A compound M_pX_q has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below. The empirical formula of the compound is

The carboxyl functional group ($$-$$COOH) is present in

As per IUPAC nomenclature, the name of the complex $$[Co{({H_2}O)_4}{(N{H_3})_2}]C{l_3}$$ is

Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. 1/V plot is shown below. The value of the van der Waals constant a (atm L² mol^$$-$$2) is

The number of aldol reactions that occur in the given transformation are

The colour of light absorbed by an aqueous solution of CuSO₄ is

The number of optically active products obtained from the complete ozonolysis of the given compound is ______.

Which of the following hydrogen halides reacts with AgNO₃(aq.) to give a precipitate that dissolves in Na₂S₂O₃ (aq.) ?

Identify the binary mixtures that can be separated into individual compounds, by differential extraction, as shown in the given scheme.

For an ideal gas, consider only P-V work in going from an initial state X to the final state Z. The final state Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is(are) correct? (Take $$\Delta$$S as change in entropy and W as work done)

Which of the following molecules, in pure form, is(are) unstable at room temperature?

The substituents R₁ and R₂ for nine peptides are listed in the table given below. How many of these peptides are positively charged at pH = 7.0 ?

When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is ____________.

then $$\varphi $$ cannot satisfy

If $$\mathop {\lim }\limits_{x \to \infty } \left( {{{{x^2} + x + 1} \over {x + 1}} - ax - b} \right) = 4$$, then

Let $$P = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix and let $$Q = [{b_{ij}}]$$, where $${b_{ij}} = {2^{i + j}}{a_{ij}}$$ for $$1 \le i,j \le 3$$. If the determinant of P is 2, then the determinant of the matrix Q is

Let $$f(x) = \left\{ {\matrix{ {{x^2}\left| {\cos {\pi \over x}} \right|,} & {x \ne 0} \cr {0,} & {x = 0} \cr } } \right.$$

x$$\in$$R, then f is

The function $$f:[0,3] \to [1,29]$$, defined by $$f(x) = 2{x^3} - 15{x^2} + 36x + 1$$, is

The value of $$6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}...} } } } \right)$$ is __________.

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the xy-plane with centre at O and constant angular speed $$\omega$$. If the angular momentum of the system, calculated about O and P are denoted by $${\overrightarrow L _O}$$ and $${\overrightarrow L _P}$$, respectively, then

A biconvex lens is formed with two planoconvex lenses as shown in the figure. Refractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both curved surface are of the same radius of curvature R = 14 cm. For this biconvex lens, for an object distance of 40 cm, the image distance will be

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed $$\omega$$, as shown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v, with respect to the rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains $$\omega$$ throughout. The magnitude of the torque (|$$\tau$$|) about O, as a function of time is best represented by which plot ?

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductively. The first and third plates are maintained at temperatures 2T and 3T, respectively. The temperature of the middle (i.e. second) plate under steady state condition is

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular frequency $$\omega$$ = ($$\pi$$/3) rad/s. Simultaneously, at t = 0, a small pebble is projected with speed v from point P at an angle of 45$$^\circ$$ as shown in the figure. Point O is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is (take g = 10 m/s²)

Young's double slit experiment is carried out by using green, red and blue light, one colour at a time. The fringe widths recorded are $$\beta$$_G, $$\beta$$_R and $$\beta$$_B, respectively. Then,

Consider the motion of a positive point charge in a region, there are simultaneous uniform electric and magnetic fields $$\overrightarrow E = {E_0}\widehat j$$ and $$\overrightarrow B = {B_0}\widehat j$$. At time t = 0, this charge has velocity $$\overrightarrow v $$ in the xy-plane, making an angle $$\theta$$ with the x-axis. Which of the following option(s) is(are) correct for time t > 0 ?

A person blows into the open end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe,

A small block of mass 0.1 kg lies on a fixed inclined plane PQ which makes an angle $$\theta$$ with the horizontal. A horizontal force of 1 N acts on the block through its centre of mass as shown in the figure. The block remains stationary if (take g = 10 m/s²)

For the resistance network shown in the figure, choose the correct option(s).

A circular wire loop of radius R is placed in the xy plane centred at the origin O. A square loop of side a(a << R) having two turns is placed with its centre at z = $$\sqrt3$$R along the axis of the circular wire loop, as shown in the figure. The plane of the square loop makes an angle of 45$$^\circ$$ with respect to z-axis. If the mutual inductance between the loops is given by $${{{\mu _0}{a^2}} \over {{2^{p/2}}R}}$$, then the value of p is ___________.

An infinitely long solid cylinder of radius R has a uniform volume charge density $$\rho$$. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression $${{23\rho R} \over {16k{\varepsilon _0}}}$$. The value of k is _____________.

A proton is fired from very far away towards a nucleus with charge Q = 120e, where e is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of the proton at its start is ____________. (Take the proton mass, $${m_p} = (5 \times 3) \times {10^{ - 27}}$$ kg; $$h/e = 4.2 \times {10^{ - 15}}$$ J.s/C; $${1 \over {4\pi {\varepsilon _0}}} = 9 \times {10^9}$$ m/F; 1 fm = 10¹⁵ m.)

A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is I_O and I_P respectively. Both these axes are perpendicular to the plane of the lamina. The ratio I_O/I_P to the nearest integer is ____________.

A cylinder cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude of the magnetic field at the point P is given by $${N \over {12}}{\mu _0}aJ$$, then the value of N is ______________.