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In the equation $$X + \frac{a}{Y^2}\left[Y - b\right] = RT$$, $$X$$ is pressure, $$Y$$ is volume, $$R$$ is universal gas constant and $$T$$ is temperature. The physical quantity equivalent to the ratio $$\frac{a}{b}$$ is:
The distance travelled by an object in time $$t$$ is given by $$s = 2.5t^2$$. The instantaneous speed of the object at $$t = 5$$ s will be :
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A passenger sitting in a train A moving at $$90$$ km h$$^{-1}$$ observes another train B moving in the opposite direction for $$8$$ s. If the velocity of the train B is $$54$$ km h$$^{-1}$$, then length of train B is:
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A vehicle of mass $$200$$ kg is moving along a levelled curved road of radius $$70$$ m with angular velocity of $$0.2$$ rad s$$^{-1}$$. The centripetal force acting on the vehicle is:
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Two planets A and B of radii $$R$$ and $$1.5R$$ have densities $$\rho$$ and $$\frac{\rho}{2}$$ respectively. The ratio of acceleration due to gravity at the surface of B to A is:
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Given below are two statements:
Statement I : For a planet, if the ratio of mass of the planet to its radius increase, the escape velocity from the planet also increase.
Statement II : Escape velocity is independent of the radius of the planet.
In the light of above statements, choose the most appropriate answer from the options given below
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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 : A spherical body of radius $$(5 \pm 0.1)$$ mm having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is 4%.
Reason R : The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius. In the light of the above statements, choose the correct answer from the options given below
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The initial pressure and volume of an ideal gas are $$P_0$$ and $$V_0$$. The final pressure of the gas when the gas is suddenly compressed to volume $$\frac{V_0}{4}$$ will be:
(Given $$\gamma$$ = ratio of specific heats at constant pressure and at constant volume.)
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The mean free path of molecules of a certain gas at STP is $$1500d$$, where $$d$$ is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at $$373$$ K is approximately:
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A particle executes SHM of amplitude $$A$$. The distance from the mean position when its kinetic energy becomes equal to its potential energy is:
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A $$10 \ \mu$$C charge is divided into two parts and placed at $$1$$ cm distance so that the repulsive force between them is maximum. The charges of the two parts are:
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In the network shown below, the charge accumulated in the capacitor in steady state will be:
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An electron is moving along the positive x-axis. If uniform magnetic field is applied parallel to the negative z-axis, then
A. The electron will experience magnetic force along positive y-axis
B. The electron will experience magnetic force along negative y-axis
C. The electron will not experience any force in magnetic field
D. The electron will continue to move along the positive x-axis
E. The electron will move along circular path in magnetic field
Choose the correct answer from the option given below:
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Given below are two statements:
Statement I : An AC circuit undergoes electrical resonance if it contains either a capacitor or an inductor.
Statement II: An AC circuit containing a pure capacitor or a pure inductor consumes high power due to its non-zero power factor. In the light of above statements, choose the correct answer from the options given below:
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Given below are two statements:
Statement I : Out of microwaves, infrared rays and ultraviolet rays, ultraviolet rays are the most effective for the emission of electrons from a metallic surface
Statement II : Above the threshold frequency, the maximum kinetic energy of photoelectrons is inversely proportional to the frequency of the incident light
In the light of above statements, choose the correct answer from the options given below
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In an electromagnetic wave, at an instant and at a particular position, the electric field is along the negative z-axis and magnetic field is along the positive x-axis. Then the direction of propagation of electromagnetic wave is:
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In a Young's double slit experiment, the ratio of amplitude of light coming from slits is $$2 : 1$$. The ratio of the maximum to minimum intensity in the interference pattern is
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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 binding energy per nucleon is practically independent of the atomic number for nuclei of mass number in the range 30 to 170.
Reason R : Nuclear force is short ranged.
In the light of the above statements, choose the correct answer from the options given below
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The output from a NAND gate having inputs A and B given below will be,
To radiate EM signal of wavelength $$\lambda$$ with high efficiency, the antennas should have a minimum size equal to:
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A car accelerates from rest of $$u$$ m s$$^{-1}$$. The energy spent in this process is $$E$$ J. The energy required to accelerate the car from $$u$$ m s$$^{-1}$$ to $$2u$$ m s$$^{-1}$$ is $$nE$$ J. The value of $$n$$ is _____.
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A light rope is wound around a hollow cylinder of mass $$5$$ kg and radius $$70$$ cm. The rope is pulled with a force of $$52.5$$ N. The angular acceleration of the cylinder will be _____ rad s$$^{-2}$$.
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Two plates A and B have thermal conductivities $$84$$ W m$$^{-1}$$ K$$^{-1}$$ and $$126$$ W m$$^{-1}$$ K$$^{-1}$$ respectively. They have same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of A and B are kept at $$100°$$C and $$0°$$C respectively, then the temperature of the surface of contact in steady state is _____ °C.
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In an experiment with sonometer when a mass of $$180$$ g is attached to the string, it vibrates with fundamental frequency of $$30$$ Hz. When a mass $$m$$ is attached, the string vibrates with fundamental frequency of $$50$$ Hz. The value of $$m$$ is _____ g.
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Three point charges $$q$$, $$-2q$$ and $$2q$$ are placed on x axis at a distance $$x = 0$$, $$x = \frac{3}{4}R$$ and $$x = R$$ respectively from origin as shown. If $$q = 2 \times 10^{-6}$$ C and $$R = 2$$ cm, the magnitude of net force experienced by the charge $$-2q$$ is _____ N.
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In the circuit shown, the energy stored in the capacitor is $$n$$ $$\mu$$J. The value of $$n$$ is _____.
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A straight wire AB of mass $$40$$ g and length $$50$$ cm is suspended by a pair of flexible leads in uniform magnetic field of magnitude $$0.40$$ T as shown in the figure. The magnitude of the current required in the wire to remove the tension in the supporting leads is _____ A. (Take $$g = 10$$ m s$$^{-2}$$.
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An insulated copper wire of 100 turns is wrapped around a wooden cylindrical core of the cross-sectional area $$24$$ cm$$^2$$. The two ends of the wire are connected to a resistor. The total resistance in the circuit is $$12 \ \Omega$$. If an externally applied uniform magnetic field in the core along its axis changes from $$1.5$$ T in one direction to $$1.5$$ T in the opposite direction, the charge flowing through a point in the circuit during the change of magnetic field will be _____ mC.
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A bi convex lens of focal length $$10$$ cm is cut in two identical parts along a plane perpendicular to the principal axis. The power of each lens after cut is _____ D.
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An atom absorbs a photon of wavelength $$500$$ nm and emits another photon of wavelength $$600$$ nm. The net energy absorbed by the atom in this process is $$n \times 10^{-4}$$ eV. The value of $$n$$ is [Assume the atom to be stationary during the absorption and emission process] (Take $$h = 6.6 \times 10^{-34}$$ J s and $$c = 3 \times 10^8$$ m s$$^{-1}$$).
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Identify the correct order of standard enthalpy of formation of sodium halides.
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Match List-I with List-II.
| List-I | List-II | ||
|---|---|---|---|
| A | Weak intermolecular forces of attraction | I | Hexamethylenediamine + adipic acid |
| B | Hydrogen bonding | II | AlEt$$_3$$ + TiCl$$_4$$ |
| C | Heavily branched polymer | III | 2-chloro-1, 3-butadiene |
| D | High density polymer | IV | Phenol + formaldehyde |
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Given below are two statements :
Statement I : SO$$_2$$ and H$$_2$$O both possess V-shaped structure
Statement II : The bond angle of SO$$_2$$ is less than that of H$$_2$$O.
In the light of the above statements, choose the most appropriate answer from the options given below:
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What happens when methane undergoes combustion in systems A and B respectively?
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Isotopes of hydrogen have almost same chemical properties, but difference in their rates of reactions.
Reason R: Isotopes of hydrogen have different enthalpy of bond dissociation.
In the light of the above statements, choose the most appropriate answer from the options given below:
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Better method for preparation of BeF$$_2$$, among the following is
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The major product for the following reaction is :
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : Order of acidic nature of the following compounds is A > B > C.
Reason R: Fluoro is a stronger electron withdrawing group than Chloro group.
In the light of the above statements, choose the correct answer from the options given below :
Given below are two statements :
Statement I : Tropolone is an aromatic compound and has $$8\pi$$ electrons.
Statement II : $$\pi$$ electrons of $$> C = O$$ group in tropolone is involved in aromaticity. In the light of the above statements choose the correct answer from the options given below:
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Which of the following are the Green house gases?
A. Water vapour
B. Ozone
C. I$$_2$$
D. Molecular hydrogen
Choose the most appropriate answer from the options given below :
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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 diameter of colloidal particles in solution should not be much smaller than wavelength of light to show Tyndall effect.
Reason R : The light scatters in all directions when the size of particles is large enough.
In the light of the above statements, choose the correct answer from the options given below :
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Given below are two statements related to Ellingham diagram :
Statement I : Ellingham diagrams can be constructed for formation of oxides, sulphides and halides of metals.
Statement II : It consists of plots of $$\Delta H°$$ vs T for formation of oxides of elements.
In the light of the above statements, choose the most appropriate answer from the options given below :
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The correct group of halide ions which can be oxidised by oxygen in acidic medium is
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The covalency and oxidation state respectively of boron in BF$$_4^-$$, are
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Which of the following complexes will exhibit maximum attraction to an applied magnetic field?
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The total number of stereoisomers for the complex $$[Cr(ox)_2ClBr]^{3-}$$ (where ox = oxalate) is
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Match List-I with List-II.
1-Bromopropane is reacted with reagents in List-I to give product in List-II
| List-I Reagent | List-II Product | ||
|---|---|---|---|
| A | KOH (alc) | I | Nitrile |
| B | KCN (alc) | II | Ester |
| C | AgNO$$_2$$ | III | Alkene |
| D | H$$_3$$CCOOAg | IV | Nitroalkane |
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Compound A from the following reaction sequence is :
In the wet tests for detection of various cations by precipitation, Ba$$^{2+}$$ cations are detected by obtaining precipitate of
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The naturally occurring amino acid that contains only one basic functional group in its chemical structure is
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$$1.1$$ g of a carbonate M$$_2$$CO$$_3$$ on treatment with excess HCl produces $$0.01$$ mol of CO$$_2$$. The molar mass of M$$_2$$CO$$_3$$ is _____ g mol$$^{-1}$$. (Nearest integer)
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The orbital angular momentum of an electron in $$3$$ s orbital is $$\frac{xh}{2\pi}$$. The value of x is _____ (nearest integer)
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$$20$$ mL of $$0.1$$ M NaOH is added to $$50$$ mL of $$0.1$$ M acetic acid solution. The pH of the resulting solution is _____ $$\times 10^{-2}$$. (Nearest integer) Given : pKa CH$$_3$$COOH $$= 4.76$$
log 2 = 0.30
log 3 = 0.48
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See the following chemical reaction:
Cr$$_2$$O$$_7^{2-}$$ + XH$$^+$$ + 6Fe$$^{2+}$$ $$\to$$ YCr$$^{3+}$$ + 6Fe$$^{3+}$$ + ZH$$_2$$O
The sum of X, Y and Z is _____.
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If the formula of Borax is Na$$_2$$ B$$_4$$O$$_x$$(OH)$$_y$$ $$\cdot$$ zH$$_2$$O, then x + y + z = _____
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$$0.400$$ g of an organic compound (X) gave $$0.376$$ g of AgBr in Carius method for estimation of bromine. % of bromine in the compound (X) is (Given: Molar mass AgBr $$= 188$$ g mol$$^{-1}$$, Br $$= 80$$ g mol$$^{-1}$$)
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Sodium metal crystallises in a body centred cubic lattice with unit cell edge length of $$4$$ A. The radius of sodium atom is _____ $$\times 10^{-1}$$ A. (Nearest integer)
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Sea water contains $$29.25$$% NaCl and $$19$$% MgCl$$_2$$ by weight of solution. The normal boiling point of the sea water is _____ °C (Nearest integer) Assume 100% ionization for both NaCl and MgCl$$_2$$. Given:
K$$_b$$H$$_2$$O $$= 0.52$$ K kg mol$$^{-1}$$. Molar mass of NaCl and MgCl$$_2$$ is $$58.5$$ and $$95$$ g mol$$^{-1}$$ respectively.
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At 298 K, the standard reduction potential for Cu$$^{2+}$$/Cu electrode is 0.34 V. Given :
K$$_{sp}$$Cu(OH)$$_2$$ $$= 1 \times 10^{-20}$$. Take $$\frac{2.303RT}{F} = 0.059$$ V. The reduction potential at pH = 14 for the above couple is $$(-) x \times 10^{-2}$$ V. The value of x is _____.
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A(g) $$\to$$ 2B(g) + C(g) is a first order reaction. The initial pressure of the system was found to be $$800$$ mm Hg which increased to $$1600$$ mm Hg after $$10$$ min. The total pressure of the system after $$30$$ min will be _____ mm Hg. (Nearest integer)
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Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^2 - \sqrt{2}x + 2 = 0$$. Then $$\alpha^{14} + \beta^{14}$$ is equal to
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Let $$S = \{z \in \mathbb{C} : \bar{z} = i(z^2 + \text{Re}(\bar{z}))\}$$. Then $$\sum_{z \in S} |z|^2$$ is equal to
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All words, with or without meaning, are made using all the letters of the word $$MONDAY$$. These words are written as in a dictionary with serial numbers. The serial number of the word $$MONDAY$$ is
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Let $$a_1, a_2, a_3, \ldots$$ be a G.P. of increasing positive numbers. Let the sum of its 6$$^{th}$$ and 8$$^{th}$$ terms be 2 and the product of its 3$$^{rd}$$ and 5$$^{th}$$ terms be $$\frac{1}{9}$$. Then $$6a_2 + a_4a_4 + a_6$$ is equal to
The coefficient of $$x^5$$ in the expansion of $$\left(2x^3 - \frac{1}{3x^2}\right)^5$$ is
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Let $$(\alpha, \beta)$$ be the centroid of the triangle formed by the lines $$15x - y = 82$$, $$6x - 5y = -4$$ and $$9x + 4y = 17$$. Then $$\alpha + 2\beta$$ and $$2\alpha - \beta$$ are the roots of the equation
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Let the centre of a circle $$C$$ be $$\alpha, \beta$$ and its radius $$r < 8$$. Let $$3x + 4y = 24$$ and $$3x - 4y = 32$$ be two tangents and $$4x + 3y = 1$$ be a normal to $$C$$. Then $$(\alpha - \beta + r)$$ is equal to
If $$\lim_{x \to 0} \frac{e^{ax} - \cos(bx) - \frac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17$$, then $$5a^2 + b^2$$ is equal to
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The statement $$(p \wedge (\sim q)) \vee ((\sim p) \wedge q) \vee ((\sim p) \wedge (\sim q))$$ is equivalent to ____
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Let for $$A = \begin{bmatrix} 1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$, $$|A| = 2$$. If $$|2 \ \text{adj}(2 \ \text{adj}(2A))| = 32^n$$, then $$3n + \alpha$$ is equal to
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If the system of equations
$$2x + y - z = 5$$
$$2x - 5y + \lambda z = \mu$$
$$x + 2y - 5z = 7$$
has infinitely many solutions, then $$(\lambda + \mu)^2 + (\lambda - \mu)^2$$ is equal to
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The range of $$f(x) = 4\sin^{-1}\left(\frac{x^2}{x^2+1}\right)$$ is
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The value of $$\frac{e^{-\frac{\pi}{4}} + \int_0^{\frac{\pi}{4}} e^{-x}\tan^{50}x \ dx}{\int_0^{\frac{\pi}{4}} e^{-x}(\tan^{49}x + \tan^{51}x) \ dx}$$
The area of the region $$\{x, y : x^2 \leq y \leq x^2 - 4, y \geq 1\}$$ is
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Let $$|\vec{a}| = 2$$, $$|\vec{b}| = 3$$ and the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$ be $$\frac{\pi}{4}$$. Then $$|(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})|^2$$ is equal to
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Let for a triangle $$ABC$$
$$\vec{AB} = -2\hat{i} + \hat{j} + 3\hat{k}$$
$$\vec{CB} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$
$$\vec{CA} = 4\hat{i} + 3\hat{j} + \delta\hat{k}$$
If $$\delta > 0$$ and the area of the triangle $$ABC$$ is $$5\sqrt{6}$$ then $$\vec{CB} \cdot \vec{CA}$$ is equal to
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The plane, passing through the points $$(0, -1, 2)$$ and $$(-1, 2, 1)$$ and parallel to the line passing through $$(5, 1, -7)$$ and $$(1, -1, -1)$$, also passes through the point
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The line, that is coplanar to the line $$\frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}$$, is
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Let N be the foot of perpendicular from the point $$P(1, -2, 3)$$ on the line passing through the points $$(4, 5, 8)$$ and $$(1, -7, 5)$$. Then the distance of N from the plane $$2x - 2y + z + 5 = 0$$ is
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The random variable $$X$$ follows binomial distribution $$B(n, p)$$, for which the difference of the mean and the variance is 1.
If $$2P(X = 2) = 3P(X = 1)$$, then $$n^2P(X > 1)$$ is equal to
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Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is _____.
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Let $$\alpha$$ denote the greatest integer $$\leq \alpha$$. Then $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{120}$$ is equal to _____.
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Let $$f(x) = \sum_{k=1}^{10} k \cdot x^k$$, $$x \in \mathbb{R}$$, if $$2f(2) + f'(2) = 119 \cdot 2^n + 1$$ then $$n$$ is equal to _____.
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The remainder, when $$7^{103}$$ is divided by $$17$$, is _____.
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The foci of a hyperbola are $$(\pm 2, 0)$$ and its eccentricity is $$\frac{3}{2}$$. A tangent, perpendicular to the line $$2x + 3y = 6$$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x- and y-axes are $$a$$ and $$b$$ respectively, then $$|6a| + |5b|$$ is equal to _____.
The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is _____.
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Let $$A = \{-4, -3, -2, 0, 1, 3, 4\}$$ and $$R = \{(a, b) \in A \times A : b = |a|$$ or $$b^2 = a + 1\}$$ be a relation on $$A$$. Then the minimum number of elements, that must be added to the relation $$R$$ so that it becomes reflexive and symmetric, is _____.
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For $$x \in (-1, 1]$$, the number of solutions of the equation $$\sin^{-1}x = 2\tan^{-1}x$$ is equal to _____.
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Let $$f_n = \int_0^{\pi/2} \sum_{k=1}^{n} \sin^{k-1}x \sum_{k=1}^{n} (2k-1)\sin^{k-1}x \cos x \ dx$$, $$n \in \mathbb{N}$$. Then $$f_{21} - f_{20}$$ is equal to _____.
If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + \frac{4x}{x^2-1}y = \frac{x+2}{(x^2-1)^{5/2}}$$, $$x \gt 1$$ such that $$y(2) = \frac{2}{9}\log_e 2 + \sqrt{3}$$ and $$y\sqrt{2} = \alpha\log_e\sqrt{\alpha} + \beta + \beta - \sqrt{\gamma}$$, $$\alpha, \beta, \gamma \in \mathbb{N}$$, then $$\alpha\beta\gamma$$ is equal to _____.
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