Which of the following statements is incorrect?

The calculated spin-only magnetic moment values in BM for $$\mathrm{[FeCl_4]^-}$$ and $$\mathrm{[Fe(CN)_6]^{3-}}$$ are

$$\mathrm{BrF_3}$$ self ionises as following

4f$$^2$$ electronic configuration is found in

The correct order of C = O bond length in ethyl propanoate (I), ethyl propenoate (II) and ethenyl propanoate (III) is

Select the molecule in which all the atoms may lie on a single plane is

The IUPAC name of

The relationship between the pair of compounds shown above are respectively,

The correct stability order of the following carbocations is

(I) $$\mathrm{{H_2}\mathop C\limits^ \oplus - CH = CH - C{H_3}}$$

(II) $$\mathrm{\mathop C\limits^ \oplus {H_2} - CH = CH - BM{e_2}}$$

(III) $$\mathrm{{H_2}\mathop C\limits^ \oplus - CH = CH - NMe}$$

(IV) $$\mathrm{{H_2}\mathop C\limits^ \oplus - CH = CH - OMe}$$

The correct order of boiling points of N-ethylethanamine (I), ethoxyethane (II) and butan-2-ol (III) is

Structure of M is,

The correct order of acidity of above compounds is

If all the nucleophilic substitution reactions at saturated carbon atoms in the above sequence of reactions follow S_N2 mechanism, then $$\mathrm{\underline E}$$ and $$\mathrm{\underline F}$$ will be respectively,

The correct option for the above reaction is

Arrange the following in order of increasing mass

I. 1 mole of N$$_2$$

II. 0.5 mole of O$$_3$$

III. $$3.011\times10^{23}$$ molecules of O$$_2$$

IV. 0.5 gram atom of O$$_2$$

Two base balls (masses : m$$_1$$ = 100 g, and m$$_2$$ = 50 g) are thrown. Both of them move with uniform velocity, but the velocity of m$$_2$$ is 1.5 times that of m$$_1$$. The ratio of de Broglie wavelengths $$\lambda$$(m$$_1$$) : $$\lambda$$(m$$_2$$) is given by

What is the edge length of the unit cell of a body centred cubic crystal of an element whose atomic radius is 75 pm?

The root mean square (rms) speed of X$$_2$$ gas is x m/s at a given temperature. When the temperature is doubled, the X$$_2$$ molecules dissociated completely into atoms. The root mean square speed of the sample of gas then becomes (in m/s)

Which of the following would give a linear plot?

(k is the rate constant of an elementary reaction and T is temp. in absolute scale)

The equivalent conductance of NaCl, HCl and CH$$_3$$COONa at infinite dilution are 126.45, 426.16 and 91 ohm$$^{-1}$$cm$$^2$$eq$$^{-1}$$ respectively at 25$$^\circ$$C. The equivalent conductance of acetic acid (at infinite dilution) would be

For the reaction A + B $$\to$$ C, we have the following data:

The order of the reaction with respect to A and B are

If in case of a radio isotope the value of half-life (T$$_{1/2}$$) and decay constant ($$\lambda$$) are identical in magnitude, then their value should be

Suppose a gaseous mixture of He, Ne, Ar and Kr is treated with photons of the frequency appropriate to ionize Ar. What ion(s) will be present in the mixture?

A solution containing 4g of polymer in 4.0 litre solution at 27$$^\circ$$C shows an osmotic pressure of 3.0 $$\times$$ 10$$^{-4}$$ atm. The molar mass of the polymer in g/mol is

The equivalent weight of KIO$$_3$$ in the given reaction is (M = molecular mass):

$$\mathrm{2Cr{(OH)_3} + 4O{H^ - } + KI{O_3} \to 2CrO_4^{2 - } + 5{H_2}O + KI}$$

At STP, the dissociation reaction of water is $$\mathrm{H_2O\rightleftharpoons H^+~(aq.)+OH^-~(aq.)}$$, and the pH of water is 7.0. The change of standard free energy ($$\Delta$$G$$^\circ$$) for the above dissociation process is given by

Na$$_2$$CO$$_3$$ is prepared by Solvay process but K$$_2$$CO$$_3$$ cannot be prepared by the same because

The molecular shapes of SF$$_4$$, CF$$_4$$ and XeF$$_4$$ are

The species in which nitrogen atom is in a state of sp hybridisation is

The correct statement about the magnetic properties of $${\left[ {Fe{{(CN)}_6}} \right]^{3 - }}$$ and $${\left[ {Fe{F_6}} \right]^{3 - }}$$ is

Nickel combines with a uninegative monodentate ligand (X$$^-$$) to form a paramagnetic complex [NiX$$_4$$]$$^{2-}$$. The hybridisation involved and number of unpaired electrons present in the complex are respectively

'$$\underline{\underline L} $$' in the above sequence of reaction is/are (where L $$\ne$$ M $$\ne$$ N)

'$$\underline G $$' in the above sequence of reactions is

Case - 1 : An ideal gas of molecular weight M at temperature T.

Case - 2 : Another ideal gas of molecular weight 2M at temperature T/2.

Identify the correct statement in context of above two cases.

63 g of a compound (Mol. Wt. = 126) was dissolved in 500 g distilled water. The density of the resultant solution as 1.126 g/ml. The molarity of the solution is

An electron in the 5d orbital can be represented by the following (n, l, m) values

The conversion(s) that can be carried out by bromine in carbon tetrachloride solvent is/are

The correct set(s) of reactions to synthesize benzoic acid starting from benzene is/are

Which statement(s) is/are applicable above critical temperature?

Which of the following mixtures act(s) as buffer solution?

$$\mathop {\lim }\limits_{x \to \infty } \left\{ {x - \root n \of {(x - {a_1})(x - {a_2})\,...\,(x - {a_n})} } \right\}$$ where $${a_1},{a_2},\,...,\,{a_n}$$ are positive rational numbers. The limit

Suppose $$f:R \to R$$ be given by $$f(x) = \left\{ \matrix{ 1,\,\,\,\,\,\,\,\,\,\,\mathrm{if}\,x = 1 \hfill \cr {e^{({x^{10}} - 1)}} + {(x - 1)^2}\sin {1 \over {x - 1}},\,\mathrm{if}\,x \ne 1 \hfill \cr} \right.$$

then

Let $$f:[1,3] \to R$$ be continuous and be derivable in (1, 3) and $$f'(x) = {[f(x)]^2} + 4\forall x \in (1,3)$$. Then

f(x) is a differentiable function and given $$f'(2) = 6$$ and $$f'(1) = 4$$, then $$L = \mathop {\lim }\limits_{h \to 0} {{f(2 + 2h + {h^2}) - f(2)} \over {f(1 + h - {h^2}) - f(1)}}$$

Let $${\cos ^{ - 1}}\left( {{y \over b}} \right) = {\log _e}{\left( {{x \over n}} \right)^n}$$, then $$A{y_2} + B{y_1} + Cy = 0$$ is possible for, where $${y_2} = {{{d^2}y} \over {d{x^2}}},{y_1} = {{dy} \over {dx}}$$

If $$I = \int {{{{x^2}dx} \over {{{(x\sin x + \cos x)}^2}}} = f(x) + \tan x + c} $$, then $$f(x)$$ is

If $$\int {{{dx} \over {(x + 1)(x - 2)(x - 3)}} = {1 \over k}{{\log }_e}\left\{ {{{|x - 3{|^3}|x + 1|} \over {{{(x - 2)}^4}}}} \right\} + c} $$, then the value of k is

the expression $${{\int\limits_0^n {[x]dx} } \over {\int\limits_0^n {\{ x\} dx} }}$$, where $$[x]$$ and $$\{ x\} $$ are respectively integral and fractional part of $$x$$ and $$n \in N$$, is equal to

The value $$\int\limits_0^{1/2} {{{dx} \over {\sqrt {1 - {x^{2n}}} }}} $$ is $$(n \in N)$$

If $${I_n} = \int\limits_0^{{\pi \over 2}} {{{\cos }^n}x\cos nxdx} $$, then I$$_1$$, I$$_2$$, I$$_3$$ ... are in

If $$y = {x \over {{{\log }_e}|cx|}}$$ is the solution of the differential equation $${{dy} \over {dx}} = {y \over x} + \phi \left( {{x \over y}} \right)$$, then $$\phi \left( {{x \over y}} \right)$$ is given by

The function $$y = {e^{kx}}$$ satisfies $$\left( {{{{d^2}y} \over {d{x^2}}} + {{dy} \over {dx}}} \right)\left( {{{dy} \over {dx}} - y} \right) = y{{dy} \over {dx}}$$. It is valid for

Given $${{{d^2}y} \over {d{x^2}}} + \cot x{{dy} \over {dx}} + 4y\cos e{c^2}x = 0$$. Changing the independent variable x to z by the substitution $$z = \log \tan {x \over 2}$$, the equation is changed to

Let $$f(x) = \left\{ {\matrix{ {x + 1,} & { - 1 \le x \le 0} \cr { - x,} & {0 < x \le 1} \cr } } \right.$$

A missile is fired from the ground level rises x meters vertically upwards in t sec, where $$x = 100t - {{25} \over 2}{t^2}$$. The maximum height reached is

If a hyperbola passes through the point P($$\sqrt2$$, $$\sqrt3$$) and has foci at ($$\pm$$ 2, 0), then the tangent to this hyperbola at P is

A, B are fixed points with coordinates (0, a) and (0, b) (a > 0, b > 0). P is variable point (x, 0) referred to rectangular axis. If the angle $$\angle$$APB is maximum, then

The average length of all vertical chords of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1,a \le x \le 2a$$, is :

The value of 'a' for which the scalar triple product formed by the vectors $$\overrightarrow \alpha = \widehat i + a\widehat j + \widehat k,\overrightarrow \beta = \widehat j + a\widehat k$$ and $$\overrightarrow \gamma = a\widehat i + \widehat k$$ is maximum, is

If the vertices of a square are $${z_1},{z_2},{z_3}$$ and $${z_4}$$ taken in the anti-clockwise order, then $${z_3} = $$

If the n terms $${a_1},{a_2},\,......,\,{a_n}$$ are in A.P. with increment r, then the difference between the mean of their squares & the square of their mean is

If $$1,{\log _9}({3^{1 - x}} + 2),{\log _3}({4.3^x} - 1)$$ are in A.P., then x equals

Reflection of the line $$\overline a z + a\overline z = 0$$ in the real axis is given by :

If one root of $${x^2} + px - {q^2} = 0,p$$ and $$q$$ are real, be less than 2 and other be greater than 2, then

The number of ways in which the letters of the word 'VERTICAL' can be arranged without changing the order of the vowels is

n objects are distributed at random among n persons. The number of ways in which this can be done so that at least one of them will not get any object is

Let $$P(n) = {3^{2n + 1}} + {2^{n + 2}}$$ where $$n \in N$$. Then

Let A be a set containing n elements. A subset P of A is chosen, and the set A is reconstructed by replacing the elements of P. A subset Q of A is chosen again. The number of ways of choosing P and Q such that Q contains just one element more than P is

Let A and B are orthogonal matrices and det A + det B = 0. Then

Let $$A = \left( {\matrix{ 2 & 0 & 3 \cr 4 & 7 & {11} \cr 5 & 4 & 8 \cr } } \right)$$. Then

If the matrix M_r is given by $${M_r} = \left( {\matrix{ r & {r - 1} \cr {r - 1} & r \cr } } \right)$$ for r = 1, 2, 3, ... then det (M₁) + det (M₂) + ... + det (M₂₀₀₈) =

Let $$\alpha,\beta$$ be the roots of the equation $$a{x^2} + bx + c = 0,a,b,c$$ real and $${s_n} = {\alpha ^n} + {\beta ^n}$$ and $$\left| {\matrix{ 3 & {1 + {s_1}} & {1 + {s_2}} \cr {1 + {s_1}} & {1 + {s_2}} & {1 + {s_3}} \cr {1 + {s_2}} & {1 + {s_3}} & {1 + {s_4}} \cr } } \right| = k{{{{(a + b + c)}^2}} \over {{a^4}}}$$ then $$k = $$

Let A, B, C are subsets of set X. Then consider the validity of the following set theoretic statement:

Let X be a nonvoid set. If $$\rho_1$$ and $$\rho_2$$ be the transitive relations on X, then

($$\circ$$ denotes the composition of relations)

Let A and B are two independent events. The probability that both A and B happen is $${1 \over {12}}$$ and probability that neither A and B happen is $${1 \over 2}$$. Then

Let S be the sample space of the random experiment of throwing simultaneously two unbiased dice and $$\mathrm{E_k=\{(a,b)\in S:ab=k\}}$$. If $$\mathrm{p_k=P(E_k)}$$, then the correct among the following is :

If $${1 \over 6}\sin \theta ,\cos \theta ,\tan \theta $$ are in G.P, then the solution set of $$\theta$$ is

(Here $$n \in N$$)

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

Let A be the point (0, 4) in the xy-plane and let B be the point (2t, 0). Let L be the midpoint of AB and let the perpendicular bisector of AB meet the y-axis M. Let N be the midpoint of LM. Then locus of N is

If $$4{a^2} + 9{b^2} - {c^2} + 12ab = 0$$, then the family of straight lines $$ax + by + c = 0$$ is concurrent at

The straight lines $$x + 2y - 9 = 0,3x + 5y - 5 = 0$$ and $$ax + by - 1 = 0$$ are concurrent if the straight line $$35x - 22y + 1 = 0$$ passes through the point

ABC is an isosceles triangle with an inscribed circle with centre O. Let P be the midpoint of BC. If AB = AC = 15 and BC = 10, then OP equals

Let O be the vertex, Q be any point on the parabola x$$^2$$ = 8y. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

The tangent at point $$(a\cos \theta ,b\sin \theta ),0 < \theta < {\pi \over 2}$$, to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ meets the x-axis at T and y-axis at T$$_1$$. Then the value of $$\mathop {\min }\limits_{0 < \theta < {\pi \over 2}} (OT)(O{T_1})$$ is

Let $$A(2\sec \theta ,3\tan \theta )$$ and $$B(2\sec \phi ,3\tan \phi )$$ where $$\theta + \phi = {\pi \over 2}$$ be two points on the hyperbola $${{{x^2}} \over 4} - {{{y^2}} \over 9} = 1$$. If ($$\alpha,\beta$$) is the point of intersection of normals to the hyperbola at A and B, then $$\beta$$ is equal to

If the lines joining the focii of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ where $$a > b$$, and an extremity of its minor axis is inclined at an angle 60$$^\circ$$, then the eccentricity of the ellipse is

If the distance between the plane $$\alpha x - 2y + z = k$$ and the plane containing the lines $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}$$ and $${{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}$$ is $$\sqrt 6 $$, then $$|k|$$ is

The angle between a normal to the plane $$2x - y + 2z - 1 = 0$$ and the X-axis is

Let $$f(x) = [{x^2}]\sin \pi x,x > 0$$. Then

If $$y = {\log ^n}x$$, where $${\log ^n}$$ means $${\log _e}{\log _e}{\log _e}\,...$$ (repeated n times), then $$x\log x{\log ^2}x{\log ^3}x\,.....\,{\log ^{n - 1}}x{\log ^n}x{{dy} \over {dx}}$$ is equal to

$$\int\limits_0^{2\pi } {\theta {{\sin }^6}\theta \cos \theta d\theta } $$ is equal to

If $$x = \sin \theta $$ and $$y = \sin k\theta $$, then $$(1 - {x^2}){y_2} - x{y_1} - \alpha y = 0$$, for $$\alpha=$$

In the interval $$( - 2\pi ,0)$$, the function $$f(x) = \sin \left( {{1 \over {{x^3}}}} \right)$$.

The average ordinate of $$y = \sin x$$ over $$[0,\pi ]$$ is :

The portion of the tangent to the curve $${x^{{2 \over 3}}} + {y^{{2 \over 3}}} = {a^{{2 \over 3}}},a > 0$$ at any point of it, intercepted between the axes

If the volume of the parallelopiped with $$\overrightarrow a \times \overrightarrow b ,\overrightarrow b \times \overrightarrow c $$ and $$\overrightarrow c \times \overrightarrow a $$ as conterminous edges is 9 cu. units, then the volume of the parallelopiped with $$(\overrightarrow a \times \overrightarrow b ) \times (\overrightarrow b \times \overrightarrow c ),(\overrightarrow b \times \overrightarrow c ) \times (\overrightarrow c \times \overrightarrow a )$$, and $$(\overrightarrow c \times \overrightarrow a ) \times (\overrightarrow a \times \overrightarrow b )$$ as conterminous edges is

Given $$f(x) = {e^{\sin x}} + {e^{\cos x}}$$. The global maximum value of $$f(x)$$

Consider a quadratic equation $$a{x^2} + 2bx + c = 0$$ where a, b, c are positive real numbers. If the equation has no real root, then which of the following is true?

Let $${a_1},{a_2},{a_3},\,...,\,{a_n}$$ be positive real numbers. Then the minimum value of $${{{a_1}} \over {{a_2}}} + {{{a_2}} \over {{a_3}}}\, + \,...\, + \,{{{a_n}} \over {{a_1}}}$$ is

Let $$A = \left( {\matrix{ 0 & 0 & 1 \cr 1 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right),B = \left( {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right)$$ and $$P\left( {\matrix{ 0 & 1 & 0 \cr x & 0 & 0 \cr 0 & 0 & y \cr } } \right)$$ be an orthogonal matrix such that $$B = PA{P^{ - 1}}$$ holds. Then

Let $$\rho$$ be a relation defined on set of natural numbers N, as $$\rho = \{ (x,y) \in N \times N:2x + y = 4\} $$. Then domain A and range B are

From the focus of the parabola $${y^2} = 12x$$, a ray of light is directed in a direction making an angle $${\tan ^{ - 1}}{3 \over 4}$$ with x-axis. Then the equation of the line along which the reflected ray leaves the parabola is

The locus of points (x, y) in the plane satisfying $${\sin ^2}x + {\sin ^2}y = 1$$ consists of

The value of $$\mathop {\lim }\limits_{n \to \infty } \left[ {\left( {{1 \over {2\,.\,3}} + {1 \over {{2^2}\,.\,3}}} \right) + \left( {{1 \over {{2^2}\,.\,{3^2}}} + {1 \over {{2^3}\,.\,{3^2}}}} \right)\, + \,...\, + \,\left( {{2 \over {{2^n}\,.\,{3^n}}} + {1 \over {{2^{n + 1}}\,.\,3n}}} \right)} \right]$$ is

The family of curves $$y = {e^{a\sin x}}$$, where 'a' is arbitrary constant, is represented by the differential equation

Let f be a non-negative function defined on $$\left[ {0,{\pi \over 2}} \right]$$. If $$\int\limits_0^x {(f'(t) - \sin 2t)dt = \int\limits_x^0 {f(t)\tan t\,dt} } ,f(0) = 1$$ then $$\int\limits_0^{{\pi \over 2}} {f(x)dx} $$ is

A balloon starting from rest is ascending from ground with uniform acceleration of 4 ft/sec$$^2$$. At the end of 5 sec, a stone is dropped from it. If T be the time to reach the stone to the ground and H be the height of the balloon when the stone reaches the ground, then

If $$f(x) = 3\root 3 \of {{x^2}} - {x^2}$$, then

If z$$_1$$ and z$$_2$$ are two complex numbers satisfying the equation $$\left| {{{{z_1} + {z_2}} \over {{z_1} - {z_2}}}} \right| = 1$$, then $${{{z_1}} \over {{z_2}}}$$ may be

A letter lock consists of three rings with 15 different letters. If N denotes the number of ways in which it is possible to make unsuccessful attempts to open the lock, then

If R and R$$^1$$ are equivalence relations on a set A, then so are the relations

Let f be a strictly decreasing function defined on R such that $$f(x) > 0,\forall x \in R$$. Let $${{{x^2}} \over {f({a^2} + 5a + 3)}} + {{{y^2}} \over {f(a + 15)}} = 1$$ be an ellipse with major axis along the y-axis. The value of 'a' can lie in the interval (s)

A rectangle ABCD has its side parallel to the line y = 2x and vertices A, B, D are on lines y = 1, x = 1 and x = $$-$$1 respectively. The coordinate of C can be

Let $$f(x) = {x^m}$$, m being a non-negative integer. The value of m so that the equality $$f'(a + b) = f'(a) + f'(b)$$ is valid for all a, b > 0 is

Which of the following statements are true?

A ray of monochromatic light is incident on the plane surface of separation between two media $$\mathrm{X}$$ and $$\mathrm{Y}$$ with angle of incidence '$$\mathrm{i}$$' in medium $\mathrm{X}$ and angle of refraction 'r' in medium Y. The given graph shows the relation between $$\sin \mathrm{i}$$ and $$\sin \mathrm{r}$$. If $$\mathrm{V}_{X}$$ and $$\mathrm{V}_{Y}$$ are the velocities of the ray in media X and Y respectively, then which of the following is true?

Three identical convex lenses each of focal length $$\mathrm{f}$$ are placed in a straight line separated by a distance $$\mathrm{f}$$ from each other. An object is located at f/2 in front of the leftmost lens. Then,

X-rays of wavelength $$\lambda$$ gets reflected from parallel planes of atoms in a crystal with spacing d between two planes as shown in the figure. If the two reflected beams interfere constructively, then the condition for maxima will be, (n is the order of interference fringe)

If the potential energy of a hydrogen atom in the first excited state is assumed to be zero, then the total energy of n = $$\infty$$ state is,

In the given circuit, find the voltage drop $$\mathrm{V_L}$$ in the load resistance $$\mathrm{R_L}$$.

Consider the logic circuit with inputs A, B, C and output Y. How many combinations of A, B and C gives the output Y = 0 ?

A particle of mass m is projected at a velocity u, making an angle $$\theta$$ with the horizontal (x-axis). If the angle of projection $$\theta$$ is varied keeping all other parameters same, then magnitude of angular momentum (L) at its maximum height about the point of projection varies with $$\theta$$ as,

A body of mass 2 kg moves in a horizontal circular path of radius 5 m. At an instant, its speed is 2$$\sqrt5$$ m/s and is increasing at the rate of 3 m/s$$^2$$. The magnitude of force acting on the body at that instant is,

In an experiment, the length of an object is measured to be 6.50 cm. This measured value can be written as 0.0650 m. The number of significant figures on 0.0650 m is

A mouse of mass m jumps on the outside edge of a rotating ceiling fan of moment of inertia I and radius R. The fractional loss of angular velocity of the fan as a result is,

Acceleration due to gravity at a height H from the surface of a planet is the same as that at a depth of H below the surface. If R be the radius of the planet, then H vs. R graph for different planets will be,

A uniform rope of length 4 m and mass 0.4 kg is held on a frictionless table in such a way that 0.6 m of the rope is hanging over the edge. The work done to pull the hanging part of the rope on to the table is, (Assume g = 10 m/s$$^2$$)

The displacement of a plane progressive wave in a medium, travelling towards positive x-axis with velocity 4 m/s at t = 0 is given by $$y = 3\sin 2\pi \left( { - {x \over 3}} \right)$$. Then the expression for the displacement at a later time t = 4 sec will be

In a simple harmonic motion, let f be the acceleration and t be the time period. If x denotes the displacement, then |fT| vs. x graph will look like,

As shown in the figure, a liquid is at same levels in two arms of a U-tube of uniform cross-section when at rest. If the U-tube moves with an acceleration 'f' towards right, the difference between liquid heights between two arms of the U-tube will be, (acceleration due to gravity = g)

Six molecules of an ideal gas have velocities 1, 3, 5, 5, 6 and 5 m/s respectively. At any given temperature, if $$\mathrm{\overline V}$$ and $$\mathrm{V_{rms}}$$ represent average and rms speed of the molecules, then

As shown in the figure, a pump is designed as horizontal cylinder with a piston having area A and an outlet orifice having an area 'a'. The piston moves with a constant velocity under the action of force F. If the density of the liquid is $$\rho$$, then the speed of the liquid emerging from the orifice is, (assume $$\mathrm{A} > >$$ a)

A given quantity of gas is taken from A to C in two ways; a) directly from A $$\to$$ C along a straight line and b) in two steps, from A $$\to$$ B and then from B $$\to$$ C. Work done and heat absorbed along the direct path A $$\to$$ C is 200 J and 280 J respectively.

If the work done along A $$\to$$ B $$\to$$ C is 80 J, then heat absorbed along this path is,

Two substances A and B of same mass are heated at constant rate. The variation of temperature $$\theta$$ of the substances with time t is shown in the figure. Choose the correct statement.

Consider a positively charged infinite cylinder with uniform volume charge density $$\rho > 0$$. An electric dipole consisting of + Q and $$-$$ Q charges attached to opposite ends of a massless rod is oriented as shown in the figure. At the instant as shown in the figure, the dipole will experience,

A thin glass rod is bent in a semicircle of radius R. A charge is non-uniformly distributed along the rod with a linear charge density $$\lambda=\lambda_0\sin\theta$$ ($$\lambda_0$$ is a positive constant). The electric field at the centre P of the semicircle is,

12 $$\mu$$C and 6 $$\mu$$C charges are given to the two conducting plates having same cross-sectional area and placed face to face close to each other as shown in the figure. The resulting charge distribution in $$\mu$$C on surfaces A, B, C and D are respectively,

A wire carrying a steady current I is kept in the x-y plane along the curve $$y=A \sin \left(\frac{2 \pi}{\lambda} x\right)$$. A magnetic field B exists in the z-direction. The magnitude of the magnetic force in the portion of the wire between x = 0 and x = $$\lambda$$ is

The figure represents two equipotential lines in x-y plane for an electric field. The x-component E$$_x$$ of the electric field in space between these equipotential lines is,

An electric dipole of dipole moment $$\vec{p}$$ is placed at the origin of the co-ordinate system along the $$\mathrm{z}$$-axis. The amount of work required to move a charge '$$\mathrm{q}$$' from the point $$(\mathrm{a}, 0, 0)$$ to the point $$(0,0, a)$$ is,

The electric field of a plane electromagnetic wave of wave number k and angular frequency $$\omega$$ is given $$\vec{E}=E_{0}(\hat{i}+\hat{j}) \sin (k z-\omega t)$$. Which of the following gives the direction of the associated magnetic field $$\vec{B}$$ ?

A charged particle in a uniform magnetic field $$\vec{B}=B_{0} \hat{k}$$ starts moving from the origin with velocity $$v=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}} ~\mathrm{m} / \mathrm{s}$$. The trajectory of the particle and the time $$t$$ at which it reaches $$2 \mathrm{~m}$$ above $$\mathrm{x}-\mathrm{y}$$ plane are,

In an experiment on a circuit as shown in the figure, the voltmeter shows 8 V reading. The resistance of the voltmeter is,

An interference pattern is obtained with two coherent sources of intensity ratio n : 1. The ratio $$\mathrm{{{{I_{\max }} - {I_{\min }}} \over {{I_{\max }} + {I_{\min }}}}}$$ will be maximum if

A circular coil is placed near a current carrying conductor, both lying on the plane of the paper. The current is flowing through the conductor in such a way that the induced current in the loop is clockwise as shown in the figure. The current in the wire is,

An amount of charge Q passes through a coil of resistance R. If the current in the coil decreases to zero at a uniform rate during time T, then the amount of heat generated in the coil will be,

A modified gravitational potential is given by $$\mathrm{V}=-\frac{\mathrm{GM}}{\mathrm{r}}+\frac{\mathrm{A}}{\mathrm{r}^{2}}$$. If the constant A is expressed in terms of gravitational constant (G), mass (M) and velocity of light (c), then from dimensional analysis, A is,

There are n elastic balls placed on a smooth horizontal plane. The masses of the balls are $$\mathrm{m}, \frac{\mathrm{m}}{2}, \frac{\mathrm{m}}{2^{2}}, \ldots \frac{\mathrm{m}}{2^{\mathrm{n}-1}}$$ respectively. If the first ball hits the second ball with velocity $$\mathrm{v}_{0}$$, then the velocity of the $$\mathrm{n}^{\text {th }}$$ ball will be,

An earth's satellite near the surface of the earth takes about 90 min per revolution. A satellite orbiting the moon also takes about $$90 \mathrm{~min}$$ per revolution. Then which of the following is true?

[where $$\rho_{\mathrm{m}}$$ is density of the moon and $$\rho_{\mathrm{e}}$$ is density of the earth.]

A bar magnet falls from rest under gravity through the centre of a horizontal ring of conducting wire as shown in figure. Which of the following graph best represents the speed (v) vs. time (t) graph of the bar magnet?

A uniform magnetic field B exists in a region. An electron of charge q and mass m moving with velocity v enters the region in a direction perpendicular to the magnetic field. Considering Bohr angular momentum quantization, which of the following statements is/are true?

A train is moving along the tracks at a constant speed u. A girl on the train throws a ball of mass m straight ahead along the direction of motion of the train with speed $$\mathrm{v}$$ with respect to herself. Then

A cyclic process is shown in p-v diagram and T-S diagram. Which of the following statements is/are true?

The figure shows two identical parallel plate capacitors A and B of capacitances C connected to a battery. The key K is initially closed. The switch is now opened and the free spaces between the plates of the capacitors are filled with a dielectric constant 3. Then which of the following statements is/are true?

A charged particle of charge q and mass m is placed at a distance 2R from the centre of a vertical cylindrical region of radius R where magnetic field varies as $$\vec{B}=\left(4 t^{2}-2 t+6\right) \hat{k}$$, where t is time. Then which of the following statements is/are true?