The increasing order of the acidity of the $$\alpha $$-hydrogen of the following compounds is :

The increasing order of basicity of the following compounds is :

A flask contains a mixture of compounds A and B. Both compounds decompose by first-order kinetics. The half-lives for A and B are 300 s and 180 s, respectively. If the concentrations of A and B are equal initially, the time required for the concentration of A to be four times that of B(in s) :
(Use ln 2 = 0.693)

The most appropriate reagent for conversion of C₂H₅CN into CH₃CH₂CH₂NH₂ is

The structure of PCl₅ in the solid state is :

The difference between the radii of 3^rd and 4^th
orbits of Li²⁺ is R₁ . The difference between the
radii of 3^rd and 4^th orbits of He⁺ is $$\Delta $$R₂ .
Ratio $$\Delta $$R₁ : $$\Delta $$R₂ is :

Which of the following derivatives of alcohols is unstable in an aqueous base?

The correct electronic configuration and spin-only
magnetic moment (BM) of Gd³⁺ (Z = 64), respectively, are :

In the following reaction sequence the major products A and B are :

Which of the following is not an essential amino acid :

Consider the following reaction:

N₂O₄(g) ⇌ 2NO₂(g); $$\Delta $$H^o = +58 kJ

For each of the following cases (a, b), the direction in which the equilibrium shifts is :
(a) Temperature is decreased.
(b) Pressure is increased by adding N₂ at constant T.

The values of the crystal field stabilization energies for a high spin d⁶ metal ion in octahedral and tetrahedral fields, respectively, are :

An oxidation-reduction reaction in which
3 electrons are transferred has a $$\Delta $$Gº of 17.37 kJ mol^–1 at
25 ^oC. The value of E^o
cell (in V) is ______ × 10^–2.
(1 F = 96,500 C mol^–1)

The total number of coordination sites
in ethylenediaminetetraacetate (EDTA^4–) is _____.

A soft drink was bottled with a partial pressure of CO₂ of 3 bar over the liquid at room temperature. The partial pressure of CO₂ over the solution approaches a value of 30 bar when 44 g of CO₂ is dissolved in 1 kg of water at room temperature. The approximate pH of the soft drink is ______ $$ \times $$ 10^–1.
(First dissociation constant of
H₂CO₃ = 4.0 $$ \times $$ 10^–7; log 2 = 0.3; density
of the soft drink = 1 g mL^–1) .

The number of chiral carbon(s) present in
peptide, Ile-Arg-Pro, is _____.

The minimum number of moles of O₂ required for complete combustion of 1 mole of propane and 2 moles of butane is _____.

The potential energy curve for the H₂ molecule as a function of internuclear distance is :

In the sixth period, the orbitals that are filled are :

If $$\alpha $$ is positive root of the equation, p(x) = x² - x - 2 = 0, then

$$\mathop {\lim }\limits_{x \to {\alpha ^ + }} {{\sqrt {1 - \cos \left( {p\left( x \right)} \right)} } \over {x + \alpha - 4}}$$ is equal to :

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ’SYLLABUS’ such that two letters are distinct and two letters are alike, is :

If the line, 2x - y + 3 = 0 is at a distance
$${1 \over {\sqrt 5 }}$$ and $${2 \over {\sqrt 5 }}$$ from the lines 4x - 2y + $$\alpha $$ = 0
and 6x - 3y + $$\beta $$ = 0, respectively, then the sum of all possible values of $$\alpha $$ and $$\beta $$ is :

The natural number m, for which the coefficient of x in the binomial expansion of

$${\left( {{x^m} + {1 \over {{x^2}}}} \right)^{22}}$$ is 1540, is .............

Let $$f(x) = x.\left[ {{x \over 2}} \right]$$, for -10< x < 10, where [t] denotes the greatest integer function. Then the number of points of discontinuity of f is equal to _____.

If
$$\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx} $$ = $$g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c$$

where c is a constant of integration, then g(0) is equal to :

If the co-ordinates of two points A and B
are $$\left( {\sqrt 7 ,0} \right)$$ and $$\left( { - \sqrt 7 ,0} \right)$$ respectively and
P is any point on the conic, 9x² + 16y² = 144, then PA + PB is equal to :

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is _________.

If $${3^{2\sin 2\alpha - 1}}$$, 14 and $${3^{4 - 2\sin 2\alpha }}$$ are the first three terms of an A.P. for some $$\alpha $$, then the sixth terms of this A.P. is:

If the minimum and the maximum values of the function $$f:\left[ {{\pi \over 4},{\pi \over 2}} \right] \to R$$, defined by
$$f\left( \theta \right) = \left| {\matrix{ { - {{\sin }^2}\theta } & { - 1 - {{\sin }^2}\theta } & 1 \cr { - {{\cos }^2}\theta } & { - 1 - {{\cos }^2}\theta } & 1 \cr {12} & {10} & { - 2} \cr } } \right|$$ are m and M respectively, then the ordered pair (m,M) is equal to :

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be :

The mean and variance of 7 observations are 8 and 16, respectively. If five observations are 2, 4, 10, 12, 14, then the absolute difference of the remaining two observations is :

If the point P on the curve, 4x² + 5y² = 20 is
farthest from the point Q(0, -4), then PQ² is equal to:

If the four complex numbers $$z,\overline z ,\overline z - 2{\mathop{\rm Re}\nolimits} \left( {\overline z } \right)$$ and $$z-2Re(z)$$ represent the vertices of a square of side 4 units in the Argand plane, then $$|z|$$ is equal to :

If the function
$$f\left( x \right) = \left\{ {\matrix{ {{k_1}{{\left( {x - \pi } \right)}^2} - 1,} & {x \le \pi } \cr {{k_2}\cos x,} & {x > \pi } \cr } } \right.$$ is
twice differentiable, then the ordered pair (k₁, k₂) is equal to :

If (a, b, c) is the image of the point (1, 2, -3) in

the line $${{x + 1} \over 2} = {{y - 3} \over { - 2}} = {z \over { - 1}}$$, then a + b + c is :

The value of $$\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $$ is:

If S is the sum of the first 10 terms of the series

$${\tan ^{ - 1}}\left( {{1 \over 3}} \right) + {\tan ^{ - 1}}\left( {{1 \over 7}} \right) + {\tan ^{ - 1}}\left( {{1 \over {13}}} \right) + {\tan ^{ - 1}}\left( {{1 \over {21}}} \right) + ....$$

then tan(S) is equal to :

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

equation $${{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0$$ satisfying y(0) = 1, then a value of y(log_e13) is :

The product of the roots of the
equation 9x² - 18|x| + 5 = 0 is :

Let $$\lambda \in $$ R . The system of linear equations
2x₁ - 4x₂ + $$\lambda $$x₃ = 1
x₁ - 6x₂ + x₃ = 2
$$\lambda $$x₁ - 10x₂ + 4x₃ = 3
is inconsistent for:

A compound microscope consists of an objective lens of focal length 1 cm and an eyepiece of focal length 5 cm with a separation of 10 cm.
The distance between an object and the objective lens, at which the strain on the eye is minimum is $${n \over {40}}$$ cm. The value of n is _____.

A particle of mass 200 MeV/c² collides with a hydrogen atom at rest. Soon after the collision the particle comes to rest, and the atom recoils and goes to its first excited state. The initial kinetic energy of the particle (in eV) is
$${N \over 4}$$. The value of N is :
(Given the mass of the hydrogen atom to be 1 GeV/c²) ______ .

Two concentric circular coils, C₁ and C₂ are
placed in the XY plane. C₁ has 500 turns, and
a radius of 1 cm. C₂ has 200 turns and radius
of 20 cm. C₂ carries a time dependent current
I(t) = (5t² – 2t + 3) A where t is in s. The emf
induced in C₁ (in mV), at the instant t = 1 s is
$${4 \over x}$$. The value of x is ___ .

Three different processes that can occur in an ideal monoatomic gas are shown in the P vs V diagram. The paths are labelled as A $$ \to $$ B, A $$ \to $$ C and A $$ \to $$ D. The change in internal energies during these process are taken as E_AB, E_AC and E_AD and the work done as W_AB, W_AC and W_AD.
The correct relation between these parameters are :

Two capacitors of capacitances C and 2C are charged to potential differences V and 2V, respectively. These are then connected in parallel in such a manner that the positive terminal of one is connected to the negative terminal of the other. The final energy of this configuration is :

A physical quantity z depends on four observables
a, b, c and d, as z = $${{{a^2}{b^{{2 \over 3}}}} \over {\sqrt c {d^3}}}$$. The percentages of error in the measurement of a, b, c and d are 2%, 1.5%, 4% and 2.5% respectively. The percentage of error in z is :

With increasing biasing voltage of a photodiode, the photocurrent magnitude :

A helicopter rises from rest on the ground vertically upwards with a constant acceleration g. A food packet is dropped from the helicopter when it is at a height h. The time taken by the packet to reach the ground is close to :
[g is the acceleration due to gravity]

A hollow spherical shell at outer radius R floats just submerged under the water surface. The inner radius of the shell is r. If the specific gravity of the shell material is $${{27} \over 8}$$ w.r.t water, the value of r is :

Assume that the displacement(s) of air is proportional to the pressure difference ($$\Delta $$p) created by a sound wave. Displacement (s) further depends on the speed of sound (v), density of air ($$\rho $$) and the frequency (f). If $$\Delta $$p ~ 10 Pa, v ~ 300 m/s, $$\rho $$ ~ 1 kg/m³ and f ~ 1000 Hz, then s will be of the order of (take the multiplicative constant to be 1) :

An electron is constrained to move along the y-axis with a speed of 0.1 c (c is the speed of light) in the presence of electromagnetic wave, whose electric field is
$$\overrightarrow E = 30\widehat j\sin \left( {1.5 \times {{10}^7}t - 5 \times {{10}^{ - 2}}x} \right)$$ V/m.
The maximum magnetic force experienced by the electron will be :
(given c = 3 $$ \times $$ 10⁸ ms^–1 and electron charge = 1.6 $$ \times $$ 10^–19 C)

A wheel is rotating freely with an angular speed $$\omega $$ on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia 3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is :

A bullet of mass 5 g, travelling with a speed of 210 m/s, strikes a fixed wooden target. One half of its kinetic energy is converted into heat in the bullet while the other half is converted into heat in the wood. The rise of temperature of the bullet if the specific heat of its material is 0.030 cal/(g – ^oC) (1 cal = 4.2 × 10⁷ ergs) close to :

A force $$\overrightarrow F = \left( {\widehat i + 2\widehat j + 3\widehat k} \right)$$ N acts at a point
$$\left( {4\widehat i + 3\widehat j - \widehat k} \right)$$ m. Then the magnitude of torque
about the point $$\left( {\widehat i + 2\widehat j + \widehat k} \right)$$ m will be $$\sqrt x $$ N m.
The value of x is _______.

A square loop of side 2$$a$$, and carrying current I, is kept in XZ plane with its centre at origin. A long wire carrying the same current I is placed parallel to the z-axis and passing through the point (0, b, 0), (b >> a). The magnitude of the torque on the loop about zaxis is given by :

Number of molecules in a volume of 4 cm³ of a perfect monoatomic gas at some temperature T and at a pressure of 2 cm of mercury is close to?
(Given, mean kinetic energy of a molecule
(at T) is 4 $$ \times $$ 10^–14 erg, g = 980 cm/s², density of
mercury = 13.6 g/cm³)

A balloon is moving up in air vertically above a point A on the ground. When it is at a height h₁, a girl standing at a distanced (point B) from A (see figure) sees it at an angle 45^o with respect to the vertical. When the balloon climbs up a further height h₂, it is seen at an angle 60^o with respect to the vertical if the girl moves further by a distance 2.464 d(point C). Then the height h₂ is (given tan 30^o = 0.5774)

A galvanometer of resistance G is converted into a voltmeter of range 0 – 1 V by connecting a resistance R₁ in series with it. The additional resistance that should be connected in series with R₁ to increase the range of the voltmeter to 0 – 2 V will be :

In a resonance tube experiment when the tube is filled with water up to a height of 17.0 cm from bottom, it resonates with a given tuning fork. When the water level is raised the next resonance with the same tuning fork occurs at a height of 24.5 cm. If the velocity of sound in air is 330 m/s, the tuning fork frequency is :

For a concave lens of focal length f, the relation between object and image distances u and v, respectively, from its pole can best be represented by (u = v is the reference line) :

The value of the acceleration due to gravity is g₁ at a height h = $${R \over 2}$$ (R = radius of the earth) from the surface of the earth. It is again equal to g₁ at a depth d below the surface of the earth. The ratio $$\left( {{d \over R}} \right)$$ equals :

An electrical power line, having a total resistance of 2 $$\Omega $$, delivers 1 kW at 220 V. The efficiency of the transmission line is approximately :

A solid sphere of radius R carries a charge Q + q distributed uniformly over its volume. A very small point like piece of it of mass m gets detached from the bottom of the sphere and falls down vertically under gravity. This piece carries charge q. If it acquires a speed v when it has fallen through a vertical height y (see figure), then :
(assume the remaining portion to be spherical).