The term that correct for the attractive forces present in a real gas in the van der Waals equation is
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The term that correct for the attractive forces present in a real gas in the van der Waals equation is
The IUPAC name of the following compound is -
Given that the abundances of isotopes $$^{54}Fe,^{56}Fe$$ and $$^{57}Fe$$ are 5%,90% and 5% respectively, the atomic mass of Fe is -
Among cellulose, poly(vinyl chloride), nylon and natural rubber, the polymer in which the intermolecular force of attraction is weakest is -
The Henry's law constant for the solubility of $$N_{2}$$ gas in water at 298 K is $$1.0\times10^{5}$$ atm. The mole fraction of $$N_{2}$$ in air is 0.8. The number of moles of $$N_{2}$$ from air dissolved in 10 moles of water at 298 K and 5 atm pressure is -
The correct acidity order of the following is -
Among the electrolytes $$Na_{2}SO_{4},\ CaCl_{2},\ Al_{2}(SO_{4})_{3}$$ and $$NH_{4}Cl$$, the most effective coagulating agent for $$Sb_{2}S_{3}sol$$ is
The reaction of $$P_{4}$$ with X leads selectively to $$P_{4}O_{6}$$. The X is
The compound(s) formed upon combustion of sodium metal in excess air is(are)
The correct statement(s) regarding defects in solids is(are) -
The compound(s) that exhibit(s) geometrical isomerism is(are)
The correct statement(s) about the compound $$H_{3}C(HO)HC-CH=CH-CH(OH)CH_{3}(X)$$ is (are)
p-Amino-N, N-dimethylaniline is added to a strongly acidic solution of X. The resulting solution is treated with a few drops of aqueous solution of Y to yield blue coloration due to the formation of methylene blue. Treatment of the aqueous solution of Y with the reagent potassium hexacyanoferrate (II) leads to the formation of an intense blue precipitate. The precipitate dissolves on excess addition of the reagent. Similarly, treatment of the solution of Y with the solution of potassium hexacyanoferrate (III) leads to a brown coloration due to the formation of Z.
The compound X is
The compound Y is -
The compound Z is -
A carbonyl compound P, which gives positive iodoform test, undergoes reaction with MeMgBr followed by dehydration to give an olefin Q. Ozonolysis fo Q leads to a dicarbonyl compound R, Which undergoes intramolecular aldol reaction to give predominantly S.
The structure of the carbonyl compound P is -
The structures of the products Q and R, respectively, are -
The structure of the product S is -
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labeled A, B, C and D, while the statements in Column II are labeled p, q, r, s and t. any given statement in Column I can have correct matching with ONE OR MORE statements (s) in column II. The appropriate bubbled corresponding to the answers to these questions have to be darkened as illustrated in the following example :
If the correct matches are A - p, s and t; B - q and r; C - p and q; and D - s and t; then the correct darkening of bubbles will look like the following.
Match each of the diatomic molecules in Column I with its property/properties in Column II
Match each of the compounds in Column I with its characteristic reaction(s) in Column II.
Tangents drawn from the point P(1, 8) to the circle $$x^{2}+y^{2}-6x-4y-11=0$$ touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is
The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
Let P(3, 2, 6) be a point in space and Q be a point on the line $$\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(-3\hat{i}+\hat{j}+5\hat{k})$$ Then the value of $$\mu$$ for which the vector $$\overrightarrow{PQ}$$ is parallel to the plane $$x-4y+3z=1$$ is
Let $$Z=\cos\theta+i\ \sin\theta$$. Then the value of $$\sum_{m=1}^{15}Im(z^{2m-1})$$ at $$\theta=2^{0}$$ is
Let $$z=x+iy$$ be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation $$Z\overline{z}^{3}+\overline{z}Z^{3}=350$$ is
If $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ and $$\overrightarrow{d}$$ are unit vectors such that $$(\overrightarrow{a}\times\overrightarrow{b}).(\overrightarrow{c}\times\overrightarrow{d})=1$$ and $$\overrightarrow{a}.\overrightarrow{c}=\frac{1}{2}$$ then
Let f be a non-negative function defined on the interval $$[0,1]$$. If
$$\int_{0}^{x}\sqrt{1 - (f'(t))^2} dt = \int_{0}^{x} f(t)dt, 0 \leq x \leq 1$$, and f(0) = 0, then
The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse $$x^2 + 9y^2 = 9$$ meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is
In a triangle ABC with fixed base BC, the vertex A moves such that
$$\cos B + \cos C = 4 \sin^2 \frac{A}{2}$$
If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then
Area of the region bounded by the curve $$y = e^x$$ and line x = 0 and y = e is
Let
$$L = \lim_{x \rightarrow 0}\frac{a - \sqrt{a^2 - x^2} - \frac{x^4}{4}}{x^4}, a>0$$. If L is finite, then
If $$\frac{\sin^4x}{2}+\frac{\cos^4x}{3} = \frac{1}{5}$$ then
Let A be the set of all $$3 \times 3$$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices in A is
The number of matrices A in A for which the system of linear equations
$$A\begin{bmatrix}x\\y\\z \end{bmatrix} = \begin{bmatrix}1\\0\\0 \end{bmatrix}$$
has a unique solution, is
The number of matrices A in A for which the system of linear equations
$$A\begin{bmatrix}x\\y\\z \end{bmatrix} = \begin{bmatrix}1\\0\\0 \end{bmatrix}$$
is inconsistent, is
A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required.
The probability that X = 3 equals
The probability that $$X \geq 3$$ equals
The conditional probability that $$X \geq 6$$ given $$X > 3$$ equals
This section contains 2 questions. Each questions contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labeled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example:
If the correct matches are A - p, s and t; B - q and r; C -p and q ; and D -s and t; then the correct darkening of bubbles will look like the following.
Match the statements/ expressions in Column I with the open intervals in Column- II
Match the conics in Column I with the statements/ expressions in Column- II
A block of base 10 cm $$\times$$ 10 cm and height 15 cm is kept on an inclined plane. The coefficient of friction between them is $$\sqrt{3}$$. The inclination $$\theta$$ of this inclined plane from the horizontal plane is gradually increased from $$0^\circ$$. Then
The x-t graph of a particle undergoing simple harmonic is shown below. The acceleration of the particle at $$t = \frac{4}{3}s$$ is-
Three concentric metallic spherical shells of radii R, 2R, 3R are given charges $$Q_1, Q_2, Q_3$$ respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, $$Q_1 : Q_2 : Q_3$$ is-
A disk of radius $$\frac{a}{4}$$ having a uniformly distributed charge 6C is placed in the x-y plane with its centre at $$\left(\frac{-a}{2}, 0, 0\right)$$. A rod of length 'a' carrying a uniformly distributed charge 8C is placed on the x-axis from $$x = \frac{a}{4}$$ to $$x = \frac{5a}{4} $$. Two point charges -7C and 3C are placed at $$\left(\frac{a}{4}, \frac{-a}{4}, 0 \right)$$ and $$\left(\frac{-3a}{4}, \frac{3a}{4}, 0 \right)$$ respectively. Consider a cubical surface formed by six surfaces $$x = \pm \frac{a}{2}, y = \pm \frac{a}{2}, z = \pm \frac{a}{2}$$. The electric flux this cubical surface is-
Look at the drawing given in the figure which has been drawn with ink of uniform linethickness. The mass of ink used to draw each of the two inner circles, and each of the two lines segments is m. The mass of the ink used to draw the outer circle is 6m. The coordinates of the centres of the different parts are : outer circle (0, 0), left inner circle (-a, a), right inner circle (a, a), vertical line (0, 0) and horizontal line (0, -a). The y-coordinate of the centre of mass of the ink in this drawing is-
The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed in a perpendicular magnetic field in the direction going into the plane of the figure. The magnitude of the field increases with time. I1 and I2 are the currents in the segments ab and cd. Then
A ball is dropped from a height of 20 m above the surface of water in a lake. The refractive index of water is $$\frac{4}{3}$$. A fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is 12.8 m above the water surface, the fish sees the speed of ball as-
Two small particles of equal masses start moving in opposite directions from a point A in a horizontal circular orbit. Their tangential velocities are v and 2v, respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many collisions, other than that at A, these two particles will again reach the point A ?
For the circuit shown in the figure
If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that
A student performed the experiment of determination of focal length of a concave mirror by u-v method using an optical bench of length 1.5 meter. The focal length of the mirror used is 24 cm. The maximum error in the location of the image can be 0.2 cm. The 5 sets of (u, v) values recorded by the student (in cm) are : (42, 56), (48, 48), (60, 40), (66, 33), (78, 39). The data set(s) that cannot come from experiment and is(are) incorrectly recorded, is (are)
$$C_v$$ and $$C_p$$ denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then
Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, $$_{1}^{2}H$$, known as deuteron and denoted by D, can be thought of as a candidate for fusion reactor. The D-D reaction is $$_{1}^{2}H + _{1}^{2}H \rightarrow _{2}^{3}He + n +$$ energy. In the core of fusion reactor, a gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of $$_{1}^{2}H$$ nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time $$t_0$$ before the particles fly away from the core. If n is the density (number/volume) of deuterons, the product nt0 is called Lawson number. In one of the criteria, a reactor is termed successful if Lawson number is greater than $$5 \times 10^{14}$$ s/cm$$^3$$.
It may be helpful to use the following. Boltzmann constant $$k = 8.6 \times 10^{-5}$$eV/K; $$\frac{e^2}{4 \pi \varepsilon_0} = 1.44 \times 10^{-9}$$ eVm.
In the core of nuclear fusion reactor, the gas becomes plasma because of -
Assume that two deuteron nuclei in the core of fusion reactor at temperature T are moving towards each other, each with kinetic energy 1.5 kT, when the separation between them is large enough to neglect Coulomb potential energy. Also neglect any interaction from other particles in the core. The minimum temperature T required for them to reach a separation of $$4 \times 10^{-15}$$ m is in the range-
Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion ?
When a particle is restricted to move along x-axis between x = 0 and x = a, where a is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends x = 0 and x = a. The wavelength of this standing wave is related to the linear momentum p of the particle according to the de Broglie relation. The energy of the particle of mass m is related to its linear momentum as $$E = \frac{p^2}{2m}$$. Thus, the energy of the particle can be denoted by a quantum number 'n' taking values 1, 2, 3, . . . . . (n = 1, called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line x = 0 to
$$x = a$$. Take $$h = 6.6 \times 10^{-34}$$J-s and $$e = 1.6 \times 10^{-19}$$ C.
The allowed energy for the particle for a particular value of n is proportional to -
If the mass of the particle is $$m = 1.0 \times 10^{-30}$$ kg and a = 6.6 nm, the energy of the particle in its ground state is closest to -
The speed of the particle, that can take discrete values, is proportional to -
This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example.
If the correct matches are A - p, s and t; B - q and r; C - p and q; and D - s and t; then the correct darkening of bubbles will look like the following.
Six point charges, each of the same magnitude q, are arranged in different manners as shown in Column II. In each case, a point M and a line PQ passing through M are shown. Let E be the electric field and V be the electric potential at M (potential at infinity is zero) due to the given charge distribution when it is at rest. Now, the whole system is set into rotation with a constant angular velocity about the line PQ. Let B be the magnetic field at M and $$\mu$$ be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current.
Column II shows five systems in which two object are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II.
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