For the above three esters, the order of rates of alkaline hydrolysis is

Ph$$ - $$ CDO$$\mathrel{\mathop{\kern0pt\longrightarrow} \limits_{Warm}^{50\% aq.NaOH}} $$ Ph$$ - $$ COO$$\mathop H\limits^\Theta $$ + an alcohol.

This alcohol is

The correct order of acidity for the following compounds is :

The reduction product of ethyl 3-oxobutanoate by NaBH₄ in methanol is

What is the major product of the following reaction?

The maximum number of electrons in an atom in which the last electron filled has the quantum numbers n = 3, l = 2 and m = $$ - $$ 1 is

In the face centered cubic lattice structure of gold the closest distance between gold atoms is ('a' being the edge length of the cubic unit cell)

The equilibrium constant for the following reactions are given at 25$$^\circ $$C

$$2A$$ $$\rightleftharpoons$$ B + C, K₁ = 1.0

$$2B$$ $$\rightleftharpoons$$ C + D, K₂ = 16

$$2C + 2D$$ $$\rightleftharpoons$$ 2P, K₃ = 25

The equilibrium constant for the reaction

P $$\rightleftharpoons$$ $$A + {1 \over 2}$$B at 25$$^\circ $$C is

Among the following, the ion which will be more effective for flocculation of Fe(OH)₃ solution is :

The mole fraction of ethanol in water is 0.08. Its molality is

5 mL of 0.1 M Pb(NO₃)₂ is mixed with 10 mL of 0.02 M KI. The amount of PbI₂ precipitated will be about

At 273 K temperature and 76 cm Hg pressure the density of a gas is 1.964 g L^-1. The gas is

Equal masses of ethane and hydrogen are mixed in an empty container at 298 K. The fraction of total pressure exerted by hydrogen is

An ideal gas expands adiabatically against vacuum. Which of the following is correct for the given process?

K_f (water) = 1.86 K kg mol^-1. The temperature at which ice begins to separate from a mixture of 10 mass % ethylene glycol is

The radius of the first Bohr orbit of a hydrogen atom is 0.53 $$\times {10^-8}$$ cm. The velocity of the electron in the first Bohr orbit is

Which of the following statements is not true for the reaction, 2F₂ + 2H₂O $$ \to $$ 4HF + O₂ ?

The number of unpaired electrons in the uranium (₉₂U) atom is

How and why does the density of liquid water change on prolonged electrolysis?

The difference between orbital angular momentum of an electron in a 4f -orbital and another electron in a 4s-orbital is

Which of the following has the largest number of atoms?

Indicate the correct IUPAC name of the coordination compound shown in the figure.

What will be the mass of one atom of ¹²C?

Bond order of He₂, $$He_2^ + $$ and $$He_2^{2 + }$$ are respectively

To a solution of a colourless efflorescent sodium salt, when dilute acid is added, a colourless gas is evolved along with formation of a white precipitate. Acidified dichromate solution turns green, when the colourless gas is passed through it. The sodium salt is

The reaction for obtaining the metal (M) from its oxide (M₂O₃) ore is given by

$${M_2}{O_3}(s) + 2Al(l)\buildrel {Heat} \over \longrightarrow A{l_2}{O_3}(l) + 2M(s)$$, (s = solid, l = liquid) in that case, M is

In the extraction of Ca by electro reduction of molten CaCl₂ some CaF₂ is added to the electrolyte for the following reason :

The total number of alkyl bromides (including stereoisomers) formed in the reaction $$M{e_3}C - CH = C{H_2} + HBr \to $$ will be

The product in the above reaction is

Which of the following compounds is asymmetric?

For a reaction 2A + B $$ \to $$ P, when concentration of B alone is doubled, t_1/2 does not change and when concentrations of both A and B is doubled, rate increases by a factor of 4. The unit of rate constant is,

A solution is saturated with SrCO₃ and SrF₂. The $$[CO_3^{2 - }]$$ is found to be 1.2 $$ \times $$ 10^-3 M. The concentration of F^- in the solution would be

Given : K_sp(SrCO₃) = 7.0 $$ \times $$ 10^-10,

K_sp(SrF₂) = 7.9 $$ \times $$ 10^-10

A homonuclear diatomic gas molecule shows 2-electron magnetic moment. The one-electron and two-electron reduced species obtained from above gas molecule can act as both oxidising and reducing agents. When the gas molecule is one-electron oxidised the bond length decreases compared to the neutral molecule. The gas molecule is

$$C{H_3} - O - C{H_2} - Cl\mathrel{\mathop{\kern0pt\longrightarrow} \limits_\Delta ^{aq{.^\Theta }OH}} C{H_3} - O - C{H_2} - OH$$

Which information below regarding this reaction is applicable?

Which of the following reactions give(s) a meso-compound as the main product?

For spontaneous polymerisation, which of the following is (are) correct?

Which of the following statement(s) is/are incorrect?

SiO₂ is attacked by which one/ones of the following?

$$Me - C \equiv C - Me\mathrel{\mathop{\kern0pt\longrightarrow} \limits_{EtOH, - 33^\circ C}^{Na/N{H_3}(liq.)}} \underline{\underline X} \,\buildrel {dil.alkaline\,KMn{O_4}} \over \longrightarrow \,\text{Product(s)}$$

The product(s) from the above reaction will be

For the following carbocations, the correct order of stability is

I. $$^ \oplus C{H_2} - COC{H_3}$$

II. $$^ \oplus C{H_2} - OC{H_3}$$

III. $$^ \oplus C{H_2} - C{H_3}$$

Let cos$$^{ - 1}\left( {{y \over b}} \right) = \log {\left( {{x \over n}} \right)^n}$$. Then

Let $$\phi (x) = f(x) + f(1 - x)$$ and $$f(x) < 0$$ in [0, 1], then

$$\int {{{f(x)\phi '(x) + \phi (x)f'(x)} \over {(f(x)\phi (x) + 1)\sqrt {f(x)\phi (x) - 1} }}dx = } $$

The value of

$$\sum\limits_{n = 1}^{10} {} \int\limits_{ - 2n - 1}^{ - 2n} {{{\sin }^{27}}} x\,dx + \sum\limits_{n = 1}^{10} {} \int\limits_{2n}^{2n + 1} {{{\sin }^{27}}} x\,dx$$ is equal to

$$\int\limits_0^2 {[{x^2}]} \,dx$$ is equal to

If the tangent to the curve y² = x³ at (m², m³) is also a normal to the curve at (m², m³), then the value of mM is

If $${x^2} + {y^2} = {a^2}$$, then $$\int\limits_0^a {\sqrt {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} dx = } $$

Let f, be a continuous function in [0, 1], then $$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{j = 0}^n {{1 \over n}} f\left( {{j \over n}} \right)$$ is

Let f be a differentiable function with $$\mathop {\lim }\limits_{x \to \infty } f(x) = 0.$$ If $$y' + yf'(x) - f(x)f'(x) = 0$$, $$\mathop {\lim }\limits_{x \to \infty } y(x) = 0$$, then (where $$y \equiv {{dy} \over {dx}})$$

If $$x\sin \left( {{y \over x}} \right)dy = \left[ {y\sin \left( {{y \over x}} \right) - x} \right]dx,\,x > 0$$ and $$y(1) = {\pi \over 2}$$, then the value of $$\cos \left( {{y \over x}} \right)$$ is

Let $$f(x) = 1 - \sqrt {({x^2})} $$, where the square root is to be taken positive, then

If the function $$f(x) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1$$ [a > 0] attains its maximum and minimum at p and q respectively such that p² = q, then a is equal to

If a and b are arbitrary positive real numbers, then the least possible value of $${{6a} \over {5b}} + {{10b} \over {3a}}$$ is

If 2 log(x + 1) $$ - $$ log(x² $$ - $$ 1) = log 2, then x =

The number of complex numbers p such that $$\left| p \right| = 1$$ and imaginary part of p⁴ is 0, is

The equation $$z\bar z + (2 - 3i)z + (2 + 3i)\bar z + 4 = 0$$ represents a circle of radius

The expression ax² + bx + c (a, b and c are real) has the same sign as that of a for all x if

In a 12 storied building, 3 persons enter a lift cabin. It is known that they will leave the lift at different floors. In how many ways can they do so if the lift does not stop at the second floor?

If the total number of m-element subsets of the set A = {a₁, a₂, ..., a_n} is k times the number of m element subsets containing a₄, then n is

Let I(n) = nⁿ, J(n) = 13.5 ......... (2n $$ - $$ 1) for all (n > 1), n $$ \in $$ N, then

If c₀, c₁, c₂, ......, c₁₅ are the binomial coefficients in the expansion

of (1 + x)¹⁵, then the value of $${{{c_1}} \over {{c_0}}} + 2{{{c_2}} \over {{c_1}}} + 3{{{c_3}} \over {{c_2}}} + ... + 15{{{c_{15}}} \over {{c_{14}}}}$$ is

Let A = $$\left( {\matrix{ {3 - t} \cr { - 1} \cr 0 \cr } \matrix{ {} \cr {} \cr {} \cr } \,\matrix{ 1 \cr {3 - t} \cr { - 1} \cr } \matrix{ {} \cr {} \cr {} \cr } \matrix{ 0 \cr 1 \cr 0 \cr } } \right)$$ and det A = 5, then

Let $$A = \left[ {\matrix{ {12} & {24} & 5 \cr x & 6 & 2 \cr { - 1} & { - 2} & 3 \cr } } \right]$$. The value of x for which the matrix A is not invertible is

Let $$A = \left( {\matrix{ a & b \cr c & d \cr } } \right)$$ be a 2 $$ \times $$ 2 real matrix with det A = 1. If the equation det (A $$ - $$ $$\lambda $$I₂) = 0 has imaginary roots (I₂ be the identity matrix of order 2), then

If $$\left| {\matrix{ {{a^2}} & {bc} & {{c^2} + ac} \cr {{a^2} + ab} & {{b^2}} & {ca} \cr {ab} & {{b^2} + bc} & {{c^2}} \cr } } \right| = k{a^2}{b^2}{c^2}$$,

then K =

If f : S $$ \to $$ R, where S is the set of all non-singular matrices of order 2 over R and $$f\left[ {\left( {\matrix{ a & b \cr c & d \cr } } \right)} \right] = ad - bc$$, then

Let the relation p be defined on R by a p b holds if and only if a $$ - $$ b is zero or irrational, then

The unit vector in ZOX plane, making angles $$45^\circ $$ and $$60^\circ $$ respectively with $$\alpha = 2\widehat i + 2\widehat j - \widehat k$$ and $$\beta = \widehat j - \widehat k$$ is

Four persons A, B, C and D throw an unbiased die, turn by turn, in succession till one gets an even number and win the game. What is the probability that A wins if A begins?

A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire to have more than 50% chance of hitting it at least once, is

$$\cos (2x + 7) = a(2 - \sin x)$$ can have a real solution for

The differential equation of the family of curves y = e^x (A cos x + B sin x) where, A, B are arbitrary constants is

The equation $$r\,\cos \left( {\theta - {\pi \over 3}} \right) = 2$$ represents

The locus of the centre of the circles which touch both the circles x² + y² = a² and x² + y² = 4ax externally is

Let each of the equations x² + 2xy + ay² = 0 and ax² + 2xy + y² = 0 represent two straight lines passing through the origin. If they have a common line, then the other two lines are given by

A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at P and Q respectively. The point O divides the segment PQ in the ratio

Area in the first quadrant between the ellipses x² + 2y² = a² and 2x² + y² = a² is

The equation of circle of radius $$\sqrt {17} $$ unit, with centre on the positive side of X-axis and through the point (0, 1) is

The length of the chord of the parabola y² = 4ax(a > 0) which passes through the vertex and makes an acute angle $$\alpha $$ with the axis of the parabola is

A double ordinate PQ of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is such that $$\Delta OPQ$$ is equilateral, O being the centre of the hyperbola. Then the eccentricity e satisfies the relation

If B and B' are the ends of minor axis and S and S' are the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over 9} = 1$$, then the area of the rhombus SBS' B' will be

The equation of the latusrectum of a parabola is x + y = 8 and the equation of the tangent at the vertex is x + y = 12. Then, the length of the latusrectum is

The equation of the plane through the point $$(2, - 1, - 3)$$ and parallel to the lines

$${{x - 1} \over 2} = {{y + 2} \over 3} = {z \over { - 4}}$$ and $${x \over 2} = {{y - 1} \over { - 3}} = {{z - 2} \over 2}$$ is

The sine of the angle between the straight line $${{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}$$ and the plane $$2x - 2y + z = 5$$ is

Let f(x) = sin x + cos ax be periodic function. Then,

The domain of $$f(x) = \sqrt {\left( {{1 \over {\sqrt x }} - \sqrt {x + 1} } \right)} $$ is

Let $$y = f(x) = 2{x^2} - 3x + 2$$. The differential of y when x changes from 2 to 1.99 is

If $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + cx} \over {1 - cx}}} \right)^{{1 \over x}}} = 4$$, then $$\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + 2cx} \over {1 - 2cx}}} \right)^{{1 \over x}}}$$ is

Let f : R $$ \to $$ R be twice continuously differentiable (or f" exists and is continuous) such that f(0) = f(1) = f'(0) = 0. Then

Let $$f(x) = {x^{13}} + {x^{11}} + {x^9} + {x^7} + {x^5} + {x^3} + x + 12$$.

Then

The area of the region

$$\{ (x,y):{x^2} + {y^2} \le 1 \le x + y\} $$ is

In open interval $$\left( {0,\,{\pi \over 2}} \right)$$

If the line y = x is a tangent to the parabola y = ax² + bx + c at the point (1, 1) and the curve passes through ($$ - $$1, 0), then

If the vectors $$\alpha = \widehat i + a\widehat j + {a^2}\widehat k,\,\beta = \widehat i + b\widehat j + {b^2}\widehat k$$ and $$\,\gamma = \widehat i + c\widehat j + {c^2}\widehat k$$ are three non-coplanar

vectors and $$\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0$$, then the value of abc is

Let z₁ and z₂ be two imaginary roots of z² + pz + q = 0, where p and q are real. The points z₁, z₂ and origin form an equilateral triangle if

If P(x) = ax² + bx + c and Q(x) = $$ - $$ax² + dx + c, where ac $$ \ne $$ 0 [a, b, c, d are all real], then P(x).Q(x) = 0 has

Let $$A = \{ x \in R: - 1 \le x \le 1\} $$ and $$f:A \to A$$ be a mapping defined by $$f(x) = x\left| x \right|$$. Then f is

Let $$f(x) = \sqrt {{x^2} - 3x + 2} $$ and $$g(x) = \sqrt x $$ be two given functions. If S be the domain of fog and T be the domain of gof, then

Let p₁ and p₂ be two equivalence relations defined on a non-void set S. Then

Consider the curve $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

Consider the curve $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

A line cuts the X-axis at A(7, 0) and the Y-axis at B(0, $$ - $$5). A variable line PQ is drawn perpendicular to AB cutting the X-axis at P(a, 0) and the Y-axis at Q(0, b). If AQ and BP intersect at R, the locus of R is

Let $$0 < \alpha < \beta < 1$$. Then, $$\mathop {\lim }\limits_{n \to \infty } \int\limits_{1/(k + \beta )}^{1/(k + \alpha )} {{{dx} \over {1 + x}}} $$ is

$$\mathop {\lim }\limits_{x \to 1} \left( {{1 \over {1nx}} - {1 \over {(x - 1)}}} \right)$$

Let $$y = {1 \over {1 + x + lnx}}$$, then

Consider the curve $$y = b{e^{ - x/a}}$$, where a and b are non-zero real numbers. Then

The area of the figure bounded by the parabola $$x = - 2{y^2},\,x = 1 - 3{y^2}$$ is

A particle is projected vertically upwards. If it has to stay above the ground for 12 sec, then

The equation $${x^{{{(\log 3x)}^2}}} - {9 \over 2}\log 3\,x + 5 = 3\sqrt 3 $$ has

In a certain test, there are n questions. In this test 2^n-i students gave wrong answers to at least i questions, where i = 1, 2, ..., n. If the total number of wrong answers given is 2047, then n is equal to

A and B are independent events. The probability that both A and B occur is $${1 \over {20}}$$ and the probability that neither of them occurs is $${3 \over {5}}$$. The probability of occurrence of A is

The equation of the straight line passing through the point (4, 3) and making intercepts on the coordinate axes whose sum is $$ - 1$$ is

Consider a tangent to the ellipse $${{{x^2}} \over 2} + {{{y^2}} \over 1} = 1$$ at any point. The locus of the mid-point of the portion intercepted between the axes is

Let $$y = {{{x^2}} \over {{{(x + 1)}^2}(x + 2)}}$$. Then $${{{d^2}y} \over {d{x^2}}}$$ is

Let $$f(x) = {1 \over 3}x\sin x - (1 - \cos \,x)$$. The smallest positive integer k such that $$\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^k}}} \ne 0$$ is

Tangent is drawn at any point P(x, y) on a curve, which passes through (1, 1). The tangent cuts X-axis and Y-axis at A and B respectively. If AP : BP = 3 : 1, then

The intensity of light emerging from one of the slits in a Young's double slit experiment is found to be 1.5 times the intensity of light emerging from the other slit. What will be the approximate ratio of intensity of an interference maximum to that of an interference minimum?

In a Fraunhofer diffraction experiment, a single slit of width 0.5 mm is illuminated by a monochromatic light of wavelength 600 nm. The diffraction pattern is observed on a screen at a distance of 50 cm from the slit. What will be the linear separation of the first order minima?

If R is the Rydberg constant in cm^-1, then hydrogen atom does not emit any radiation of wavelength in the range of

A nucleus X emits a $$\beta $$-particle to produce a nucleus Y. If their atomic masses are M_x and M_y respectively, then the maximum energy of the $$\beta $$-particle emitted is (where, m_e is the mass of an electron and c is the velocity of light)

For nuclei with mass number close to 119 and 238, the binding energies per nucleon are approximately 7.6 MeV and 8.6 MeV, respectively. If a nucleus of mass number 238 breaks into two nuclei of nearly equal masses, what will be the approximate amount of energy released in the process of fission?

A common emitter transistor amplifier is connected with a load resistance of 6 k$$\Omega $$ . When a small AC signal of 15 mV is added to the base-emitter voltage, the alternating base current is 20$$\mu $$A and the alternating collector current is 1.8 mA. What is the voltage gain of the amplifier?

In the circuit shown, the value of $$\beta $$ of the transistor is 48. If the supplied base current is 200 $$\mu $$A, what is the voltage at the terminal Y?

The frequency v of the radiation emitted by an atom when an electron jumps from one orbit to another is given by v = k$$\delta $$E, where k is a constant and $$\delta $$E is the change in energy level due to the transition. Then, dimension of k is

Consider the vectors $$A = \hat i + \hat j - \hat k$$ ,$$B = 2\hat i - \hat j + \hat k$$ and $$C = {1 \over {\sqrt 5 }}\left( {\hat i - 2\hat j + 2\hat k} \right)$$ . What is the value of C. (A$$ \times $$ B) ?

A fighter plane, flying horizontally with a speed 360 km/h at an altitude of 500 m drops a bomb for a target straight ahead of it on the ground. The bomb should be dropped at what approximate distance ahead of the target? Assume that acceleration due to gravity (g) is 10 ms^-2. Also, neglect air drag.

A block of mass m rests on a horizontal table with a coefficient of static friction $$\mu $$. What minimum force must be applied on the block to drag it on the table?

A tennis ball hits the floor with a speed v at an angle $$\theta $$ with the normal to the floor. If the collision is inelastic and the coefficient of restitution is $$\varepsilon $$, what will be the angle of reflection?

The bob of a swinging second pendulum (one whose time period is 2 s) has a small speed v₀ at its lowest point. It height from this lowest point 2.25 s after passing through it, is given by

A steel and a brass wire, each of length 50 cm and cross-sectional area 0.005 cm² hang from a ceiling and are 15 cm apart. Lower ends of the wires are attached to a light horizontal bar. A suitable downward load is applied to the bar, so that each of the wires extends in length by 0.1 cm. At what distance from the steel wire, the load must be applied?

[Young's modulus of steel = 2 $$ \times $$ 10¹² dyne/cm² and that of brass = 1 $$ \times $$ 10¹² dyne/cm²]

Which of the following diagrams correctly shows the relation between the terminal velocity v_T of a spherical body falling in a liquid and viscosity $$\eta $$ of the liquid?

An ideal gas undergoes the cyclic process abca as shown in the given p - V diagram


It rejects 50J of heat during ab and absorbs 80J of heat during ca. During bc, there is no transfer of heat and 40J of work is done by the gas. What should be the area of the closed curved abca?

A container AB in the shape of a rectangular parallelopiped of length 5 m is divided internally by a movable partition P as shown in the figure.


The left compartment is filled with a given mass of an ideal gas of molar mass 32 while the right compartment is filled with an equal mass of another ideal gas of molar mass 18 at same temperature. What will be the distance of P from the left wall A when equilibrium is established?

When 100 g of boiling water at 100$$^\circ $$ C is added into a calorimeter containing 300 g of cold water at 10$$^\circ $$ C, temperature of the mixture becomes 20$$^\circ $$ C. Then, a metallic block of mass 1 kg at 10$$^\circ $$ C is dipped into the mixture in the calorimeter. After reaching thermal equilibrium, the final temperature becomes 19$$^\circ $$ C. What is the specific heat of the metal in CGS unit?

As shown in the figure, a point charge q₁ = + 1 $$ \times $$ 10^-6 C is placed at the origin in xy-plane and another point charge q₂ = + 3 $$ \times $$ 10^-6 C is placed at the coordinate (10, 0).


In that case, which of the following graph(s) shows most correctly the electric field vector in E_x in x-direction?

Four identical point masses, each of mass m and carrying charge + q are placed at the corners of a square of sides a on a frictionless plane surface. If the particles are released simultaneously, the kinetic energy of the system when they are infinitely far apart is

A very long charged solid cylinder of radius a contains a uniform charge density p. Dielectric constant of the material of the cylinder is K. What will be the magnitude of electric field at a radial distance x (x < a) from the axis of the cylinder?

A galvanometer can be converted to a voltmeter of full scale deflection V₀ by connecting a series resistance R₁ and can be converted to an ammeter of full scale deflection I₀ by connecting a shunt resistance R₂. What is the current flowing through the galvanometer at its full scale deflection?

As shown in the figure, a single conducting wire is bent to form a loop in the form of a circle of radius r concentrically inside a square of side a, where a : r = 8 : $$\pi $$. A battery B drives a current through the wire. If the battery B and the gap G are of negligible sizes, determine the strength of magnetic field at the common centre O.

As shown in the figure, a wire is bent to form a D-shaped closed loop, carrying current I, where the curved part is a semi-circle of radius R. The loop is placed in a uniform magnetic field B, which is directed into the plane of the paper. The magnetic force felt by the closed loop is

What will be the equivalent resistance between the terminals A and B of the infinite resistive network shown in the figure?

When a DC voltage is applied at the two ends of a circuit kept in a closed box, it is observed that the current gradually increases from zero to a certain value and then remains constant. What do you think that the circuit contains?

Consider the circuit shown.


If all the cells have negligible internal resistance, what will be the current through the 2$$\Omega $$ resistor when steady state is reached?

Consider a conducting wire of length L bent in the form of a circle of radius R and another conductor of length a (a < < R) is bent in the form of a square. The two loops are then placed in same plane such that the square loop is exactly at the centre of the circular loop. What will be the mutual inductance between the two loops?

An object, is placed 60 cm in front of a convex mirror of focal length 30 cm. A plane mirror is now placed facing the object in between the object and the convex mirror such that it covers lower half of the convex mirror. What should be the distance of the plane mirror from the object, so that there will be no parallax between the images formed by the two mirrors?

A thin convex lens is placed just above an empty vessel of depth 80 cm. The image of a coin kept at the bottom of the vessel is thus formed 20 cm above the lens. If now water is poured in the vessel upto a height of 64 cm, what will be the approximate new position of the image? Assume that refractive index of water is 4/3.

A conducting circular loop of resistance 20$$\Omega $$ and cross-sectional area 20 $$ \times $$ 10^-2 m² is placed perpendicular to a spatially uniform magnetic field B, which varies with time t as B = 2sin(50$$\pi $$t) T. Find the net charge flowing through the loop in 20 ms starting from t = 0.

A pair of parallel metal plates are kept with a separation d. One plate is at a potential + V and the other is at ground potential. A narrow beam of electrons enters the space between the plates with a velocity v₀ and in a direction parallel to the plates. What will be the angle of the beam with the plates after it travels an axial distance L?

A metallic block of mass 20 kg is dragged with a uniform velocity of 0.5 ms^-1 on a horizontal table for 2.1 s. The coefficient of static friction between the block and the table is 0.10. What will be the maximum possible rise in temperature of the metal block, if the specific heat of the block is 0.1 CGS unit? Assume g = 10 ms^-2 and uniform rise in temperature throughout the whole block. [Ignore absorption of heat by the table]

Consider an engine that absorbs 130 cal of heat from a hot reservoir and delivers 30 cal heat to a cold reservoir in each cycle. The engine also consumes 2 J energy in each cycle to overcome friction. If the engine works at 90 cycles per minute, what will be the maximum power delivered to the load? [Assume the thermal equivalent of heat is 4.2 J/cal]

Two pith balls, each carrying charge + q are hung from a hook by two springs. It is found that when each charge is tripled, angle between the strings double. What was the initial angle between the strings?

A point source of light is used in an experiment of photoelectric effects. If the distance between the source and the photoelectric surface is doubled, which of the following may result?

Two metallic spheres of equal outer radii are found to have same moment of inertia about their respective diameters. Then, which of the following statement(s) is/are true?

A simple pendulum of length l is displaced, so that its taught string is horizontal and then released. A uniform bar pivoted at one end is simultaneously released from its horizontal position. If their motions are synchronous, what is the length of the bar?

A 400$$\Omega $$ resistor, a 250 mH inductor and a 2.5 $$\mu $$F capacitor are connected in series with an AC source of peak voltage 5 V and angular frequency 2kHZ. What is the peak value of the electrostatic energy of the capacitor?

A charged particle moves with constant velocity in a region, where no effect of gravity is felt but an electrostatic field E together with a magnetic field B may be present. Then, which of the following cases are possible?