If $$R$$, $$X_L$$ and $$X_C$$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:
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If $$R$$, $$X_L$$ and $$X_C$$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:
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The initial speed of a projectile fired from ground is $$u$$. At the highest point during its motion, the speed of projectile is $$\dfrac{\sqrt{3}}{2}u$$. The time of flight of the projectile is:
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As shown in figure, a 70 kg garden roller is pushed with a force of $$\vec{F} = 200$$ N at an angle of 30° with horizontal. The normal reaction on the roller is (Given $$g = 10$$ m s$$^{-2}$$)
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100 balls each of mass $$m$$ moving with speed $$v$$ simultaneously strike a wall normally and reflected back with same speed, in time $$t$$ s. The total force exerted by the balls on the wall is
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At a certain depth $$d$$ below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height $$3R$$ above earth surface. Where $$R$$ is Radius of earth (Take $$R = 6400$$ km). The depth $$d$$ is equal to
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Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following {Assume negligible air friction}
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If 1000 droplets of water of surface tension 0.07 N m$$^{-1}$$, having same radius 1 mm each, combine to form a single drop. In the process the released surface energy is- (Take $$\pi = \dfrac{22}{7}$$)
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The pressure of a gas changes linearly with volume from $$A$$ to $$B$$ as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be
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The correct relation between $$\gamma = \dfrac{C_P}{C_V}$$ and temperature $$T$$ is:
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The maximum potential energy of a block executing simple harmonic motion is 25 J. A is amplitude of oscillation. At $$\dfrac{A}{2}$$, the kinetic energy of the block is
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Which of the following correctly represents the variation of electric potential $$(V)$$ of a charged spherical conductor of radius $$(R)$$ with radial distance $$(r)$$ from the centre?
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The drift velocity of electrons for a conductor connected in an electrical circuit is $$V_d$$. The conductor is now replaced by another conductor with same material and same length but double the area of cross-section. The applied voltage remains same. The new drift velocity of electrons will be
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A bar magnet with a magnetic moment 5.0 A m$$^2$$ is placed in parallel position relative to a magnetic field of 0.4 T. The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is ________.
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A rod with circular cross-section area 2 cm$$^2$$ and length 40 cm is wound uniformly with 400 turns of an insulated wire. If a current of 0.4 A flows in the wire windings, the total magnetic flux produced inside the windings is $$4\pi \times 10^{-6}$$ Wb. The relative permeability of the rod is
(Given: Permeability of vacuum $$\mu_0 = 4\pi \times 10^{-7}$$ N A$$^{-2}$$)
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Two polaroids $$A$$ and $$B$$ are placed in such a way that the pass-axis of polaroids are perpendicular to each other. Now, another polaroid $$C$$ is placed between $$A$$ and $$B$$ bisecting angle between them. If intensity of unpolarised light is $$I_0$$ then intensity of transmitted light after passing through polaroid $$B$$ will be:
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Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R
Assertion A: The beam of electrons shows wave nature and exhibit interference and diffraction.
Reason R: Davisson Germer Experimentally verified the wave nature of electrons.
In the light of the above statements. Choose the most appropriate answer from the options given below:
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If a source of electromagnetic radiation having power 15 kW produces $$10^{16}$$ photons per second, the radiation belongs to a part of spectrum is:
(Take Planck constant $$h = 6 \times 10^{-34}$$ J s)
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A free neutron decays into a proton but a free proton does not decay into neutron. This is because
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The effect of increase in temperature on the number of electrons in conduction band $$(n_e)$$ and resistance of a semiconductor will be as:
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The amplitude of 15sin(1000$$\pi t$$) is modulated by 10sin(4$$\pi t$$) signal. The amplitude modulated signal contains frequency(ies) of
(A) 500 Hz
(B) 2 Hz
(C) 250 Hz
(D) 498 Hz
(E) 502 Hz
Choose the correct answer from the options given below:
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The speed of a swimmer is 4 km h$$^{-1}$$ in still water. If the swimmer makes his strokes normal to the flow of river of width 1 km, he reaches a point 750 m down the stream on the opposite bank. The speed of the river water is ______ km h$$^{-1}$$.
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A lift of mass $$M = 500$$ kg is descending with speed of 2 m s$$^{-1}$$. Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of 2 m s$$^{-2}$$. The kinetic energy of the lift at the end of fall through to a distance of 6 m will be ______ kJ.
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A solid sphere of mass 1 kg rolls without slipping on a plane surface. Its kinetic energy is $$7 \times 10^{-3}$$ J. The speed of the centre of mass of the sphere is ______ cm s$$^{-1}$$.
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A thin rod having a length of 1 m and area of cross-section $$3 \times 10^{-6}$$ m$$^2$$ is suspended vertically from one end. The rod is cooled from 210°C to 160°C. After cooling, a mass $$M$$ is attached at the lower end of the rod such that the length of rod again becomes 1 m. Young's modulus and coefficient of linear expansion of the rod are $$2 \times 10^{11}$$ N m$$^{-2}$$ and $$2 \times 10^{-5}$$ K$$^{-1}$$, respectively. The value of $$M$$ is ______ kg. (Take $$g = 10$$ m s$$^{-2}$$)
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In the figure given below, a block of mass $$M = 490$$ g placed on a frictionless table is connected with two springs having same spring constant ($$K = 2$$ N m$$^{-1}$$). If the block is horizontally displaced through $$X$$ m then the number of complete oscillations it will make in $$14\pi$$ seconds will be ______.
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Expression for an electric field is given by $$\vec{E} = 4000 \; x^2 \; \hat{i}$$ V m$$^{-1}$$. The electric flux through the cube of side 20 cm when placed in electric field (as shown in the figure) is ______ V cm.
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Two identical cells, when connected either in parallel or in series gives same current in an external resistance 5 $$\Omega$$. The internal resistance of each cell will be ______ $$\Omega$$.
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An inductor of 0.5 mH, a capacitor of 20 $$\mu$$F and resistance of 20 $$\Omega$$ are connected in series with a 220 V ac source. If the current is in phase with the emf, the amplitude of current of the circuit is $$\sqrt{x}$$ A. The value of $$x$$ is-
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In a medium the speed of light wave decreases to 0.2 times to its speed in free space. The ratio of relative permittivity to the refractive index of the medium is $$x$$:1. The value of $$x$$ is ______.
(Given speed of light in free space $$= 3 \times 10^8$$ m s$$^{-1}$$ and for the given medium $$\mu_r = 1$$)
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For hydrogen atom, $$\lambda_1$$ and $$\lambda_2$$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $$\lambda_1$$ and $$\lambda_2$$ is $$\dfrac{x}{32}$$. The value of $$x$$ is ______.
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Which transition in the hydrogen spectrum would have the same wavelength as the Balmer type transition from n = 4 to n = 2 of He$$^+$$ spectrum
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The correct increasing order of the ionic radii is
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Match List I with List II
List I List II
A. XeF$$_4$$ I. See-saw
B. SF$$_4$$ II. Square planar
C. NH$$_4^+$$ III. Bent T-shaped
D. BrF$$_3$$ IV. Tetrahedral
Choose the correct answer from the options given below:
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$$H_2O_2$$ acts as a reducing agent in
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Match items of column I and II
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Choose the correct set of reagents for the following conversion
trans Ph - CH = CH - CH$$_3$$ $$\rightarrow$$ cis Ph - CH = CH - CH$$_3$$
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Which one of the following statements is correct for electrolysis of brine solution?
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Adding surfactants in non polar solvent, the micelles structure will look like
(a)
(b)
(c)
(d)
The methods NOT involved in concentration of ore are
(A) Liquation
(B) Leaching
(C) Electrolysis
(D) Hydraulic washing
(E) Froth floatation
Choose the correct answer from the options given below:
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Identify X, Y and Z in the following reaction. (Equation not balanced)
$$ClO^. + NO_2 \rightarrow X \xrightarrow{H_2O} Y + Z$$
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When Cu$$^{2+}$$ ion is treated with KI, a white precipitate, X appears in solution. The solution is titrated with sodium thiosulphate, the compound Y is formed. X and Y respectively are
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The correct order of basicity of oxides of vanadium is
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Nd$$^{2+}$$ = ______
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Cobalt chloride when dissolved in water forms pink colored complex X which has octahedral geometry. This solution on treating with conc HCl forms deep blue complex, Y which has a Z geometry. X, Y and Z, respectively, are
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The correct order of melting point of dichlorobenzenes is
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An organic compound 'A' with empirical formula C$$_6$$H$$_6$$O gives sooty flame on burning. Its reaction with bromine solution in low polarity solvent results in high yield of B. B is
Consider the following reaction
Propanal + Methanal $$\xrightarrow{(i)$$ dil. NaOH $$}{\xrightarrow{(ii) \Delta \; (iii)$$ NaCN $$\; (iv) \; H_3O^+}}$$ Product B (C$$_5$$H$$_8$$O$$_3$$)
The correct statement for product B is. It is
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Which of the following artificial sweeteners has the highest sweetness value in comparison to cane sugar?
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A protein 'X' with molecular weight of 70,000 u, on hydrolysis gives amino acids. One of these amino acid is
On complete combustion, 0.492 g of an organic compound gave 0.792 g of CO$$_2$$. The % of carbon in the organic compound is (Nearest integer)
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Zinc reacts with hydrochloric acid to give hydrogen and zinc chloride. The volume of hydrogen gas produced at STP from the reaction of 11.5 g of zinc with excess HCl is L (Nearest integer)
(Given: Molar mass of Zn is 65.4 g mol$$^{-1}$$ and Molar volume of H$$_2$$ at STP = 22.7 L)
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The enthalpy change for the conversion of $$\dfrac{1}{2}$$Cl$$_2$$(g) to Cl$$^-$$(aq) is (-) ______ kJmol$$^{-1}$$ (Nearest integer)
Given: $$\Delta_{dis}H^0_{Cl_2(g)} = 240$$ kJmol$$^{-1}$$
$$\Delta_{eg}H^o_{Cl(g)} = -350$$ kJmol$$^{-1}$$
$$\Delta_{hyd}H^o_{Cl^-(g)} = -380$$ kJmol$$^{-1}$$
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For reaction: $$SO_2(g) + \dfrac{1}{2}O_2(g) \rightleftharpoons SO_3(g)$$ $$K_P = 2 \times 10^{12}$$ at 27°C and 1 atm pressure. The $$K_c$$ for the same reaction is ______ $$\times 10^{13}$$. (Nearest integer)
(Given $$R = 0.082$$ L atm K$$^{-1}$$ mol$$^{-1}$$)
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The total pressure of a mixture of non-reacting gases X(0.6 g) and Y(0.45 g) in a vessel is 740 mm of Hg. The partial pressure of the gas X is mm of Hg. (Nearest Integer)
(Given: molar mass X = 20 and Y = 45 g mol$$^{-1}$$)
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At 27°C, a solution containing 2.5 g of solute in 250.0 mL of solution exerts an osmotic pressure of 400 Pa. The molar mass of the solute is g mol$$^{-1}$$ (Nearest integer)
(Given: R = 0.083 L bar$$^{-1}$$ mol$$^{-1}$$)
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The logarithm of equilibrium constant for the reaction $$Pd^{2+} + 4Cl^- \rightleftharpoons PdCl_4^{2-}$$ is (Nearest integer)
Given: $$\dfrac{2.303RT}{F} = 0.06$$ V
$$Pd^{2+}_{(aq)} + 2e^- \rightleftharpoons Pd(s)$$ $$E^o = 0.83$$ V
$$PdCl_4^{2-}(aq) + 2e^- \rightleftharpoons Pd(s) + 4Cl^-(aq)$$ $$
$$E^o = 0.65$$ V
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$$A \rightarrow B$$
The rate constants of the above reaction at 200 K and 300 K are 0.03 min$$^{-1}$$ and 0.05 min$$^{-1}$$ respectively. The activation energy for the reaction is J (Nearest integer)
(Given: ln 10 = 2.3, R = 8.3 J K$$^{-1}$$ mol$$^{-1}$$, log 5 = 0.70, log 3 = 0.48, log 2 = 0.30)
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The oxidation state of phosphorus in hypophosphoric acid is
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How many of the transformation given below would result in aromatic amines?
The number of real roots of the equation $$\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}$$, is:
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For all $$z \in C$$ on the curve $$C_1$$: $$|z| = 4$$, let the locus of the point $$z + \dfrac{1}{z}$$ be the curve $$C_2$$. Then
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If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296, respectively, then the sum of common ratios of all such GPs is
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Let a circle $$C_1$$ be obtained on rolling the circle $$x^2 + y^2 - 4x - 6y + 11 = 0$$ upwards 4 units on the tangent T to it at the point (3, 2). Let $$C_2$$ be the image of $$C_1$$ in T. Let $$A$$ and $$B$$ be the centers of circles $$C_1$$ and $$C_2$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the x-axis. Then the area of the trapezium AMNB is:
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If the maximum distance of normal to the ellipse $$\dfrac{x^2}{4} + \dfrac{y^2}{b^2} = 1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is:
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Consider:
S1: $$p \Rightarrow q \lor p \land \sim q$$ is a tautology.
S2: $$\sim p \Rightarrow \sim q \land \sim p \lor q$$ is a contradiction.
Then
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Let $$R$$ be a relation on $$N \times N$$ defined by $$a, b R c, d$$ if and only if $$ad(b - c) = bc(a - d)$$. Then $$R$$ is
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Let $$A = \begin{matrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{matrix}$$. Then the sum of the diagonal elements of the matrix $$A + I^{11}$$ is equal to:
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For the system of linear equations
$$x + y + z = 6$$
$$\alpha x + \beta y + 7z = 3$$
$$x + 2y + 3z = 14$$
which of the following is NOT true?
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If $$\sin^{-1}\dfrac{\alpha}{17} + \cos^{-1}\dfrac{4}{5} - \tan^{-1}\dfrac{77}{36} = 0$$, $$0 < \alpha < 13$$, then $$\sin^{-1}\sin\alpha + \cos^{-1}\cos\alpha$$ is equal to
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Let $$y = fx$$ represent a parabola with focus $$(-\dfrac{1}{2}, 0)$$ and directrix $$y = -\dfrac{1}{2}$$. Then
$$S = \{x \in \mathbb{R}: \tan^{-1}(\sqrt{fx}) + \sin^{-1}(\sqrt{fx+1}) = \dfrac{\pi}{2}\}$$:
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If the domain of the function $$f(x) = \dfrac{x}{1 + x^2}$$, where $$x$$ is greatest integer $$\le x$$, is $$[2, 6)$$, then its range is
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Let $$y = f(x) = \sin^3\left(\frac{\pi}{3} \cos\left(\frac{\pi}{3\sqrt{2}}(-4x^3 + 5x^2 + 1)^{\frac{3}{2}}\right)\right)$$. Then, at $$x = 1$$,
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A wire of length 20 m is to be cut into two pieces. A piece of length $$\ell_1$$ is bent to make a square of area $$A_1$$ and the other piece of length $$\ell_2$$ is made into a circle of area $$A_2$$. If $$2A_1 + 3A_2$$ is minimum then $$\pi\ell_1 : \ell_2$$ is equal to:
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Let $$\alpha \in (0, 1)$$ and $$\beta = \log_e(1 - \alpha)$$. Let $$P_n(x) = x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \ldots + \dfrac{x^n}{n}$$, $$x \in (0, 1)$$. Then the integral $$\int_0^{\alpha} \dfrac{t^{50}}{1-t} dt$$ is equal to
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The value of $$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \dfrac{2 + 3\sin x}{\sin x(1 + \cos x)} dx$$ is equal to
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Let a differentiable function $$f$$ satisfy $$f(x) + \int_3^x \dfrac{f(t)}{t} dt = \sqrt{x+1}$$, $$x \ge 3$$. Then $$12f(8)$$ is equal to:
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Let $$\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$$, and $$\vec{b}$$ and $$\vec{c}$$ be two nonzero vectors such that $$\vec{a} + \vec{b} + \vec{c} = \vec{a} + \vec{b} - \vec{c}$$ and $$\vec{b} \cdot \vec{c} = 0$$. Consider the following two statements:
A: $$\vec{a} + \lambda\vec{c} \ge \vec{a}$$ for all $$\lambda \in \mathbb{R}$$.
B: $$\vec{a}$$ and $$\vec{c}$$ are always parallel.
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Let the shortest distance between the lines L: $$\dfrac{x-5}{-2} = \dfrac{y-\lambda}{0} = \dfrac{z+\lambda}{1}$$, $$\lambda \ge 0$$ and L$$_1$$: $$x+1 = y-1 = 4-z$$ be $$2\sqrt{6}$$. If $$(\alpha, \beta, \gamma)$$ lies on L, then which of the following is NOT possible?
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A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is
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Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is ______.
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Let $$a_1, a_2, \ldots, a_n$$ be in A.P. If $$a_5 = 2a_7$$ and $$a_{11} = 18$$, then $$12\left(\dfrac{1}{\sqrt{a_{10}} + \sqrt{a_{11}}} + \dfrac{1}{\sqrt{a_{11}} + \sqrt{a_{12}}} + \ldots + \dfrac{1}{\sqrt{a_{17}} + \sqrt{a_{18}}}\right)$$ is equal to ______.
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Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to ______.
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Let $$\alpha > 0$$, be the smallest number such that the expansion of $$\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}$$ has a term $$\beta x^{-\alpha}$$, $$\beta \in N$$. Then $$\alpha$$ is equal to ______.
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The remainder on dividing $$5^{99}$$ by 11 is ______.
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If the variance of the frequency distribution
is 3, then $$\alpha$$ is equal to ______.
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Let for $$x \in R$$, $$f(x) = \dfrac{x+x}{2}$$ and $$g(x) = \begin{cases} x, & x < 0 \\ x^2, & x \ge 0 \end{cases}$$. Then area bounded by the curve $$y = f \circ g(x)$$ and the lines $$y = 0, 2y - x = 15$$ is equal to ______.
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Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$\vec{a} = \sqrt{14}$$, $$\vec{b} = \sqrt{6}$$ and $$\vec{a} \times \vec{b} = \sqrt{48}$$. Then $$(\vec{a} \cdot \vec{b})^2$$ is equal to ______.
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Let the line $$L$$: $$\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z-3}{1}$$ intersect the plane $$2x + y + 3z = 16$$ at the point $$P$$. Let the point $$Q$$ be the foot of perpendicular from the point $$R(1, -1, -3)$$ on the line $$L$$. If $$\alpha$$ is the area of triangle $$PQR$$, then $$\alpha^2$$ is equal to ______.
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Let $$\theta$$ be the angle between the planes $$P_1 = \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$$ and $$P_2 = \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$$. Let L be the line that meets $$P_2$$ at the point (4, -2, 5) and makes angle $$\theta$$ with the normal of $$P_2$$. If $$\alpha$$ is the angle between L and $$P_2$$ then $$\tan^2\theta \cot^2\alpha$$ is equal to ______.
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