The exothermic formation of ClF₃ is represented by the equation:
Cl₂ (g) + 3F₂ (g) $$\leftrightharpoons$$ 2ClF₃ (g); $$\Delta H$$ = -329 kJ
Which of the following will increase the quantity of ClF₃ in an equilibrium mixture of Cl₂, F₂ and ClF₃?

If the bond dissociation energies of XY, X₂ and Y₂ (all diatomic molecules) are in the ratio of 1:1:0.5 and $$\Delta H_f$$ for the formation of XY is -200 kJ mole^-1. The bond dissociation energy of X₂ will be :

Consider an endothermic reaction, X $$\to$$ Y with the activation energies E_b and E_f for the backward and forward reactions, respectively. In general :

Consider the reaction: N₂ + 3H₂ $$\to$$ 2NH₃ 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?

If $$\alpha$$ is the degree of dissociation of Na₂SO₄, the vant Hoff’s factor (i) used for calculating the molecular mass is :

The solubility product of a salt having general formula MX₂, in water is: 4 $$\times$$ 10^-12 . The concentration of M²⁺ ions in the aqueous solution of the salt is :

What is the conjugate base of OH^-?

Hydrogen ion concentration in mol / L in a solution of pH = 5.4 will be :

An amount of solid NH4HS is placed in a flask already containing ammonia gas at a certain temperature and 0.50 atm. Pressure. Ammonium hydrogen sulphide decomposes to yield NH₃ and H₂S 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 NH₄HS decomposition at this temperature is :

For the reaction 2NO₂ (g) $$\leftrightharpoons$$ 2NO (g) + O₂ (g), (K_c = 1.8 $$\times$$ 10^-6 at 184^oC) (R = 0.0831 kJ/(mol. K))
When K_p and K_c are compared at 184^oC , it is found that :

Which one of the following species is diamagnetic in nature?

Due to the presence of an unpaired electron, free radicals are:

Reaction of one molecule of HBr with one molecule of 1,3-butadiene at 40^oC gives predominantly

Which types of isomerism is shown by 2,3-dichlorobutane?

Acid catalyzed hydration of alkenes except ethene leads to the formation of

2 methylbutane on reacting with bromine in the presence of sunlight gives mainly

Of the five isomeric hexanes, the isomer which can give two monochlorinated compounds is

Benzene and toluene form nearly ideal solutions. At 20 ^oC, the vapour pressure of benzene is 75 torr and that of toluene is 22 torr. The partial vapour pressure of benzene at 20 ^oC for a solution containing 78 g of benzene and 46 g of toluene in torr is

Equimolar solutions in the same solvent have

Elimination of bromine from 2-bromobutane results in the formation of-

Among the following acids which has the lowest pK_a value?

The oxidation state of Cr in [Cr(NH₃)₄Cl₂]⁺ is :

Amongst the following the most basic compound is

The IUPAC name of the coordination compound K₃[Fe(CN)₆] is

Tertiary alkyl halides are practically inert to substitution by S_N2 mechanism because of

The decreasing order of nucleophilicity among the nucleophiles

Pick out the isoelectronic structure from the following :

$$\eqalign{ & \left( i \right)\,\,\,\,\,\,C{H_3}^ + \cr & \left( {ii} \right)\,\,\,\,{H_3}{O^ + } \cr & \left( {iii} \right)\,\,\,N{H_3} \cr & \left( {iv} \right)\,\,\,\,C{H_3}^ - \cr} $$

The oxidation state of chromium in the final product formed the reaction between $$K{\rm I}$$ and acidified potassium dichromate solution is :

$$p$$-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 :

Calomel $$\left( {H{g_2}C{l_2}} \right)$$ on reaction with ammonium hydroxide gives :

The reaction



is fastest when $$X$$ is

On heating mixture of $$C{u_2}O$$ and $$C{u_2}S$$ will give :

A schematic plot of $$ln$$ $${K_{eq}}$$ versus inverse of temperature for a reaction is shown below


The reaction must be

Which of the following factors may be regarded as the main cause of lanthanide contraction?

Aluminium oxide may be electrolysed at 1000^oC to furnish aluminium metal (Atomic mass = 27 amu; 1 Faraday = 96,500 Coulombs). The cathode reaction is Al³⁺ + 3e^- $$\to$$ Al^o To prepare 5.12 kg of aluminium metal by this method would require

For a spontaneous reaction the ∆G , equilibrium constant (K) and $$E_{cell}^o$$ will be respectively

Calculate $$ \wedge _{HOAc}^\infty $$ Using appropriate molar conductances of the electrolytes listed above at infinite dilution in H₂O at 25^oC

The highest electrical conductivity of the following aqueous solutions is of :

A reaction involving two different reactants can never be

t_1/4 can be taken as the time taken for the concentration of a reactant to drop to $$3 \over 4$$ of its initial value. If the rate constant for a first order reaction is K, the t_1/4 can be written as

The photon of hard gamma radiation knocks a proton out of $${}_{12}^{24}Mg$$ nucleus to form

Hydrogen bomb is based on the principle of

The lanthanide contraction is responsible for the fact that :

The best reagent to convert pent -3- en-2-ol into pent -3-en-2-one is

In both DNA and RNA, heterocyclic base and phosphate ester linkages are at-

The value of the ‘spin only’ magnetic moment for one of the following configurations is 2.84 BM. The correct one is :

Which one of the following methods is neither meant for the synthesis nor for separation of amines?

Alkyl halides react with dialkyl copper reagents to give

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 NH₃ along with a solid residue. The solid residue give violet colour with alkaline copper sulphate solution. The compound is

Which of the following compounds shows optical isomerism?

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)

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

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of the substance will

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?

In a multi-electron atom, which of the following orbitals described by the three quantum members will have the same energy in the absence of magnetic 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

Of the following sets which one does NOT contain isoelectronic species?

In which of the following arrangements the order is NOT according to the property indicated against it?

Which of the following oxides is amphoteric in character?

Lattice energy of an ionic compounds depends upon

If $$f$$ is a real valued differentiable function satisfying

$$\left| {f\left( x \right) - f\left( y \right)} \right|$$ $$ \le {\left( {x - y} \right)^2}$$, $$x, y$$ $$ \in R$$
and $$f(0)$$ = 0, then $$f(1)$$ equals

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

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 given constant and f(0) = 1, f(2a - x) is equal to

Let $$f:( - 1,1) \to B$$, be a function defined by
$$f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}$$,
then $$f$$ is both one-one and onto when B is the interval

Suppose $$f(x)$$ is differentiable at x = 1 and

$$\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5$$, then $$f'\left( 1 \right)$$ equals

Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$a{x^2} + bx + c = 0$$, then

$$\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}$$ is equal to

Let x₁, x₂,...........,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

If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals

If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately :

The value of $$a$$ for which the sum of the squares of the roots of the equation $${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is :

If the cube roots of unity are 1, $$\omega \,,\,{\omega ^2}$$ then the roots of the equation $${(x - 1)^3}$$ + 8 = 0, are :

The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,} $$ is

The system of equations

$$\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr {x + y + \alpha \,z = \alpha - 1} \cr } $$

has no solutions, if $$\alpha $$ is :

$$\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx} $$ is equal to

A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness that 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 $${a^2} + {b^2} + {c^2} = - 2$$ and

f$$\left( x \right) = \left| {\matrix{ {1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr {\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \cr {\left( {1 + {a^2}} \right)x} & {\left( {1 + {b^2}} \right)x} & {1 + {c^2}x} \cr } } \right|,$$

then f$$(x)$$ is a polynomial of degree :

Area of the greatest rectangle that can be inscribed in the
ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is :

If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant $$$\Delta = \left| {\matrix{ {\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr {\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr {\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr } } \right|$$$
is equal to :

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 :

The differential equation representing the family of curves $${y^2} = 2c\left( {x + \sqrt c } \right),$$ where $$c>0,$$ is a parameter, is of order and degree as follows:

The area enclosed between the curve $$y = {\log _e}\left( {x + e} \right)$$ and the coordinate axes 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 $$x = {\pi \over 4}$$ and $$x = \beta > {\pi \over 4}$$ is $$\left( {\beta \sin \beta + {\pi \over 4}\cos \beta + \sqrt 2 \beta } \right).$$ Then $$f\left( {{\pi \over 2}} \right)$$ is

The value of integral, $$\int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx $$ is

If $${I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } } $$ and $${I_4} = \int\limits_1^2 {{2^{{x^3}}}dx} $$ then

If $$x{{dy} \over {dx}} = y\left( {\log y - \log x + 1} \right),$$ then the solution of the equation is :

Let $$A$$ and $$B$$ two events such that $$P\left( {\overline {A \cup B} } \right) = {1 \over 6},$$ $$P\left( {A \cap B} \right) = {1 \over 4}$$ and $$P\left( {\overline A } \right) = {1 \over 4},$$ where $${\overline A }$$ stands for complement of event $$A$$. Then events $$A$$ and $$B$$ are :

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 :

A random variable $$X$$ has Poisson distribution with mean $$2$$.
Then $$P\left( {X > 1.5} \right)$$ equals :

For any vector $${\overrightarrow a }$$ , the value of $${\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}$$ is equal to :

The angle between the lines $$2x=3y=-z$$ and $$6x=-y=-4z$$ is :

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 :

If $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$, then $$z$$ lies on :

If $${z_1}$$ and $${z_2}$$ are two non-zero complex numbers such that $$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$, then arg $${z_1}$$ - arg $${z_2}$$ is equal to :

In a triangle $$PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$$ and $$ \tan \left( {{Q \over 2}} \right)$$ are the roots of $$a{x^2} + bx + c = 0,\,\,a \ne 0$$ then

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

If $$x$$ is so small that $${x^3}$$ and higher powers of $$x$$ may be neglected, then $${{{{\left( {1 + x} \right)}^{{3 \over 2}}} - {{\left( {1 + {1 \over 2}x} \right)}^3}} \over {{{\left( {1 - x} \right)}^{{1 \over 2}}}}}$$ may be approximated as

If the coefficient of $${x^7}$$ in $${\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}$$ equals the coefficient of $${x^{ - 7}}$$ in $${\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}$$, then $$a$$ and $$b$$ satisfy the relation

If the letter of the 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

If the coefficients of r^th, (r+1)^th, and (r + 2)^th terms in the binomial expansion of $${{\rm{(1 + y )}}^m}$$ are in A.P., then m and r satisfy the equation

If $$x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,$$ where a, b, c are in A.P and $$\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| < 1$$ then x, y, z are in

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 $$(a, b)$$ $$ \ne $$ $$(0, 0)$$ is :

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 :

If the circles $${x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0$$ and $${x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, = 0$$ intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for :

If the pair of lines $$a{x^2} + 2\left( {a + b} \right)xy + b{y^2} = 0$$ 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 :

An ellipse has $$OB$$ as semi minor axis, $$F$$ and $$F$$' its focii and theangle $$FBF$$' is a right angle. Then the eccentricity of the ellipse is :

The value of $$a$$ for which the sum of the squares of the roots of the equation
$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is

Let $$f:R \to R$$ be a differentiable function having $$f\left( 2 \right) = 6$$,
$$f'\left( 2 \right) = \left( {{1 \over {48}}} \right)$$. Then $$\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {{{4{t^3}} \over {x - 2}}dt} $$ equals :

Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is :

If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals

If $${\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,$$ then $$4{x^2} - 4xy\cos \alpha + {y^2}$$ is equal to :

Let $$a, b$$ and $$c$$ be distinct non-negative numbers. If the vectors $$a\widehat i + a\widehat j + c\widehat k,\,\,\widehat i + \widehat k$$ and $$c\widehat i + c\widehat j + b\widehat k$$ lie in a plane, then $$c$$ is :

If non zero numbers $$a, b, c$$ are in $$H.P.,$$ then the straight line $${x \over a} + {y \over b} + {1 \over c} = 0$$ always passes through a fixed point. That point is :

If $$C$$ is the mid point of $$AB$$ and $$P$$ is any point outside $$AB,$$ then :

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 :

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \mathrm{s}$. Then the lizard will catch the insect after :

A photocell is illuminated by a small bright source placed $$1$$ $$m$$ away. When the same source of light is placed $${1 \over 2}$$ $$m$$ away, the number of electrons emitted by photo-cathode would

If the kinetic energy of a free electron doubles, it's 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?

In a full wave rectifier circuit operating from $$50$$ $$Hz$$ mains frequency, the fundamental frequency in the ripple would be

A Young's double slit experiment uses a monochromatic source. The shape of the interference fringes formed on a screen 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 -

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

A circuit has a resistance of $$12$$ $$ohm$$ and an impedance of $$15$$ $$ohm$$. The power factor of the circuit will be

Two voltmeters, one of copper and another of silver, are joined in parallel. When a total charge $$q$$ flows through the voltmeters, 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 voltmeter is

Two thin, long, parallel wires, separated by a distance $$'d'$$ carry a current of $$'i'$$ $$A$$ in the same direction. They will

The resistance of hot tungsten filament is about $$10$$ times the cold resistance. What will be resistance of $$100$$ $$W$$ and $$200$$ $$V$$ lamp when not in use ?

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

In the circuit, the galvanometer $$G$$ shows zero deflection. If the batteries $$A$$ and $$B$$ have negligible internal resistance, the value of the resistor $$R$$ will be -

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

The phase difference between the alternating current and $$emf$$ is $${\pi \over 2}.$$ Which of the following cannot be the constituent of the circuit?

Two sources of equal $$emf$$ are connected to an external resistance $$R.$$ The internal resistance of the two sources are $${R_1}$$ and $${R_2}\left( {{R_1} > {R_1}} \right).$$ If the potential difference across the source having internal resistance $${R_2}$$ is zero, 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

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

A thin glass (refractive index $$1.5$$) lens has optical power of $$-5$$ $$D$$ in air. Its optical power in a liquid medium with refractive index $$1.6$$ will be

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 $$=500$$ $$nm$$ ]

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 fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is $${4 \over 3}$$ and the fish is $$12$$ $$cm$$ below the surface, the radius of this circle in $$cm$$ 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 $$B,l$$ and $$v$$ where $$l$$ is the width of each tube, will be

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

A nuclear transformation is denoted by $$X\left( {n,\alpha } \right)\matrix{ 7 \cr 3 \cr } Li.$$ Which of the following is the nucleus of element $$X$$ ?

A magnetic needle is kept in a non-uniform magnetic field. It experiences :

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

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?

Average density of the earth

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 center is

A body $$A$$ of mass $$M$$ while falling vertically downloads under gravity breaks into two-parts; a body $$B$$ of mass $${1 \over 3}$$ $$M$$ and a body $$C$$ of mass $${2 \over 3}$$ $$M.$$ The center of mass of bodies $$B$$ and $$C$$ taken together shifts compared to that 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 the spring of spring constant $$k$$ and compresses it by length $$L.$$ The maximum momentum of the block after collision is

A T shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $$'\,\,\overrightarrow F \,\,'$$ 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 $$C.$$

A system goes from $$A$$ to $$B$$ via two processes $$I$$ and $$II$$ as shown in figure. If $$\Delta {U_1}$$ and $$\Delta {U_2}$$ are the changes in internal energies in the processes $$I$$ and $$II$$ respectively, then

The figure shows a system of two concentric spheres of radii $${r_1}$$ and $${r_2}$$ are kept at temperatures $${T_1}$$ and $${T_2}$$, respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

Which of the following is incorrect regarding the first law of thermodynamics?

A gaseous mixture consists of $$16$$ $$g$$ of helium and $$16$$ $$g$$ of oxygen. The ratio $${{Cp} \over {{C_v}}}$$ of the mixture is

The temperature-entropy diagram of a reversible engine cycle is given in the figure. Its efficiency is

The function $${\sin ^2}\left( {\omega t} \right)$$ represents

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

Two simple harmonic motions are represented by the equations $${y_1} = 0.1\,\sin \left( {100\pi t + {\pi \over 3}} \right)$$ and $${y_2} = 0.1\,\cos \,\pi t.$$ The phase difference of the velocity of particle $$1$$ with respect to the velocity of particle $$2$$ is

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

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

Two thin wire 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

When two tuning forks (fork $$1$$ and fork $$2$$) are sounded simultaneously, $$4$$ beats per second are heated. Now, some tape is attached on the prong of the fork $$2.$$ When the tuning forks are sounded again, $$6$$ beats per second are heard. If the frequency of fork $$1$$ is $$200$$ $$Hz$$, then what was the original frequency of fork $$2$$ ?

A charged ball $$B$$ hangs from a silk thread $$S,$$ which makes angle $$\theta $$ with a large charged conducting sheet $$P,$$ as shown in the figure. The surface charge density $$\sigma $$ of the sheet is proportional to f

If a simple harmonic motion is represented by $${{{d^2}x} \over {d{t^2}}} + \alpha x = 0.$$ its time period is

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

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

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 center of the coils will be
$$\left( {\mu = 4\pi \times {{10}^{ - 7}}Wb/A.m} \right)$$

An energy source will supply a constant current into the load if its internal resistance is

A mass $$'m'$$ moves with a velocity $$'v'$$ and collides inelastically with another identical mass. After collision the $${1^{st}}$$ mass moves with velocity $${v \over {\sqrt 3 }}$$ in a direction perpendicular to the initial direction of motion. Find the speed of the $${2^{nd}}$$ mass after collision.

Out of the following pair, which one does NOT have identical dimensions is

Consider a car moving on a straight road with a speed of $$100$$ $$m/s$$. The distance at which car can be stopped is $$\left[ {{\mu _k} = 0.5} \right]$$

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?

A particle of mass 0.3 kg 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?

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

The relation between time t and distance x is t = ax² + bx where a and b are constants. The acceleration is

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

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, $${{{F_1}} \over {{F_2}}}\,$$ is

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 $${f \over 2}$$ to come to rest. If the total distance traversed is 15 S, then

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 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

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

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

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

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 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 $$G$$ $$ = 6.67 \times {10^{ - 11}}\,\,N{m^2}/k{g^2}$$)