The respective number of significant figures for the numbers $$23.023$$, $$0.0003$$ and $$2.1 \times 10^{-3}$$ are
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The respective number of significant figures for the numbers $$23.023$$, $$0.0003$$ and $$2.1 \times 10^{-3}$$ are
A particle is moving with velocity $$\vec{v} = K(y\hat{i} + x\hat{j})$$, where $$K$$ is a constant. The general equation for its path is
A small particle of mass $$m$$ is projected at an angle $$\theta$$ with the x-axis with an initial velocity $$v_0$$ in the $$x-y$$ plane as shown in the figure. At a time $$t < \frac{v_0 \sin\theta}{g}$$, the angular momentum of the particle is where $$\hat{i}, \hat{j}$$ and $$\hat{k}$$ are unit vectors along x, y and z-axis respectively.
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Two fixed frictionless inclined plane making an angle $$30^\circ$$ and $$60^\circ$$ with the vertical are shown in the figure. Two block $$A$$ and $$B$$ are placed on the two planes. What is the relative vertical acceleration of $$A$$ with respect to $$B$$?
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The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $$U(x) = \frac{a}{x^{12}} - \frac{b}{x^6}$$, where $$a$$ and $$b$$ are constants and $$x$$ is the distance between the atoms. If the dissociation energy of the molecule is $$D = [U(x = \infty) - U_{\text{at equilibrium}}]$$, $$D$$ is
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Out of the four choices given after the statements, choose the one that best describes the two statements.
Statement-1 : Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.
Statement-2 : Principle of conservation of momentum holds true for all kinds of collisions.
The figure shows the position - time $$(x - t)$$ graph of one-dimensional motion of a body of mass $$0.4$$ kg. The magnitude of each impulse is
A point P moves in counter-clockwise direction on a circular path as shown in the figure. The movement of 'P' is such that it sweeps out a length $$s = t^3 + 5$$, where $$s$$ is in metres and $$t$$ is in seconds. The radius of the path is $$20$$ m. The acceleration of 'P' when $$t = 2$$ s is nearly
For a particle in uniform circular motion the acceleration $$\vec{a}$$ at a point $$P(R, \theta)$$ on the circle of radius R is (here $$\theta$$ is measured from the $$x$$-axis)
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A ball is made of a material of density $$\rho$$ where $$\rho_{\text{oil}} < \rho < \rho_{\text{water}}$$ with $$\rho_{\text{oil}}$$ and $$\rho_{\text{water}}$$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?
Two conductors have the same resistance at $$0^\circ$$C but their temperature coefficients of resistance are $$\alpha_1$$ and $$\alpha_2$$. The respective temperature coefficients of their series and parallel combinations are nearly
A diatomic ideal gas is used in a Car engine as the working substance. If during the adiabatic expansion part of the cycle, volume of the gas increases from $$V$$ to $$32V$$ the efficiency of the engine is
The equation of a wave on a string of linear mass density $$0.04 \text{ kg m}^{-1}$$ is given by $$y = 0.02(\text{m}) \sin\left[2\pi\left(\frac{t}{0.04(\text{s})} - \frac{x}{0.50(\text{m})}\right)\right]$$. The tension in the string is
A thin semi-circular ring of radius $$r$$ has a positive charge $$q$$ distributed uniformly over it. The net field $$\vec{E}$$ at the centre $$O$$ is
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Let there be a spherically symmetric charge distribution with charge density varying as $$\rho(r) = \rho_0 \left(\frac{5}{4} - \frac{r}{R}\right)$$ upto $$r = R$$, and $$\rho(r) = 0$$ for $$r > R$$, where $$r$$ is the distance from the origin. The electric field at a distance $$r$$ ($$r < R$$) from the origin is given by
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Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $$30^\circ$$ with each other. When suspended in a liquid of density $$0.8$$ g cm$$^{-3}$$, the angle remains the same. If density of the material of the sphere is $$16$$ g cm$$^{-3}$$, the dielectric constant of the liquid is
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Let $$C$$ be the capacitance of a capacitor discharging through a resistor R. Suppose $$t_1$$ is the time taken for the energy stored in the capacitor to reduce to half its initial value and $$t_2$$ is the time taken for the charge to reduce to one-fourth its initial value. Then the ratio $$t_1/t_2$$ will be
Two long parallel wires are at a distance $$2d$$ apart. They carry steady equal current flowing out of the plane of the paper as shown. The variation of the magnetic field along the line $$XX'$$ is given by
A rectangular loop has a sliding connector PQ of length $$\ell$$ and resistance $$R\Omega$$ and it is moving with a speed $$v$$ as shown. The set-up is placed in a uniform magnetic field going into the plane of the paper. The three currents $$I_1, I_2$$ and $$I$$ are
In the circuit shown below, the key K is closed at $$t = 0$$. The current through the battery is
In a series LCR circuit $$R = 200\,\Omega$$ and the voltage and the frequency of the main supply is $$220$$ V and $$50$$ Hz respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $$30^\circ$$. On taking out the inductor from the circuit the current leads the voltage by $$30^\circ$$. The power dissipated in the LCR circuit is
If a source of power $$4$$ kW produces $$10^{20}$$ photons/second, the radiation belong to a part of the spectrum called
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An initially parallel cylindrical beam travels in a medium of refractive index $$\mu(I) = \mu_0 + \mu_2 I$$, where $$\mu_0$$ and $$\mu_2$$ are positive constants and $$I$$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. As the beam enters the medium, it will
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An initially parallel cylindrical beam travels in a medium of refractive index $$\mu(I) = \mu_0 + \mu_2 I$$, where $$\mu_0$$ and $$\mu_2$$ are positive constants and $$I$$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The initial shape of the wave front of the beam is
An initially parallel cylindrical beam travels in a medium of refractive index $$\mu(I) = \mu_0 + \mu_2 I$$, where $$\mu_0$$ and $$\mu_2$$ are positive constants and $$I$$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The speed of light in the medium is
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Statement-1 : When ultraviolet light is incident on a photocell, its stopping potential is $$V_0$$ and the maximum kinetic energy of the photoelectrons is $$K_{\max}$$. When the ultraviolet light is replaced by X-rays, both $$V_0$$ and $$K_{\max}$$ increase. Statement-2 : Photoelectrons are emitted with speeds ranging from zero to a maximum value because of the range of frequencies present in the incident light. Of the four choices given after the statements, choose the one that best describes the two statements.
A nucleus of mass $$M + \Delta m$$ is at rest and decays into two daughter nuclei of equal mass $$\frac{M}{2}$$ each. Speed of light is $$c$$. The binding energy per nucleon for the parent nucleus is $$E_1$$ and that for the daughter nuclei is $$E_2$$. Then
A nucleus of mass $$M + \Delta m$$ is at rest and decays into two daughter nuclei of equal mass $$\frac{M}{2}$$ each. Speed of light is $$c$$. The speed of daughter nuclei is
A radioactive nucleus (initial mass number $$A$$ and atomic number $$Z$$) emits 3 $$\alpha$$-particles and 2 positrons. The ratio of number of neutrons to that of protons in the final nucleus will be
The combination of gates shown below yields
Ionisation energy of $$\text{He}^+$$ is $$19.6 \times 10^{-18}$$ Jatom$$^{-1}$$. The energy of the first stationary state ($$n = 1$$) of $$\text{Li}^{2+}$$ is
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The correct sequence which shows decreasing order of the ionic radii of the elements is
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The standard enthalpy of formation of $$\text{NH}_3$$ is $$-46.0$$ kJ mol$$^{-1}$$. If the enthalpy of formation of $$\text{H}_2$$ from its atoms is $$-436$$ kJ mol$$^{-1}$$ and that of $$\text{N}_2$$ is $$-712$$ kJ mol$$^{-1}$$, the average bond enthalpy of N$$-$$H bond in $$\text{NH}_3$$ is
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The energy required to break one mole of Cl$$-$$Cl bonds in $$\text{Cl}_2$$ is $$242$$ kJ mol$$^{-1}$$. The longest wavelength of light capable of breaking a single Cl$$-$$Cl bond is ($$c = 3 \times 10^8$$ ms$$^{-1}$$ and $$N_A = 6.02 \times 10^{23}$$ mol$$^{-1}$$)
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For a particular reversible reaction at temperature $$T$$, $$\Delta H$$ and $$\Delta S$$ were found to be both $$+ve$$. If $$T_e$$ is the temperature at equilibrium, the reaction would be spontaneous when
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In aqueous solution the ionization constants for carbonic acid are $$K_1 = 4.2 \times 10^{-7}$$ and $$K_2 = 4.8 \times 10^{-11}$$. Select the correct statement for a saturated $$0.034$$M solution of the carbonic acid.
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Solubility product of silver bromide is $$5.0 \times 10^{-13}$$. The quantity of potassium bromide (molar mass taken as $$120$$ g mol$$^{-1}$$) to be added to $$1$$ litre of $$0.05$$ M solution of silver nitrate to start the precipitation of AgBr is
At $$25^\circ$$C, the solubility product of $$\text{Mg(OH)}_2$$ is $$1.0 \times 10^{-11}$$. At which pH, will $$\text{Mg}^{2+}$$ ions start precipitating in the form of $$\text{Mg(OH)}_2$$ from a solution of $$0.001$$M $$\text{Mg}^{2+}$$ ions?
$$29.5$$ mg of an organic compound containing nitrogen was digested according to Kjeldahl's method and the evolved ammonia was absorbed in $$20$$ mL of $$0.1$$ M HCl solution. The excess of the acid required $$15$$ mL of $$0.1$$ M NaOH solution for complete neutralization. The percentage of nitrogen in the compound is
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The correct order of increasing basicity of the given conjugate bases ($$R = CH_3$$) is
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Out of the following, the alkene that exhibits optical isomerism is
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The main product of the following reaction is $$\text{C}_6\text{H}_5\text{CH}_2\text{CH(OH)CH(CH}_3)_2 \xrightarrow{\text{conc. H}_2\text{SO}_4} ?$$
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The edge length of a face centered cubic cell of an ionic substance is $$508$$ pm. If the radius of the cation is $$110$$ pm, the radius of the anion is
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Percentage of free space in cubic close packed structure and in body centred packed structure are respectively
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If $$10^{-4}$$ dm$$^3$$ of water is introduced into a $$1.0$$ dm$$^3$$ flask at $$300$$ K, how many moles of water are in the vapour phase when equilibrium is established? (Given: Vapour pressure of $$\text{H}_2\text{O}$$ at $$300$$ K is $$3170$$ Pa; $$R = 8.314$$ J K$$^{-1}$$ mol$$^{-1}$$)
If sodium sulphate is considered to be completely dissociated into cations and anions in aqueous solution, the change in freezing point of water $$(\Delta T_f)$$, when $$0.01$$ mol of sodium sulphate is dissolved in $$1$$ kg of water, is ($$K_f = 1.86$$ K kg mol$$^{-1}$$)
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On mixing, heptane and octane form an ideal solution. At $$373$$ K, the vapour pressures of the two liquid components (heptane and octane) are $$105$$ kPa and $$45$$ kPa respectively. Vapour pressure of the solution obtained by mixing $$25.0$$ g of heptane and $$35$$ g of octane will be (molar mass of heptane $$= 100$$ g mol$$^{-1}$$ and of octane $$= 114$$ g mol$$^{-1}$$).
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The Gibbs energy for the decomposition of $$\text{Al}_2\text{O}_3$$ at $$500^\circ$$C is as follows: $$\frac{2}{3} \text{Al}_2\text{O}_3 \rightarrow \frac{4}{3} \text{Al} + \text{O}_2$$, $$\Delta_r G = +966$$ kJ mol$$^{-1}$$. The potential difference needed for electrolytic reduction of $$\text{Al}_2\text{O}_3$$ at $$500^\circ$$C is at least
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The correct order of $$E^0_{M^{2+}/M}$$ values with negative sign for the four successive elements Cr, Mn, Fe and Co is
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The time for half life period of a certain reaction $$A \to$$ products is $$1$$ hour. When the initial concentration of the reactant '$$A$$' is $$2.0$$ mol L$$^{-1}$$, how much time does it take for its concentration to come from $$0.50$$ to $$0.25$$ mol L$$^{-1}$$ if it is a zero order reaction?
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Consider the reaction: $$\text{Cl}_2(\text{aq}) + \text{H}_2\text{S}(\text{aq}) \to \text{S}(\text{s}) + 2\text{H}^+(\text{aq}) + 2\text{Cl}^-(\text{aq})$$. The rate equation for this reaction is rate $$= k[\text{Cl}_2][\text{H}_2\text{S}]$$. Which of these mechanisms is/are consistent with this rate equation? (A) $$\text{Cl}_2 + \text{H}_2\text{S} \to \text{H}^+ + \text{Cl}^- + \text{Cl}^+ + \text{HS}^-$$ (slow); $$\text{Cl}^+ + \text{HS}^- \to \text{H}^+ + \text{Cl}^- + \text{S}$$ (fast). (B) $$\text{H}_2\text{S} \leftrightarrow \text{H}^+ + \text{HS}^-$$ (fast equilibrium); $$\text{Cl}_2 + \text{HS}^- \to 2\text{Cl}^- + \text{H}^+ + \text{S}$$ (slow).
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Three reactions involving $$\text{H}_2\text{PO}_4^-$$ are given below: (i) $$\text{H}_3\text{PO}_4 + \text{H}_2\text{O} \to \text{H}_3\text{O}^+ + \text{H}_2\text{PO}_4^-$$ (ii) $$\text{H}_2\text{PO}_4^- + \text{H}_2\text{O} \to \text{HPO}_4^{2-} + \text{H}_3\text{O}^+$$ (iii) $$\text{H}_2\text{PO}_4^- + \text{OH}^- \to \text{H}_3\text{PO}_4 + \text{O}^{2-}$$. In which of the above does $$\text{H}_2\text{PO}_4^-$$ act as an acid?
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A solution containing $$2.675$$ g of $$\text{CoCl}_3 \cdot 6\text{NH}_3$$ (molar mass $$= 267.5$$ g mol$$^{-1}$$) is passed through a cation exchanger. The chloride ions obtained in solution were treated with excess of $$\text{AgNO}_3$$ to give $$4.78$$ g of AgCl (molar mass $$= 143.5$$ g mol$$^{-1}$$). The formula of the complex is (At. Mass of Ag $$= 108$$ u)
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Which one of the following has an optical isomer? (en = ethylenediamine)
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Consider the following bromides: 
The correct order of $$S_N1$$ reactivity is
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From amongst the following alcohols the one that would react fastest with conc. HCl and anhydrous $$\text{ZnCl}_2$$, is
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One mole of a symmetrical alkene on ozonolysis gives two moles of an aldehyde having a molecular mass of $$44$$ u. The alkene is
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In the chemical reactions shown below, the compounds 'A' and 'B' respectively are
The polymer containing strong intermolecular forces e.g. hydrogen bonding, is
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Biuret test is not given by
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If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 - x + 1 = 0$$, then $$\alpha^{2009} + \beta^{2009} =$$
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The number of complex numbers $$z$$ such that $$|z - 1| = |z + 1| = |z - i|$$ equals
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There are two urns. Urn A has $$3$$ distinct red balls and urn B has $$9$$ distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is
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A person is to count $$4500$$ currency notes. Let $$a_n$$ denote the number of notes he counts in the $$n^{\text{th}}$$ minute. If $$a_1 = a_2 = \ldots = a_{10} = 150$$ and $$a_{10}, a_{11}, \ldots$$ are in A.P. with common difference $$-2$$, then the time taken by him to count all notes is
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Let $$S_1 = \sum_{j=1}^{10} j(j-1)\,^{10}C_j$$, $$S_2 = \sum_{j=1}^{10} j\,^{10}C_j$$ and $$S_3 = \sum_{j=1}^{10} j^2\,^{10}C_j$$. Statement-1: $$S_3 = 55 \times 2^9$$. Statement-2: $$S_1 = 90 \times 2^8$$ and $$S_2 = 10 \times 2^8$$.
Let $$\cos(\alpha + \beta) = \frac{4}{5}$$ and let $$\sin(\alpha - \beta) = \frac{5}{13}$$, where $$0 \le \alpha, \beta \le \frac{\pi}{4}$$, then $$\tan 2\alpha =$$
The line $$L$$ given by $$\frac{x}{5} + \frac{y}{b} = 1$$ passes through the point $$(13, 32)$$. The line $$K$$ is parallel to $$L$$ and has the equation $$\frac{x}{c} + \frac{y}{3} = 1$$. Then the distance between $$L$$ and $$K$$ is
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The circle $$x^2 + y^2 = 4x + 8y + 5$$ intersects the line $$3x - 4y = m$$ at two distinct points if
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If two tangents drawn from a point $$P$$ to the parabola $$y^2 = 4x$$ are at right angles, then the locus of $$P$$ is
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Let $$f: R \to R$$ be a positive increasing function with $$\lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1$$. Then $$\lim_{x \to \infty} \frac{f(2x)}{f(x)} =$$
For two data sets, each of size $$5$$, the variances are given to be $$4$$ and $$5$$ and the corresponding means are given to be $$2$$ and $$4$$, respectively. The variance of the combined data set is
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For a regular polygon, let $$r$$ and $$R$$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is
Let $$S$$ be a non-empty subset of $$R$$. Consider the following statement: P: There is a rational number $$x \in S$$ such that $$x > 0$$. Which of the following statements is the negation of the statement $$P$$?
Consider the following relations: $$R = \{(x, y) \mid x, y \text{ are real numbers and } x = wy \text{ for some rational number } w\}$$; $$S = \left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p \text{ and } q \text{ are integers such that } n, q \ne 0 \text{ and } qm = pn\right\}$$. Then
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The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is
Let $$A$$ be a $$2 \times 2$$ matrix with non-zero entries and let $$A^2 = I$$, where $$I$$ is $$2 \times 2$$ identity matrix. Define $$\text{Tr}(A) = $$ sum of diagonal elements of $$A$$ and $$|A| = $$ determinant of matrix $$A$$. Statement-1: $$\text{Tr}(A) = 0$$. Statement-2: $$|A| = 1$$.
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Consider the system of linear equations: $$x_1 + 2x_2 + x_3 = 3$$; $$2x_1 + 3x_2 + x_3 = 3$$; $$3x_1 + 5x_2 + 2x_3 = 1$$. The system has
Let $$f: R \to R$$ be a continuous function defined by $$f(x) = \frac{1}{e^x + 2 e^{-x}}$$. Statement-1: $$f(c) = \frac{1}{3}$$, for some $$c \in R$$. Statement-2: $$0 < f(x) \le \frac{1}{2\sqrt{2}}$$, for all $$x \in R$$.
Let $$f: (-1, 1) \to R$$ be a differentiable function with $$f(0) = -1$$ and $$f'(0) = 1$$. Let $$g(x) = [f(2f(x) + 2)]^2$$. Then $$g'(0) =$$
The equation of the tangent to the curve $$y = x + \frac{4}{x^2}$$, that is parallel to the $$x$$-axis, is
Let $$f: R \to R$$ be defined by $$f(x) = \begin{cases} k - 2x, & \text{if } x \le -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$$. If $$f$$ has a local minimum at $$x = -1$$, then a possible value of $$k$$ is
Let $$p(x)$$ be a function defined on $$R$$ such that $$p'(x) = p'(1 - x)$$, for all $$x \in [0, 1]$$, $$p(0) = 1$$ and $$p(1) = 41$$. Then $$\int_0^1 p(x) \, dx$$ equals
The area bounded by the curves $$y = \cos x$$ and $$y = \sin x$$ between the ordinates $$x = 0$$ and $$x = \frac{3\pi}{2}$$ is
Solution of the differential equation $$\cos x \, dy = y(\sin x - y)\,dx$$, $$0 < x < \frac{\pi}{2}$$ is
Let $$\vec{a} = \hat{j} - \hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} - \hat{k}$$. Then vector $$\vec{b}$$ satisfying $$\vec{a} \times \vec{b} + \vec{c} = \vec{0}$$ and $$\vec{a} \cdot \vec{b} = 3$$ is
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If the vectors $$\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$$, $$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$$ and $$\vec{c} = \lambda \hat{i} + \hat{j} + \mu \hat{k}$$ are mutually orthogonal, then $$(\lambda, \mu) =$$
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Statement-1: The point $$A(3, 1, 6)$$ is the mirror image of the point $$B(1, 3, 4)$$ in the plane $$x - y + z = 5$$.
Statement-2: The plane $$x - y + z = 5$$ bisects the line segment joining $$A(3, 1, 6)$$ and $$B(1, 3, 4)$$.
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A line $$AB$$ in three-dimensional space makes angles $$45^\circ$$ and $$120^\circ$$ with the positive $$x$$-axis and the positive $$y$$-axis respectively. If $$AB$$ makes an acute angle $$\theta$$ with the positive $$z$$-axis, then $$\theta$$ equals
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Four numbers are chosen at random (without replacement) from the set $$\{1, 2, 3, \ldots, 20\}$$. Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $$\frac{1}{85}$$. Statement-2: If the four chosen numbers form an AP, then the set of all possible values of common difference is $$\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 5\}$$.
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An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is
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