Production of iron in blast furnace follows the following equation

Fe₃O₄(s) + 4CO(g) $\to$ 3Fe(l) + 4CO₂(g)

when 4.640 kg of Fe₃O₄ and 2.520 kg of CO are allowed to react then the amount of iron (in g) produced is :

[Given : Molar Atomic mass (g mol^$-$1) : Fe = 56, Molar Atomic mass (g mol^$-$1) : O = 16, Molar Atomic mass (g mol^$-$1) : C = 12]

The solubility of AgCl will be maximum in which of the following?

The electronic configuration of Pt (atomic number 78) is :

Which of the following statements are correct?

(A) The electronic configuration of Cr is [Ar] 3d⁵ 4s¹.

(B) The magnetic quantum number may have a negative value.

(C) In the ground state of an atom, the orbitals are filled in order of their increasing energies.

(D) The total number of nodes are given by n $-$ 2.

Choose the most appropriate answer from the options given below :

Consider the above reactions, the product A and product B respectively are

Given below are two statements :

Statement I : The esterification of carboxylic acid with an alcohol is a nucleophilic acyl substitution.

Statement II : Electron withdrawing groups in the carboxylic acid will increase the rate of esterification reaction.

Choose the most appropriate option :

Let $\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$, $\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$ and $\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$ where $\alpha ,\,\beta \in R$, be three vectors. If the projection of $\overrightarrow a$ on $\overrightarrow c$ is ${{10} \over 3}$ and $\overrightarrow b \times \overrightarrow c = - 6\widehat i + 10\widehat j + 7\widehat k$, then the value of $\alpha + \beta$ is equal to :

The area enclosed by y² = 8x and y = $\sqrt2$ x that lies outside the triangle formed by y = $\sqrt2$ x, x = 1, y = 2$\sqrt2$, is equal to:

Let $\alpha$ and $\beta$ be the roots of the equation x² + (2i $-$ 1) = 0. Then, the value of |$\alpha$⁸ + $\beta$⁸| is equal to :

The domain of the function ${\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)$ is :

Let PQ be a focal chord of the parabola y² = 4x such that it subtends an angle of ${\pi \over 2}$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$, ${a^2} > {b^2}$. If e is the eccentricity of the ellipse E, then the value of ${1 \over {{e^2}}}$ is equal to :

Let $S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}$. Let $|z - 4i|$ attains minimum and maximum values, respectively, at z₁ $\in$ S and z₂ $\in$ S. If $5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha$ + $\beta$ is equal to ___________.

A body of mass M at rest explodes into three pieces, in the ratio of masses 1 : 1 : 2. Two smaller pieces fly off perpendicular to each other with velocities of 30 ms^$-$1 and 40 ms^$-$1 respectively. The velocity of the third piece will be :

A wire of length L is hanging from a fixed support. The length changes to L₁ and L₂ when masses 1 kg and 2 kg are suspended respectively from its free end. Then the value of L is equal to :

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R :

Assertion A : The photoelectric effect does not takes place, if the energy of the incident radiation is less than the work function of a metal.

Reason R : Kinetic energy of the photoelectrons is zero, if the energy of the incident radiation is equal to the work function of a metal.

In the light of the above statements, choose the most appropriate answer from the options given below.

A charge particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charge particle increases to 4 times its initial value. What will be the ratio of new radius to the original radius of circular path of the charge particle :

A block of metal weighing 2 kg is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of 1 kgs^$-$1 and at a speed of 10 ms^$-$1. Then, the initial acceleration of the block, in ms^$-$2, will be :

A longitudinal wave is represented by $x = 10\sin 2\pi \left( {nt - {x \over \lambda }} \right)$ cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to :

Two coils require 20 minutes and 60 minutes respectively to produce same amount of heat energy when connected separately to the same source. If they are connected in parallel arrangement to the same source; the time required to produce same amount of heat by the combination of coils, will be ___________ min.

$\sqrt {{d_1}}$ and $\sqrt {{d_2}}$ are the impact parameters corresponding to scattering angles 60$^\circ$ and 90$^\circ$ respectively, when an $\alpha$ particle is approaching a gold nucleus. For d₁ = x d₂, the value of x will be ____________.

The activation energy of one of the reactions in a biochemical process is 532611 J mol^$-$1. When the temperature falls from 310 K to 300 K, the change in rate constant observed is k₃₀₀ = x $\times$ 10^$-$3 k₃₁₀. The value of x is ____________.

[Given : $\ln 10 = 2.3$, R = 8.3 J K^$-$1 mol^$-$1]

An acidified manganate solution undergoes disproportionation reaction. The spin-only magnetic moment value of the product having manganese in higher oxidation state is _____________ B.M. (Nearest integer)

Kjeldahl's method was used for the estimation of nitrogen in an organic compound. The ammonia evolved from 0.55 g of the compound neutralised 12.5 mL of 1 M H₂SO₄ solution. The percentage of nitrogen in the compound is _____________. (Nearest integer)

Observe structures of the following compounds

The total number of structures/compounds which possess asymmetric carbon atoms is ______________.

C₆H₁₂O₆ \buildrel \text{Zymase} \over \longrightarrow \( A \)\mathrel{\mathop{\kern0pt\longrightarrow} \limits_\Delta ^\text{NaOI}} B + CHI₃

The number of carbon atoms present in the product B is _______________.

Let the solution curve of the differential equation

$x{{dy} \over {dx}} - y = \sqrt {{y^2} + 16{x^2}}$, $y(1) = 3$ be $y = y(x)$. Then y(2) is equal to:

Let $A = [{a_{ij}}]$ be a square matrix of order 3 such that ${a_{ij}} = {2^{j - i}}$, for all i, j = 1, 2, 3. Then, the matrix A² + A³ + ...... + A¹⁰ is equal to :

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is :

Let y = y(x) be the solution of the differential equation ${{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 < x < {\pi \over 2}$ with $y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {32}}$. If $y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over {18}}{e^{ - {{\tan }^{ - 1}}(\alpha )}}$, then the value of 3$\alpha$² is equal to ___________.

Let c, k $\in$ R. If $f(x) = (c + 1){x^2} + (1 - {c^2})x + 2k$ and $f(x + y) = f(x) + f(y) - xy$, for all x, y $\in$ R, then the value of $|2(f(1) + f(2) + f(3) + \,\,......\,\, + \,\,f(20))|$ is equal to ____________.

Let $H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$, a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt 2 + \sqrt {14} )$. If the eccentricity H is ${{\sqrt {11} } \over 2}$, then the value of a² + b² is equal to __________.

Let b₁b₂b₃b₄ be a 4-element permutation with b_i $\in$ {1, 2, 3, ........, 100} for 1 $\le$ i $\le$ 4 and b_i $\ne$ b_j for i $\ne$ j, such that either b₁, b₂, b₃ are consecutive integers or b₂, b₃, b₄ are consecutive integers. Then the number of such permutations b₁b₂b₃b₄ is equal to ____________.

A particle of mass 500 gm is moving in a straight line with velocity v = b x^5/2. The work done by the net force during its displacement from x = 0 to x = 4 m is : (Take b = 0.25 m^$-$3/2 s^$-$1).

For a series LCR circuit, I vs $\omega$ curve is shown :

(a) To the left of $\omega$_r, the circuit is mainly capacitive.

(b) To the left of $\omega$_r, the circuit is mainly inductive.

(c) At $\omega$_r, impedance of the circuit is equal to the resistance of the circuit.

(d) At $\omega$_r, impedance of the circuit is 0.

Choose the most appropriate answer from the options given below :

Two vectors $\overrightarrow A$ and $\overrightarrow B$ have equal magnitudes. If magnitude of $\overrightarrow A$ + $\overrightarrow B$ is equal to two times the magnitude of $\overrightarrow A$ $-$ $\overrightarrow B$, then the angle between $\overrightarrow A$ and $\overrightarrow B$ will be :

The escape velocity of a body on a planet 'A' is 12 kms^$-$1. The escape velocity of the body on another planet 'B', whose density is four times and radius is half of the planet 'A', is :

The intensity of the light from a bulb incident on a surface is 0.22 W/m². The amplitude of the magnetic field in this light-wave is ______________ $\times$ 10^$-$9 T.

(Given : Permittivity of vacuum $\in$₀ = 8.85 $\times$ 10^$-$12 C² N^$-$1-m^$-$2, speed of light in vacuum c = 3 $\times$ 10⁸ ms^$-$1)

A parallel beam of light is allowed to fall on a transparent spherical globe of diameter 30 cm and refractive index 1.5. The distance from the centre of the globe at which the beam of light can converge is _____________ mm.

Two isomers 'A' and 'B' with molecular formula C₄H₈ give different products on oxidation with KMnO₄ in acidic medium. Isomer 'A' on reaction with KMnO₄/H⁺ results in effervescence of a gas and gives ketone. The compound 'A' is

In the given conversion the compound A is :

Sugar moiety in DNA and RNA molecules respectively are

Given below are two statements :

Statement I : Phenols are weakly acidic.

Statement II : Therefore they are freely soluble in NaOH solution and are weaker acids than alcohols and water.

Choose the most appropriate option :

1.2 mL of acetic acid is dissolved in water to make 2.0 L of solution. The depression in freezing point observed for this strength of acid is 0.0198$^\circ$C. The percentage of dissociation of the acid is ___________. (Nearest integer)

[Given : Density of acetic acid is 1.02 g mL^$-$1, Molar mass of acetic acid is 60 g mol^$-$1, K_f(H₂O) = 1.85 K kg mol^$-$1]

The probability that a randomly chosen 2 $\times$ 2 matrix with all the entries from the set of first 10 primes, is singular, is equal to :

If the system of linear equations

2x + y $-$ z = 7

x $-$ 3y + 2z = 1

x + 4y + $\delta$z = k, where $\delta$, k $\in$ R has infinitely many solutions, then $\delta$ + k is equal to:

Let a set A = A₁ $\cup$ A₂ $\cup$ ..... $\cup$ A_k, where A_i $\cap$ A_j = $\phi$ for i $\ne$ j, 1 $\le$ j, j $\le$ k. Define the relation R from A to A by R = {(x, y) : y $\in$ A_i if and only if x $\in$ A_i, 1 $\le$ i $\le$ k}. Then, R is :

The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle ${\pi \over 4}$ at the origin, is equal to :

If the constant term in the expansion of

${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$ is 2^k.l, where l is an odd integer, then the value of k is equal to:

$\int_0^5 {\cos \left( {\pi \left( {x - \left[ {{x \over 2}} \right]} \right)} \right)dx}$,

where [t] denotes greatest integer less than or equal to t, is equal to:

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be ${24 \over 5}$ and ${194 \over 25}$ respectively. If the mean and variance of the first 4 observation are ${7 \over 2}$ and a respectively, then (4a + x₅) is equal to:

Two balls A and B are placed at the top of 180 m tall tower. Ball A is released from the top at t = 0 s. Ball B is thrown vertically down with an initial velocity 'u' at t = 2 s. After a certain time, both balls meet 100 m above the ground. Find the value of 'u' in ms^$-$1. [use g = 10 ms^$-$2] :

A spherical shell of 1 kg mass and radius R is rolling with angular speed $\omega$ on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin O is ${a \over 3}$ R²$\omega$. The value of a will be :

A cylinder of fixed capacity of 44.8 litres contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by 20.0$^\circ$C will be :

(Given gas constant R = 8.3 JK^$-$1-mol^$-$1)

A parallel plate capacitor filled with a medium of dielectric constant 10, is connected across a battery and is charged. The dielectric slab is replaced by another slab of dielectric constant 15. Then the energy of capacitor will :

Using Young's double slit experiment, a monochromatic light of wavelength 5000 $\mathop A\limits^o$ produces fringes of fringe width 0.5 mm. If another monochromatic light of wavelength 6000 $\mathop A\limits^o$ is used and the separation between the slits is doubled, then the new fringe width will be :

As per the given figure, two plates A and B of thermal conductivity K and 2 K are joined together to form a compound plate. The thickness of plates are 4.0 cm and 2.5 cm respectively and the area of cross-section is 120 cm² for each plate. The equivalent thermal conductivity of the compound plate is $\left( {1 + {5 \over \alpha }} \right)$ K, then the value of $\alpha$ will be ______________.

A body is performing simple harmonic with an amplitude of 10 cm. The velocity of the body was tripled by air jet when it is at 5 cm from its mean position. The new amplitude of vibration is $\sqrt{x}$ cm. The value of x is _____________.

The variation of applied potential and current flowing through a given wire is shown in figure. The length of wire is 31.4 cm. The diameter of wire is measured as 2.4 cm. The resistivity of the given wire is measured as x $\times$ 10^$-$3 $\Omega$ cm. The value of x is ____________. [Take $\pi$ = 3.14]

17.0 g of NH₃ completely vapourises at $-$33.42$^\circ$C and 1 bar pressure and the enthalpy change in the process is 23.4 kJ mol^$-$1. The enthalpy change for the vapourisation of 85 g of NH₃ under the same conditions is _________ kJ.

A dilute solution of sulphuric acid is electrolysed using a current of 0.10 A for 2 hours to produce hydrogen and oxygen gas. The total volume of gases produced a STP is _____________ cm³. (Nearest integer)

[Given : Faraday constant F = 96500 C mol^$-$1 at STP, molar volume of an ideal gas is 22.7 L mol^$-$1]

The number of terminal oxygen atoms present in the product B obtained from the following reaction is _____________.

FeCr₂O₄ + Na₂CO₃ + O₂ $\to$ A + Fe₂O₃ + CO₂

A + H⁺ $\to$ B + H₂O + Na⁺

Let $f:R \to R$ be a function defined by :

f(x) = \left\{ {\matrix{ {\max \,\{ {t^3} - 3t\} \,t \le x} & ; & {x \le 2} \cr {{x^2} + 2x - 6} & ; & {2 < x < 3} \cr {[x - 3] + 9} & ; & {3 \le x \le 5} \cr {2x + 1} & ; & {x > 5} \cr } } \right.

where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and $I = \int\limits_{ - 2}^2 {f(x)\,dx}$. Then the ordered pair (m, I) is equal to :

$50\tan \left( {3{{\tan }^{ - 1}}\left( {{1 \over 2}} \right) + 2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right) + 4\sqrt 2 \tan \left( {{1 \over 2}{{\tan }^{ - 1}}(2\sqrt 2 )} \right)$ is equal to ____________.

In van der Waal equation $\left[ {P + {a \over {{V^2}}}} \right]$ [V $-$ b] = RT; P is pressure, V is volume, R is universal gas constant and T is temperature. The ratio of constants ${a \over b}$ is dimensionally equal to :

A positive charge particle of 100 mg is thrown in opposite direction to a uniform electric field of strength 1 $\times$ 10⁵ NC^$-$1. If the charge on the particle is 40 $\mu$C and the initial velocity is 200 ms^$-$1, how much distance it will travel before coming to the rest momentarily :

For the network shown below, the value of V_B $-$ V_A is ____________ V.