The energy of one mole of photons of radiation of wavelength 300 nm is

(Given : h = 6.63 $\times$ 10^$-$34 J s, N_A = 6.02 $\times$ 10²³ mol^$-$1, c = 3 $\times$ 10⁸ m s^$-$1)

At 25$^\circ$C and 1 atm pressure, the enthalpies of combustion are as given below :

The enthalpy of formation of ethane is

Arrange the following carbocations in decreasing order of stability.

Hex-4-ene-2-ol on treatment with PCC gives 'A'. 'A' on reaction with sodium hypoiodite gives 'B', which on further heating with soda lime gives 'C'. The compound 'C' is :

The conversion of propan-1-ol to n-butylamine involves the sequential addition of reagents. The correct sequential order of reagents is

In the flame test of a mixture of salts, a green flame with blue centre was observed. Which one of the following cations may be present?

0.2 g of an organic compound was subjected to estimation of nitrogen by Dumas method in which volume of N₂ evolved (at STP) was found to be 22.400 mL. The percentage of nitrogen in the compound is _________. [nearest integer]

(Given : Molar mass of N₂ is 28 g mol^$-$1. Molar volume of N₂ at STP : 22.4 L)

Let f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.

where [t] denotes greatest integer $\le$ t. If m is the number of points where $f$ is not continuous and n is the number of points where $f$ is not differentiable, then the ordered pair (m, n) is :

If $y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2} < {x^3} < {{3\pi } \over 2}$, then

Let S = \left\{ {\left( {\matrix{ { - 1} & a \cr 0 & b \cr } } \right);a,b \in \{ 1,2,3,....100\} } \right\}\( and let \){T_n} = \{ A \in S:{A^{n(n + 1)}} = I\} \(. Then the number of elements in \)\bigcap\limits_{n = 1}^{100} {{T_n}} is ___________.

A stone of mass m, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is

Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :

Potential energy as a function of r is given by $U = {A \over {{r^{10}}}} - {B \over {{r^5}}}$, where r is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be :

A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be :

In the given circuit, the value of current I_L will be ____________ mA. (When R_L = 1k$\Omega$)

Metals generally melt at very high temperature. Amongst the following, the metal with the highest melting point will be :

Which of the following reagents / reactions will convert 'A' to 'B' ?

A company dissolves 'x' amount of CO₂ at 298 K in 1 litre of water to prepare soda water. X = __________ $\times$ 10^$-$3 g. (nearest integer)

(Given : partial pressure of CO₂ at 298 K = 0.835 bar.

Henry's law constant for CO₂ at 298 K = 1.67 kbar.

Atomic mass of H, C and O is 1, 12, and 6 g mol^$-$1, respectively)

PCl₅ dissociates as

PCl₅(g) $\rightleftharpoons$ PCl₃(g) + Cl₂(g)

5 moles of PCl₅ are placed in a 200 litre vessel which contains 2 moles of N₂ and is maintained at 600 K. The equilibrium pressure is 2.46 atm. The equilibrium constant K_p for the dissociation of PCl₅ is __________ $\times$ 10^$-$3. (nearest integer)

(Given : R = 0.082 L atm K^$-$1 mol^$-$1; Assume ideal gas behaviour)

In alanylglycyl leucyl alanyl valine, the number of peptide linkages is ___________.

The sum of all the real roots of the equation

$({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0$ is

Let the system of linear equations

x + y + $\alpha$z = 2

3x + y + z = 4

x + 2z = 1

have a unique solution (x$^ *$, y$^ *$, z$^ *$). If ($\alpha$, x$^ *$), (y$^ *$, $\alpha$) and (x$^ *$, $-$y$^ *$) are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

Let S = {z $\in$ C : |z $-$ 3| $\le$ 1 and z(4 + 3i) + $\overline z$(4 $-$ 3i) $\le$ 24}. If $\alpha$ + i$\beta$ is the point in S which is closest to 4i, then 25($\alpha$ + $\beta$) is equal to ___________.

The remainder on dividing 1 + 3 + 3² + 3³ + ..... + 3²⁰²¹ by 50 is _________.

Identify the pair of physical quantities that have same dimensions:

The distance between Sun and Earth is R. The duration of year if the distance between Sun and Earth becomes 3R will be :

A proton, a deutron and an $\alpha$-particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is :

Given below are two statements :

Statement I : The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor.

Statement II : In ac circuit, the average power delivered by the source never becomes zero.

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

An object of mass 5 kg is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of 10 N throughout the motion. The ratio of time of ascent to the time of descent will be equal to : [Use g = 10 ms^$-$2].

A long cylindrical volume contains a uniformly distributed charge of density $\rho$. The radius of cylindrical volume is R. A charge particle (q) revolves around the cylinder in a circular path. The kinetic energy of the particle is :

An electric bulb is rated as 200 W. What will be the peak magnetic field at 4 m distance produced by the radiations coming from this bulb? Consider this bulb as a point source with 3.5% efficiency.

Two light beams of intensities in the ratio of 9 : 4 are allowed to interfere. The ratio of the intensity of maxima and minima will be :

In Bohr's atomic model of hydrogen, let K, P and E are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level :

A circular coil of 1000 turns each with area 1m² is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of 0.07T. The maximum voltage generation will be ___________ V.

The correct order of bond orders of ${C_2}^{2 - }$, ${N_2}^{2 - }$ and ${O_2}^{2 - }$

For a first order reaction, the time required for completion of 90% reaction is 'x' times the half life of the reaction. The value of 'x' is

(Given : ln 10 = 2.303 and log 2 = 0.3010)

Given below are two statements :

Statement I : The presence of weaker $\pi$-bonds make alkenes less stable than alkanes.

Statement II : The strength of the double bond is greater than that of carbon-carbon single bond.

In the light of the above statements, choose the correct answer from the options given below :

Options:

Consider the above reaction. The number of $\pi$ electrons present in the product 'P' is __________.

Let $x * y = {x^2} + {y^3}$ and $(x * 1) * 1 = x * (1 * 1)$.

Then a value of $2{\sin ^{ - 1}}\left( {{{{x^4} + {x^2} - 2} \over {{x^4} + {x^2} + 2}}} \right)$ is :

Let x, y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is

Let the maximum area of the triangle that can be inscribed in the ellipse ${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6\sqrt 3$. Then the eccentricity of the ellipse is :

The number of distinct real roots of the equation

x⁷ $-$ 7x $-$ 2 = 0 is

A random variable X has the following probability distribution :

The value of P(1 < X < 4 | X $\le$ 2) is equal to :

If the shortest distance between the lines ${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }$ and ${{x - 2} \over 1} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is ${1 \over {\sqrt 3 }}$, then the sum of all possible value of $\lambda$ is :

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____________.

The sum of all the elements of the set $\{ \alpha \in \{ 1,2,.....,100\} :HCF(\alpha ,24) = 1\}$ is __________.

Let a circle C : (x $-$ h)² + (y $-$ k)² = r², k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ___________.

What will be the most suitable combination of three resistors A = 2$\Omega$, B = 4$\Omega$, C = 6$\Omega$ so that $\left( {{{22} \over 3}} \right)$$\Omega$ is equivalent resistance of combination?

The soft-iron is a suitable material for making an electromagnet. This is because soft-iron has

If the charge on a capacitor is increased by 2 C, the energy stored in it increases by 44%. The original charge on the capacitor is (in C)

A ray of light is incident at an angle of incidence 60$^\circ$ on the glass slab of refractive index $\sqrt3$. After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is 4$\sqrt3$ cm. The thickness of the glass slab is __________ cm.

A monoatomic gas performs a work of ${Q \over {4}}$ where Q is the heat supplied to it. The molar heat capacity of the gas will be ______________ R during this transformation. Where R is the gas constant.

120 g of an organic compound that contains only carbon and hydrogen gives 330 g of CO₂ and 270 g of water on complete combustion. The percentage of carbon and hydrogen, respectively are

Transition metal complex with highest value of crystal field splitting ($\Delta$₀) will be :

The resistance of a conductivity cell containing 0.01 M KCl solution at 298 K is 1750 $\Omega$. If the conductivity of 0.01 M KCl solution at 298 K is 0.152 $\times$ 10^$-$3 S cm^$-$1, then the cell constant of the conductivity cell is ____________ $\times$ 10^$-$3 cm^$-$1.

Manganese (VI) has ability to disproportionate in acidic solution. The difference in oxidation states of two ions it forms in acidic solution is ____________.

The value of the integral

$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}}$ is equal to

Let the area of the triangle with vertices A(1, $\alpha$), B($\alpha$, 0) and C(0, $\alpha$) be 4 sq. units. If the points ($\alpha$, $-$$\alpha$), ($-$$\alpha$, $\alpha$) and ($\alpha$², $\beta$) are collinear, then $\beta$ is equal to :

Let $\widehat a$ and $\widehat b$ be two unit vectors such that $|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$. If $\theta$ $\in$ (0, $\pi$) is the angle between $\widehat a$ and $\widehat b$, then among the statements :

(S1) : $2|\widehat a \times \widehat b| = |\widehat a - \widehat b|$

(S2) : The projection of $\widehat a$ on ($\widehat a$ + $\widehat b$) is ${1 \over 2}$

Let $\lambda$$^ *$ be the largest value of $\lambda$ for which the function ${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$ is increasing for all x $\in$ R. Then ${f_{{\lambda ^ * }}}(1) + {f_{{\lambda ^ * }}}( - 1)$ is equal to :

The area (in sq. units) of the region enclosed between the parabola y² = 2x and the line x + y = 4 is __________.

Let the hyperbola $H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$ and the ellipse $E:3{x^2} + 4{y^2} = 12$ be such that the length of latus rectum of H is equal to the length of latus rectum of E. If ${e_H}$ and ${e_E}$ are the eccentricities of H and E respectively, then the value of $12\left( {e_H^2 + e_E^2} \right)$ is equal to ___________.

Let P₁ be a parabola with vertex (3, 2) and focus (4, 4) and P₂ be its mirror image with respect to the line x + 2y = 6. Then the directrix of P₂ is x + 2y = ____________.

Two identical charged particles each having a mass 10 g and charge 2.0 $\times$ 10^$-$7C are placed on a horizontal table with a separation of L between them such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is 0.25, find the value of L. [Use g = 10 ms^$-$2]

A 100 g of iron nail is hit by a 1.5 kg hammer striking at a velocity of 60 ms^$-$1. What will be the rise in the temperature of the nail if one fourth of energy of the hammer goes into heating the nail?

[Specific heat capacity of iron = 0.42 Jg^$-$1 $^\circ$C^$-$1]

The light of two different frequencies whose photons have energies 3.8 eV and 1.4 eV respectively, illuminate a metallic surface whose work function is 0.6 eV successively. The ratio of maximum speeds of emitted electrons for the two frequencies respectively will be :

A body is projected from the ground at an angle of 45$^\circ$ with the horizontal. Its velocity after 2s is 20 ms^$-$1. The maximum height reached by the body during its motion is __________ m. (use g = 10 ms^$-$2)

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by $y = (10\cos \pi x\sin {{2\pi t} \over T})$ cm

The amplitude of the particle at $x = {4 \over 3}$ cm will be ___________ cm.