[E=a\theta +b{{\theta }^{2}}\]in volts where the ratio a/b is\[700{}^\circ C\]. If the cold junction is kept at\[0{}^\circ C,\]then the neutral temperature is
A)
\[700{}^\circ C\]
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B)
\[350{}^\circ C\]
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C)
\[1400{}^\circ C\]
doneclear
D)
no neutral temperature is possible for this thermocouple
question_answer53) The electrochemical equivalent of metal is\[3.3\times {{10}^{-7}}\,kg/C\]. The mass of the metal liberated at the cathode when a 3 A current is passed for 2 s, will be
question_answer54) A current\[i\]ampere flows along an infinitely long straight thin walled tube, then the magnetic induction at any point inside the tube is
question_answer55) A long wire carries a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is B. It is then bent into a circular loop of n turns. The magnetic field at the centre of the coil will be
question_answer56) The magnetic field due to a current carrying circular loop of radius 3 cm at a point on the axis at a distance of 4 cm from, the centre is 54\[\mu T\]. What will be its value at tile centre of the loop?
question_answer57) Two long conductors, separated by a distance d. carry currents\[{{I}_{1}}\]and\[{{I}_{2}}\]in the same direction. They exert a force F on each other. Now, the current in one of them is increased to two times and its direction is reversed. The distance is also increased to 3d. The new value of the force between them is
question_answer58) The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is 2 s. The magnet is cut along its length into three equal parts and three parts are then placed on each other with their like poles together. The time period of this combination will be
question_answer60) In an LCR series AC circuit, the voltage across each of the components. L, C and R is 50 V. The voltage across the LC combination will be
question_answer61) A coil having n turns and resistance\[R\,\Omega \]. is connected with a galvanometer of resistance\[4R\,\Omega \]. This combination is moved in time t seconds from a magnetic field\[{{W}_{1}}\]weber to\[{{W}_{2}}\] weber. The induced current in the circuit is
question_answer62) In a uniform magnetic field of induction B, a wire in the form of semi-circle, of radius r rotates about the diameter of the circle with angular frequency\[\omega \]. If the total resistance of the circuit is R, the mean power generated per period of rotation is
question_answer63) In an LCR circuit, capacitance is changed from C to 2 C. For the resonant frequency to remain unchanged, the inductance should be changed from L to
question_answer64) A metal conductor of length 1m rotates vertically about one of its ends at angular velocity 5 rad/s. If the horizontal component of the earth's magnetic field is\[0.2\times {{10}^{-4}}T\]then the emf developed between the two ends of the conductor is
question_answer65) According to Einstein's photoelectric equation, the plot of the kinetic energy of the emitted photoelectrons from a metal\[Vs\]the frequency, of the incident radiation gives a straight line whose slope
A)
depends on the nature of the metal used
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B)
depends on the intensity of the radiation
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C)
depends both on the intensity of the radiation and the metal used
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D)
is the same for all metals and independent of the intensity of the radiation
question_answer66) The work function of a substance is\[4.0\text{ }eV\]. The longest wavelength of light that can cause photoelectron emission from this substance is approximately
question_answer67) A charged oil drop is suspended in uniform field of\[3\times {{10}^{4}}V/m,\]so that it neither falls nor rises. The charge on the drop will be (take the mass of the charge \[=9.9\times {{10}^{-15}}kg\text{ }and\text{ }g=10\text{ }m/{{s}^{2}}\])
question_answer68) A nucleus disintegrates into two nuclear parts which have their velocities in the ratio 2 : 1. The ratio of their nuclear sizes will be
question_answer69) The binding energy per nucleon of deuteron\[(_{1}^{2}H)\]and helium nucleus\[(_{2}^{4}He)\]is,\[1.1\,MeV\]and \[7\text{ }MeV,\]respectively. If two deuteron nuclei react to form a single helium nucleus, then the energy released is
question_answer70) An\[\alpha -\]particle of energy\[5\text{ }MeV\]is scattered through\[180{}^\circ \,\]by a fixed uranium nucleus. The distance of the closest approach is of the order of
question_answer72) For a transistor amplifier in common emitter configuration for load impedance of \[1\,k\Omega ({{h}_{fe}}=50\]and \[{{h}_{oe}}\,=25\,\,\,\mu A/V\]), the current gain is
question_answer77) Consider the ground state of\[Cr\]atom\[(Z=24)\]. The numbers of electrons with the azimuthal quantum numbers, \[l=1\] and 2 are, respectively
question_answer79) The wavelength of the radiation emitted, when in a hydrogen atom electron falls from infinity to stationary state 1, would be (Rydberg constant\[=1.097\times {{10}^{7}}\text{ }{{m}^{-1}})\]
question_answer81) Which one of the following sets of ions represents the collection of isoelectronic species? (At.\[nos.\,F=9,Cl=17,Na=11,Mg=12,\] \[Al=13,K=19,Ca=20,Sc=21)\]
question_answer84) The formation of the oxide ion\[{{O}^{2-}}(g)\]requires first an exothermic and then an exothermic step as shown below \[O(g)+{{e}^{-}}={{O}^{-}}(g);\] \[\Delta {{H}^{o}}=-142\,kJ\,mo{{l}^{-1}}\] \[{{O}^{-}}{{(g)}^{-}}+{{e}^{-}}={{O}^{2}}(g);\] \[\Delta {{H}^{o}}=844\,kJ\,mo{{l}^{-1}}\] This is because
A)
oxygen is more electronegative
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B)
oxygen has high electron affinity
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C)
\[{{O}^{-}}\]ion will tend to resist the addition of another electron
doneclear
D)
\[{{O}^{-}}\]ion has comparatively larger size than oxygen atom
question_answer88) As the temperature is raised from\[20{}^\circ C\]to \[40{}^\circ C,\]the average kinetic energy of neon atoms changes by a factor of which of the following?
question_answer94) \[6.02\times {{10}^{20}}\]molecules of urea are present in 100 mL of its solution. The concentration of urea solution is (Avogadro constant, \[{{N}_{A}}\,=6.02\,\times {{10}^{23}}\,mo{{l}^{-1}}\])
question_answer95) To neutralize completely 20 mL of 0.1M aqueous solution of phosphorous acid \[({{H}_{3}}P{{O}_{3}}),\]the volume of 0.1M aqueous KOH solution required is
question_answer96) For which of the following parameters the structural isomers\[{{C}_{2}}{{H}_{2}}OH\]and\[C{{H}_{3}}OC{{H}_{3}}\] would be expected to have the same values? (Assume ideal behaviour)
A)
Heat of vaporization,
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B)
Vapour pressure at the same temperature
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C)
Boiling points
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D)
Gaseous densities at the same temperature and pressure
question_answer99) What type of crystal defect is indicated in the diagram below?\[N{{a}^{+}},C{{l}^{-}},N{{a}^{+}},C{{l}^{-}},N{{a}^{+}},C{{l}^{-}}\] \[C{{l}^{-}}+C{{l}^{-}}N{{a}^{+}}+N{{a}^{+}}\]\[N{{a}^{+}}C{{l}^{-}}+C{{l}^{-}}N{{a}^{+}}C{{l}^{-}}\]\[C{{l}^{-}}N{{a}^{+}}C{{l}^{-}}N{{a}^{+}}+N{{a}^{+}}\]
question_answer100) An ideal gas expands in volume from \[1\times {{10}^{-3}}{{m}^{3}}\]to\[1\times {{10}^{-2}}{{m}^{3}}\]at\[300\text{ }K\]against a constant pressure of\[1\times {{10}^{5}}N{{m}^{-2}}\]. The work done is
question_answer102) In a first order reaction, the concentration of the reactant, decreases from 0.8 M to 0,4 M in 15 min, The time taken for the concentration to change from 0.1 M to 0.025 M is
question_answer105) The equilibrium constant for the reaction \[{{N}_{2}}(g)+{{O}_{2}}(g)2NO(g)\] at temperature T is \[4\times {{10}^{-4}}\]. The value of\[{{K}_{c}}\] for the reaction \[NO(g)\frac{1}{2}{{N}_{2}}(g)+\frac{1}{2}{{O}_{2}}(g)\]at the same temperature is
question_answer106) The rate equation for the reaction \[A+B\xrightarrow{{}}C\]is found to be rate\[=k[A][B]\] The correct statement in relation to this reaction is that the
A)
unit of k must be\[{{s}^{-1}}\]
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B)
\[{{t}_{1/2}}\]is a constant
doneclear
C)
rate of formation of C is twice the rate of disappearance of A
doneclear
D)
value of k is independent of the initial concentrations of A and B
question_answer107) Consider the following\[E{}^\circ \]values \[E{{{}^\circ }_{F{{e}^{3+}}/F{{e}^{2+}}}}=+0.77\,V\] \[E{{{}^\circ }_{S{{n}^{2+}}/Sn}}=-0.14\,V\] Under standard conditions the potential for the reaction \[Sn(s)+2F{{e}^{3+}}(aq)\xrightarrow{{}}2F{{e}^{2+}}(aq)\] \[+S{{n}^{2+}}(aq)\]is
question_answer108) The molar solubility (in\[mol\text{ }{{L}^{-1}}\]) of a sparingly soluble salt\[M{{X}_{4}}\]is 's'. The corresponding solubility product is\[{{K}_{sp}}\]. s as given in terms of \[{{K}_{sp}}\]by the relation
question_answer109) The standard emf of a cell, involving one electron change is found to be 0.591 V at\[25{}^\circ C\]. The equilibrium constant of the reaction is \[(F=96500\text{ }C\text{ }mo{{l}^{-1}})\]
question_answer110) The enthalpies of combustion of carbon and carbon monoxide are\[-393.5\]and\[-283\text{ }kJ\text{ }mo{{l}^{-1}}\]respectively. The enthalpy of formation of carbon monoxide per mole is
question_answer112) In a cell that utilises the reaction, \[Zn(s)+2{{H}^{+}}(aq)\xrightarrow{{}}Z{{n}^{2+}}(aq)+{{H}_{2}}(g)\] addition of\[{{H}_{2}}S{{O}_{4}}\]to cathode compartment, will
A)
lower the E and shift equilibrium to the left
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B)
lower the E and shift the equilibrium to the right
doneclear
C)
increase the E and shift the equilibrium to the right
doneclear
D)
increase the E and shift the equilibrium to the left
question_answer118) Aluminium chloride exists as dimer,\[A{{l}_{2}}C{{l}_{6}}\]in solid state as well as in solution of non-polar solvents such as benzene. When dissolved in water, it gives
question_answer119) The soldiers of Napolean army while at Alps during freezing winter suffered a serious problem as regards to the tin buttons of their uniforms. White metallic tin buttons got converted to grey powder. This transformation is related to
A)
a change in the crystalline structure of tin
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B)
an interaction with nitrogen of the air at very low temperatures
doneclear
C)
a change in the partial pressure of oxygen in the air
doneclear
D)
an interaction with water vapour contained in the humid air
question_answer120) The\[E_{{{M}^{3+}}/{{M}^{2+}}}^{o}\]values for\[Cr,Mn,Fe\]and\[Co\]are\[-0.41+1.57.+0.77\]and\[+1.97\text{ }V\]respectively. For which one of these metals the change in oxidation state from +2 to +3 is easiest?
question_answer121) Excess of\[KI\]reacts with\[CuS{{O}_{4}}\]solution and then\[N{{a}_{2}}{{S}_{2}}{{O}_{3}}\]solution is added to it. Which of the statements is incorrect for this reaction?
question_answer122) Among the properties (A) reducing (B) oxidizing (C) completing, the set of properties shown by\[C{{N}^{-}}\]ion towards metal species is
question_answer125) Coordination compounds have great importance in biological systems. In this context which of the following statements is incorrect?
A)
Chlorophylls are green pigments in plants and contain calcium
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B)
Haemoglobin is the red pigment of blood and contains iron
doneclear
C)
Cyanocobalamin is vitamin \[{{B}_{12}}\] and contains cobalt
question_answer128) The correct order of magnetic moments (spin only values in BM) among the following is (At. nos.\[Mn=25,\text{ }Fe=26,\text{ }Co=27\])
question_answer129) Consider the following nuclear reactions \[_{92}^{238}M\to _{y}^{x}N+2_{2}^{4}He;_{y}^{x}N\to _{B}^{A}L+2{{\beta }^{+}}\] The number of neutrons in the element L is
question_answer130) The half-life of a radioisotope is four hours. If the initial mass of the isotope was 200 g, the mass remaining after 24 h undecayed is
question_answer132) The ammonia evolved from the treatment of 0.30 g of an organic compound for the estimation of nitrogen was passed in 100 mL of 0.1 M sulphuric acid. The excess of acid required 20 mL of 0.5 M sodium hydroxide solution for complete neutralization. The organic compound is
question_answer139) Consider the acidity of the carboxylic acids (i) \[PhCOOH\] (ii) \[o-N{{O}_{2}}{{C}_{6}}{{H}_{4}}COOH\] (iii) \[p-N{{O}_{2}}{{C}_{6}}{{H}_{4}}COOH\] (iv) \[m-N{{O}_{2}}{{C}_{6}}{{H}_{4}}COOH\] Which of the following order is correct?
question_answer149) Insulin production and its action in human body are responsible for the level of diabetes. This compound belongs to which of the following categories?
question_answer164) If one root of the equation\[{{x}^{2}}+px+12=0\]is 4, while the equation\[{{x}^{2}}+px+q=0\]has equal roots, then the value of 'q' is
question_answer165) The coefficient of the middle term in the binomial expansion in powers of\[x\]of\[{{(1+\alpha x)}^{4}}\] and of\[{{(1-ax)}^{6}}\]is the same, if a equals
question_answer167) If\[{{S}_{n}}=\sum\limits_{r=0}^{n}{\frac{1}{^{n}{{C}_{r}}}}\]and\[{{t}_{n}}=\sum\limits_{r=0}^{n}{\frac{r}{^{n}{{C}_{r}}}}\]then\[\frac{{{t}_{n}}}{{{S}_{n}}}\]is equal to
question_answer168) Let\[{{T}_{r}}\]be the rth term of an AP whose first term is a and common difference is d. If for some positive integers \[m,n,m\ne n,{{T}_{m}}=\frac{1}{n}\]and\[{{T}_{n}}=\frac{1}{m},\]then\[a-d\]equals
question_answer169) The sum of the first\[n\]terms of the series \[{{1}^{2}}+{{2.2}^{2}}+{{3}^{2}}+{{2.4}^{2}}+{{5}^{2}}+{{2.6}^{2}}+....\]is \[\frac{n{{(n+1)}^{2}}}{2},\] when n is even. When n is odd, the sum is
question_answer172) If\[u=\sqrt{{{a}^{2}}{{\cos }^{2}}\theta +{{b}^{2}}{{\sin }^{2}}\theta }\] \[+\sqrt{{{a}^{2}}{{\sin }^{2}}\theta +{{b}^{2}}{{\cos }^{2}}\theta },\] then the difference between the maximum and minimum values of\[{{u}^{2}}\]is given by
question_answer173) The sides of a triangle are\[\sin \alpha ,\text{ }\cos \alpha \]and\[\sqrt{1+\sin \alpha \cos \alpha }\]for some\[0<\alpha <\frac{\pi }{2}\]. Then, the greatest angle of the triangle is
question_answer174) A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is \[60{}^\circ \]and when he retires 40 m away from the tree, the angle of elevation becomes\[30{}^\circ \]. The breadth of the river is
question_answer178) If\[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{a}{x}+\frac{b}{{{x}^{2}}} \right)}^{2x}}={{e}^{2}},\]then the values of a and b are
question_answer182) A function\[y=f(x)\]has a second order derivative\[f'\,'=6(x-1)\]. If its graph passes through the point (2, 1) and at that point, the tangent to the graph is\[y=3x-5,\]then the function is
question_answer191) If\[f(x)=\frac{{{e}^{x}}}{1+{{e}^{x}}},{{I}_{1}}=\int_{f(-a)}^{f(a)}{x}g\{x(1-x)\}dx\]and\[{{I}_{2}}=\int_{f(-a)}^{f(a)}{g\{x(1-x)\}dx},\] then the value of\[\frac{{{I}_{2}}}{{{I}_{1}}}\]is
question_answer195) Let A (2, - 3) and B (- 2,1) be vertices of a\[\Delta ABC\] If the centroid of this triangle moves on the line \[2x+3y=1,\]then the locus of the vertex C is the line
question_answer196) The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is -1, is
question_answer199) If a circle passes through the point [a, b) and cuts the circle\[{{x}^{2}}+{{y}^{2}}=4\]orthogonally, then the locus of its centre is
question_answer200) A variable circle passes through the fixed point \[A(p,\text{ }q)\]and touches x-axis. The locus of the other end of the diameter through A is
question_answer201) If the lines\[2x+3y+1=0\]and\[3x-y-4=0\]lie along diameters of a circle of circumference\[10\pi ,\]then the equation of the circle is
question_answer203) If\[a\ne 0\]and the line\[2bx+3cy+4d=0\]passes through the points of intersection of the parabolas\[{{y}^{2}}=4\text{ }ax\]and\[{{x}^{2}}=4\text{ }ay,\]then
question_answer204) The eccentricity of an ellipse with its centre at the origin, is\[\frac{1}{2}\]. If one of the directories is\[x=4,\]then the equation of the ellipse is
question_answer205) A line makes the same angle\[\theta \]with each of the x and z-axes. If the angle\[\beta ,\]which it makes with y-axis, is such that\[{{\sin }^{2}}\beta =3{{\sin }^{2}}\theta ,\]then\[\cos \theta \]equals
question_answer207) A line with direction cosines proportional to 2,1,2 meets each of the lines\[x=y+a=z\]and \[x+a=2y=2z\]. The coordinates of each of the points of intersection are given by
question_answer208) If the straight lines \[x=1+s,y=-3-\lambda s,\] \[z=1+\lambda \,s\]and\[x=\frac{t}{2},y=1+t,z=2-t,\]with parameters s and t respectively, are coplanar, then\[\lambda ,\]equals
question_answer209) The intersection of the spheres \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+7x-2y-z=13\] and \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-3x+3y+4z=8\] is the same as the intersection of one of the sphere and the plane
question_answer210) Let a, b and c be three non-zero vectors such that no two of these are collinear. If the vector \[a+2b\]is collinear with c and\[b+3c\]is collinear with a\[(\lambda \]being some non-zero scalar), then \[a+2b+6c\]equals
question_answer211) A particle is acted upon by constant forces \[4\hat{i}+\hat{j}-3\hat{k}\]and\[3\hat{i}+\hat{j}-\hat{k}\]which displace it from a point\[\hat{i}+2\hat{j}-3\hat{k}\]to the point \[5\hat{i}+4\hat{j}+\hat{k}\]. The work done in standard units by the forces is given by
question_answer212) If a, b, c are non-coplanar vectors and\[\lambda \]is a real number, then the vectors\[a+2b+3c,\lambda b+4c\]and\[(2\lambda -1)c\]are non-coplanar for
question_answer213) Let\[u,v,w\]be such that\[|u|=1,|v|=2,\]\[|w|=3.\]If the projection v along u is equal to that of w along u and\[v,w\]are perpendicular to each other, then\[|u-v+w|\]equals
question_answer214) Let a,b and c be non-zero vectors such that \[(a\times b)\times c=\frac{1}{3}|b||c|a.\]If\[\theta \]is the acute angle between the vectors b and c, then\[\sin \theta \]equals
question_answer215) Consider the following statements (1) Mode can be computed from histogram. (2) Median is not independent of change of scale. (3) Variance is independent of change of origin and scale. (4) Which of these is/are correct?
question_answer216) In a series of\[2n\]observations, half of them equal\[a\]and remaining half equal\[-a\]. If the standard deviation of the observations is 2, then \[|a|\] equals
question_answer217) The probability that A speaks truth is 4/5 while this probability for B is 3/4. The probability that they contradict each other when asked to speak on a fact, is
question_answer220) With two forces acting at a point, the maximum effect is obtained when their resultant is 4N. If they act at right angles, then their resultant is 3N. Then, the forces are
question_answer221) In a right angled \[\Delta ABC,\,\,\angle A={{90}^{\text{o}}}\] and sides a,b,c are respectively, 5 cm, 4 cm and 3 cm. If a force F has moments 0, 9 and 16 (in N cm) units respectively about vertices A, B and C, the magnitude of F is
question_answer222) Three forces P, Q and R acting along\[IA,IB\]and \[IC,\]where\[I\]is the incentre of a\[\Delta ABC,\]are in equilibrium. Then, P : Q : R is
question_answer223) A particle moves towards East from a point A to a point B at the rate of 4 km/h and then towards North from B to C at rate of 5 km/h. If AB = 12 km and BC = 5 km, then its average speed for its journey from A to C and resultant average velocity direct from A to C are respectively
question_answer224) A velocity 1/4 m/s is resolved into two components along OA and OB making angles \[30{}^\circ \]and\[45{}^\circ \]respectively with the given velocity, Then, the component along OB is
question_answer225) If\[{{t}_{1}}\]and\[{{t}_{2}}\]are the times of flight of two particles having the same initial velocity u and range R on the horizontal, then\[t_{1}^{2}+t_{2}^{2}\]is equal to