A particle is moving eastwards with a velocity of $$5$$ m/s. In $$10$$ seconds the velocity changes to $$5$$ m/s northwards. The average acceleration in this time is
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A particle is moving eastwards with a velocity of $$5$$ m/s. In $$10$$ seconds the velocity changes to $$5$$ m/s northwards. The average acceleration in this time is
Out of the following pair, which one does NOT have identical dimensions is
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The relation between time $$t$$ and distance $$x$$ is $$t = ax^2 + bx$$ where $$a$$ and $$b$$ are constants. The acceleration is
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A car starting from rest accelerates at the rate $$f$$ through a distance $$S$$, then continues at constant speed for time $$t$$ and then decelerates at the rate $$f/2$$ to come to rest. If the total distance traversed is $$15S$$, then
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A parachutist after bailing out falls $$50$$ m without friction. When parachute opens, it decelerates at $$2$$ m/s$$^2$$. He reaches the ground with a speed of $$3$$ m/s. At what height, did he bail out?
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Two points $$A$$ and $$B$$ move from rest along a straight line with constant acceleration $$f$$ and $$f'$$ respectively. If $$A$$ takes $$m$$ sec. more than $$B$$ and describes '$$n$$' units more than $$B$$ in acquiring the same speed then
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$$A$$ and $$B$$ are two like parallel forces. A couple of moment $$H$$ lies in the plane of $$A$$ and $$B$$ and is contained with them. The resultant of $$A$$ and $$B$$ after combining is displaced through a distance
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A projectile can have the same range $$R$$ for two angles of projection. If $$t_1$$ and $$t_2$$ be the times of flights in the two cases, then the product of the two time of flights is proportional to
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A particle is projected from a point $$O$$ with velocity $$u$$ at an angle of $$60^\circ$$ with the horizontal. When it is moving in a direction at right angles to its direction at $$O$$, its velocity then is given by
A smooth block is released at rest on a $$45^\circ$$ incline and then slides a distance $$d$$. The time taken to slide is $$n$$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is
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The upper half of an inclined plane with inclination $$\phi$$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by
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A block is kept on a frictionless inclined surface with angle of inclination $$\alpha$$. The incline is given an acceleration $$a$$ to keep the block stationary. Then $$a$$ is equal to?
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A particle of mass $$0.3$$ kg is subjected to a force $$F = -kx$$ with $$k = 15$$ N/m. What will be its initial acceleration if it is released from a point $$20$$ cm away from the origin?
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Consider a car moving on a straight road with a speed of $$100$$ m/s. The distance at which car can be stopped is $$[\mu_k = 0.5]$$
An annular ring with inner and outer radii $$R_1$$ and $$R_2$$ is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, $$F_1/F_2$$ is
A bullet fired into a fixed target loses half of its velocity after penetrating $$3$$ cm. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?
A spherical ball of mass $$20$$ kg is stationary at the top of a hill of height $$100$$ m. It rolls down a smooth surface to the ground, then climbs up another hill of height $$30$$ m and finally rolls down to a horizontal base at a height of $$20$$ m above the ground. The velocity attained by the ball is
A body of mass $$m$$ is accelerated uniformly from rest to a speed $$v$$ in a time $$T$$. The instantaneous power delivered to the body as a function of time is given by
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A body $$A$$ of mass $$M$$ while falling vertically downwards under gravity breaks into two parts; a body $$B$$ of mass $$1/3 M$$ and a body $$C$$ of mass $$2/3 M$$. The centre of mass of bodies $$B$$ and $$C$$ taken together shifts compared to that of body $$A$$ towards
The block of mass $$M$$ moving on the frictionless horizontal surface collides with a spring of spring constant $$K$$ and compresses it by length $$L$$. The maximum momentum of the block after collision is
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A mass '$$m$$' moves with a velocity $$v$$ and collides inelastically with another identical mass. After collision the $$1^{st}$$ mass moves with velocity $$v/\sqrt{3}$$ in a direction perpendicular to the initial direction of motion. Find the speed of the $$2^{nd}$$ mass after collision 
The moment of inertia of a uniform semicircular disc of mass $$M$$ and radius $$r$$ about a line perpendicular to the plane of the disc through the centre is
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A 'T' shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $$F$$ is applied at the point $$P$$ parallel to $$AB$$, such that the object has only the translational motion without rotation. Find the location of $$P$$ with respect to $$C$$ 
Average density of the earth
The change in the value of $$g$$ at a height '$$h$$' above the surface of the earth is the same as at a depth '$$d$$' below the surface of earth. When both '$$d$$' and '$$h$$' are much smaller than the radius of earth, then which one of the following is correct?
A particle of mass $$10$$ g is kept on the surface of a uniform sphere of mass $$100$$ kg and radius $$10$$ cm. Find the work to be done against the gravitational force between them to take the particle far away from the sphere (you may take $$G = 6.67 \times 10^{-11}$$ Nm$$^2$$/kg$$^2$$)
If $$S$$ is stress and $$Y$$ is Young's modulus of material of a wire, the energy stored in the wire per unit volume is
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A $$20$$ cm long capillary tube is dipped in water. The water rises up to $$8$$ cm. If the entire arrangement is put in a freely falling elevator the length of water column in the capillary tube will be
The figure shows a system of two concentric spheres of radii $$r_1$$ and $$r_2$$ and kept at temperatures $$T_1$$ and $$T_2$$ respectively. The radial rate of flow of heat in a substance between the two concentric sphere is proportional to 
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Which of the following is incorrect regarding the first law of thermodynamics?
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The temperature-entropy diagram of a reversible engine cycle is given in the figure. Its efficiency is 
A system goes from $$A$$ to $$B$$ via two processes I and II as shown in the figure. If $$\Delta U_1$$ and $$\Delta U_2$$ are the changes in internal energies in the processes I and II respectively, the 
A gaseous mixture consists of $$16$$ g of helium and $$16$$ g of oxygen. The ratio $$\frac{C_p}{C_v}$$ of the mixture is
The function $$\sin^2(\omega t)$$ represents
Two simple harmonic motions are represented by the equation $$y_1 = 0.1 \sin\left(100\pi t + \frac{\pi}{3}\right)$$ and $$y_2 = 0.1 \cos \pi t$$. The phase difference of the velocity of particle 1 w.r.t. the velocity of the particle 2 is
If a simple harmonic motion is represented by $$\frac{d^2 x}{dt^2} + \alpha x = 0$$, its time period is
The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillation bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would
When two tuning forks (fork 1 and fork 2) are sounded simultaneously, $$4$$ beats per second are heard. Now, some tape is attached on the prong of the fork $$2$$. When the tuning forks are sounded again, $$6$$ beats per seconds are heard. If the frequency of fork 1 is $$200$$ Hz, then what was the original frequency of fork 2?
An observer moves towards a stationary source of sound, with a velocity one fifth of the velocity of sound. What is the percentage increase in the apparent frequency?
A charged ball $$B$$ hangs from a silk thread $$S$$ which makes an angle $$\theta$$ with a large charged conducting sheet $$P$$, as show in the figure. The surface charge density $$\sigma$$ of the sheet is proportional to
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Two point charges $$+8q$$ and $$-2q$$ are located at $$x = 0$$ and $$x = L$$ respectively. The location of a point on the $$x$$ axis at which the net electric field due to these two point charges is zero is
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Two thin wires rings each having a radius $$R$$ are placed at a distance $$d$$ apart with their axes coinciding. The charges on the two rings are $$+q$$ and $$-q$$. The potential difference between the centres of the two rings is
A fully charged capacitor has a capacitance '$$C$$'. It is discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat capacity '$$s$$' and mass '$$m$$'. If the temperature of the block is raised by '$$\Delta T$$'. The potential difference $$V$$ across the capacitance is
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A parallel plate capacitor is made by stacking $$n$$ equally spaced plates connected alternatively. If the capacitance between any two adjacent plates is $$C$$ then the resultant capacitance is
A moving coil galvanometer has $$150$$ equal divisions. Its current sensitivity is $$10$$ divisions per milliampere and voltage sensitivity is $$2$$ divisions per millivolt. In order that each division reads $$1$$ volt, the resistance in ohms needed to be connected in series with the coil will be
Two voltameters one of copper and another of silver, are joined in parallel. When a total charge $$q$$ flows through the voltameters, equal amount of metals are deposited. If the electrochemical equivalents of copper and silver are $$z_1$$ and $$z_2$$ respectively the charge which flows through the silver voltameter is
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In the circuit, the galvanometer $$G$$ shows zero deflection. If the batteries $$A$$ and $$B$$ have negligible resistance, the value of the resistor $$R$$ will be 
Two sources of equal emf are connected to an external resistance $$R$$. The internal resistance of the two sources are $$R_1$$ and $$R_2$$ ($$R_2 > R_1$$). If the potential difference across the source having internal resistance $$R_2$$ is zero, then
A heater coil is cut into two equal parts and only one part is now used in the heater. The heat generated will now be
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An energy source will supply a constant current into the load if its internal resistance is
In a potentiometer experiment the balancing with a cell is at length $$240$$ cm. On shunting the cell with a resistance of $$2 \Omega$$ the balancing length becomes $$120$$ cm. The internal resistance of the cell is
The resistance of hot tungsten filament is about $$10$$ times the cold resistance. What will be the resistance of $$100$$ W and $$200$$ V lamp when not in use?
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A magnetic needle is kept in a non-uniform magnetic field. It experiences
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Two thin, long parallel wires separated by a distance '$$d$$' carry a current of '$$i$$' A in the same direction. They will
Two concentric coils each of radius equal to $$2\pi$$ cm are placed at right angles to each other. $$3$$ Ampere and $$4$$ ampere are the currents flowing in each coil respectively. The magnetic induction in Weber/m$$^2$$ at the centre of the coils will be ($$\mu_0 = 4\pi \times 10^{-7}$$ Wb/A-m)
A uniform electric field and a uniform magnetic field are acting along the same direction in a certain region. If an electron is projected along the direction of the fields with a certain velocity then
A charged particle of mass $$m$$ and charge $$q$$ travels on a circular path of radius $$r$$ that is perpendicular to a magnetic field $$B$$. The time taken by the particle to complete one revolution is
One conducting U tube can slide inside another as shown in figure, maintaining electrical contacts between the tubes. The magnetic field $$B$$ is perpendicular to the plane of the figure. If each tube moves towards the other at a constant speed $$V$$, then the emf induced in the circuit in terms of $$B$$, $$\ell$$ and $$V$$ where $$\ell$$ is the width of each tube will be 
A coil of inductance $$300$$ mH and resistance $$2 \Omega$$ is connected to a source of voltage $$2$$ V. The current reaches half of its steady state value in
The self inductance of the motor of an electric fan is $$10$$ H. In order to impart maximum power at $$50$$ Hz, it should be connected to a capacitance of
A circuit has a resistance of $$12 \Omega$$ and an impedance of $$15 \Omega$$. The power factor of the circuit will be
The phase difference between the alternating current and emf is $$\pi/2$$. Which of the following cannot be the constituent of the circuit?
A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is $$4/3$$ and the fish is $$12$$ cm below the surface, the radius of this circle in cm is
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A thin glass (refractive index $$1.5$$) lens has optical power of $$-5D$$ in air. Its optical power in a liquid medium with refractive index $$1.6$$ will be
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A Young's double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen is
Two point white dots are $$1$$ mm apart on a black paper. They are viewed by eye of pupil diameter $$3$$ mm. Approximately, what is the maximum distance at which these dots can be resolved by the eye? [Take wavelength of light $$= 500$$ nm]
When an unpolarized light of intensity $$I_0$$ is incident on a polarizing sheet, the intensity of the light which does not get transmitted is
If $$I_0$$ is the intensity of the principal maximum in the single slit diffraction pattern, then what will be its intensity when the slit width is doubled?
A photocell is illuminated by a small bright source placed $$1$$ m away. When the same source of light is placed $$\frac{1}{2}$$ m away, the number of electrons emitted by photo cathode would
If the kinetic energy of a free electron doubles, its deBroglie wavelength changes by the factor
The diagram shows the energy levels for an electron in a certain atom. Which transition shown represents the emission of a photon with the most energy? 
The intensity of gamma radiation from a given source is $$I$$. On passing through $$36$$ mm of lead, it is reduced to $$\frac{I}{8}$$. The thickness of lead which will reduce the intensity to $$\frac{I}{2}$$ will be
Starting with a sample of pure $$^{66}$$Cu, $$7/8$$ of it decays into Zn in $$15$$ minutes. The corresponding half-life is
If radius of $$^{27}_{13}$$Al nucleus is estimated to be $$3.6$$ Fermi then the radius of $$^{125}_{52}$$Te nucleus be nearly
A nuclear transformation is denoted by $$X(n, \alpha) \to {}^7_3 \text{Li}$$. Which of the following is the nucleus of element $$X$$?
The electrical conductivity of a semiconductor increases when electromagnetic radiation of wavelength shorter than $$2480$$ nm is incident on it. The band gap (in eV) for the semiconductor is
In a common base amplifier, the phase difference between the input signal voltage and output voltage is
In a full wave rectifier circuit operating from $$50$$ Hz mains frequency, the fundamental frequency in the ripple would be
Two solutions of a substance (non electrolyte) are mixed in the following manner. $$480$$ ml of $$1.5$$ M first solution + $$520$$ mL of $$1.2$$ M second solution. What is the molarity of the final mixture?
If we consider that $$\frac{1}{6}$$, in place of $$\frac{1}{12}$$, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will
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An organic compound having molecular mass $$60$$ is found to contain C $$= 20\%$$, H $$= 6.67\%$$ and N $$= 46.67\%$$ while rest is oxygen. On heating it gives NH$$_3$$ alongwith a solid residue. The solid residue give violet colour with alkaline copper sulphate solution. The compound is
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In a multi-electron atom, which of the following orbitals described by the three quantum numbers will have the same energy in the absence of magnetic acid and electric fields? (a) $$n = 1, l = 0, m = 0$$ (b) $$n = 2, l = 0, m = 0$$ (c) $$n = 2, l = 1, m = 1$$ (d) $$n = 3, l = 2, m = 1$$ (e) $$n = 3, l = 2, m = 0$$
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Of the following sets which one does NOT contain isoelectronic species?
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Which of the following statements in relation to the hydrogen atom is correct?
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In which of the following arrangements the order is NOT according to the property indicated against it?
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The photon of hard gamma radiation knocks a proton out of $$^{24}_{12}$$Mg nucleus to form
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Which of the following oxides is amphoteric in character?
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Which one of the following species is diamagnetic in nature?
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Lattice energy of an ionic compounds depends upon
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The molecular shapes of SF$$_4$$, CF$$_4$$ and XeF$$_4$$ are
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Which one of the following statements is NOT true about the effect of an increase in temperature on the distribution of molecular speeds in a gas?
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Benzene and toluene form nearly ideal solutions. At $$20^\circ$$C, the vapour pressure of benzene is $$75$$ torr and that of toluene is $$22$$ torr. The partial vapour pressure of benzene at $$20^\circ$$C for a solution containing $$78$$ g of benzene and $$46$$ g of toluene in torr is
Consider an endothermic reaction, $$X \longrightarrow Y$$ with the activation energies $$E_b$$ and $$E_f$$ for the backward and forward reactions, respectively. In general
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Consider the reaction: $$\text{N}_2 + 3\text{H}_2 \longrightarrow 2\text{NH}_3$$ carried out at constant temperature and pressure. If $$\Delta H$$ and $$\Delta U$$ are the enthalpy and internal energy changes for the reaction, which of the following expressions is true?
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The exothermic formation of ClF$$_3$$ is represented by the equation: $$\text{Cl}_{2(g)} + 3\text{F}_{2(g)} \rightleftharpoons 2\text{ClF}_{3(g)}; \Delta_r H = -329$$ kJ. Which of the following will increase the quantity of ClF$$_3$$ in an equilibrium mixture of Cl$$_2$$, F$$_2$$ and ClF$$_3$$?
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If the bond dissociation energies of $$XY$$, $$X_2$$ and $$Y_2$$ (all diatomic molecules) are in the ratio of $$1 : 1 : 0.5$$ and $$\Delta_f H$$ for the formation of $$XY$$ is $$-200$$ kJ mole$$^{-1}$$. The bond dissociation energy of $$X_2$$ will be
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For the reaction $$2\text{NO}_{2(g)} \rightleftharpoons 2\text{NO}_{(g)} + \text{O}_{2(g)}$$, $$(K_c = 1.8 \times 10^{-6} \text{ at } 184^\circ\text{C})$$, $$(R = 0.0831 \text{ kJ}/(\text{mol} \cdot \text{K}))$$. When $$K_p$$ and $$K_c$$ are compared at $$184^\circ$$C, it is found that
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A schematic plot of $$\ln K_{eq}$$ versus inverse of temperature for a reaction is shown below. The reaction must be
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An amount of solid NH$$_4$$HS is placed in a flask already containing ammonia gas at a certain temperature and $$0.50$$ atm pressure. Ammonium hydrogen sulphide decomposes to yield NH$$_3$$ and H$$_2$$S gases in the flask. When the decomposition reaction reaches equilibrium, the total pressure in the flask rises to $$0.84$$ atm. The equilibrium constant for NH$$_4$$HS decomposition at this temperature is
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The solubility product of a salt having general formula $$MX_2$$, in water is $$4 \times 10^{-12}$$. The concentration of M$$^{2+}$$ ions in the aqueous solution of the salt is
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Hydrogen ion concentration in mol/L in a solution of pH $$= 5.4$$ will be
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What is the conjugate base of OH$$^-$$?
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Calomel (Hg$$_2$$Cl$$_2$$) on reaction with ammonium hydroxide gives
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Hydrogen bomb is based on the principle of
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Based on lattice energy and other considerations which one of the following alkali metal chlorides is expected to have the highest melting point?
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The number and type of bonds between two carbon atoms in calcium carbide are
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Heating an aqueous solution of aluminium chloride to dryness will give
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In silicon dioxide
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The structure of diborane (B$$_2$$H$$_6$$) contains
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Due to the presence of an unpaired electron, free radicals are:
The best reagent to convert pent-3-en-2-ol into pent-3-en-2-one is
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Consider the nucleophiles:
The decreasing order of nucleophilicity among the nucleophiles is
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Of the five isomeric hexanes, the isomer which can give two monochlorinated compounds is
Which types of isomerism is shown by 2,3-dichlorobutane?
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An ionic compound has a unit cell consisting of A ions at the corners of a cube and B ions on the centres of the faces of the cube. The empirical formula for this compound would be
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If $$\alpha$$ is the degree of dissociation of Na$$_2$$SO$$_4$$, the vant Hoff's factor ($$i$$) used for calculating the molecular mass is
Equimolar solutions in the same solvent have
For a spontaneous reaction the $$\Delta G$$, equilibrium constant ($$K$$) and $$E^\circ_{cell}$$ will be respectively
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The highest electrical conductivity of the following aqueous solutions is of
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Aluminium oxide may be electrolysed at $$1000^\circ$$C to furnish aluminium metal (Atomic mass $$= 27$$ amu; $$1$$ Faraday $$= 96{,}500$$ Coulombs). The cathode reaction is $$\text{Al}^{3+} + 3e^- \longrightarrow \text{Al}^\circ$$. To prepare $$5.12$$ kg of aluminium metal by this method would require
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Given the molar conductances at infinite dilution: 
Calculate $$\Lambda^\infty_{HOAc}$$ using appropriate molar conductances of the electrolytes listed above at infinite dilution in H$$_2$$O at $$25^\circ$$C
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A reaction involving two different reactants can never be
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$$t_{1/4}$$ can be taken as the time taken for the concentration of a reactant to drop to $$\frac{3}{4}$$ of its initial value. If the rate constant for a first order reaction is $$K$$, the $$t_{1/4}$$ can be written as
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The volume of a colloidal particle, $$V_C$$ as compared to the volume of a solute particle in a true solution $$V_s$$, could be
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The disperse phase in colloidal iron (III) hydroxide and colloidal gold is positively and negatively charged, respectively, which of the following statements is NOT correct?
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During the process of electrolytic refining of copper, some metals present as impurity settle as 'anode mud'. These are
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The number of hydrogen atom(s) attached to phosphorus atom in hypophosphorous acid is
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The correct order of the thermal stability of hydrogen halides (H-X) is
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Heating mixture of Cu$$_2$$O and Cu$$_2$$S will give
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The oxidation state of chromium in the final product formed by the reaction between KI and acidified potassium dichromate solution is
The lanthanide contraction is responsible for the fact that
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Which of the following factors may be regarded as the main cause of lanthanide contraction?
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The oxidation state of Cr in $$[\text{Cr}(\text{NH}_3)_4 \text{Cl}_2]^+$$ is
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The IUPAC name of the coordination compound $$\text{K}_3[\text{Fe}(\text{CN})_6]$$ is
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Which of the following compounds shows optical isomerism?
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Which one of the following cyano complexes would exhibit the lowest value of paramagnetic behaviour? (At. No. Cr = 24, Mn = 25, Fe = 26, Co = 27)
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The value of the 'spin only' magnetic moment for one of the following configurations is $$2.84$$ BM. The correct one is
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2-methylbutane on reacting with bromine in the presence of sunlight gives mainly
Tertiary alkyl halides are practically inert to substitution by $$S_N 2$$ mechanism because of
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Reaction of one molecule of HBr with one molecule of 1,3-butadiene at $$40^\circ$$C gives predominantly
Alkyl halides react with dialkyl copper reagents to give
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Elimination of bromine from 2-bromobutane results in the formation of-
Acid catalyzed hydration of alkenes except ethene leads to the formation of
p-cresol reacts with chloroform in alkaline medium to give the compound A which adds hydrogen cyanide to form, the compound B. The latter on acidic hydrolysis gives chiral carboxylic acid. The structure of the carboxylic acid is
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Reaction of cyclohexanone with dimethylamine in the presence of catalytic amount of an acid forms a compound if water during the reaction is continuously removed. The compound formed is generally known as
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Among the following acids which has the lowest p$$K_a$$ value
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Which one of the following methods is neither meant for the synthesis nor for separation of amines?
Amongst the following the most basic compound is
Which of the following is a polyamide?
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Which of the following is fully fluorinated polymer?
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Which one of the following types of drugs reduces fever?
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In both DNA and RNA, heterocyclic base and phosphate ester linkages are at-
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The value of $$\alpha$$ for which the sum of the squares of the roots of the equation $$x^2 - (a - 2)x - a - 1 = 0$$ assume the least value is
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If roots of the equation $$x^2 - bx + c = 0$$ be two consecutive integers, then $$b^2 - 4c$$ equals
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If both the roots of the quadratic equation $$x^2 - 2kx + k^2 + k - 5 = 0$$ are less than $$5$$, then $$k$$ lies in the interval
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If the cube roots of unity are $$1, \omega, \omega^2$$ then the roots of the equation $$(x - 1)^3 + 8 = 0$$ are
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If $$z_1$$ and $$z_2$$ are two non-zero complex numbers such that $$|z_1 + z_2| = |z_1| + |z_2|$$ then $$\arg z_1 - \arg z_2$$ is equal to
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If $$\omega = \frac{z}{z - \frac{1}{3} i}$$ and $$|\omega| = 1$$, then $$z$$ lies on
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If the letters of word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
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If $$x = \sum_{n=0}^\infty a^n, y = \sum_{n=0}^\infty b^n, z = \sum_{n=0}^\infty c^n$$ where $$a, b, c$$ are in A.P. and $$|a| < 1, |b| < 1, |c| < 1$$, then $$x, y, z$$ are in
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If in a triangle $$ABC$$, the altitudes from the vertices $$A, B, C$$ on opposite sides are in H.P., then $$\sin A, \sin B, \sin C$$ are in
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If non-zero numbers $$a, b, c$$ are in H.P., then the straight line $$\frac{x}{a} + \frac{y}{b} + \frac{1}{c} = 0$$ always passes through a fixed point. That point is
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The sum of the series $$1 + \frac{1}{4 \cdot 2!} + \frac{1}{16 \cdot 4!} + \frac{1}{64 \cdot 6!} + \ldots$$ ad inf. is
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If $$A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$$ and $$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$, then which one of the following holds for all $$n \geq 1$$, by the principle of mathematical induction
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If the coefficients of $$r$$th, $$(r + 1)$$th and $$(r + 2)$$th terms in the binomial expansion of $$(1 + y)^m$$ are in A.P., then $$m$$ and $$r$$ satisfy the equation
The value of $$^{50}C_4 + \sum_{r=1}^6 {}^{56-r}C_3$$ is
If the coefficient of $$x^7$$ in $$\left[ax^2 + \left(\frac{1}{bx}\right)\right]^{11}$$ equals the coefficient of $$x^{-7}$$ in $$\left[ax^2 - \left(\frac{1}{bx}\right)\right]^{11}$$, then $$a$$ and $$b$$ satisfy the relation
If a vertex of a triangle is $$(1, 1)$$ and the mid-points of two sides through this vertex are $$(-1, 2)$$ and $$(3, 2)$$, then the centroid of the triangle is
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If the circles $$x^2 + y^2 + 2ax + cy + a = 0$$ and $$x^2 + y^2 - 3ax + dy - 1 = 0$$ intersect in two distinct points $$P$$ and $$Q$$ then the line $$5x + by - a = 0$$ passes through $$P$$ and $$Q$$ for
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A circle touches the $$x$$-axis and also touches the circle with centre at $$(0, 3)$$ and radius $$2$$. The locus of the centre of the circle is
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If a circle passes through the point $$(a, b)$$ and cuts the circle $$x^2 + y^2 = p^2$$ orthogonally, then the equation of the locus of its centre is
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If the pair of lines $$ax^2 + 2(a + b)xy + by^2 = 0$$ lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then
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Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the locus $$y^2 = 8x$$. The locus of mid point of $$PQ$$ is
An ellipse has $$OB$$ as semi minor axis, $$F$$ and $$F'$$ its focii and the angle $$FBF'$$ is a right angle. Then the eccentricity of the ellipse is
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The locus of a point $$P(\alpha, \beta)$$ moving under the condition that the line $$y = \alpha x + \beta$$ is a tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ is
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$$\lim_{n \to \infty} \left[\frac{1}{n^2} \sec^2 \frac{1}{n^2} + \frac{2}{n^2} \sec^2 \frac{4}{n^2} + \ldots + \frac{1}{n^2} \sec^2 1\right]$$ equals
Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$ax^2 + bx + c = 0$$, then $$\lim_{x \to \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}$$ is equal to
If in a frequently distribution, the mean and median are $$21$$ and $$22$$ respectively, then its mode is approximately
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Let $$x_1, x_2, \ldots, x_n$$ be $$n$$ observations such that $$\sum x_i^2 = 400$$ and $$\sum x_i = 80$$. Then a possible value of $$n$$ among the following is
A lizard, at an initial distance of $$21$$ cm behind an insect, moves from rest with an acceleration of $$2$$ cm/s$$^2$$ and pursues the insect which is crawling uniformly along a straight line at a speed of $$20$$ cm/s. Then the lizard will catch the insect after
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$$ABC$$ is a triangle. Forces $$\vec{P}, \vec{Q}, \vec{R}$$ acting along $$IA, IB$$ and $$IC$$ respectively are in equilibrium, where $$I$$ is the incentre of $$\triangle ABC$$. Then $$P : Q : R$$ is
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In a triangle $$PQR$$, $$\angle R = \frac{\pi}{2}$$. If $$\tan\left(\frac{P}{2}\right)$$ and $$\tan\left(\frac{Q}{2}\right)$$ are the roots of $$ax^2 + bx + c = 0, a \neq 0$$ then
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In a triangle $$ABC$$, let $$\angle C = \frac{\pi}{2}$$. If $$r$$ is the inradius and $$R$$ is the circumradius of the triangle $$ABC$$, then $$2(r + R)$$ equals
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Let $$R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$$ be a relation on the set $$A = \{3, 6, 9, 12\}$$. The relation is
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If $$A^2 - A + I = 0$$, then the inverse of $$A$$ is
The system of equations $$\alpha x + y + z = \alpha - 1$$, $$x + \alpha y + z = \alpha - 1$$, $$x + y + \alpha z = \alpha - 1$$ has no solution, if $$\alpha$$ is
If $$a^2 + b^2 + c^2 = -2$$ and $$f(x) = \begin{vmatrix} 1 + a^2 x & (1 + b^2)x & (1 + c^2)x \\ (1 + a^2)x & 1 + b^2 x & (1 + c^2)x \\ (1 + a^2)x & (1 + b^2)x & 1 + c^2 x \end{vmatrix}$$ then $$f(x)$$ is a polynomial of degree
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If $$a_1, a_2, a_3, \ldots, a_n, \ldots$$ are in G.P., then the determinant $$\Delta = \begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}$$ is equal to
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If $$\cos^{-1} x - \cos^{-1} \frac{y}{2} = \alpha$$, then $$4x^2 - 4xy \cos \alpha + y^2$$ is equal to
Let $$f : (-1, 1) \to B$$, be a function defined by $$f(x) = \tan^{-1} \frac{2x}{1 - x^2}$$, then $$f$$ is both one-one and onto when $$B$$ is the interval
A real valued function $$f(x)$$ satisfies the functional equation $$f(x - y) = f(x)f(y) - f(a - x) f(a + y)$$ where $$a$$ is a given constant and $$f(0) = 1$$, $$f(2a - x)$$ is equal to
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Suppose $$f(x)$$ is differentiable at $$x = 1$$ and $$\lim_{h \to 0} \frac{1}{h} f(1 + h) = 5$$, then $$f'(1)$$ equals
Area of the greatest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ is
The normal to the curve $$x = a(\cos\theta + \theta \sin\theta), y = a(\sin\theta - \theta \cos\theta)$$ at any point '$$\theta$$' is such that
A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval $$\to$$ Function
Let $$f$$ be differentiable for all $$x$$. If $$f(1) = -2$$ and $$f'(x) \geq 2$$ for $$x \in [1, 6]$$, then
If $$f$$ is a real-valued differentiable function satisfying $$|f(x) - f(y)| \leq (x - y)^2, x, y \in R$$ and $$f(0) = 0$$, then $$f(1)$$ equals
If $$x$$ is so small that $$x^3$$ and higher powers of $$x$$ may be neglected, then $$\frac{(1 + x)^{3/2} - \left(1 + \frac{1}{2} x\right)^3}{(1 - x)^{1/2}}$$
A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness than melts at a rate of $$50$$ cm$$^3$$/min. When the thickness of ice is $$5$$ cm, then the rate at which the thickness of ice decreases, is
If the equation $$a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x = 0, a_1 \neq 0, n \geq 2$$, has a positive root $$x = \alpha$$, then the equation $$na_n x^{n-1} + (n - 1) a_{n-1} x^{n-2} + \ldots + a_1 = 0$$ has a positive root, which is
$$\int \left\{\frac{(\log x - 1)}{(1 + (\log x)^2)}\right\}^2 dx$$ is equal to
If $$I_1 = \int_0^1 2^{x^2} dx, I_2 = \int_0^1 2^{x^3} dx, I_3 = \int_1^2 2^{x^2} dx$$ and $$I_4 = \int_1^2 2^{x^3} dx$$ then
Let $$f : R \to R$$ be a differentiable function having $$f(2) = 6, f'(2) = \left(\frac{1}{48}\right)$$. Then $$\lim_{x \to 2} \int_6^{f(x)} \frac{4t^3}{x - 2} dt$$ equals
The value of $$\int_{-\pi}^\pi \frac{\cos^2 x}{1 + a^x} dx, a > 0$$, is
The area enclosed between the curve $$y = \log_e(x + e)$$ and the coordinate axes is
The parabolas $$y^2 = 4x$$ and $$x^2 = 4y$$ divide the square region bounded by the lines $$x = 4, y = 4$$ and the coordinate axes. If $$S_1, S_2, S_3$$ are respectively the areas of these parts numbered from top to bottom; then $$S_1 : S_2 : S_3$$ is
Let $$f(x)$$ be a non-negative continuous function such that the area bounded by the curve $$y = f(x)$$, $$x$$-axis and the ordinates $$x = \frac{\pi}{4}$$ and $$x = \beta > \frac{\pi}{4}$$ is $$\left(\beta \sin \beta + \frac{\pi}{4} \cos \beta + \sqrt{2} \beta\right)$$. Then $$f\left(\frac{\pi}{2}\right)$$ is
The differential equation representing the family of curves $$y^2 = 2c(x + \sqrt{c})$$, where $$c > 0$$, is a parameter, is of order and degree as follows:
If $$x \frac{dy}{dx} = y(\log y - \log x + 1)$$, then the solution of the equation is
If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB$$, then
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For any vector $$\vec{a}$$ the value of $$(\vec{a} \times \hat{i})^2 + (\vec{a} \times \hat{j})^2 + (\vec{a} \times \hat{k})^2$$ is equal to
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If $$\vec{a}, \vec{b}, \vec{c}$$ are non-coplanar vectors and $$\lambda$$ is a real number then $$[\lambda(\vec{a} + \vec{b}) \, \lambda^2 \vec{b} \, \lambda \vec{c}] = [\vec{a} \, \vec{b} + \vec{c} \, \vec{b}]$$ for
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Let $$\vec{a} = \hat{i} - \hat{k}, \vec{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}$$ and $$\vec{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k}$$. Then $$[\vec{a}, \vec{b}, \vec{c}]$$ depends on
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The resultant $$R$$ of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is
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The line parallel to the $$x$$-axis and passing through the intersection of the lines $$ax + 2by + 3b = 0$$ and $$bx - 2ay - 3a = 0$$, where $$(a, b) \neq (0, 0)$$ is
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If the angle $$\theta$$ between the line $$\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$$ and the plane $$2x - y + \sqrt{\lambda} z + 4 = 0$$ is such that $$\sin\theta = \frac{1}{3}$$ the value of $$\lambda$$ is
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The angle between the lines $$2x = 3y = -z$$ and $$6x = -y = -4z$$ is
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If the plane $$2ax - 3ay + 4az + 6 = 0$$ passes through the midpoint of the line joining the centres of the spheres $$x^2 + y^2 + z^2 + 6x - 8y - 2z = 13$$ and $$x^2 + y^2 + z^2 - 10x + 4y - 2z = 8$$, then $$a$$ equals
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The distance between the line $$\vec{r} = 2\hat{i} - 2\hat{j} + 3\hat{k} + \lambda(\hat{i} - \hat{j} + 4\hat{k})$$ and the plane $$\vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5$$ is
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Let $$a, b$$ and $$c$$ be distinct non-negative numbers. If the vectors $$a\hat{i} + a\hat{j} + c\hat{k}, \hat{i} + \hat{k}$$ and $$c\hat{i} + c\hat{j} + b\hat{k}$$ lie in a plane, then $$c$$ is
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The plane $$x + 2y - z = 4$$ cuts the sphere $$x^2 + y^2 + z^2 - x + z - 2 = 0$$ in a circle of radius
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Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is
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A random variable $$X$$ has Poisson distribution with mean $$2$$. Then $$P(X > 1.5)$$ equals
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Let $$A$$ and $$B$$ be two events such that $$P(\overline{A \cup B}) = \frac{1}{6}, P(A \cap B) = \frac{1}{4}$$ and $$P(\bar{A}) = \frac{1}{4}$$, where $$\bar{A}$$ stands for complement of event $$A$$. Then events $$A$$ and $$B$$ are
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