If a rocket runs on a fuel (C₁₅H₃₀) and liquid oxygen, the weight of oxygen required and CO₂ released for every litre of fuel respectively are :

(Given : density of the fuel is 0.756 g/mL)

Consider the following pairs of electrons

(A) (a) n = 3, $l$ = 1, m₁ = 1, m_s = + ${1 \over 2}$

      (b) n = 3, 1 = 2, m₁ = 1, m_s = + ${1 \over 2}$

(B) (a) n = 3, $l$ = 2, m₁ = $-$2, m_s = $-$${1 \over 2}$

      (b) n = 3, $l$ = 2, m₁ = $-$1, m_s = $-$${1 \over 2}$

(C) (a) n = 4, $l$ = 2, m₁ = 2, m_s = + ${1 \over 2}$

      (b) n = 3, $l$ = 2, m₁ = 2, m_s = + ${1 \over 2}$

The pairs of electrons present in degenerate orbitals is/are :

Match List - I with List - II :

Choose the most appropriate answer from the options given below :

For a reaction at equilibrium

A(g) $\rightleftharpoons$ B(g) + ${1 \over 2}$ C(g)

the relation between dissociation constant (K), degree of dissociation ($\alpha$) and equilibrium pressure (p) is given by :

Given below are the oxides:

Na₂O, As₂O₃, N₂O, NO and Cl₂O₇

Number of amphoteric oxides is :

The most stable trihalide of nitrogen is :

In the given reaction sequence, the major product 'C' is :

Two statements are given below :

Statement I : The melting point of monocarboxylic acid with even number of carbon atoms is higher than that of with odd number of carbon atoms acid immediately below and above it in the series.

Statement II : The solubility of monocarboxylic acids in water decreases with increase in molar mass.

Choose the most appropriate option :

Which of the following is an example of conjugated diketone?

The major product of the above reactions is :

A polysaccharide 'X' on boiling with dil H₂SO₄ at 393 K under 2-3 atm pressure yields 'Y'. 'Y' on treatment with bromine water gives gluconic acid. 'X' contains $\beta$-glycosidic linkages only. Compound 'X' is :

During the qualitative analysis of salt with cation y²⁺, addition of a reagent (X) to alkaline solution of the salt gives a bright red precipitate. The reagent (X) and the cation (y²⁺) present respectively are :

2O₃(g) $\rightleftharpoons$ 3O₂(g)

At 300 K, ozone is fifty percent dissociated. The standard free energy change at this temperature and 1 atm pressure is ($-$) ____________ J mol^$-$1. (Nearest integer)

[Given : ln 1.35 = 0.3 and R = 8.3 J K^$-$1 mol^$-$1]

The osmotic pressure of blood is 7.47 bar at 300 K. To inject glucose to a patient intravenously, it has to be isotonic with blood. The concentration of glucose solution in gL^$-$1 is _____________.

(Molar mass of glucose = 180 g mol^$-$1, R = 0.083 L bar K^$-$1 mol^$-$1) (Nearest integer)

The cell potential for the following cell

Pt |H₂(g)|H⁺ (aq)|| Cu²⁺ (0.01 M)|Cu(s)

is 0.576 V at 298 K. The pH of the solution is __________. (Nearest integer)

(Given : $E_{C{u^{2 + }}/Cu}^o = 0.34$ V and ${{2.303\,RT} \over F} = 0.06$ V)

The rate constants for decomposition of acetaldehyde have been measured over the temperature range 700 - 1000 K. The data has been analysed by plotting ln k vs ${{{{10}^3}} \over T}$ graph. The value of activation energy for the reaction is ___________ kJ mol^$-$1. (Nearest integer)

(Given : R = 8.31 J K^$-$1 mol^$-$1)

The difference in oxidation state of chromium in chromate and dichromate salts is ___________.

In the cobalt-carbonyl complex : [Co₂(CO)₈], number of Co-Co bonds is "X" and terminal CO ligands is "Y". X + Y = ___________.

A 0.166 g sample of an organic compound was digested with conc. H₂SO₄ and then distilled with NaOH. The ammonia gas evolved was passed through 50.0 mL of 0.5 N H₂SO₄. The used acid required 30.0 mL of 0.25 N NaOH for complete neutralization. The mass percentage of nitrogen in the organic compound is ____________.

Number of electrophilic centres in the given compound is _______________.

The major product 'A' of the following given reaction has _____________ sp² hybridized carbon atoms.

Let $A = \{ z \in C:1 \le |z - (1 + i)| \le 2\}$

and $B = \{ z \in A:|z - (1 - i)| = 1\}$. Then, B :

The remainder when 3²⁰²² is divided by 5 is :

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random, are found to be 1 red and 1 black. If the probability that both balls come from Bag A is ${6 \over {11}}$, then n is equal to __________.

The number of values of $\alpha$ for which the system of equations :

x + y + z = $\alpha$

$\alpha$x + 2$\alpha$y + 3z = $-$1

x + 3$\alpha$y + 5z = 4

is inconsistent, is

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of

the equation 3x² + $\lambda$x $-$ 1 = 0 is 15, then 6($\alpha$³ + $\beta$³)² is equal to :

The set of all values of k for which

${({\tan ^{ - 1}}x)^3} + {({\cot ^{ - 1}}x)^3} = k{\pi ^3},\,x \in R$, is the interval :

For the function

$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$, which one of the following is NOT correct?

The sum of absolute maximum and absolute minimum values of the function $f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$ in the interval [0, 1] is :

If $\{ {a_i}\} _{i = 1}^n$, where n is an even integer, is an arithmetic progression with common difference 1, and $\sum\limits_{i = 1}^n {{a_i} = 192} ,\,\sum\limits_{i = 1}^{n/2} {{a_{2i}} = 120}$, then n is equal to :

If x = x(y) is the solution of the differential equation

$y{{dx} \over {dy}} = 2x + {y^3}(y + 1){e^y},\,x(1) = 0$; then x(e) is equal to :

Let $\widehat a$, $\widehat b$ be unit vectors. If $\overrightarrow c$ be a vector such that the angle between $\widehat a$ and $\overrightarrow c$ is ${\pi \over {12}}$, and $\widehat b = \overrightarrow c + 2\left( {\overrightarrow c \times \widehat a} \right)$, then ${\left| {6\overrightarrow c } \right|^2}$ is equal to :

The domain of the function

$f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}$ is :

The number of one-one functions f : {a, b, c, d} $\to$ {0, 1, 2, ......, 10} such

that 2f(a) $-$ f(b) + 3f(c) + f(d) = 0 is ___________.

In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answer, $-$2 marks for each wrong answer and 0 mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets 5 marks is ____________.

Let $A\left( {{3 \over {\sqrt a }},\sqrt a } \right),\,a > 0$, be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If $D(3\cos \theta ,a\sin \theta )$ is a point in the fourth quadrant such that the maximum area of $\Delta$ACD is 12 square units, then a is equal to ____________.

Let a line having direction ratios, 1, $-$4, 2 intersect the lines ${{x - 7} \over 3} = {{y - 1} \over { - 1}} = {{z + 2} \over 1}$ and ${x \over 2} = {{y - 7} \over 3} = {z \over 1}$ at the points A and B. Then (AB)² is equal to ___________.

The number of points where the function

f(x) = \left\{ {\matrix{ {|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr {[4{x^2} - 1]} & {if} & { - 1 < x < 1} \cr {|x + 1| + |x - 2|} & {if} & {x \ge 1} \cr } } \right.

[t] denotes the greatest integer $\le$ t, is discontinuous is _____________.

Let $f(\theta ) = \sin \theta + \int\limits_{ - \pi /2}^{\pi /2} {(\sin \theta + t\cos \theta )f(t)dt}$. Then the value of $\left| {\int_0^{\pi /2} {f(\theta )d\theta } } \right|$ is _____________.

Let $\mathop {Max}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \alpha$ and $\mathop {Min}\limits_{0\, \le x\, \le 2} \left\{ {{{9 - {x^2}} \over {5 - x}}} \right\} = \beta$.

If $\int\limits_{\beta - {8 \over 3}}^{2\alpha - 1} {Max\left\{ {{{9 - {x^2}} \over {5 - x}},x} \right\}dx = {\alpha _1} + {\alpha _2}{{\log }_e}\left( {{8 \over {15}}} \right)}$ then ${\alpha _1} + {\alpha _2}$ is equal to _____________.

Let S be the region bounded by the curves y = x³ and y² = x. The curve y = 2|x| divides S into two regions of areas R₁, R₂. If max {R₁, R₂} = R₂, then ${{{R_2}} \over {{R_1}}}$ is equal to ______________.

If the shortest distance between the lines

$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\widehat j} \right)$

and $\overrightarrow r = \left( { - \widehat j + 2\widehat k} \right) + \mu \left( {\widehat i - \widehat j + \widehat k} \right)$ is $\sqrt {{2 \over 3}}$, then the integral value of a is equal to ___________.

The bulk modulus of a liquid is 3 $\times$ 10¹⁰ Nm^$-$2. The pressure required to reduce the volume of liquid by 2% is :

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In an uniform magnetic field, speed and energy remains the same for a moving charged particle.

Reason (R) : Moving charged particle experiences magnetic force perpendicular to its direction of motion.

Two identical cells each of emf 1.5 V are connected in parallel across a parallel combination of two resistors each of resistance 20 $\Omega$. A voltmeter connected in the circuit measures 1.2 V. The internal resistance of each cell is :

Identify the pair of physical quantities which have different dimensions:

A projectile is projected with velocity of 25 m/s at an angle $\theta$ with the horizontal. After t seconds its inclination with horizontal becomes zero. If R represents horizontal range of the projectile, the value of $\theta$ will be :

[use g = 10 m/s²]

A block of mass 10 kg starts sliding on a surface with an initial velocity of 9.8 ms^$-$1. The coefficient of friction between the surface and block is 0.5. The distance covered by the block before coming to rest is :

[use g = 9.8 ms^$-$2]

A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N. If the maximum speed with which the stone can revolve is ${K \over \pi }$ rev./min. The value of K is :

(Assume the string is massless and unstretchable)

A vertical electric field of magnitude 4.9 $\times$ 10⁵ N/C just prevents a water droplet of a mass 0.1 g from falling. The value of charge on the droplet will be :

(Given : g = 9.8 m/s²)

A particle experiences a variable force $\overrightarrow F = \left( {4x\widehat i + 3{y^2}\widehat j} \right)$ in a horizontal x-y plane. Assume distance in meters and force is newton. If the particle moves from point (1, 2) to point (2, 3) in the x-y plane, then Kinetic Energy changes by :

The approximate height from the surface of earth at which the weight of the body becomes ${1 \over 3}$ of its weight on the surface of earth is :

[Radius of earth R = 6400 km and $\sqrt 3$ = 1.732]

A resistance of 40 $\Omega$ is connected to a source of alternating current rated 220 V, 50 Hz. Find the time taken by the current to change from its maximum value to the rms value :

The equations of two waves are given by :

y₁ = 5 sin 2$\pi$(x - vt) cm

y₂ = 3 sin 2$\pi$(x $-$ vt + 1.5) cm

These waves are simultaneously passing through a string. The amplitude of the resulting wave is :

A plane electromagnetic wave travels in a medium of relative permeability 1.61 and relative permittivity 6.44. If magnitude of magnetic intensity is 4.5 $\times$ 10^$-$2 Am^$-$1 at a point, what will be the approximate magnitude of electric field intensity at that point?

(Given : Permeability of free space $\mu$₀ = 4$\pi$ $\times$ 10^$-$7 NA^$-$2, speed of light in vacuum c = 3 $\times$ 10⁸ ms^$-$1)

Choose the correct option from the following options given below :

Nucleus A is having mass number 220 and its binding energy per nucleon is 5.6 MeV. It splits in two fragments 'B' and 'C' of mass numbers 105 and 115. The binding energy of nucleons in 'B' and 'C' is 6.4 MeV per nucleon. The energy Q released per fission will be :

A parallel plate capacitor is formed by two plates each of area 30$\pi$ cm² separated by 1 mm. A material of dielectric strength 3.6 $\times$ 10⁷ Vm^$-$1 is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is 7 $\times$ 10^$-$6C, the value of dielectric constant of the material is :

[Use ${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}$ Nm² C^$-$2]

The magnetic field at the centre of a circular coil of radius r, due to current I flowing through it, is B. The magnetic field at a point along the axis at a distance ${r \over 2}$ from the centre is :

Two metallic blocks M₁ and M₂ of same area of cross-section are connected to each other (as shown in figure). If the thermal conductivity of M₂ is K then the thermal conductivity of M₁ will be :

[Assume steady state heat conduction]

0.056 kg of Nitrogen is enclosed in a vessel at a temperature of 127$^\circ$C. Th amount of heat required to double the speed of its molecules is ____________ k cal.

Take R = 2 cal mole^$-$1 K^$-$1)

Two identical thin biconvex lens of focal length 15 cm and refractive index 1.5 are in contact with each other. The space between the lenses is filled with a liquid of refractive index 1.25. The focal length of the combination is __________ cm.

As shown in the figure an inductor of inductance 200 mH is connected to an AC source of emf 220 V and frequency 50 Hz. The instantaneous voltage of the source is 0 V when the peak value of current is ${{\sqrt a } \over \pi }$ A. The value of $a$ is ___________.

Sodium light of wavelengths 650 nm and 655 nm is used to study diffraction at a single slit of aperture 0.5 mm. The distance between the slit and the screen is 2.0 m. The separation between the positions of the first maxima of diffraction pattern obtained in the two cases is ___________ $\times$ 10^$-$5 m.

When light of frequency twice the threshold frequency is incident on the metal plate, the maximum velocity of emitted electron is v₁. When the frequency of incident radiation is increased to five times the threshold value, the maximum velocity of emitted electron becomes v₂. If v₂ = x v₁, the value of x will be __________.

From the top of a tower, a ball is thrown vertically upward which reaches the ground in 6 s. A second ball thrown vertically downward from the same position with the same speed reaches the ground in 1.5 s. A third ball released, from the rest from the same location, will reach the ground in ____________ s.

A ball of mass 100 g is dropped from a height h = 10 cm on a platform fixed at the top of a vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed by a distance ${h \over 2}$. The spring constant is _____________ Nm^$-$1.

(Use g = 10 ms^$-$2)

A metre scale is balanced on a knife edge at its centre. When two coins, each of mass 10 g are put one on the top of the other at the 10.0 cm mark the scale is found to be balanced at 40.0 cm mark. The mass of the metre scale is found to be x $\times$ 10^$-$2 kg. The value of x is ___________.