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A physical quantity $$z$$ depends on four observables $$a$$, $$b$$, $$c$$ and $$d$$, as $$z = \frac{a^2 b^{2/3}}{\sqrt{cd^3}}$$. The percentage 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:
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A balloon is moving up in air vertically above a point $$A$$ on the ground. When it is at a height $$h_1$$, a girl standing at a distance $$d$$ (point B) from $$A$$ (see figure) sees it at an angle $$45^\circ$$ with respect to the vertical. When the balloon climbs up a further height $$h_2$$, it is seen at an angle $$60^\circ$$ with respect to the vertical if the girl moves further by a distance $$2.464\,d$$ (point C). Then the height $$h_2$$ is (given $$\tan 30^\circ = 0.5774$$):
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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]:
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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:
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The value of the acceleration due to gravity is $$g_1$$ at a height $$h = \frac{R}{2}$$ ($$R$$ = radius of the earth) from the surface of the earth. It is again equal to $$g_1$$ at a depth $$d$$ below the surface the earth. The ratio $$\left(\frac{d}{R}\right)$$ equals:
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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 $$\frac{27}{8}$$ with respect to water, the value of $$r$$ is:
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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 processes 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:
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Number of molecules in a volume of $$4\,\text{cm}^3$$ of a perfect monoatomic gas at some temperature T and at a pressure of $$2\,\text{cm}$$ of mercury is close to? (Given, mean kinetic energy of a molecule (at T) is $$4 \times 10^{-14}\,\text{erg}$$, $$g = 980\,\text{cm s}^{-2}$$, density of mercury $$= 13.6\,\text{g cm}^{-3}$$)
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A bullet of mass $$5\,\text{gram}$$, travelling with a speed of $$210\,\text{m s}^{-1}$$, strikes a fixed wooden target. One half of its kinetic energy is converted into heat in the wood. The rise of temperature of the bullet if the specific heat of its material is $$0.030\,(\text{gram}\,^\circ\text{C})^{-1}$$ ($$1\,\text{calorie} = 4.2 \times 10^7\,\text{ergs}$$) close to:
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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 $$\Delta p \sim 10\,\text{Pa}$$, $$v \sim 300\,\text{m/s}$$, $$\rho \sim 1\,\text{kg/m}^3$$, $$f \sim 1000\,\text{Hz}$$, then $$s$$ will be of the order of (take the multiplicative constant to be 1):
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In a resonance tube experiment when the tube is filled with water up to a height of $$17.0\,\text{cm}$$, 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 $$24.5\,\text{cm}$$. If the velocity of sound in air is $$330\,\text{m s}^{-1}$$, the tuning fork frequency is:
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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)
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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:
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A galvanometer of resistance $$G$$ is converted into a voltmeter of range $$0 - 1\,\text{V}$$ by connecting a resistance $$R$$ in series with it. The additional resistance that should be connected in series with $$R_1$$ to increase the range of the voltmeter to $$0 - 2\,\text{V}$$ will be:
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An electrical power line, having a total resistance of $$2\,\Omega$$, delivers $$1\,\text{kW}$$ at $$220\,\text{V}$$. The efficiency of the transmission line is approximately:
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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 $$\vec{E} = 30\hat{j}\sin(1.5 \times 10^7 t - 5 \times 10^{-2}x)\,\text{V m}^{-1}$$, where $$t$$ in in seconds and $$x$$ is in meters. The maximum magnetic force experienced by the electron will be: (given $$c = 3 \times 10^8\,\text{m s}^{-1}$$ and electron charge $$= 1.6 \times 10^{-19}\,\text{Coulombs}$$)
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A square loop of side $$2a$$, 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 $$z$$-axis is given by:
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For a concave lens of focal length $$f$$, the relation between object and image distance $$u$$ and $$v$$, respectively, from its pole can best be represented by ($$u = v$$ is the reference line):
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Activities of three radioactive substances $$A$$, $$B$$ and $$C$$ are represented by the curves $$A$$, $$B$$ and $$C$$, in the figure. Then their half-lives $$T_{1/2}(A) : T_{1/2}(B) : T_{1/2}(C)$$ are in the ratio:
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With increasing biasing voltage of a photo diode, the photocurrent magnitude:
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A particle of mass $$200\,\text{MeV c}^{-2}$$ 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 $$\frac{N}{4}$$. The value of $$N$$ is: (Given the mass of the hydrogen atom to be $$1\,\text{GeV c}^{-2}$$)........
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A force $$\vec{F} = \left(\hat{i} + 2\hat{j} + 3\hat{k}\right)\,\text{N}$$ acts at a point $$\left(4\hat{i} + 3\hat{j} - \hat{k}\right)\,\text{m}$$. Then the magnitude of torque about the point $$\left(\hat{i} + 2\hat{j} + \hat{k}\right)\,\text{m}$$ will be $$\sqrt{x}\,\text{N-m}$$. The value of $$x$$ is..........
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Two concentric circular coils, $$C_1$$ and $$C_2$$, are placed in the $$XY$$ plane. $$C_1$$ has 500 turns, and a radius of $$1\,\text{cm}$$. $$C_2$$ has 200 turns and radius of $$20\,\text{cm}$$. $$C_2$$ carries a time dependent current $$I(t) = (5t^2 - 2t + 3)\,\text{A}$$ where $$t$$ is in s. The emf induced in $$C_1$$ (in mV) at the instant $$t = 1\,\text{s}$$ is $$\frac{4}{x}$$. The value of $$x$$ is..........
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A compound microscope consists of an objective lens of focal length $$1\,\text{cm}$$ and an eye piece of focal length $$5\,\text{cm}$$ with a separation of $$10\,\text{cm}$$. The distance between an object and the objective lens, at which the strain on the eye is minimum is $$\frac{n}{40}\,\text{cm}$$. The value of $$n$$ is.......
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A beam of electrons of energy $$E$$ scatters from a target having atomic spacing of $$1\,\text{\AA}$$. The first maximum intensity occurs at $$\theta = 60^\circ$$. Then $$E$$ (in eV) is......... (Planck's constant $$h = 6.64 \times 10^{-34}\,\text{Js}$$, $$1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$$, electron mass $$m = 9.1 \times 10^{-31}\,\text{kg}$$)
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The difference between the radii of 3rd and 4th orbits of $$\text{Li}^{2+}$$ is $$\Delta R_1$$. The difference between the radii of 3rd and 4th orbits of $$\text{He}^+$$ is $$\Delta R_2$$. Ratio $$\Delta R_1 : \Delta R_2$$ is:
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In the sixth period, the orbitals that are filled are:
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The potential energy curve for the $$\text{H}_2$$ molecule as a function of internuclear distance is:
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Consider the following reaction:
$$\text{N}_2\text{O}_4(g) = 2\text{NO}_2(g);\;\Delta H^0 = +58\,\text{k}$$
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 $$\text{N}_2$$ at constant T.
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The equation that represents the water-gas shift reaction is:
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The increasing order of the acidity of the $$\alpha$$-hydrogen of the following compounds is:
In the following reaction sequence the major products A and B are:
The condition that indicates a polluted environment is:
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A diatomic molecule $$\text{X}_2$$ has a body-centred cubic (bcc) structure with a cell edge of $$300\,\text{pm}$$. The density of the molecule is $$6.17\,\text{g cm}^{-3}$$. The number of molecules present in $$200\,\text{g}$$ of $$\text{X}_2$$ is: (Avogadro constant $$(N_A) = 6 \times 10^{23}\,\text{mol}^{-1}$$)
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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 $$300\,\text{s}$$ and $$180\,\text{s}$$, 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) is: (Use $$\ln 2 = 0.693$$)
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Identify the correct molecular picture showing what happens at the critical micellar concentration (CMC) of an aqueous solution of a surfactant.
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An Ellingham diagram provides information about:
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The structure of $$\text{PCl}_5$$ in the solid state is:
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The correct electronic configuration and spin-only magnetic moment (BM) of $$\text{Gd}^{3+}$$ ($$Z = 64$$), respectively, are:
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The values of the crystal field stabilization energies for a high spin $$d^6$$ metal ion in octahedral and tetrahedral fields, respectively, are:
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Which of the following derivatives of alcohols is unstable in an aqueous base?
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The most appropriate reagent for conversion of $$\text{C}_2\text{H}_5\text{CN}$$ into $$\text{CH}_3\text{CH}_2\text{CH}_2\text{NH}_2$$ is:
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The increasing order of basicity of the following compounds is:
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If a person is suffering from the deficiency of nor-adrenaline, what kind of drug can be suggested?
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Which of the following is not an essential amino acid?
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A soft drink was bottled with a partial pressure of $$\text{CO}_2$$ of $$3\,\text{bar}$$ over the liquid at room temperature. The partial pressure of $$\text{CO}_2$$ over the solution approaches a value of $$30\,\text{bar}$$ when $$44\,\text{g}$$ of $$\text{CO}_2$$ is dissolved in $$1\,\text{kg}$$ of water at room temperature. The approximate pH of the soft drink is __________ $$\times 10^{-1}$$. (First dissociation constant of $$\text{H}_2\text{CO}_3 = 4.0 \times 10^{-7}$$; $$\log 2 = 0.3$$; density of the soft drink $$= 1\,\text{g mL}^{-1}$$)
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The minimum number of moles of $$\text{O}_2$$ required for complete combustion of 1 mole of propane and 2 moles of butane is...............
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An oxidation-reduction reaction in which 3 electrons are transferred has a $$\Delta G^0$$ of $$17.37\,\text{kJ mol}^{-1}$$ at $$25^\circ\text{C}$$. The value of $$E^0_{\text{cell}}$$ (in V) is __________ $$\times 10^{-2}$$. ($$1\text{F} = 96500\,\text{C mol}^{-1}$$)
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The total number of coordination sites in ethylenediaminetetraacetate ($$\text{EDTA}^{4-}$$) is..................
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The number of chiral carbon(s) present in peptide, Ile-Arg-Pro, is...............
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The product of the roots of the equation $$9x^2 - 18|x| + 5 = 0$$ is:
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If the four complex numbers $$z$$, $$\overline{z}$$, $$\overline{z} - 2\,\text{Re}(\overline{z})$$ and $$z - 2\,\text{Re}(z)$$ represent the vertices of a square of side 4 units in the Argand plane, then $$|z|$$ is equal to:
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If $$2^{10} + 2^9 \cdot 3^1 + 2^8 \cdot 3^2 + \ldots + 2 \cdot 3^9 + 3^{10} = S - 2^{11}$$, then $$S$$ is equal to:
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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 term of this A.P. is:
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If the common tangent to the parabolas, $$y^2 = 4x$$ and $$x^2 = 4y$$ also touches the circle, $$x^2 + y^2 = c^2$$, then $$c$$ is equal to:
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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, $$9x^2 + 16y^2 = 144$$, then $$PA + PB$$ is equal to:
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If the point $$P$$ on the curve, $$4x^2 + 5y^2 = 20$$ is farthest from the point $$Q(0, -4)$$, then $$PQ^2$$ is equal to:
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If $$\alpha$$ is the positive root of the equation, $$p(x) = x^2 - x - 2 = 0$$, then $$\lim_{x \to \alpha^+} \frac{\sqrt{1 - \cos(p(x))}}{x + \alpha - 4}$$ is equal to:
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The negation of the Boolean expression $$x \leftrightarrow \sim y$$ is equivalent to:
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The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14 then the absolute difference of the remaining two observations is:
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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:
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If the minimum and the maximum values of the function $$f : \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \to R$$, defined by$$f(\theta) = \begin{vmatrix} -\sin^2\theta & -1 - \sin^2\theta & 1 \\ -\cos^2\theta & -1 - \cos^2\theta & 1 \\ 12 & 10 & -2 \end{vmatrix}$$ are $$m$$ and $$M$$ respectively, then the ordered pair $$(m, M)$$ is equal to:
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Let $$\lambda \in \mathbb{R}$$. The system of linear equations
$$2x_1 - 4x_2 + \lambda x_3 = 1$$
$$x_{1} - 6x_{2} + x_{3} = 2$$
$$\lambda x_1 - 10x_2 + 4x_3 = 3$$
is inconsistent for:
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If $$S$$ is the sum of the first 10 terms of the series, $$\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) + \tan^{-1}\left(\frac{1}{21}\right) + \ldots$$, then $$\tan(S)$$ is equal to:
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If the function $$f(x) = \begin{cases} k_1(x - \pi)^2 - 1, & x \leq \pi \\ k_2 \cos x, & x > \pi \end{cases}$$ is twice differentiable, then the ordered pair $$(k_1, k_2)$$ is equal to:
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If $$\int (e^{2x} + 2e^x - e^{-x} - 1)e^{(e^x + e^{-x})}\,dx = g(x)e^{(e^x + e^{-x})} + c$$, where $$c$$ is a constant of integration, then $$g(0)$$ is:
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The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{1 + e^{\sin x}}\,dx$$ is:
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If $$y = y(x)$$ is the solution of the differential equation $$\frac{5 + e^x}{2 + y} \cdot \frac{dy}{dx} + e^x = 0$$ satisfying $$y(0) = 1$$, then value of $$y(\log_e 13)$$ is:
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If the volume of a parallelepiped, whose coterminous edges are given by the vectors $$\vec{a} = \hat{i} + \hat{j} + n\hat{k}$$, $$\vec{b} = 2\hat{i} + 4\hat{j} - n\hat{k}$$ and $$\vec{c} = \hat{i} + n\hat{j} + 3\hat{k}$$ $$(n \geq 0)$$ is 158 cubic units, then:
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If $$(a, b, c)$$ is the image of the point $$(1, 2, -3)$$ in the line, $$\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$$, then $$a + b + c$$ is equal to:
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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
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The natural number $$m$$, for which the coefficient of $$x$$ in the binomial expansion of $$\left(x^m + \frac{1}{x^2}\right)^{22}$$ is 1540, is
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If the line, $$2x - y + 3 = 0$$ is at a distance $$\frac{1}{\sqrt{5}}$$ and $$\frac{2}{\sqrt{5}}$$ 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 __________.
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Let $$f(x) = x \cdot \left[\frac{x}{2}\right]$$, for $$-10 < x < 10$$, where $$[t]$$ denotes the greatest integer function. Then the number of points of discontinuity of $$f(x)$$ is equal to
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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
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