One of the products of the following reaction is P.


Structure of P is

For the reaction below, the product is Q.


The compound Q is

Cyclopentanol on reaction with NaH followed by CS₂ and CH₃I produces a/an

The compound, which evolves carbon dioxide on treatment with aqueous solution of sodium bicarbonate at 25$$^\circ$$C, is

The indicated atom is not a nucleophilic site in

The charge carried by 1 millimole of Mⁿ⁺ ions is 193 coulombs. The value of n is

Which of the following mixtures will have the lowest pH at 298 K?

Consider the following two first order reactions occurring at 298 K with same initial concentration of A :

(1) A $$\to$$ B; rate constant, k = 0.693 min^$$-$$1

(2) A $$\to$$ C; half-life, t_1/2 = 0.693 min

Choose the correct option.

For the equilibrium, H₂O(l) $$\rightleftharpoons$$ H₂O(v), which of the following is correct?

For a van der Waals' gas, the term $$\left( {{{ab} \over {{V^2}}}} \right)$$ represents some

In the equilibrium, H₂ + I₂ $$\rightleftharpoons$$ 2HI, if at a given temperature the concentration of the reactants are increased, the value of the equilibrium constant, K_C, will

If electrolysis of aqueous CuSO₄ solution is carried out using Cu-electrodes, the reaction taking place at the anode is

Which one of the following electronic arrangements is absurd?

The quantity hv/K_B corresponds to

In the crystalline solid MSO₄ . nH₂O of molar mass 250 g mol^$$-$$1, the percentage of anhydrous salt is 64 by weight. The value of n is

At S.T.P. the volume of 7.5 g of a gas is 5.6 L. The gas is

The half-life period of $${}_{53}{I^{125}}$$ is 60 days. The radioactivity after 180 days will be

Consider, the radioactive disintegration

$${}_{82}{A^{210}}\buildrel {} \over \longrightarrow B\buildrel {} \over \longrightarrow C\buildrel {} \over \longrightarrow {}_{82}{A^{206}}$$

The sequence of emission can be

The second ionization energy of the following elements follows the order

The melting points of (i) BeCl₂ (ii) CaCl₂ and (iii) HgCl₂ follows the order

Which of these species will have non-zero magnetic moment?

The first electron affinity of C, N and O will be of the order

The H - N - H angle in ammonia is 107.6$$^\circ$$ while the H - P - H angle in phosphine is 93.5$$^\circ$$. Relative to phosphine, the p-character of the lone-pair on ammonia is expected to be

The reactive species in chlorine bleach is

The conductivity measurement of a coordination compound of cobalt (III) shows that it dissociates into 3 ions in solution. The compound is

In the Bayer's process, the leaching of alumina is done by using

Which atomic species cannot be used as a nuclear fuel?

The molecule/molecules that has/have delocalised lone pair(s) of electrons is/are

The conformations of n-butane, commonly known as eclipsed, gauche and anti-conformations can be interconverted by

The correct order of the addition reaction rates of halogen acids with ethylene is

The total number of isomeric linear dipeptides which can be synthesised from racemic alanine is

The kinetic study of a reaction like vA $$\to$$ P at 300 K provides the following curve, where concentration is taken in mol dm^$$-$$3 and time in min.


Identify the correct order (n) and rate constant (k)

At constant pressure, the heat of formation of a compound is not dependent on temperature, when

A copper coin was electroplated with Zn and then heated at high temperature until there is a change in colour. What will be the resulting colour?

Oxidation of allyl alcohol with a peracid gives a compound of molecular formula C₃H₆O₂, which contains an asymmetric carbon atom. The structure of the compound is

Haloform reaction with I₂ and KOH will be respond by

Identify the correct statement(s) :

Compounds with spin only magnetic moment equivalent to five unpaired electrons are

Which of the following chemicals may be used to identify three unlabelled beakers containing conc. NaOH, conc. H₂SO₄ and water?

The compound (s), capable of producing achiral compound on heating at 100$$^\circ$$ is/are

$$\mathop {\lim }\limits_{x \to {0^ + }} ({x^n}\ln x),\,n > 0$$

If $$\int {\cos x\log \left( {\tan {x \over 2}} \right)} dx$$ = $$\sin x\log \left( {\tan {x \over 2}} \right)$$ + f(x), then f(x) is equal to (assuming c is a arbitrary real constant).

y = $$\int {\cos \left\{ {2{{\tan }^{ - 1}}\sqrt {{{1 - x} \over {1 + x}}} } \right\}} dx$$ is an equation of a family of

The value of the integration

$$\int\limits_{ - {\pi \over 4}}^{\pi /4} {\left( {\lambda |\sin x| + {{\mu \sin x} \over {1 + \cos x}} + \gamma } \right)} dx$$

The value of $$\mathop {\lim }\limits_{x \to 0} {1 \over x}\left[ {\int\limits_y^a {{e^{{{\sin }^2}t}}dt - } \int\limits_{x + y}^a {{e^{{{\sin }^2}t}}dt} } \right]$$ is equal to

If $$\int {{2^{{2^x}}}.\,{2^x}dx} = A\,.\,{2^{{2^x}}} + C$$, then A is equal to

The value of the integral $$\int\limits_{ - 1}^1 {\left\{ {{{{x^{2015}}} \over {{e^{|x|}}({x^2} + \cos x)}} + {1 \over {{e^{|x|}}}}} \right\}} dx$$ is equal to

$$\mathop {\lim }\limits_{n \to \infty } {3 \over n}\left[ {1 + \sqrt {{n \over {n + 3}}} + \sqrt {{n \over {n + 6}}} + \sqrt {{n \over {n + 9}}} + ... + \sqrt {{n \over {n + 3(n - 1)}}} } \right]$$

The general solution of the differential equation $$\left( {1 + {e^{{x \over y}}}} \right)dx + \left( {1 - {x \over y}} \right){e^{x/y}}dy = 0$$ is (C is an arbitrary constant)

General solution of $${(x + y)^2}{{dy} \over {dx}} = {a^2},a \ne 0$$ is (C is an arbitrary constant)

Let P(4, 3) be a point on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal at P intersects the X-axis at (16, 0), then the eccentricity of the hyperbola is

If the radius of a spherical balloon increases by 0.1%, then its volume increases approximately by

The three sides of a right angled triangle are in GP (geometric progression). If the two acute angles be $$\alpha$$ and $$\beta$$, then tan$$\alpha$$ and tan$$\beta$$ are

If $$\log _2^6 + {1 \over {2x}} = {\log _2}\left( {{2^{{1 \over x}}} + 8} \right)$$, then the value of x are

Let z be a complex number such that the principal value of argument, arg z > 0. Then, arg z $$-$$ arg($$-$$ z) is

The general value of the real angle $$\theta$$, which satisfies the equation, $$(\cos \theta + i\sin \theta )(\cos 2\theta + i\sin 2\theta )...(\cos n\theta + i\sin n\theta ) = 1$$ is given by, (assuming k is an integer)

Let a, b, c be real numbers such that a + b + c < 0 and the quadratic equation ax² + bx + c = 0 has imaginary roots. Then,

A candidate is required to answer 6 out of 12 questions which are divided into two parts A and B, each containing 6 questions and he/she is not permitted to attempt more than 4 questions from any part. In how many different ways can he/she make up his/her choice of 6 questions?

There are 7 greeting cards, each of a different colour and 7 envelopes of same 7 colours as that of the cards. The number of ways in which the cards can be put in envelopes, so that exactly 4 of the cards go into envelopes of respective colour is,

7²ⁿ + 16n $$-$$1 (n$$ \in $$ N) is divisible by

The number of irrational terms in the expansion of $${\left( {{3^{{1 \over 8}}} + {5^{{1 \over 4}}}} \right)^{84}}$$ is

Let A be a square matrix of order 3 whose all entries are 1 and let I₃ be the identity matrix of order 3. Then, the matrix $$A - 3{I_3}$$ is

If M is any square matrix of order 3 over R and if M' be the transpose of M, then adj(M') $$-$$ (adj M)' is equal to

If $$A = \left( {\matrix{ 5 & {5x} & x \cr 0 & x & {5x} \cr 0 & 0 & 5 \cr } } \right)$$ and $$|A{|^2} = 25$$, then | x | is equal to

Let A and B be two square matrices of order 3 and AB = O₃, where O₃ denotes the null matrix of order 3. Then,

Let P and T be the subsets of k, y-plane defined by

P = {(x, y) : x > 0, y > 0 and x² + y² = 1}

T = {(x, y) : x > 0, y > 0 and x⁸ + y⁸ < 1}

Then, P $$ \cap $$ T is

Let $$f:R \to R$$ be defined by $$f(x) = {x^2} - {{{x^2}} \over {1 + {x^2}}}$$ for all $$x \in R$$. Then,

Let the relation $$\rho $$ be defined on R as a$$\rho $$b if 1 + ab > 0. Then,

A problem in mathematics is given to 4 students whose chances of solving individually are $${{1 \over 2}}$$, $${{1 \over 3}}$$, $${{1 \over 4}}$$ and $${{1 \over 5}}$$. The probability that the problem will be solved at least by one student is

If X is a random variable such that $$\sigma$$(X) = 2.6, then $$\sigma$$(1 $$-$$ 4X) is equal to

If $${e^{\sin x}} - {e^{-\sin x}} - 4 = 0$$, then the number of real values of x is

The angles of a triangle are in the ratio 2 : 3 : 7 and the radius of the circumscribed circle is 10 cm. The length of the smallest side is

A variable line passes through a fixed point $$({x_1},{y_1})$$ and meets the axes at A and B. If the rectangle OAPB be completed, the locus of P is, (O being the origin of the system of axes).

A straight line through the point (3, $$-$$2) is inclined at an angle 60$$^\circ$$ to the line $$\sqrt 3 x + y = 1$$. If it intersects the X-axis, then its equation will be

A variable line passes through the fixed point $$(\alpha ,\beta )$$. The locus of the foot of the perpendicular from the origin on the line is

If the point of intersection of the lines 2ax + 4ay + c = 0 and 7bx + 3by $$-$$ d = 0 lies in the 4th quadrant and is equidistant from the two axes, where a, b, c and d are non-zero numbers, then ad : bc equals to

A variable circle passes through the fixed point A(p, q) and touches X-axis. The locus of the other end of the diameter through A is

If P(0, 0), Q(1, 0) and R$$\left( {{1 \over 2},{{\sqrt 3 } \over 2}} \right)$$ are three given points, then the centre of the circle for which the lines PQ, QR and RP are the tangents is

For the hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$, which of the following remains fixed when $$\alpha$$ varies?

S and T are the foci of an ellipse and B is the end point of the minor axis. If STB is equilateral triangle, the eccentricity of the ellipse is

The equation of the directrices of the hyperbola $$3{x^2} - 3{y^2} - 18x + 12y + 2 = 0$$ is

P is the extremity of the latusrectum of ellipse $$3{x^2} + 4{y^2} = 48$$ in the first quadrant. The eccentric angle of P is

The direction ratios of the normal to the plane passing through the points (1, 2, $$-$$3), ($$-$$1, $$-$$2, 1) and parallel to $${{x - 2} \over 2} = {{y + 1} \over 3} = {z \over 4}$$ is

The equation of the plane, which bisects the line joining the points (1, 2, 3) and (3, 4, 5) at right angles is

The limit of the interior angle of a regular polygon of n sides as n $$ \to $$ $$\infty $$ is

Let f(x) > 0 for all x and f'(x) exists for all x. If f is the inverse function of h and $${h'(x) = {1 \over {1 + \log x}}}$$. Then, f'(x) will be

Consider the function f(x) = cos x². Then,

$$\mathop {\lim }\limits_{x \to {0^ + }} {({e^x} + x)^{1/x}}$$

Let f(x) be a derivable function, f'(x) > f(x) and f(0) = 0. Then,

Let $$f:[1,3] \to R$$ be a continuous function that is differentiable in (1, 3) an

f'(x) = | f(x) |² + 4 for all x$$ \in $$ (1, 3). Then,

Let $$a = \min \{ {x^2} + 2x + 3:x \in R\} $$ and $$b = \mathop {\lim }\limits_{\theta \to 0} {{1 - \cos \theta } \over {{\theta ^2}}}$$. Then $$\sum\limits_{r = 0}^n {{a^r}{b^{n - r}}} $$ is

Let a > b > 0 and I(n) = a^1/n $$-$$ b^1/n, J(n) = (a $$-$$ b)^1/n for all n $$ \ge $$ 2, then

Let $$\widehat \alpha $$, $$\widehat \beta $$, $$\widehat \gamma $$ be three unit vectors such that $$\widehat \alpha \, \times \,(\widehat \beta \times \widehat \gamma ) = {1 \over 2}(\widehat \beta + \widehat \gamma )$$ where $$\widehat \alpha \, \times \,(\widehat \beta \times \widehat \gamma ) = $$$$(\widehat \alpha \,.\,\widehat \gamma )\widehat \beta - (\widehat \alpha \,.\,\widehat \beta )\widehat \gamma $$. If $$\widehat \beta $$ is not parallel to $$\widehat \gamma $$, then the angle between $$\widehat \alpha $$ and $$\widehat \beta $$ is

The position vectors of the points A, B, C and D are $$3\widehat i - 2\widehat j - \widehat k$$, $$2\widehat i - 3\widehat j + 2\widehat k$$, $$5\widehat i - \widehat j + 2\widehat k$$ and $$4\widehat i - \widehat j - \lambda \widehat k$$, respectively. If the points A, B, C and D lie on a plane, the value of $$\lambda$$ is

A particle starts at the origin and moves 1 unit horizontally to the right and reaches P₁, then it moves $${1 \over 2}$$ unit vertically up and reaches P₂, then it moves $${1 \over 4}$$ unit horizontally to right and reaches P₃, then it moves $${1 \over 8}$$ unit vertically down and reaches P₄, then it moves $${1 \over 16}$$ unit horizontally to right and reaches P₅ and so on. Let P_n = (x_n, y_n) and $$\mathop {\lim }\limits_{n \to \infty } {x_n} = \alpha $$ and $$\mathop {\lim }\limits_{n \to \infty } {y_n} = \beta $$. Then, ($$\alpha$$, $$\beta$$) is

For any non-zero complex number z, the minimum value of | z | + | z $$-$$ 1 | is

The system of equations

$$\eqalign{ & \lambda x + y + 3z = 0 \cr & 2x + \mu y - z = 0 \cr & 5x + 7y + z = 0 \cr} $$

has infinitely many solutions in R. Then,

Let f : X $$ \to $$ Y and A, B are non-void subsets of Y, then (where the symbols have their usual interpretation)

Let S, T, U be three non-void sets and f : S $$ \to $$ T, g : T $$ \to $$ U be so that gof : s $$ \to $$ U is surjective. Then,

The polar coordinate of a point P is $$\left( {2, - {\pi \over 4}} \right)$$. The polar coordinate of the point Q which is such that line joining PQ is bisected perpendicularly by the initial line, is

The length of conjugate axis of a hyperbola is greater than the length of transverse axis. Then, the eccentricity e is

The value of $$\mathop {\lim }\limits_{x \to {0^ + }} {x \over p}\left[ {{q \over x}} \right]$$ is

Let $$f(x) = {x^4} - 4{x^3} + 4{x^2} + c,\,c \in R$$. Then

The graphs of the polynomial x² $$-$$ 1 and cos x intersect

A point is in motion along a hyperbola $$y = {{10} \over x}$$ so that its abscissa x increases uniformly at a rate of 1 unit per second. Then, the rate of change of its ordinate when the point passes through (5, 2)

Let $${I_n} = \int\limits_0^1 {{x^n}} {\tan ^{ - 1}}xdx$$. If $${a_n}{I_{n + 2}} + {b_n}{I_n} = {c_n}$$ for all n $$ \ge $$ 1, then

Two particles A and B move from rest along a straight line with constant accelerations f and h, respectively. If A takes m seconds more than B and describes n units more than that of B acquiring the same speed, then

The area bounded by y = x + 1 and y = cos x and the X-axis, is

Let x₁, x₂ be the roots of $${x^2} - 3x + a = 0$$ and x₃, x₄ be the roots of $${x^2} - 12x + b = 0$$. If $${x_1} < {x_2} < {x_3} < {x_4}$$ and $${x_1},{x_2},{x_3},{x_4}$$ are in GP, then ab equals

If $$\theta \in R$$ and $${{1 - i\cos \theta } \over {1 + 2i\cos \theta }}$$ is real number, then $$\theta $$ will be (when I : Set of integers)

Let $$A = \left[ {\matrix{ 3 & 0 & 3 \cr 0 & 3 & 0 \cr 3 & 0 & 3 \cr } } \right]$$. Then, the roots of the equation det $$(A - \lambda {I_3})$$ = 0 (where I₃ is the identity matrix of order 3) are

Straight lines x $$-$$ y = 7 and x + 4y = 2 intersect at B. Points A and C are so chosen on these two lines such that AB = AC. The equation of line AC passing through (2, $$-$$7) is

Equation of a tangent to the hyperbola 5x² $$-$$ y² = 5 and which passes through an external point (2, 8) is

Let f and g be differentiable on the interval I and let a, b $$ \in $$ I, a < b. Then,

Consider the function $$f(x) = {{{x^3}} \over 4} - \sin \pi x + 3$$

A ray of light is reflected by a plane mirror. $${\widehat e_0}$$, $$\widehat e$$ and $$\widehat n$$ be the unit vectors along the incident ray, reflected ray and the normal to the reflecting surface respectively.

Which of the following gives an expression for $$\widehat e$$?

A parent nucleus X undergoes $$\alpha$$-decay with a half-life of 75000 yrs. The daughter nucleus Y undergoes $$\beta$$-decay with a half-life of 9 months. In a particular sample, it is found that the rate of emission of $$\beta$$-particles is nearly constant (over several months) at $${10^7}/h$$. What will be the number of $$\alpha$$-particles emitted in an hour?

A proton and an electron initially at rest are accelerated by the same potential difference. Assuming that a proton is 2000 times heavier than an electron, what will be the relation between the de Broglie wavelength of the proton ($$\lambda$$_p) and that of electron ($$\lambda$$_e)?

To which of the following the angular velocity of the electron in the n-th Bohr orbit is proportional?

In the circuit shown, what will be the current through the 6V zener?

Each of the two inputs A and B can assume values either 0 or 1. Then which of the following will be equal to $$\overline {A\,} $$ . $$\overline {B\,} $$?

The correct dimensional formula for impulse is given by

The density of the material of a cube can be estimated by measuring its mass and the length of one of its sides. If the maximum error in the measurement of mass and length are 0.3% and 0.2% respectively, the maximum error in the estimation of the density of the cube is approximately

Two weights of the mass m₁ and m₂ (> m₁) are joined by an inextensible string of negligible mass passing over a fixed frictionless pulley. The magnitude of the acceleration of the loads is

A body starts from rest, under the action of an engine working at a constant power and moves along a straight line. The displacement s is given as a function of time (t) as

Two particles are simultaneously projected in the horizontal direction from a point P at a certain height. The initial velocities of the particles are oppositely directed to each other and have magnitude v each. The separation between the particles at a time when their position vectors (drawn from the point P) are mutually perpendicular, is

Assume that the earth moves around the sun in a circular orbit of radius R and there exists a planet which also move around the sun in a circular orbit with an angular speed twice as large as that of the earth. The earth of the orbit of the planet is

A compressive force is applied to a uniform rod of rectangular cross-section so that its length decreases by 1%. If the Poisson's ratio for the material of the rod be 0.2, which of the following statements is correct?

"The volume approximately ................."

A small spherical body of radius r and density $$\rho $$ moves with the terminal velocity v in a fluid of coefficient of viscosity $$\eta $$ and density $$\sigma$$. What will be the net force on the body?

Two black bodies A and B have equal surface areas are maintained at temperatures 27$$^\circ$$C and 177$$^\circ$$C respectively. What will be the ratio of the thermal energy radiated per second by A to that by B?

What will be the molar specific heat at constant volume of an ideal gas consisting of rigid diatomic molecules?

Consider the given diagram. An ideal gas is contained in a chamber (left) of volume V and is at an absolute temperature T. It is allowed to rush freely into the right chamber of volume V which is initially vacuum. The whole system is thermally isolated. What will be the final temperature of the equilibrium has been attained?

Five identical capacitors, of capacitance 20$$\mu$$F each, are connected to a battery of 150V, in a combination as shown in the diagram. What is the total amount of charge stored?

Eleven equal point charges, all of them having a charge +Q, are placed at all the hour positions of a circular clock of radius r, except at the 10 h position. What is the electric field strength at the centre of the clock?

A negative charge is placed at the midpoint between two fixed equal positive charges, separated by a distance 2d. If the negative charge is given a small displacement x(x < < d) perpendicular to the line joining the positive charges, how the force (F) developed on it will approximately depend on x?

To which of the following quantities, the radius of the circular path of a charged particle moving at right angles to a uniform magnetic field is directly proportional?

An electric current 'I' enters and leaves a uniform circular wire of radius r through diametrically opposite points. A particle carrying a charge q moves along the axis of the circular wire with speed v. What is the magnetic force experienced by the particle when it passes through the centre of the circle?

A current 'I' is flowing along an infinite, straight wire, in the positive Z-direction and the same current is flowing along a similar parallel wire 5 m apart, in the negative Z-direction. A point P is at a perpendicular distance 3m from the first wire and 4m from the second. What will be magnitude of the magnetic field B of P?

A square conducting loop is placed near an infinitely long current carrying wire with one edge parallel to the wire as shown in the figure. If the current in the straight wire is suddenly halved, which of the following statements will be true?


"The loop will .........".

What is the current I shown in the given circuit?

When the value of R in the balanced Wheatstone bridge, shown in the figure, is increased from 5$$\Omega $$ to 7$$\Omega $$, the value of S has to be increased by 3$$\Omega $$ in order to maintain the balance. What is the initial values of S?

When a 60 mH inductor and a resistor are connected in series with an AC voltage source, the voltage leads the current by 60$$^\circ$$. If the inductor is replaced by a 0.5 $$\mu$$F capacitor, the voltage lags behind the current by 30$$^\circ$$. What is the frequency of the AC supply?

A point object is placed on the axis of a thin convex lens of focal length 0.05 m at a distance of 0.2 m from the lens and its image is formed on the axis. If the object is now made to oscillate along the axis with a small amplitude of A cm, then what is the amplitude of oscillation of the image?

[you may assume, $${1 \over {1 + x}} \approx 1 - x$$, where x < < 1]

In Young's experiment for the interference of light, the separation between the slits is d and the distance of the screen from the slits is D. If D is increased by 0.5% and d is decreased by 0.3% then for the light of a given wavelength , which one of the following is true? "The fringe width ..............

When the frequency of the light used is changed from $$4 \times {10^{14}}{s^{ - 1}}$$ to $$5 \times {10^{14}}{s^{ - 1}}$$, the angular width of the principal (central) maximum in a single slit Fraunhoffer diffraction pattern changes by 0.6 radian. What is the width of the slit (assume that the experiment is performed in vacuum)?

A capacitor of capacitance C is connected in series with a resistance R and DC source of emf E through a key. The capacitor starts charging when the key is closed. By the time the capacitor has been fully charged, what amount of energy is dissipated in the resistance R?

A horizontal fire hose with a nozzle of cross-sectional area $${5 \over {\sqrt {21} }} \times {10^{ - 3}}{m^2}$$ delivers a cubic metre of water in 10s. What will be the maximum possible increase in the temperature of water while it hits a rigid wall (neglecting the effect of gravity)?

Two identical blocks of ice move in opposite directions with equal speed and collide with each other. What will be the minimum speed required to make both the blocks melt completely, if the initial temperatures of the blocks were $$-$$8$$^\circ$$C each? (Specific heat of ice is 2100 Jkg^$$-$$1K^$$-$$1 and latent heat of fusion of ice is 3.36 $$ \times $$ 10⁵ Jkg^$$-$$1)

A particle with charge q moves with a velocity v in a direction perpendicular to the directions of uniform electric and magnetic fields, E and B respectively, which are mutually perpendicular to each other. Which one of the following gives the condition for which the particle moves undeflected in its original trajectory?

A parallel plate capacitor in series with a resistance of 100$$\Omega $$, an inductor of 20 mH and an AC voltage source of variable frequency shows resonance at a frequency of $${{1250} \over \pi }$$ Hz. If this capacitor is charged by a DC voltage source to a voltage 25 V, what amount of charge will be stored in each plate of the capacitor?

Electrons are emitted with kinetic energy T from a metal plate by an irradiation of light of intensity J and frequency v. Then, which of the following will be true?

The initial pressure and volume of a given mass of an ideal gas with $$\left( {{{{C_p}} \over {{C_V}}} = \gamma } \right)$$, taken in a cylinder fitted with a piston, are p₀ and V₀ respectively. At this stage the gas has the same temperature as that of the surrounding medium which is T₀. It is adiabatically compressed to a volume equal to $${{{V_0}} \over 2}$$. Subsequently the gas is allowed to come to thermal equilibrium with the surroundings. What is the heat released to the surrounding?

A projectile thrown with an initial velocity of 10 ms^$$-$$1 at an angle $$\alpha$$ with the horizontal, has a range of 5 m. Taking g = 10 ms^$$-$$2 and neglecting air resistance, what will be the estimated value of $$\alpha$$?

In the circuit shown in the figure all the resistance are identical and each has the value r$$\Omega $$. The equivalent resistance of the combination between the points A and B will remain unchanged even when the following pairs of points marked in the figure are connected through a resistance R.

A metallic loop is placed in a uniform magnetic field B with the plane of the loop perpendicular to B. Under which condition(s) given an emf will be induced in the loop? "If the loop is ............"