Which of the following units denotes the dimensions $$ML^2/Q^2$$, where Q denotes the electric charge?
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Which of the following units denotes the dimensions $$ML^2/Q^2$$, where Q denotes the electric charge?
A particle located at $$x = 0$$ at time $$t = 0$$, starts moving along the positive $$x$$-direction with a velocity '$$v$$' that varies as $$v = \alpha\sqrt{x}$$. The displacement of the particle varies with time as
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A body falling from rest under gravity passes a certain point $$P$$. It was at a distance of $$400\,m$$ from $$P$$, $$4\,s$$ prior to passing through $$P$$. If $$g = 10\,m/s^2$$, then the height above the point $$P$$ from where the body began to fall is
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A mass of $$M\,kg$$ is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of $$45^\circ$$ with the initial vertical direction is
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A player caught a cricket ball of mass $$150\,g$$ moving at a rate of $$20\,m/s$$. If the catching process is completed in $$0.1\,s$$, the force of the blow exerted by the ball on the hand of the player is equal to
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A particle of mass $$100\,g$$ is thrown vertically upwards with a speed of $$5\,m/s$$. The work done by the force of gravity during the time the particle goes up is
A ball of mass $$0.2\,kg$$ is thrown vertically upwards by applying a force by hand. If the hand moves $$0.2\,m$$ while applying the force and the ball goes upto $$2\,m$$ height further, find the magnitude of the force. Consider $$g = 10\,m/s^2$$
The potential energy of a $$1\,kg$$ particle free move along the x-axis is given by $$$V(x) = \left(\frac{x^4}{4} - \frac{x^2}{2}\right)J$$$ The total mechanical energy of the particle $$2\,J$$. Then, the maximum speed (in $$m/s$$) is
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A bomb of mass $$16\,kg$$ at rest explodes into two pieces of masses of $$4\,kg$$ and $$12\,kg$$. The velocity of the $$12\,kg$$ mass is $$4\,ms^{-1}$$. The kinetic energy of the other mass is
Consider a two particle system with particles having masses $$m_1$$ and $$m_2$$. If the first particle is pushed towards the centre of mass through a distance $$d$$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?
A force of $$-F\hat{k}$$ acts on $$O$$, the origin of the coordinate system. The torque about the point $$(1, -1)$$ is
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A thin circular ring of mass $$m$$ and radius $$R$$ is rotating about its axis with a constant angular velocity $$\omega$$. Two objects each of mass $$M$$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity $$\omega' =$$
Four point masses, each of value $$m$$, are placed at the corners of a square $$ABCD$$ of side $$\ell$$. The moment of inertia of system about an axis passing through $$A$$ and parallel to $$BD$$ is
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A wire elongates by $$\ell\,mm$$ when a load $$W$$ is hanged from it. If the wire goes over a pulley and two weights $$W$$ each are hung at the two ends, the elongation of the wire will be (in $$mm$$)
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If the terminal speed of a sphere of gold (density $$= 19.5\,kg/m^3$$) is $$0.2\,m/s$$ in a viscous liquid (density $$= 1.5\,kg/m^3$$) of the same size in the same liquid.
Assuming the sun to be a spherical body of radius $$R$$ at a temperature of $$T\,K$$, evaluate the total radiant power, incident on Earth, at a distance $$r$$ from the Sun.
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The work of $$146\,kJ$$ is performed in order to compress one kilo mole of gas adiabatically and in this process the temperature of the gas increases by $$7^\circ C$$. The gas is $$(R = 8.3\,J\,mol^{-1}\,K^{-1})$$
Two rigid boxes containing different ideal gases are placed on a table. Box A contains one mole of nitrogen at temperature $$T_0$$, while Box $$B$$ contains one mole of helium at temperature $$(7/3)T_0$$. The boxes are then put into thermal contact with each other and heat flows between them until the gases reach a common final temperature. (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $$T_f$$, in terms of $$T_0$$ is
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Starting from the origin, a body oscillates simple harmonically with a period of $$2\,s$$. After what time will its kinetic energy be 75% of the total energy?
The maximum velocity of a particle, executing simple harmonic motion with an amplitude $$7\,mm$$, is $$4.4\,m/s$$. The period of oscillation is
A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $$\omega$$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time
A whistle producing sound waves of frequencies $$9500\,Hz$$ and above is approaching a stationary person with speed $$v\,ms^{-1}$$. The velocity of sound in air is $$300\,ms^{-1}$$. If the person can hear frequencies upto a maximum of $$10{,}000\,Hz$$, the maximum value of $$v$$ upto which he can hear the whistle is
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A string is stretched between fixed points separated by $$75\,cm$$. It is observed to have resonant frequencies of $$420\,Hz$$ and $$315\,Hz$$. There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is
A electric dipole is placed at an angle of $$30^\circ$$ to a non-uniform electric field. The dipole will experience
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Two insulating plates are both uniformly charged in such a way that the potential difference between them is $$V_2 - V_1 = 20\,V$$. (i.e. plate 2 is at a higher potential). The plates are separated by $$d = 0.1\,m$$ and can be treated as infinitely large. An electron is released from rest on the inner surface of plate 1. What is its speed when it hits plate 2? $$(e = 1.6\times 10^{-19}\,C, m_e = 9.11\times 10^{-31}\,kg)$$
Two spherical conductors $$A$$ and $$B$$ of radii $$1\,mm$$ and $$2\,mm$$ are separated by a distance of $$5\,cm$$ and are uniformly charged. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surface of spheres $$A$$ and $$B$$ is
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A material '$$B$$' has twice the specific resistance of '$$A$$'. A circular wire made of '$$B$$' has twice the diameter of a wire made of '$$A$$'. Then for the two wires to have the same resistance, the ratio $$\ell_A/\ell_B$$ of their respective lengths must be
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The Kirchhoff's first law $$\left(\sum i = 0\right)$$ and second law $$\left(\sum iR = \sum E\right)$$, where the symbols have their usual meanings, are respectively based on
In a Wheatstone's bridge, there resistances $$P, Q$$ and $$R$$ connected in the three arms and the fourth arm is formed by two resistances $$S_1$$ and $$S_2$$ connected in parallel. The condition for bridge to be balanced will be
Consider the circuit shown below. The current $$I$$ drawn from the 5 volt source will be
The resistance of a bulb filament is $$100\,\Omega$$ at a temperature of $$100^\circ C$$. If its temperature coefficient of resistance be $$0.005$$ per $$^\circ C$$, its resistance will become $$200\,\Omega$$ at a temperature of
A thermocouple is made from two metals, Antimony and Bismuth. If one junction of the couple is kept hot and the other is kept cold then, an electric current will
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An electric bulb is rated $$220$$ volt $$- 100$$ watt. The power consumed by it when operated on $$110$$ volt will be
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Needles $$N_1, N_2$$ and $$N_3$$ are made of a ferromagnetic, a paramagnetic and a diamagnetic substance respectively. A magnet when brought close to them will
In a region, steady and uniform electric and magnetic fields are present. These two fields are parallel to each other. A charged particle is released from rest in this region. The path of the particle will be a
A long solenoid has 200 turns per $$cm$$ and carries a current $$i$$. The magnetic field at its centre is $$6.28 \times 10^{-2}\,Weber/m^2$$. Another long solenoid has 100 turns per $$cm$$ and it carries a current $$i/3$$. The value of the magnetic field at its centre is
In an AC generator, a coil with $$N$$ turns, all of the same area $$A$$ and total resistance $$R$$, rotates with frequency $$\omega$$ in a magnetic field $$B$$. The maximum value of emf generated in the coil is
The flux linked with a coil at any instant '$$t$$' is given by $$\phi = 10t^2 - 50t + 250$$. The induced emf at $$t = 3\,s$$ is
In a series resonant LCR circuit, the voltage across $$R$$ is $$100$$ volts and $$R = 1\,k\Omega$$ with $$C = 2\,\mu F$$. The resonant frequency $$\omega$$ is $$200\,rad/s$$. At resonance the voltage across $$L$$ is
An inductor $$(L = 100\,mH)$$, a resistor $$(R = 100\,\Omega)$$ and a battery $$(E = 100\,V)$$ are initially connected in series as shown in the figure. After a long time the battery is disconnected after short circuiting the points $$A$$ and $$B$$.
The current in the circuit $$1\,mm$$ after the circuit is
The rms value of the electric field of the light coming from the Sun is $$720\,N/C$$. The average total energy density of the electromagnetic wave is
The refractive index of glass is $$1.520$$ for red light and $$1.525$$ for blue light. Let $$D_1$$ and $$D_2$$ be the angles of minimum deviation for red and blue light respectively in a prism of this glass. Then
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The time by a photoelectron to come out after the photon strikes is approximately
The threshold frequency for a metallic surface corresponds to an energy of $$6.2\,eV$$, and the stopping potential for a radiation incident on this surface $$5\,V$$. The incident radiation lies in
The anode voltage of a photocell is kept fixed. The wavelength $$\lambda$$ of the light falling on the cathode is gradually changed. The plate current $$I$$ of the photocell varies as follows :
An alpha nucleus of energy $$\dfrac{1}{2}mv^2$$ bombards a heavy nuclear target of charge $$Ze$$. Then the distance of closest approach for the alpha nucleus will be proportional to
The energy spectrum of $$\beta$$-particles [number $$N(E)$$ as a function of $$\beta$$-energy $$E$$] emitted from a radioactive source is
When $$_3Li^7$$ nuclei are bombarded by protons, and the resultant nuclei are $$_4Be^8$$, the emitted particles will be
A solid which is transparent to visible light and whose conductivity increases with temperature is formed by
The 'rad' is the correct unit used to report the measurement of
If the binding energy per nucleon in $$_3^7Li$$ and $$_2^4He$$ nuclei are $$5.60\,MeV$$ and $$7.06\,MeV$$ respectively, then in the reaction $$p + _3^7Li \to 2\,_2^4He$$ energy of proton must be
If the ratio of the concentration of electrons that of holes in a semiconductor is $$\dfrac{7}{5}$$ and the ratio of currents is $$\dfrac{7}{4}$$, then what is the ratio of their drift velocities?
In a common base mode of a transistor, the collector current is $$5.488\,mA$$ for an emitter current of $$5.60\,mA$$. The value of the base current amplification factor $$(\beta)$$ will be
If the lattice constant of this semiconductor is decreased, then which of the following is correct?
In the following, which one of the diodes is reverse biased?
The circuit has two oppositely connect ideal diodes in parallel. What is the current following in the circuit?
How many moles of magnesium phosphate, $$Mg_3(PO_4)_2$$ will contain $$0.25$$ mole of oxygen atoms?
Density of a $$2.05\,M$$ solution of acetic acid in water is $$1.02\,g/mL$$. The molality of the solution is
According to Bohr's theory, the angular momentum of an electron in $$5^{th}$$ orbit is
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In the transformation of $$_{92}^{238}U$$ to $$_{92}^{234}U$$, if one emission is an $$\alpha$$-particle, what should be the other emission(s)?
Uncertainty in the position of an electron (mass $$= 9.1 \times 10^{-31}\,kg$$) moving with a velocity $$300\,ms^{-1}$$, accurate upto $$0.001\%$$, will be
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Which one of the following sets of ions represents a collection of isoelectronic species?
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The "spin-only" magnetic moment [in units of Bohr magneton, $$(\mu_B)$$] of $$Ni^{2+}$$ in aqueous solution would be (Atomic number of $$Ni = 28$$)
The increasing order of the first ionization enthalpies of the elements B, P, S and F (lowest first) is
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The decreasing values of bond angles from $$NH_3$$ $$(106^\circ)$$ to $$SbH_3$$ $$(101^\circ)$$ down group-15 of the periodic table is due to
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Following statements regarding the periodic trends of chemical reactivity of the alkali metals and the halogens are given. Which of these statements gives the correct picture?
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Which of the following molecules/ions does not contain unpaired electrons?
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Among the following mixtures, dipole-dipole as the major interaction, is present in
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A metal, $$M$$ forms chlorides in its $$+2$$ and $$+4$$ oxidation states. Which of the following statements about these chlorides is correct?
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In which of the following molecules/ions are all the bonds not equal?
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Phosphorus pentachloride dissociates as follows, in a closed reaction vessel, $$PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)$$ If total pressure at equilibrium of the reaction mixture is $$P$$ and degree of dissociation of $$PCl_5$$ is $$x$$, the partial pressure of $$PCl_3$$ will be
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The standard enthalpy of formation $$(\Delta_fH^\circ)$$ at $$298\,K$$ for methane, $$CH_4(g)$$, is $$-74.8\,kJ\,mol^{-1}$$. The additional information required to determine the average energy for $$C - H$$ bond formation would be
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An ideal gas is allowed to expand both reversibly and irreversibly in an isolated system. If $$T_i$$ is the initial temperature and $$T_f$$ is the final temperature, which of the following statements is correct?
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The enthalpy changes for the following processes are listed below: $$Cl_2(g) = 2Cl(g),\ 242.3\,kJ\,mol^{-1}$$; $$I_2(g) = 2I(g),\ 151.0\,kJ\,mol^{-1}$$; $$ICl(g) = I(g) + Cl(g),\ 211.3\,kJ\,mol^{-1}$$; $$I_2(s) = I_2(g),\ 62.76\,kJ\,mol^{-1}$$. Given that the standard states for iodine and chlorine are $$I_2(s)$$ and $$Cl_2(g)$$, the standard enthalpy of formation for $$ICl(g)$$ is
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$$(\Delta H - \Delta U)$$ for the formation of carbon monoxide (CO) from its elements at $$298\,K$$ is $$(R = 8.314\,J\,K^{-1}\,mol^{-1})$$
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The equilibrium constant for the reaction $$$SO_3(g) \rightleftharpoons SO_2(g) + \frac{1}{2}O_2(g)$$$ is $$K_c = 4.9 \times 10^{-2}$$. The value of $$K_c$$ for the reaction $$$2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g)$$$ will be
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Which of the following statements is true?
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The ionic mobility of alkali metal ions in aqueous solution is maximum for
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The IUPAC name of the compound shown below is
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The increasing order of the rate of HCN addition to compounds A - D is (A) HCHO (B) $$CH_3COCH_3$$ (C) $$PhCOCH_3$$ (D) PhCOPh
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The increasing order of stability of the following free radicals is
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Increasing order of stability among the three main conformations (i.e. Eclipse, Anti, Gauche) of 2-fluoroethanol is
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The alkene formed as a major product in the above elimination reaction is
Total volume of atoms present in a face-centre cubic unit cell of a metal is ($$r$$ is atomic radius)
$$18\,g$$ of glucose $$(C_6H_{12}O_6)$$ is added to $$178.2\,g$$ of water. The vapour pressure of water for this aqueous solution at $$100^\circ C$$ is
The molar conductivities $$\Lambda^\circ_{NaOAc}$$ and $$\Lambda^\circ_{HCl}$$ at infinite dilution in water at $$25^\circ C$$ are $$91.0$$ and $$426.2\,S\,cm^2/mol$$ respectively. To calculate $$\Lambda^\circ_{HOAc}$$, the additional value required is
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Given the data at $$25^\circ C$$, $$Ag + I^- \longrightarrow AgI + e^-;\ E^\circ = 0.152\,V$$; $$Ag \longrightarrow Ag^+ + e^-;\ E^\circ = -0.800\,V$$. What is the value of $$\log K_{sp}$$ for $$AgI$$? $$\left(2.303\dfrac{RT}{F} = 0.059\,V\right)$$
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Resistance of a conductivity cell filled with a solution of an electrolyte of concentration $$0.1\,M$$ is $$100\,\Omega$$. The conductivity of this solution is $$1.29\,S\,m^{-1}$$. Resistance of the same cell when filled with $$0.2\,M$$ of the same solution is $$520\,\Omega$$. The molar conductivity of $$0.02\,M$$ solution of the electrolyte will be
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A reaction was found to be second order with respect to the concentration of carbon monoxide. If the concentration of carbon monoxide is doubled, with everything else kept the same, the rate of reaction will
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Rate of a reaction can be expressed by Arrhenius equation as: $$$k = Ae^{-E/RT}$$$ In this equation, E represents
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The following mechanism has been proposed for the reaction of NO with $$Br_2$$ to form NOBr : $$$NO(g) + Br_2(g) \rightleftharpoons NOBr_2(g)$$$ $$$NOBr_2(g) + NO(g) \longrightarrow 2NOBr(g)$$$ If the second step is the rate determining step, the order of the reaction with respect to $$NO(g)$$ is
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In Langmuir's model of adsorption of a gas on a solid surface
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Which of the following chemical reactions depicts the oxidizing behaviour of $$H_2SO_4$$?
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What products are expected from the disproportionation reaction of hypochlorous acid?
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Lanthanoid contraction is caused due to
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The IUPAC name for the complex $$[Co(NO_2)(NH_3)_5]Cl_2$$ is
Nickel $$(Z = 28)$$ combines with a uninegative monodentate ligand $$X^-$$ to form a paramagnetic complex $$[NiX_4]^{2-}$$. The number of unpaired electron(s) in the nickel and geometry of this complex ion are, respectively
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In $$Fe(CO)_5$$, the $$Fe - C$$ bond possesses
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How many EDTA (ethylenediaminetetraacetic acid) molecules are required to make an octahedral complex with a $$Ca^{2+}$$ ion?
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HBr reacts with $$CH_2 = CH - OCH_3$$ under anhydrous conditions at room temperature to give
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$$$CH_3Br + Nu^- \longrightarrow CH_3 - Nu + Br^-$$$ The decreasing order of the rate of the above reaction with nucleophiles $$(Nu^-)$$ A to D is $$[Nu^- = (A)PhO^-, (B) AcO^-, (C) HO^-, (D) CH_3O^-]$$
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Fluorobenzene $$(C_6H_5F)$$ can be synthesized in the laboratory
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Question:
Consider the reaction of bromoethane $$\left(CH_3CH_2Br\right)$$ with two different cyanide reagents in an aqueous ethanolic solution:
Reaction 1:
$$CH_3CH_2Br + KCN \rightarrow \text{Product A}$$
Reaction 2:
$$CH_3CH_2Br + AgCN \rightarrow \text{Product B}$$
Identify the major products A and B, respectively.
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Reaction of trans-2-phenyl-1-bromocyclopentane on reaction with alcoholic KOH produces
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The structure of the compound that gives a tribromo derivative on treatment with bromine water is
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Phenyl magnesium bromide reacts with methanol to give
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Among the following the one that gives positive iodoform test upon reaction with $$I_2$$ and NaOH is
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The electrophile involved in the above reaction is
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The correct order of increasing acid strength of the compounds is (a) $$CH_3CO_2H$$ (b) $$MeOCH_2CO_2H$$ (c) $$CF_3CO_2H$$
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The term anomers of glucose refers to
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The pyrimidine bases present in DNA are
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If the roots of the quadratic equation $$x^2 + px + q = 0$$ are $$\tan 30^\circ$$ and $$\tan 15^\circ$$, respectively then the value of $$2 + q - p$$ is
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All the values of $$m$$ for which both roots of the equations $$x^2 - 2mx + m^2 - 1 = 0$$ are greater than $$-2$$ but less than $$4$$, lie in the interval
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If $$z^2 + z + 1 = 0$$, where $$z$$ is a complex number, then the value of $$$\left(z + \frac{1}{z}\right)^2 + \left(z^2 + \frac{1}{z^2}\right)^2 + \left(z^3 + \frac{1}{z^3}\right)^2 + \cdots + \left(z^6 + \frac{1}{z^6}\right)^2$$$ is
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At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is
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The value of $$\sum_{k=1}^{10}\left(\sin\dfrac{2k\pi}{11} + i\cos\dfrac{2k\pi}{11}\right)$$ is
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Let $$a_1, a_2, a_3, \ldots$$ be terms of an A.P. If $$\dfrac{a_1 + a_2 + \cdots + a_p}{a_1 + a_2 + \cdots + a_q} = \dfrac{p^2}{q^2}, p \ne q$$, then $$\dfrac{a_6}{a_{21}}$$ equals
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If $$a_1, a_2, \ldots, a_n$$ are in H.P., then the expression $$a_1a_2 + a_2a_3 + \ldots + a_{n-1}a_n$$ is equal to
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If the expansion in powers of $$x$$ of the function $$\dfrac{1}{(1 - ax)(1 - bx)}$$ is $$a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots$$, then $$a_n$$ is
For natural numbers $$m, n$$ if $$(1 - y)^m(1 + y)^n = 1 + a_1y + a_2y^2 + \ldots$$, and $$a_1 = a_2 = 10$$ then $$(m, n)$$ is
The number of values of $$x$$ in the interval $$[0, 3\pi]$$ satisfying the equation $$2\sin^2 x + 5\sin x - 3 = 0$$ is
If $$0 < x < \pi$$ and $$\cos x + \sin x = \dfrac{1}{2}$$, then $$\tan x$$ is
A straight line through the point $$A(3, 4)$$ is such that its intercept between the axes is bisected at $$A$$. Its equation is
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The two lines $$x = ay + b, z = cy + d$$; and $$x = a'y + b', z = c'y + d'$$ are perpendicular to each other if
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If $$(a, a^2)$$ falls inside the angle made by the lines $$y = \dfrac{x}{2}, x > 0$$ and $$y = 3x, x > 0$$, then $$a$$ belongs to
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If the lines $$3x - 4y - 7 = 0$$ and $$2x - 3y - 5 = 0$$ are two diameters of a circle of area $$49\pi$$ square units, the equation of the circle is
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Let $$C$$ be the circle with centre $$(0, 0)$$ and radius $$3$$ units. The equation of the locus of the mid points of the chords of the circle $$C$$ that subtend an angle of $$\dfrac{2\pi}{3}$$ at its centre is
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The locus of the vertices of the family of parabolas $$y = \dfrac{a^3 x^2}{3} + \dfrac{a^2 x}{2} - 2a$$ is
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Angle between the tangents to the curve $$y = x^2 - 5x + 6$$ at the points $$(2, 0)$$ and $$(3, 0)$$ is
In an ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is
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Suppose a population $$A$$ has $$100$$ observations $$101, 102, \ldots, 200$$, and another population $$B$$ has $$100$$ observations $$151, 152, \ldots, 250$$. If $$V_A$$ and $$V_B$$ represent the variances of the two populations, respectively, then $$\dfrac{V_A}{V_B}$$ is
A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $$x$$. The maximum area enclosed by the park is
Let $$W$$ denote the words in the English dictionary. Define the relation $$R$$ by : $$R = \{(x, y) \in W \times W \mid$$ the words $$x$$ and $$y$$ have at least one letter in common $$\}$$. Then $$R$$ is
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If $$A$$ and $$B$$ are square matrices of size $$n \times n$$ such that $$A^2 - B^2 = (A - B)(A + B)$$, then which of the following will be always true?
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Let $$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}, a, b \in N$$. Then
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The set of points where $$f(x) = \dfrac{x}{1 + |x|}$$ is differentiable is
If $$x^m \cdot y^n = (x + y)^{m+n}$$, then $$\dfrac{dy}{dx}$$ is
If $$x$$ is real, the maximum value of $$\dfrac{3x^2 + 9x + 17}{3x^2 + 9x + 7}$$ is
The function $$f(x) = \dfrac{x}{2} + \dfrac{2}{x}$$ has a local minimum at
The value of the integral, $$\displaystyle\int_3^6 \dfrac{\sqrt{x}}{\sqrt{9 - x} + \sqrt{x}}\,dx$$ is
$$\displaystyle\int_0^\pi xf(\sin x)\,dx$$ is equal to
$$\displaystyle\int_{-3\pi/2}^{-\pi/2}\left[(x + \pi)^3 + \cos^2(x + 3\pi)\right]dx$$ is equal to
The value of $$\displaystyle\int_1^a [x]f'(x)\,dx, a > 1$$, where $$[x]$$ denotes the greatest integer not exceeding $$x$$ is
The differential equation whose solution is $$Ax^2 + By^2 = 1$$, where $$A$$ and $$B$$ are arbitrary constants is of
$$ABC$$ is a triangle, right angled at $$A$$. The resultant of the forces acting along $$\overrightarrow{AB}, \overrightarrow{AC}$$ with magnitudes $$\dfrac{1}{AB}$$ and $$\dfrac{1}{AC}$$ respectively is the force along $$\overrightarrow{AD}$$, where $$D$$ is the foot of the perpendicular from $$A$$ onto $$BC$$. The magnitude of the resultant is
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If $$(\overline{a} \times \overline{b}) \times \overline{c} = \overline{a} \times (\overline{b} \times \overline{c})$$, where $$\overline{a}, \overline{b}$$ and $$\overline{c}$$ are any three vectors such that $$\overline{a} \cdot \overline{b} \ne 0, \overline{b} \cdot \overline{c} \ne 0$$, then $$\overline{a}$$ and $$\overline{c}$$ are
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A particle has two velocities of equal magnitude inclined to each other at an angle $$\theta$$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $$\theta$$ is
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The values of $$a$$, for which the points $$A, B, C$$ with position vectors $$2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}$$ and $$a\hat{i} - 3\hat{j} + \hat{k}$$ respectively are the vertices of a right-angled triangle with $$C = \dfrac{\pi}{2}$$ are
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The image of the point $$(-1, 3, 4)$$ in the plane $$x - 2y = 0$$ is
At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of $$5$$ phone calls during $$10$$-minute time intervals. The probability that there is at the most one phone call during a $$10$$-minute time period is
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