The d-electron configuration of [Ru(en)₃ ]Cl₂
and [Fe(H₂O)₆]Cl₂ , respectively are :

Consider the following reaction :
The product 'P' gives positive ceric ammonium nitrate test. This is because of the presence of which of these –OH group(s)?

For the reaction
2A + 3B + $${3 \over 2}$$C $$ \to $$ 3P, which statement is correct ?

Complex A has a composition of H₁₂O₆Cl₃Cr. If the complex on treatment with conc.H₂SO₄ loses 13.5% of its original mass, the correct molecular formula of A is :
[Given: atomic mass of Cr = 52 amu and Cl = 35 amu]

Consider the following molecules and statements related to them :
(a) (B) is more likely to be crystalline than (A)
(b) (B) has higher boiling point than (A)
(c) (B) dissolves more readily than (A) in water

Identify the correct option from below :

The compound A in the following reaction is :

The incorrect statement is :

The increasing order of the reactivity of the following compound in nucleophilic addition reaction is :

Propanal, Benzaldehyde, Propanone, Butanone

The decreasing order of reactivity of the following compounds towards nucleophilic substitution (S_N2) is :

The major product in the following reaction is :

100 mL of 0.1 M HCl is taken in a beaker and to it 100 mL of 0.1 M NaOH is added in steps of 2 mL and the pH is continuously measured. Which of the following graphs correctly depicts the change in pH?

Three isomers A, B and C (mol. formula C₈H₁₁N) give the following results :
R has lower boiling point than S

A, B and C, respectively are :

The five successive ionization enthalpies of an element are 800, 2427, 3658, 25024 and 32824 kJ mol^–1. The number of valence electrons in the element is :

Consider the hypothetical situation where the azimuthal quantum number, $$l$$, takes values 0, 1, 2, ....., n + 1, where n is the principal quantum number. Then, the element with atomic number :

Among the statements (I – IV), the correct ones are:
(I) Be has smaller atomic radius compared to Mg.
(II) Be has higher ionization enthalpy than Al.
(III) Charge/radius ratio of Be is greater than that of Al.
(IV) Both Be and Al form mainly covalent compounds.

The strengths of 5.6 volume hydrogen peroxide (of density 1 g/mL) in terms of mass percentage and molarity (M), respectively, are: (Take molar mass of hydrogen peroxide as 34 g/mol)

The number of groups present in a tripeptide Asp–Glu–Lys is ____.

6.023 $$ \times $$ 10²² molecules are present in 10 g of a substance 'x'. The molarity of a solution containing 5 g of substance 'x' in 2 L solution is _____ × 10^-3

If 250 cm³ of an aqueous solution containing 0.73 g of a protein A is isotonic with one litre of another aqueous solution containing 1.65 g of a protein B, at 298 K, the ratio of the molecular masses of A and B is ______ × 10^–2 (to the nearest integer).

The volume (in mL) of 0.1 N NaOH required to neutralise 10 mL of 0.1 N phosphinic acid is ___________.

An acidic solution of dichromate is electrolyzed for 8 minutes using 2A current. As per the following equation
Cr₂O₇^2- + 14H⁺ + 6e^– $$ \to $$ 2Cr³⁺ + 7H₂O
The amount of Cr³⁺ obtained was 0.104 g. The efficiency of the process(in%) is (Take : F = 96000 C, At. mass of chromium = 52) ______.

Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If
T = {x $$ \in $$ R | f(x) = f(0)}, then the sum of squares of all the elements of T is :

Let a, b c $$ \in $$ R be such that a² + b² + c² = 1. If
$$a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\theta + {{4\pi } \over 3}} \right)$$,
where $${\theta = {\pi \over 9}}$$, then the angle between the vectors $$a\widehat i + b\widehat j + c\widehat k$$ and $$b\widehat i + c\widehat j + a\widehat k$$ is :

Let the latus ractum of the parabola y² = 4x be the common chord to the circles C₁ and C₂ each of them having radius 2$$\sqrt 5 $$. Then, the distance between the centres of the circles C₁ and C₂ is :

Let R₁ and R₂ be two relation defined as follows :
R₁ = {(a, b) $$ \in $$ R² : a² + b² $$ \in $$ Q} and
R₂ = {(a, b) $$ \in $$ R² : a² + b² $$ \notin $$ Q},
where Q is the set of all rational numbers. Then :

If the value of the integral
$$\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx$$

is $${k \over 6}$$, then k is equal to :

Let e₁ and e₂ be the eccentricities of the ellipse,
$${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) and the hyperbola,
$${{{x^2}} \over {16}} - {{{y^2}} \over {{b^2}}} = 1$$ respectively satisfying e₁e₂ = 1. If $$\alpha $$
and $$\beta $$ are the distances between the foci of the
ellipse and the foci of the hyperbola
respectively, then the ordered pair ($$\alpha $$, $$\beta $$) is equal to :

Let S be the set of all integer solutions, (x, y, z), of the system of equations
x – 2y + 5z = 0
–2x + 4y + z = 0
–7x + 14y + 9z = 0
such that 15 $$ \le $$ x² + y² + z² $$ \le $$ 150. Then, the number of elements in the set S is equal to ______ .

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that 4^th A.M. is equal to 2^nd G.M., then m is equal to _________ .

The total number of 3-digit numbers, whose sum of digits is 10, is __________.

The probability that a randomly chosen 5-digit number is made from exactly two digits is :

If x³dy + xy dx = x²dy + 2y dx; y(2) = e and
x > 1, then y(4) is equal to :

$$\mathop {\lim }\limits_{x \to a} {{{{\left( {a + 2x} \right)}^{{1 \over 3}}} - {{\left( {3x} \right)}^{{1 \over 3}}}} \over {{{\left( {3a + x} \right)}^{{1 \over 3}}} - {{\left( {4x} \right)}^{{1 \over 3}}}}}$$ ($$a$$ $$ \ne $$ 0) is equal to :

If the surface area of a cube is increasing at a rate of 3.6 cm²/sec, retaining its shape; then the rate of change of its volume (in cm³/sec), when the length of a side of the cube is 10 cm, is :

If a $$\Delta $$ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates :

Let x_i (1 $$ \le $$ i $$ \le $$ 10) be ten observations of a random variable X. If
$$\sum\limits_{i = 1}^{10} {\left( {{x_i} - p} \right)} = 3$$ and $$\sum\limits_{i = 1}^{10} {{{\left( {{x_i} - p} \right)}^2}} = 9$$
where 0 $$ \ne $$ p $$ \in $$ R, then the standard deviation of these observations is :

If z₁ , z₂ are complex numbers such that
Re(z₁) = |z₁ – 1|, Re(z₂) = |z₂ – 1| , and
arg(z₁ - z₂) = $${\pi \over 6}$$, then Im(z₁ + z₂ ) is equal to :

If $$\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx$$ = A(x)$${\tan ^{ - 1}}\left( {\sqrt x } \right)$$ + B(x) + C,
where C is a constant of integration, then the ordered pair (A(x), B(x)) can be :

If the term independent of x in the expansion of
$${\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}$$ is k, then 18 k is equal to :

The set of all real values of $$\lambda $$ for which the quadratic equations,
($$\lambda $$² + 1)x² – 4$$\lambda $$x + 2 = 0 always have exactly one root in the interval (0, 1) is :

An massless equilateral triangle EFG of side ‘a’ (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is $${N \over {20}}$$ ma² where N is an integer. The value of N is _____.

A galvanometer coil has 500 turns and each turn has an average area of 3 $$ \times $$ 10^–4 m² . If a torque of 1.5 Nm is required to keep this coil parallel to a magnetic field when a current of 0.5 A is flowing through it, the strength of the field (in T) is ______.

A block starts moving up an inclined plane of inclination 30^o with an initial velocity of v₀ . It comes back to its initial position with velocity $${{{v_0}} \over 2}$$. The value of the coefficient of kinetic friction between the block and the inclined plane is close to $${I \over {1000}}$$. The nearest integer to I is____.

When an object is kept at a distance of 30 cm from a concave mirror, the image is formed at a distance of 10 cm from the mirror. If the object is moved with a speed of 9 cms^–1, the speed (in cms^–1) with which image moves at that instant is ____.

A calorimeter of water equivalent 20 g contains 180 g of water at 25^oC. ‘m’ grams of steam at 100^oC is mixed in it till the temperature of the mixure is 31^oC. The value of ‘m’ is close to :
(Latent heat of water = 540 cal g^–1, specific heat of water = 1 cal g^{–1 o}C^–1)

Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to :

Two light waves having the same wavelength $$\lambda $$ in vacuum are in phase initially. Then the first wave travels a path L₁ through a medium of refractive index n₁ while the second wave travels a path of length L₂ through a medium of refractive index n₂ . After this the phase difference between the two waves is :

A perfectly diamagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. The whole system is placed in a uniform magnetic field $$\overrightarrow B $$ . Then the field inside the paramagnetic substance is :

Concentric metallic hollow spheres of radii R and 4R hold charges Q₁ and Q₂ respectively. Given that surface charge densities of the concentric spheres are equal, the potential difference V(R) – V(4R) is :

To raise the temperature of a certain mass of gas by 50^oC at a constant pressure, 160 calories of heat is required. When the same mass of gas is cooled by 100^oC at constant volume, 240 calories of heat is released. How many degrees of freedom does each molecule of this gas have (assume gas to be ideal)?

A uniform magnetic field B exists in a direction perpendicular to the plane of a square loop made of a metal wire. The wire has a diameter of 4 mm and a total length of 30 cm. The magnetic field changes with time at a steady rate $${{dB} \over {dt}}$$ = 0.032 Ts^–1. The induced current in the loop is close to (Resistivity of the metal wire is 1.23 $$ \times $$ 10^–8 $$\Omega $$m)

Amount of solar energy received on the earth’s surface per unit area per unit time is defined a solar constant. Dimension of solar constant is :

The radius R of a nucleus of mass number A can be estimated by the formula
R = (1.3 $$ \times $$ 10^–15)A^1/3 m.
It follows that the mass density of a nucleus is of the order of :

(M_prot. $$ \cong $$ M_neut $$ \simeq $$ 1.67 $$ \times $$ 10^–27 kg)

A block of mass 1.9 kg is at rest at the edge of a table, of height 1 m. A bullet of mass 0.1 kg collides with the block and sticks to it. If the velocity of the bullet is 20 m/s in the horizontal direction just before the collision then the kinetic energy just before the combined system strikes the floor, is [Take g = 10 m/s² . Assume there is no rotational motion and loss of energy after the collision is negligable.]

Two resistors 400$$\Omega $$ and 800$$\Omega $$ are connected in series across a 6 V battery. The potential difference measured by a voltmeter of 10 k$$\Omega $$ across 400 $$\Omega $$ resistor is close to :

A uniform rod of length ‘$$l$$’ is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $$\omega $$ the rod makes an angle $$\theta $$ with it (see figure). To find $$\theta $$ equate the rate of change of angular momentum (direction going into the paper) $${{m{l^2}} \over {12}}{\omega ^2}\sin \theta \cos \theta $$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces F_H and F_V about the CM. The value of $$\theta $$ is then such that :

Which of the following will NOT be observed when a multimeter (operating in resistance measuring mode) probes connected across a component, are just reversed?

If a semiconductor photodiode can detect a photon with a maximum wavelength of 400 nm, then its band gap energy is :

Planck’s constant h = 6.63 $$ \times $$ 10^–34 J.s. Speed of light c = 3 $$ \times $$ 10⁸ m/s

A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The displacement (s) - time (t) graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale) :

Two sources of light emit X-rays of wavelength 1 nm and visible light of wavelength 500 nm, respectively. Both the sources emit light of the same power 200 W. The ratio of the number density of photons of X-rays to the number density of photons of the visible light of the given wavelengths is :

The mass density of a planet of radius R varies with the distance r from its centre as
$$\rho $$(r) = $${\rho _0}\left( {1 - {{{r^2}} \over {{R^2}}}} \right)$$.
Then the gravitational field is maximum at :

A block of mass m attached to a massless spring is performing oscillatory motion of amplitude ‘A’ on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining system become fA. The value of f is :