Deficiency of vitamin K causes :

In the following molecule,


Hybridisation of Carbon a, b, and c respectively are :

Consider the given reaction, percentage yield of :

Given below are two statements :

Statement I : C₂H₅OH and AgCN both can generate nucleophile.

Statement II : KCN and AgCN both will generate nitrile nucleophile with all reaction conditions.

Choose the most appropriate option :

The oxide that shows magnetic property is :

In the reaction of hypobromite with amide, the carbonyl carbon is lost as :

The secondary valency and the number of hydrogen bonded water molecule(s) in CuSO₄ . 5H₂O, respectively, are :

The first ionization energy of magnesium is smaller as compared to that of elements X and Y, but higher than that of Z. The elements X, Y and Z, respectively, are :

Given below are two statements :

Statement I : Bohr's theory accounts for the stability and line spectrum of Li⁺ ion.

Statement II : Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic field.

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

The oxidation states of nitrogen in NO, NO₂, N₂O and NO$$_3^ - $$ are in the order of :

An organic compound "A" on treatment with benzene sulphonyl chloride gives compound B. B is soluble in dil. NaOH solution. Compound A is :

Main products formed during a reaction of 1-methoxy naphthalene with hydroiodic acid are :

Consider the above reaction, the product 'X' and 'Y' respectively are :

In Tollen's test for aldehyde, the overall number of electron(s) transferred to the Tollen's reagent formula [Ag(NH₃)₂]⁺ per aldehyde group to form silver mirror is ___________. (Round off to the Nearest Integer).

10.0 mL of Na₂CO₃ solution is titrated against 0.2 M HCl solution. The following titre values were obtained in 5 readings :

4.8 mL, 4.9 mL, 5.0 mL, 5.0 mL and 5.0 mL

Based on these readings, and convention of titrimetric estimation the concentration of Na₂CO₃ solution is ___________ mM.

(Round off to the Nearest Integer).

The number of species below that have two lone pairs of electrons in their central atom is _________. (Round off to the Nearest Integer).

SF₄, BF$$_4^ - $$, ClF₃, AsF₃, PCl₅, BrF₅, XeF₄, SF₆

Consider the above reaction where 6.1 g of Benzoic acid is used to get 7.8 g of m-bromo benzoic acid. The percentage yield of the product is __________

(Round off to the Nearest Integer).

[Given : Atomic masses : C : 12.0 u, H : 1.0 u, O : 16.0 u, Br : 80.0 u]

The molar conductivities at infinite dilution of barium chloride, sulphuric acid and hydrochloric acid are 280, 860 and 426 S cm² mol^$$-$$1 respectively. The molar conductivity at infinite dilution of barium sulphate is _________ S cm² mol^$$-$$1. (Round off to the Nearest Integer ).

The gas phase reaction $$2A(g) \rightleftharpoons {A_2}(g)$$ at 400 K has $$\Delta$$G^o = + 25.2 kJ mol^-1.

The equilibrium constant K_C for this reaction is ________ $$\times$$ 10^$$-$$2. (Round off to the Nearest Integer).

[Use : R = 8.3 J mol^$$-$$1 K^$$-$$1, ln 10 = 2.3 log₁₀ 2 = 0.30, 1 atm = 1 bar]

[antilog ($$-$$0.3) = 0.501]

A reaction has a half life of 1 min. The time required for 99.9% completion of the reaction is _________ min. (Round off to the Nearest Integer). [Use : ln 2 = 0.69; ln 10 = 2.3]

The solubility of CdSO₄ in water is 8.0 $$\times$$ 10^$$-$$4 mol L^$$-$$1. Its solubility in 0.01 M H₂SO₄ solution is __________ $$\times$$ 10^$$-$$6 mol L^$$-$$1. (Round off to the Nearest Integer). (Assume that solubility is much less than 0.01 M)

A solute A dimerizes in water. The boiling point of a 2 molal solution of A is 100.52$$^\circ$$C. The percentage association of A is __________. (Round off to the Nearest Integer).

[Use : K_b for water = 0.52 K kg mol^$$-$$1 Boiling point of water = 100$$^\circ$$C]

Let y = y(x) be the solution of the differential equation

$${{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x} \right)$$, 0 < x < 2.1, with y(2) = 0. Then the value of $${{dy} \over {dx}}$$ at x = 1 is equal to :

Let the system of linear equations

4x + $$\lambda$$y + 2z = 0

2x $$-$$ y + z = 0

$$\mu$$x + 2y + 3z = 0, $$\lambda$$, $$\mu$$$$\in$$R.

has a non-trivial solution. Then which of the following is true?

The area bounded by the curve 4y² = x²(4 $$-$$ x)(x $$-$$ 2) is equal to :

If 15sin⁴$$\alpha$$ + 10cos⁴$$\alpha$$ = 6, for some $$\alpha$$$$\in$$R, then the value of

27sec⁶$$\alpha$$ + 8cosec⁶$$\alpha$$ is equal to :

Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two non-zero vectors perpendicular to each other and $$|\overrightarrow a | = |\overrightarrow b |$$. If $$|\overrightarrow a \times \overrightarrow b | = |\overrightarrow a |$$, then the angle between the vectors $$\left( {\overrightarrow a + \overrightarrow b + \left( {\overrightarrow a \times \overrightarrow b } \right)} \right)$$ and $${\overrightarrow a }$$ is equal to :

Let a complex number be w = 1 $$-$$ $${\sqrt 3 }$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$ arg(w) = $${\pi \over 2}$$. Then the area of the triangle with vertices origin, z and w is equal to :

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

$$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr b,\,if\,x\, = 0 \hfill \cr {{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill \cr} \right.$$

If f is continuous at x = 0, then the value of a + b is equal to :

Let g(x) = $$\int_0^x {f(t)dt} $$, where f is continuous function in [ 0, 3 ] such that $${1 \over 3}$$ $$ \le $$ f(t) $$ \le $$ 1 for all t$$\in$$ [0, 1] and 0 $$ \le $$ f(t) $$ \le $$ $${1 \over 2}$$ for all t$$\in$$ (1, 3]. The largest possible interval in which g(3) lies is :

Let in a series of 2n observations, half of them are equal to a and remaining half are equal to $$-$$a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :

In a triangle ABC, if $$|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10$$, then the projection of the vector $$\overrightarrow {AB} $$ on $$\overrightarrow {AC} $$ is equal to :

Define a relation R over a class of n $$\times$$ n real matrices A and B as

"ARB iff there exists a non-singular matrix P such that PAP^$$-$$1 = B".

Then which of the following is true?

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of $$\Delta$$ABC, then (R + r) is equal to :

Let f : R $$-$$ {3} $$ \to $$ R $$-$$ {1} be defined by f(x) = $${{x - 2} \over {x - 3}}$$.

Let g : R $$ \to $$ R be given as g(x) = 2x $$-$$ 3. Then, the sum of all the values of x for which f^$$-$$1(x) + g^$$-$$1(x) = $${{13} \over 2}$$ is equal to :

Let S₁ be the sum of first 2n terms of an arithmetic progression. Let S₂ be the sum of first 4n terms of the same arithmetic progression. If (S₂ $$-$$ S₁) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :

If $$\sum\limits_{r = 1}^{10} {r!({r^3} + 6{r^2} + 2r + 5) = \alpha (11!)} $$, then the value of $$\alpha$$ is equal to ___________.

If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x³) + x g(x³) is divisible by x² + x + 1, then P(1) is equal to ___________.

The term independent of x in the expansion of

$${\left[ {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right]^{10}}$$, x $$\ne$$ 1, is equal to ____________.

Let y = y(x) be the solution of the differential equation

xdy $$-$$ ydx = $$\sqrt {({x^2} - {y^2})} dx$$, x $$ \ge $$ 1, with y(1) = 0. If the area bounded by the line x = 1, x = e^$$\pi$$, y = 0 and y = y(x) is $$\alpha$$e^2$$\pi$$ + $$\beta$$, then the value of 10($$\alpha$$ + $$\beta$$) is equal to __________.

Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3} \cr } } \right]$$. Then the value of n$$\in$$N for which Pⁿ = 5I $$-$$ 8P is equal to ____________.

Let f : R $$ \to $$ R satisfy the equation f(x + y) = f(x) . f(y) for all x, y $$\in$$R and f(x) $$\ne$$ 0 for any x$$\in$$R. If the function f is differentiable at x = 0 and f'(0) = 3, then

$$\mathop {\lim }\limits_{h \to 0} {1 \over h}(f(h) - 1)$$ is equal to ____________.

Let P(x) be a real polynomial of degree 3 which vanishes at x = $$-$$3. Let P(x) have local minima at x = 1, local maxima at x = $$-$$1 and $$\int\limits_{ - 1}^1 {P(x)dx} $$ = 18, then the sum of all the coefficients of the polynomial P(x) is equal to _________.

A plane electromagnetic wave propagating along y-direction can have the following pair of electric field $$\left( {\overrightarrow E } \right)$$ and magnetic field $$\left( {\overrightarrow B } \right)$$ components.

If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately : [Take g = 10 ms^$$-$$2, the radius of earth, R = 6400 $$\times$$ 10³ m, Take $$\pi$$ = 3.14]

An object of mass m₁ collides with another object of mass m₂, which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses m₂ : m₁ is :

In a series LCR circuit, the inductive reactance (X_L) is 10$$\Omega$$ and the capacitive reactance (X_C) is 4$$\Omega$$. The resistance (R) in the circuit is 6$$\Omega$$. The power factor of the circuit is :

A particle of mass m moves in a circular orbit under the central potential field, $$U(r) = - {C \over r}$$, where C is a positive constant. The correct radius $$-$$ velocity graph of the particle's motion is :

The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is $${\overrightarrow L }$$. The magnitude of the areal velocity of the planet is :

Three rays of light, namely red (R), green (G) and blue (B) are incident on the face PQ of a right angled prism PQR as shown in the figure.


The refractive indices of the material of the prism for red, green and blue wavelength are 1.27, 1.42 and 1.49 respectively. The colour of the ray(s) emerging out of the face PR is :

Consider a uniform wire of mass M and length L. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the center is :

The velocity $$-$$ displacement graph of a particle is shown in the figure.


The acceleration $$-$$ displacement graph of the same particle is represented by :

Consider a sample of oxygen behaving like an ideal gas. At 300 K, the ratio of root mean square (rms) velocity to the average velocity of gas molecule would be :

(Molecular weight of oxygen is 32g/mol; R = 8.3 J K^$$-$$1 mol^$$-$$1)

The time taken for the magnetic energy to reach 25% of its maximum value, when a solenoid of resistance R, inductance L is connected to a battery, is :

A solid cylinder of mass m is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is :


[The coefficient of static friction, $$\mu$$_s' is 0.4]

The function of time representing a simple harmonic motion with a period of $${\pi \over \omega }$$ is :

For an adiabatic expansion of an ideal gas, the fractional change in its pressure is equal to (where $$\gamma$$ is the ratio of specific heats) :

A proton and an $$\alpha$$-particle, having kinetic energies K_p and K_$$\alpha$$ respectively, enter into a magnetic field at right angles.

The ratio of the radii of trajectory of proton to that of $$\alpha$$-particle is 2 : 1. The ratio of K_p : K_$$\alpha$$ is :

An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is S₁ and that of the other part is S₂. Given that S₁ > S₂. If the piston is removed then the total entropy of the system will be :

Which of the following statements are correct?

(A) Electric monopoles do not exist whereas magnetic monopoles exist.

(B) Magnetic field lines due to a solenoid at its ends and outside cannot be completely straight and confined.

(C) Magnetic field lines are completely confined within a toroid.

(D) Magnetic field lines inside a bar magnet are not parallel.

(E) $$\chi $$ = $$-$$1 is the condition for a perfect diamagnetic material, where x is its magnetic susceptibility.

Choose the correct answer from the options given below :

Two wires of same length and thickness having specific resistances 6$$\Omega$$ cm and 3$$\Omega$$ cm respectively are connected in parallel. The effective resistivity is $$\rho$$$$\Omega$$ cm. The value of $$\rho$$, to the nearest integer, is ____________.

Consider a 72 cm long wire AB as shown in the figure. The galvanometer jockey is placed at P on AB at a distance x cm from A. The galvanometer shows zero deflection.


The value of x, to the nearest integer, is ___________.

A galaxy is moving away from the earth at a speed of 286 kms^$$-$$1. The shift in the wavelength of a redline at 630 nm is x $$\times$$ 10^$$-$$10 m. The value of x, to the nearest integer, is ____________. [Take the value of speed of light c, as 3 $$\times$$ 10⁸ ms^$$-$$1]

Consider a water tank as shown in the figure. It's cross-sectional area is 0.4 m². The tank has an opening B near the bottom whose cross-section area is 1 cm². A load of 24 kg is applied on the water at the top when the height of the water level is 40 cm above the bottom, the velocity of water coming out the opening B is v ms^-1.

The value of v, to the nearest integer, is ___________. [Take value of g to be 10 ms^-2]

The radius of a sphere is measured to be (7.50 $$\pm$$ 0.85) cm. Suppose the percentage error in its volume is x.

The value of x, to the nearest x, is __________.

The projectile motion of a particle of mass 5 g is shown in the figure.


The initial velocity of the particle is $$5\sqrt 2 $$ ms^-1 and the air resistance is assumed to be negligible. The magnitude of the change in momentum between the points A and B is x $$\times$$ 10^-2 kgms^-1. The value of x, to the nearest integer, is __________.

An infinite number of point charges, each carrying 1 $$\mu$$C charge, are placed along the y-axis at y = 1 m, 2 m, 4 m, 8 m ...............

The total force on a 1C point charge, placed at the origin, is x $$\times$$ 10³ N.

The value of x, to the nearest integer, is __________. [Take $${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}$$ Nm²/C²]