If $$\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$$, then the angle between $$A$$ and $$B$$ is
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If $$\vec{A} \times \vec{B} = \vec{B} \times \vec{A}$$, then the angle between $$A$$ and $$B$$ is
Which one of the following represents the correct dimensions of the coefficient of viscosity?
A ball is released from the top of a tower of height $$h$$ metres. It takes $$T$$ seconds to reach the ground. What is the position of the ball in $$T/3$$ seconds?
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An automobile travelling with 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
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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 person starts running with a constant speed $$v_0/2$$ to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection?
A projectile can have the same range $$R$$ for two angles of projection. If $$T_1$$ and $$T_2$$ be the time of flights in the two cases, then the product of the two time of flights is directly proportional to
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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
A machine gun fires a bullet of mass $$40$$ g with a velocity $$1200 \text{ ms}^{-1}$$. 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?
Two masses $$m_1 = 5$$ kg and $$m_2 = 4.8$$ kg tied to a string are hanging over a light frictionless pulley. What is the acceleration of the masses when lift free to move $$(g = 9.8 \text{ m/s}^2)$$?
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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 $$g = 10 \text{ m/s}^2$$)
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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
Which of the following statements is false for a particle moving in a circle with a constant angular speed?
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?
A force $$\vec{F} = (5\hat{i} + 3\hat{j} + 2\hat{k})N$$ is applied over a particle which displaces it from origin to the point $$\vec{r} = (2\hat{i} - \hat{j})$$ m. The work done on the particle in joules is
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
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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
A wire fixed at the upper end stretches by length $$\ell$$ by applying a force $$F$$. The work done in stretching is
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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?
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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 $$I_A$$ and $$I_B$$ such that
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A satellite of mass $$m$$ revolves around the earth of radius $$R$$ at a height $$x$$ from its surface. If $$g$$ is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is
The time period of an earth satellite in circular orbit is independent of
If $$g$$ is the acceleration due to gravity on the earth's surface, the gain in the potential energy of object of mass $$m$$ raised from the surface of the earth to a height equal to the radius $$R$$ of the earth is
Suppose the gravitational force varies inversely as the $$n$$th power of distance. Then the time period planet in circular orbit of radius $$R$$ around the sun will be proportional to
Spherical balls of radius $$R$$ are falling in a viscous fluid of viscosity $$\eta$$ with a velocity $$v$$. The retarding viscous force acting on the spherical ball is
If two soap bubbles of different radii are connected by a tube,
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 earth to what it was previously will be
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The temperature of 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 $$f$$ equal to
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The thermistors are usually made of
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Time taken by a $$836$$ W heater to heat one litre of water from $$10^\circ$$C to $$40^\circ$$C is
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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
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Which of the following statements is correct for any thermodynamic system?
One mole of ideal monoatomic gas $$(\gamma = 5/3)$$ is mixed with one mole of diatomic gas $$(\gamma = 7/5)$$. What is $$\gamma$$ for the mixture? $$\gamma$$ denotes the ratio of specific heat at constant pressure, to that at constant volume.
Two thermally insulated vessels 1 and 2 are filled with air at temperatures $$(T_1, T_2)$$, volume $$(V_1, V_2)$$ and pressure $$(P_1, P_2)$$ respectively. If the valve joining two vessels is opened, the temperature inside the vessel at equilibrium will be
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 $$t_0$$ in air. Neglecting frictional force of water and given that the density of the bob is $$\left(\frac{4}{3}\right) \times 1000 \text{ kg/m}^3$$. What relationship between $$t$$ and $$t_0$$ is true?
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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
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The total energy of particle, executing simple harmonic motion is
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 $$\cos\omega t$$ $$(\omega \neq \omega_0)$$ is applied to the oscillator. The time displacement of the oscillator will be proportional to
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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
The displacement $$y$$ of a particle in a medium can be expressed as $$y = 10^{-6} \sin(110t + 20x + \pi/4)$$ m, where $$t$$ is in seconds and $$x$$ in meter. The speed of the wave is
Two spherical conductor $$B$$ 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 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
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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 $$q$$ were given a speed $$2v$$, the closest distances of approach would be
Four charges equal to $$-Q$$ are placed at the four corners of a square and a charge $$q$$ is at its centre. If the system is in equilibrium the value of $$q$$ is
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A charged oil drop is suspended in a 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 and $$g = 10 \text{ m/s}^2$$)
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An $$\alpha$$-particle of energy $$5$$ MeV is scattered through $$180^\circ$$ by a fixed uranium nucleus. The distance of the closest approach is of the order of
The total current supplied to the circuit by the battery is
The resistance of the series combination of two resistances is $$S$$. When they are joined in parallel through total resistance is $$P$$. If $$S = nP$$, then the minimum possible value of $$n$$ is
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An electric current is passed through a circuit containing two wires of the same material, connected in parallel. If the length 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
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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 $$Y$$?
The electrochemical equivalent of a metal is $$3.3 \times 10^{-7}$$ kg per coulomb. The mass of the metal liberated at the cathode when a $$3$$ A current is passed for $$2$$ seconds will be
A piece of copper and another of germanium are cooled from room temperature to $$77$$ K, the resistance of
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
The materials suitable for making electromagnets should have
A current $$I$$ ampere flows along an infinitely long straight thin-walled tube, then the magnetic induction at any point inside the tube is
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
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 the centre of the Loop?
Two long conductors, separated by a distance $$d$$ carry current $$I_1$$ and $$I_2$$ in the same direction. They exert a force $$F$$ on each other. Now the current in one of them increased to two times and its direction reversed. The distance is also increased to $$3d$$. The new value of the force between them is
A coil having $$n$$ turns and 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
In a uniform magnetic field of induction $$B$$ a wire in the form of semicircle of radius $$r$$ rotates about the diameter of the circle with angular frequency $$\omega$$. The axis of rotation is perpendicular to the field. If the total resistance of the circuit is $$R$$ the mean power generated per period of rotation is
A metal conductor of length $$1$$ m rotates vertically about one of its ends at angular velocity $$5$$ radians per second. If the horizontal component of earth's magnetic field is $$0.3 \times 10^{-4}$$ T, then the e.m.f. developed between the two ends of the conductor is
Alternating current can not be measured by D.C. ammeter because
In an LCR series a.c. circuit, the voltage across each of the components, $$L, C$$ and $$R$$ is $$50$$ V. The voltage across the LC combination will be
In a LCR circuit capacitance is changed from $$C$$ to $$2C$$. For the resonant frequency to remain unchanged, the inductance should be changed from $$L$$ to
An electromagnetic wave of frequency $$v = 3.0$$ MHz passes from vacuum into a dielectric medium with permittivity $$\varepsilon = 4.0$$. Then
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 $$n$$
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A plane 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 form 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?
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The angle of incidence at which reflected light totally polarized for reflection from air to glass (refractive index $$n$$), is
The maximum number of possible interference maxima for slit-separation equal to twice the wavelength in Young's double-slit experiment is
A radiation of energy $$E$$ falls normally on a perfectly reflecting surface. The momentum transferred to the surface is
According to Einstein's photoelectric equation, the plot of the kinetic energy of the emitted photo electrons from a metal $$V_s$$ the frequency, of the incident radiation gives straight line whose slope
The work function of a substance is $$4.0$$ eV. Then longest wavelength of light that can cause photoelectron emission from this substance approximately
The manifestation of band structure in solids is due to
A nucleus disintegrates into two nuclear parts which have their velocities in the ratio $$2 : 1$$. The ratio of their nuclear sizes will be
The binding energy per nucleon of deuteron $$\binom{2}{1}H$$ and helium nucleus $$\binom{4}{2}He$$ is $$1.1$$ MeV and $$7$$ MeV respectively. If two deuteron nuclei react to form a single helium nucleus, then the energy released is
When npn transistor is used as amplifier
For a transistor amplifier in common emitter configuration having load impedance of $$1$$ k$$\Omega$$ ($$h_{fe} = 50$$ and $$h_{oe} = 25$$) the current gain is
When p-n junction diode is forward biased
$$6.02 \times 10^{20}$$ molecules of urea are present in $$100$$ mL of its solution. The concentration of urea solution is
To neutralize completely $$20$$ mL of $$0.1$$ M aqueous solution of phosphorous acid $$(H_3PO_3)$$, the volume of $$0.1$$ M aqueous KOH solution required is
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Which of the following sets of quantum numbers is correct for an electron in $$4f$$ orbital?
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Consider the ground state of Cr atom $$(Z = 24)$$. The number of electrons with the azimuthal quantum numbers $$l = 1$$ and $$2$$ are respectively
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The wavelength of the radiation emitted, when in hydrogen atom electron falls from infinity to stationary state $$1$$, would be (Rydberg constant $$= 1.097 \times 10^7$$ m$$^{-1}$$)
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Which one the following sets of ions represents the collection of isoelectronic species?
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Which one the following ions has the highest value of ionic radius?
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Consider the following nuclear reactions $$_{92}^{238}M \rightarrow {}_y^x N + {}_2^4 He$$; $$_y^N N \rightarrow {}_B^{A} L + 2\beta^+$$. The number of neutrons in the element $$L$$ is
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The correct order of bond angles (smallest first) in $$H_2S, NH_3, BF_3$$ and $$SiH_4$$ is
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The bond order in NO is $$2.5$$ while that in $$NO^+$$ is $$3$$. Which of the following statements is true for these two species?
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The states of hybridization of boron and oxygen atoms in boric acid $$(H_3BO_3)$$ are respectively
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Which one of the following has the regular tetrahedral structure?
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The maximum number of $$90^\circ$$ angles between bond pair of electrons is observed in
Which one of the following aqueous solutions will exhibit highest boiling point?
Which one the following does not have $$sp^2$$ hybridized carbon?
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?
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In Vander Waals equation of state of the gas law, the constant '$$b$$' is a measure of
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The formation of the oxide ion $$O^{2-}(g)$$ requires first an exothermic and then an endothermic step as shown below: $$O(g) + e^- \rightarrow O^-(g)$$, $$\Delta H^\circ = -142$$ kJ mol$$^{-1}$$; $$O^-(g) + e^- \rightarrow O^{2-}(g)$$, $$\Delta H^\circ = 844$$ kJ mol$$^{-1}$$. This is because
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An ideal gas expands in volume from $$1 \times 10^{-3}$$ m$$^3$$ to $$1 \times 10^{-2}$$ m$$^3$$ at $$300$$ K against a constant pressure of $$1 \times 10^5$$ N m$$^{-2}$$. The work done is
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The enthalpies of combustion of carbon and carbon monoxide are $$-393.5$$ and $$-283$$ kJ mol$$^{-1}$$ respectively. The enthalpy of formation of carbon monoxide per mole is
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What is the equilibrium expression for the reaction $$P_{4(s)} + 5O_{2(g)} \rightleftharpoons P_4O_{10(s)}$$?
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For the reaction, $$CO(g) + Cl_2(g) \rightleftharpoons COCl_2(g)$$ the $$\frac{K_p}{K_c}$$ is equal to
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The equilibrium constant for the reaction $$N_2(g) + O_2(g) \rightleftharpoons 2NO(g)$$ at temperature $$T$$ is $$4 \times 10^{-4}$$. The value of $$K_c$$ for the reaction $$NO(g) \rightleftharpoons \frac{1}{2}N_2(g) + \frac{1}{2}O_2(g)$$ at the same temperature is
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Among $$Al_2O_3, SiO_2, P_2O_3$$ and $$SO_2$$ the correct order of acid strength is
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The conjugate base of $$H_2PO_4^-$$ is
The molar solubility product is $$K_{sp}$$. '$$s$$' is given in terms of $$K_{sp}$$ by the relation
Excess of KI reacts with $$CuSO_4$$ solution and then $$Na_2S_2O_3$$ solution is added to it. Which of the statements is incorrect for this reaction?
Among the properties (a) reducing (b) oxidising (c) complexing, the set of properties shown by $$CN^-$$ ion towards metal species is
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Beryllium and aluminium exhibit many properties which are similar. But the two elements differ in
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Aluminium chloride exists as dimer, $$Al_2Cl_6$$ in solid state as well as in solution of non-polar solvents such as benzene. When dissolved in water, it gives
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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
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For which of the following parameters the structural isomers $$C_2H_5OH$$ and $$CH_3OCH_3$$ would be expected to have the same values? (Assume ideal behaviour)
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The compound formed in the positive test for nitrogen with the Lassaigne solution of an organic compound is
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
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The IUPAC name of the compound
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Which of the following will have meso-isomer also?
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Rate of the reaction
is fastest when Z is
Amongst the following compounds, the optically active alkane having lowest molecular mass is
Which of the following compound is not chiral?
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Which one of the following has the minimum boiling point?
The smog is essentially caused by the presence of
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What type of crystal defect is indicated in the diagram below? Na$$^+$$ Cl$$^-$$ Na$$^+$$ Cl$$^-$$ Na$$^+$$ Cl$$^-$$ Cl$$^-$$ $$\square$$ Cl$$^-$$ $$\square$$ Na$$^+$$ $$\square$$ Na$$^+$$ Na$$^+$$ Cl$$^-$$ $$\square$$ Cl$$^-$$ Na$$^+$$ Cl$$^-$$ Cl$$^-$$ Na$$^+$$ Cl$$^-$$ Na$$^+$$ $$\square$$ Na$$^+$$
Which of the following liquid pairs shows a positive deviation from Raoult's law?
In hydrogen-oxygen fuel cell, combustion of hydrogen occurs to
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Consider the following $$E^\circ$$ values: $$E^\circ_{Fe^{3+}/Fe^{2+}} = 0.77$$ V; $$E^\circ_{Sn^{2+}/Sn} = -0.14$$ V. Under standard conditions the potential for the reaction $$Sn(s) + 2Fe^{3+}(aq) \rightarrow 2Fe^{2+}(aq) + Sn^{2+}(aq)$$ is
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The standard e.m.f 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 ($$F = 96{,}500$$ C mol$$^{-1}$$; $$R = 8.314$$ J K$$^{-1}$$ mol$$^{-1}$$)
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The limiting molar conductivities $$\Lambda^\circ$$ for NaCl, KBr and KCl are $$126, 152$$ and $$150$$ S cm$$^2$$ mol$$^{-1}$$ respectively. The $$\Lambda^\circ$$ for NaBr is
In a cell that utilises the reaction $$Zn(s) + 2H^+(aq) \rightarrow Zn^{2+}(aq) + H_2(g)$$ addition of $$H_2SO_4$$ to cathode compartment, will
The $$E^\circ_{M^{3+}/M^{2+}}$$ values for Cr, Mn, Fe and Co are $$-0.41, +1.57, +0.77$$ and $$+1.97$$ V respectively. For which one of these metals the change in oxidation state form $$+2$$ to $$+3$$ is easiest?
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In first order reaction, the concentration of the reactant decreases from $$0.8$$ M to $$0.4$$ M in $$15$$ minutes. The time taken for the concentration to change from $$0.1$$ M to $$0.025$$ M is
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The rate equation for the reaction $$2A + B \rightarrow C$$ is found to be: rate $$= k[A][B]$$. The correct statement in relation to this reaction is that the
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The half-life of a radioisotope is four hours. If the initial mass of the isotope was $$200$$ g, the mass remaining after $$24$$ hours undecayed is
Which one of the following ores is best concentrated by froth-flotation method?
Which among the following factors is the most important in making fluorine the strongest oxidizing halogen?
Which one the following statement regarding helium is incorrect?
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One mole of magnesium nitride on the reaction with an excess of water gives
Of the following outer electronic configurations of atoms, the highest oxidation state is achieved by which one of them?
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Cerium $$(Z = 58)$$ is an important member of the lanthanoids. Which of the following statements about cerium is incorrect?
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The coordination number of central metal atom in a complex is determined by
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Which one of the following complexes in an outer orbital complex?
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Coordination compound have great importance in biological systems. In this context which of the following statements is incorrect?
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Which one the following has largest number of isomers?
The correct order of magnetic moments (spin only values in B.M.) among is (Atomic numbers: Mn = 25, Fe = 26, Co = 27)
The compound formed on heating chlorobenzene with chloral in the presence concentrated sulphuric acid is
Acetyl bromide reacts with excess of $$CH_3MgI$$ followed by treatment with a saturated solution of $$NH_4Cl$$ given
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Among the following compound which can be dehydrated very easily is
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Which one of the following reduced with zinc and hydrochloric acid to give the corresponding hydrocarbon?
Consider the acidity of the carboxylic acids:
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On mixing ethyl acetate with aqueous sodium chloride, the composition of the resultant solution is
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Which of the following undergoes reaction with $$50\%$$ sodium hydroxide solution to give the corresponding alcohol and acid?
Which of the following is the strongest base?
Identify the correct statements regarding enzymes
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Insulin production and its action in human body are responsible for the level of diabetes. This compound belongs to which of the following categories?
Which one of the following statements is false?
Which base is present in RNA but not in DNA?
Let two numbers have arithmetic mean $$9$$ and geometric mean $$4$$. Then these numbers are the roots of the quadratic equation
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If $$(1 - p)$$ is a root of quadratic equation $$x^2 + px + (1 - p) = 0$$, then its roots are
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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
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Let $$z, w$$ be complex numbers such that $$\bar{z} + i\bar{w} = 0$$ and $$\arg zw = \pi$$. Then $$\arg z$$ equals
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If $$z = x - iy$$ and $$z^{1/3} = p + iq$$, then $$\frac{\left(\frac{x}{p} + \frac{y}{q}\right)}{(p^2 + q^2)}$$ is equal to
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If $$|z^2 - 1| = |z|^2 + 1$$, then $$z$$ lies on
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How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?
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The number of ways of distributing $$8$$ identical balls in $$3$$ distinct boxes so that none of the boxes is empty is
Let $$T_r$$ be the $$r$$th term of an A.P. whose first term is $$a$$ and common difference is $$d$$. If for some positive integers $$m, n, m \neq n, T_m = \frac{1}{n}$$ and $$T_n = \frac{1}{m}$$, then $$a - d$$ equals
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The sum of the first $$n$$ terms of the series $$1^2 + 2 \cdot 2^2 + 3^2 + 2 \cdot 4^2 + 5^2 + 2 \cdot 6^2 + \ldots$$ is $$\frac{n(n+1)^2}{2}$$ when $$n$$ is even. When $$n$$ is odd the sum is
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The sum of series $$\frac{1}{2!} + \frac{1}{4!} + \frac{1}{6!} + \ldots$$ is
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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
Let $$S(K) = 1 + 3 + 5 + \ldots + (2K - 1) = 3 + K^2$$. Then which of the following is true?
The coefficient of the middle term in the binomial expansion in powers of $$x$$ of $$(1 + \alpha x)^4$$ and of $$(1 - \alpha x)^6$$ is the same if $$\alpha$$ equals
The coefficient of $$x^n$$ in expansion of $$(1 + x)(1 - x)^n$$ is
If $$S_n = \sum_{r=0}^n \frac{1}{{}^n C_r}$$ and $$t_n = \sum_{r=0}^n \frac{r}{{}^n C_r}$$, then $$\frac{t_n}{S_n}$$ is equal to
Let $$\alpha, \beta$$ be such that $$\pi < \alpha - \beta < 3\pi$$. If $$\sin\alpha + \sin\beta = -\frac{21}{65}$$ and $$\cos\alpha + \cos\beta = -\frac{27}{65}$$, then the value of $$\cos\frac{\alpha - \beta}{2}$$ is
Let $$A(2, -3)$$ and $$B(-2, 1)$$ be vertices of a triangle $$ABC$$. If the centroid of this triangle moves on the line $$2x + 3y = 1$$, then the locus of the vertex $$C$$ is the line
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The equation of the straight line passing through the point $$(4, 3)$$ and making intercepts on the co-ordinate axes whose sum is $$-1$$ is
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If one of the lines given by $$6x^2 - xy + 4cy^2 = 0$$ is $$3x + 4y = 0$$, then $$c$$ equals
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A line makes the same angle $$\theta$$, with each of the $$x$$ and $$z$$ axis. If the angle $$\beta$$, which it makes with $$y$$-axis, is such that $$\sin^2 \beta = 3 \sin^2 \theta$$, then $$\cos^2 \theta$$ equals
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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
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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
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The intercept on the line $$y = x$$ by the circle $$x^2 + y^2 - 2x = 0$$ is $$AB$$. Equation of the circle on $$AB$$ as a diameter is
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A variable circle passes through the fixed point $$A(p, q)$$ and touches $$x$$-axis. The locus of the other end of the diameter through $$A$$ is
If $$a \neq 0$$ and the line $$2bx + 3cy + 4d = 0$$ passes through the points of intersection of the parabolas $$y^2 = 4ax$$ and $$x^2 = 4ay$$, then
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The eccentricity of an ellipse, with its centre at the origin, is $$\frac{1}{2}$$. If one of the directrices is $$x = 4$$, then the equation of the ellipse is
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If $$\lim_{x \to \infty}\left(1 + \frac{a}{x} + \frac{b}{x^2}\right)^{2x} = e^2$$, then the values of $$a$$ and $$b$$, are
Let $$f(x) = \frac{1 - \tan x}{4x - \pi}, x \neq \frac{\pi}{4}, x \in \left[0, \frac{\pi}{2}\right]$$. If $$f(x)$$ is continuous in $$\left[0, \frac{\pi}{2}\right]$$, then $$f\left(\frac{\pi}{4}\right)$$ is
$$\lim_{n \to \infty} \sum_{r=1}^n \frac{1}{n} e^{r/n}$$ is
Consider the following statements:
(a) Mode can be computed from histogram.
(b) Median is not independent of change of scale.
(c) Variance is independent of change of origin and scale.
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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
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 $$40$$ meter away from the tree the angle of elevation becomes $$30^\circ$$. The breadth of the river is
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 the 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
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The sides of a triangle are $$\sin\alpha, \cos\alpha$$ and $$\sqrt{1 + \sin\alpha \cos\alpha}$$ for some $$0 < \alpha < \frac{\pi}{2}$$. Then the greatest angle of the triangle is
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Let $$R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$$ be a relation on the set $$A = \{1, 2, 3, 4\}$$. The relation $$R$$ is
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Let $$A = \begin{pmatrix} 0 & 0 & -1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$$. The only correct statement about the matrix $$A$$ is
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Let $$A = \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{pmatrix}$$ and $$10 B = \begin{pmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{pmatrix}$$. If $$B$$ is the inverse of matrix $$A$$, then $$\alpha$$ is
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If $$a_1, a_2, a_3, \ldots, a_n, \ldots$$ are in G.P., then the value of the determinant $$\begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}$$ is
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The range of the function $$f(x) = {}^{7-x} P_{x-3}$$ is
If $$f: R \to S$$, defined by $$f(x) = \sin x - \sqrt{3}\cos x + 1$$, is onto, then the interval of $$S$$ is
The graph of the function $$y = f(x)$$ is symmetrical about the line $$x = 2$$, then
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The domain of the function $$f(x) = \frac{\sin^{-1}(x - 3)}{\sqrt{9 - x^2}}$$ is
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If $$x = e^{y + e^{y + \ldots \infty}}, x > 0$$, then $$\frac{dy}{dx}$$ is
A point on the parabola $$y^2 = 18x$$ at which the ordinate increases at twice the rate of the abscissa is
A function $$y = f(x)$$ has a second order derivative $$f''(x) = 6(x - 1)$$. 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
The normal to the curve $$x = a(1 + \cos\theta), y = a\sin\theta$$ at '$$\theta$$' always passes through the fixed point
If $$2a + 3b + 6c = 0$$, then at least one root of the equation $$ax^2 + bx + c = 0$$ lies in the interval
If the sum of the slopes of the lines given by $$x^2 - 2cxy - 7y^2 = 0$$ is four times their product, then $$c$$ has the value
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If $$\int \frac{\sin x}{\sin(x - \alpha)} dx = Ax + B \log \sin(x - \alpha) + C$$, then value of $$(A, B)$$ is
$$\int \frac{dx}{\cos x - \sin x}$$ is equal to
The value of $$\int_{-2}^3 |1 - x^2| dx$$ is
The value of $$I = \int_0^{\pi/2} \frac{(\sin x + \cos x)^2}{\sqrt{1 + \sin 2x}} dx$$ is
If $$\int_0^\pi x f(\sin x) dx = A \int_0^{\pi/2} f(\sin x) dx$$, then $$A$$ is
If $$f(x) = \frac{e^x}{1 + e^x}$$, $$I_1 = \int_{f(-a)}^{f(a)} xg\{x(1 - x)\} dx$$ and $$I_2 = \int_{f(-a)}^{f(a)} g\{x(1 - x)\} dx$$, then the value of $$\frac{I_2}{I_1}$$ is
The area of the region bounded by the curves $$y = |x - 2|, x = 1, x = 3$$ and the $$x$$-axis is
The differential equation for the family of curves $$x^2 + y^2 - 2ay = 0$$, where $$a$$ is an arbitrary constant is
The solution of the differential equation $$y \, dx + (x + x^2 y) dy = 0$$ is
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 co-planar then $$\lambda$$ equals
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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero vectors such that no two of these are collinear. If the vector $$\vec{a} + 2\vec{b}$$ is collinear with $$\vec{c}$$ and $$\vec{b} + 3\vec{c}$$ is collinear with $$\vec{a}$$ ($$\lambda$$ being some non-zero scalar) then $$\vec{a} + 2\vec{b} + 6\vec{c}$$ equals
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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
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If $$\vec{a}, \vec{b}, \vec{c}$$ are non-coplanar vectors and $$\lambda$$ is a real number, then the vectors $$\vec{a} + 2\vec{b} + 3\vec{c}, \lambda \vec{b} + 4\vec{c}$$ and $$(2\lambda - 1)\vec{c}$$ are non-coplanar for
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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be non-zero vectors such that $$(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$$. If $$\theta$$ is the acute angle between the vectors $$\vec{b}$$ and $$\vec{c}$$, then $$\sin\theta$$ equals
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With two forces acting at a point, the maximum effect is obtained when their resultant is $$4$$ N. If they act at right angles, then their resultant is $$3$$ N. Then the forces are
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In a right angle $$\triangle ABC, \angle A = 90^\circ$$ and sides $$a, b, c$$ are respectively, $$5$$ cm, $$4$$ cm and $$3$$ cm. If a force $$\vec{F}$$ has moments $$0, 9$$ and $$16$$ in N cm. units respectively about vertices $$A, B$$ and $$C$$, then magnitude of $$\vec{F}$$ is
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Three forces $$\vec{P}, \vec{Q}$$ and $$\vec{R}$$ acting along $$IA, IB$$ and $$IC$$, where $$I$$ is the incentre of a $$\triangle ABC$$, are in equilibrium. Then $$\vec{P} : \vec{Q} : \vec{R}$$ is
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A velocity $$\frac{1}{4}$$ 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
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Distance between two parallel planes $$2x + y + 2z = 8$$ and $$4x + 2y + 4z + 5 = 0$$ is
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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 co-ordinates of each of the point of intersection are given by
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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
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Let $$\vec{u}, \vec{v}, \vec{w}$$ be such that $$|\vec{u}| = 1, |\vec{v}| = 2, |\vec{w}| = 3$$. If the projection $$\vec{v}$$ along $$\vec{u}$$ is equal to that of $$\vec{w}$$ along $$\vec{u}$$ and $$\vec{v}, \vec{w}$$ are perpendicular to each other then $$|\vec{u} - \vec{v} + \vec{w}|$$ equals
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The probability that $$A$$ speaks truth is $$\frac{4}{5}$$, while this probability for $$B$$ is $$\frac{3}{4}$$. The probability that they contradict each other when asked to speak on a fact is
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A random variable $$X$$ has the probability distribution: $$X: 1, 2, 3, 4, 5, 6, 7, 8$$; $$p(X): 0.15, 0.23, 0.12, 0.10, 0.20, 0.08, 0.07, 0.05$$. For the events $$E = \{X$$ is a prime number$$\}$$ and $$F = \{X < 4\}$$, the probability $$P(E \cup F)$$ is
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The mean and the variance of a binomial distribution are $$4$$ and $$2$$ respectively. Then the probability of $$2$$ successes is
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