Identify the pair of physical quantities that have same dimensions:
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Identify the pair of physical quantities that have same dimensions:
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An object of mass $$5$$ kg is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of $$10$$ N throughout the motion. The ratio of time of ascent to the time of descent will be equal to : [Use $$g = 10$$ m s$$^{-2}$$]
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A stone of mass $$m$$, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is
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Potential energy as a function of $$r$$ is given by $$U = \frac{A}{r^{10}} - \frac{B}{r^5}$$, where $$r$$ is the interatomic distance, $$A$$ and $$B$$ are positive constants. The equilibrium distance between the two atoms will be :
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A fly wheel is accelerated uniformly from rest and rotates through $$5$$ rad in the first second. The angle rotated by the fly wheel in the next second, will be :
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The distance between Sun and Earth is $$R$$. The duration of year if the distance between Sun and Earth becomes $$3R$$ will be :
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A $$100$$ g of iron nail is hit by a $$1.5$$ kg hammer striking at a velocity of $$60$$ ms$$^{-1}$$. What will be the rise in the temperature of the nail if one fourth of energy of the hammer goes into heating the nail? [Specific heat capacity of iron $$= 0.42$$ J g$$^{-1}$$ °C$$^{-1}$$]
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A Carnot engine takes $$5000$$ kcal of heat from a reservoir at $$727°$$C and gives heat to a sink at $$127°$$C. The work done by the engine is
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Two massless springs with spring constants $$2k$$ and $$9k$$, carry $$50$$ g and $$100$$ g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :
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Two light beams of intensities in the ratio of $$9 : 4$$ are allowed to interfere. The ratio of the intensity of maxima and minima will be :
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Two identical charged particles each having a mass $$10$$ g and charge $$2.0 \times 10^{-7}$$ C are placed on a horizontal table with a separation of $$L$$ between them such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is $$0.25$$, find the value of $$L$$. [Use $$g = 10$$ ms$$^{-2}$$]
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A long cylindrical volume contains a uniformly distributed charge of density $$\rho$$. The radius of cylindrical volume is $$R$$. A charge particle ($$q$$) revolves around the cylinder in a circular path. The kinetic energy of the particle is :
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If the charge on a capacitor is increased by $$2C$$, the energy stored in it increases by $$44\%$$. The original charge on the capacitor is (in $$C$$)
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What will be the most suitable combination of three resistors $$A = 2$$ $$\Omega$$, $$B = 4$$ $$\Omega$$, $$C = 6$$ $$\Omega$$ so that $$\left(\frac{22}{3}\right)\Omega$$ is equivalent resistance of combination?
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The soft-iron is a suitable material for making an electromagnet. This is because soft-iron has
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A proton, a deuteron and an $$\alpha$$-particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is :
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Given below are two statements :
Statement-I: The reactance of an ac circuit is zero. It is possible that the circuit contains a capacitor and an inductor.
Statement-II: In ac circuit, the average power delivered by the source never becomes zero.
In the light of the above statements, choose the correct answer from the options given below
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An electric bulb is rated as $$200$$ W. What will be the peak magnetic field at $$4$$ m distance produced by the radiations coming from this bulb? Consider this bulb as a point source with $$3.5\%$$ efficiency.
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The light of two different frequencies whose photons have energies $$3.8$$ eV and $$1.4$$ eV respectively, illuminate a metallic surface whose work function is $$0.6$$ eV successively. The ratio of maximum speeds of emitted electrons for the two frequencies respectively will be :
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In Bohr's atomic model of hydrogen, let $$K$$, $$P$$ and $$E$$ are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level :
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A body is projected from the ground at an angle of $$45°$$ with the horizontal. Its velocity after $$2$$ s is $$20$$ m s$$^{-1}$$. The maximum height reached by the body during its motion is ______ m. (use $$g = 10$$ m s$$^{-2}$$)
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In an experiment to verify Newton's law of cooling, a graph is plotted between the temperature difference $$(\Delta T)$$ of the water and surroundings and time as shown in figure. The initial temperature of water is taken as $$80°$$C. The value of $$t_2$$ as mentioned in the graph will be ______
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A monoatomic gas performs a work of $$\frac{Q}{4}$$ where $$Q$$ is the heat supplied to it. The molar heat capacity of the gas will be ______ $$R$$.
Where $$R$$ is the gas constant.
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Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by $$y = \left(10 \cos \pi x \sin \frac{2\pi t}{T}\right)$$ cm. The amplitude of the particle at $$x = \frac{4}{3}$$ cm will be ______ cm.
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A potentiometer wire of length $$10$$ m and resistance $$20$$ $$\Omega$$ is connected in series with a $$25$$ V battery and an external resistance $$30$$ $$\Omega$$. A cell of emf $$E$$ in secondary circuit is balanced by $$250$$ cm long potentiometer wire. The value of $$E$$ (in volt) is $$\frac{x}{10}$$. The value of $$x$$ is ______.
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A circular coil of $$1000$$ turns each with area $$1$$ m$$^2$$ is rotated about its vertical diameter at the rate of one revolution per second in a uniform horizontal magnetic field of $$0.07$$ T. The maximum voltage generation will be ______ V.
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A ray of light is incident at an angle of incidence $$60°$$ on the glass slab of refractive index $$\sqrt{3}$$. After refraction, the light ray emerges out from other parallel faces and lateral shift between incident ray and emergent ray is $$4\sqrt{3}$$ cm. The thickness of the glass slab is ______ cm.
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A sample contains $$10^{-2}$$ kg each of two substances $$A$$ and $$B$$ with half lives $$4$$ s and $$8$$ s respectively. The ratio of their atomic weights is $$1 : 2$$. The ratio of the amounts of $$A$$ and $$B$$ after $$16$$ s is $$\frac{x}{100}$$. The value of $$x$$ is ______.
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In the given circuit, the value of current $$I_L$$ will be ______ mA. (When $$R_L = 1$$ k$$\Omega$$)
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An antenna is placed in a dielectric medium of dielectric constant $$6.25$$. If the maximum size of that antenna is $$5.0$$ mm, it can radiate a signal of minimum frequency of ______ GHz.
(Given $$\mu_r = 1$$ for dielectric medium)
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$$120$$ g of an organic compound which contains only carbon and hydrogen on complete combustion gives $$330$$ g of $$CO_2$$ and $$270$$ g of water. The percentage of carbon and hydrogen in the organic compound are respectively
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The energy of one mole of photons of radiation of wavelength $$300$$ nm is (Given :
$$h = 6.63 \times 10^{-34}$$ J s, $$N_A = 6.02 \times 10^{23}$$ mol$$^{-1}$$, $$c = 3 \times 10^{8}$$ m s$$^{-1}$$)
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Metals generally melt at very high temperatures, Among the following which one has the highest melting point?
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The correct order of bond orders of $$C_2^{2-}$$, $$N_2^{2-}$$ and $$O_2^{2-}$$ is
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At $$25°$$C and 1 atm pressure, the enthalpies of combustion are as given below:
| Substance | $$H_2$$ | C (graphite) | $$C_2H_6(g)$$ |
|---|---|---|---|
| $$\frac{\Delta_c H^\ominus}{kJ mol^{-1}}$$ | $$-286.0$$ | $$-394.0$$ | $$-1560.0$$ |
The enthalpy of formation of ethane is
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Which one of the following compounds is used as a chemical in certain type of fire extinguishers
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Arrange the following carbocations in decreasing order of stability.
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Given below are two statements.
Statement I: The presence of weaker $$\pi$$-bonds make alkenes less stable than alkanes
Statement II: The strength of the double bond is greater than that of carbon-carbon single bond.
In the light of the above statements, choose the correct answer from the options given below.
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Which of the following reagents / reactions will convert 'A' to 'B'?
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Some gases are responsible for heating of atmosphere (green house effect). Identify from the following the gaseous species which does not cause it.
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In the industrial production of which of the following, molecular hydrogen is obtained as a bye product.
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For a first order reaction, the time required for completion of $$90\%$$ reaction is '$$x$$' times the half life of the reaction. The value of '$$x$$' is
(Given: $$\ln 10 = 2.303$$ and $$\log 2 = 0.3010$$)
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Which of the following chemical reactions represents Hall-Heroult Process?
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$$PCl_5$$ is well known but $$NCl_5$$ is not. Because,
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Transition metal complex with highest value of crystal field splitting $$(\Delta_0)$$ will be
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Hex-4-ene-2-ol on treatment with PCC gives 'A'. 'A' on reaction with sodium hypoiodite gives 'B', which on further heating with soda lime gives 'C'. The compound 'C' is
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The conversion of propan-1-ol to n-butylamine involves the sequential addition of reagents. The correct sequential order of reagents is
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Which of the following is not a condensation polymer?
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The structure shown below is of which well-known drug molecule?
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In the flame test of a mixture of salts, a green flame with blue centre was observed. Which one of the following cations may be present?
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At $$300$$ K, a sample of $$3.0$$ g of gas A occupies the same volume as $$0.2$$ g of hydrogen at $$200$$ K at the same pressure. The molar mass of gas A is ______ g mol$$^{-1}$$. (nearest integer) Assume that the behaviour of gases as ideal.
(Given: The molar mass of hydrogen ($$H_2$$) gas is $$2.0$$ g mol$$^{-1}$$.)
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$$PCl_5$$ dissociates as $$PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g)$$. $$5$$ moles of $$PCl_5$$ are placed in a $$200$$ litre vessel which contains $$2$$ moles of $$N_2$$ and is maintained at $$600$$ K. The equilibrium pressure is $$2.46$$ atm. The equilibrium constant $$K_p$$ for the dissociation of $$PCl_5$$ is ______ $$\times 10^{-3}$$. (nearest integer) (Given: $$R = 0.082$$ L atm K$$^{-1}$$ mol$$^{-1}$$; Assume ideal gas behaviour)
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Manganese (VI) has ability to disproportionate in acidic solution. The difference in oxidation states of two ions it forms in acidic solution is ______
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$$0.2$$ g of an organic compound was subjected to estimation of nitrogen by Dumas method in which volume of $$N_2$$ evolved (at STP) was found to be $$22.400$$ mL. The percentage of nitrogen in the compound is ______ [nearest integer] (Given: Molar mass of $$N_2$$ is $$28$$ g mol$$^{-1}$$, Molar volume of $$N_2$$ at STP: $$22.4$$ L)
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A company dissolves '$$x$$' amount of $$CO_2$$ at $$298$$ K in $$1$$ litre of water to prepare soda water. $$X = $$ ______ $$\times 10^{-3}$$ g. (nearest integer)
(Given: partial pressure of $$CO_2$$ at $$298$$ K $$= 0.835$$ bar. Henry's law constant for $$CO_2$$ at $$298$$ K $$= 1.67$$ kbar. Atomic mass of H, C and O is $$1, 12,$$ and $$6$$ g mol$$^{-1}$$, respectively)
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The resistance of a conductivity cell containing $$0.01$$ MKCl solution at $$298$$ K is $$1750$$ $$\Omega$$. If the conductivity of $$0.01$$ MKCl solution at $$298$$ K is $$0.152 \times 10^{-3}$$ S cm$$^{-1}$$, then the cell constant of the conductivity cell is ______ $$\times 10^{-3}$$ cm$$^{-1}$$
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When $$200$$ mL of $$0.2$$ M acetic acid is shaken with $$0.6$$ g of wood charcoal, the final concentration of acetic acid after adsorption is $$0.1$$ M. The mass of acetic acid adsorbed per gram of carbon is ______ g.
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(a) Baryte, (b) Galena, (c) Zinc blende and (d) Copper pyrites. How many of these minerals are sulphide based? ______
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Consider the above reaction. The number of $$\pi$$ electrons present in the product 'P' is ______
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In alanylglycylleucylalanylvaline the number of peptide linkages is: ______
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The sum of all real roots of equation $$\left(e^{2x} - 4\right)\left(6e^{2x} - 5e^x + 1\right) = 0$$ is
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Let $$x, y > 0$$. If $$x^3 y^2 = 2^{15}$$, then the least value of $$3x + 2y$$ is
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The number of solutions of the equation $$\cos\left(x + \frac{\pi}{3}\right) \cos\left(\frac{\pi}{3} - x\right) = \frac{1}{4}\cos^2 2x, x \in [-3\pi, 3\pi]$$ is:
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Let the area of the triangle with vertices $$A(1, \alpha)$$, $$B(\alpha, 0)$$ and $$C(0, \alpha)$$ be $$4$$ sq. units. If the points $$(\alpha, -\alpha)$$, $$(-\alpha, \alpha)$$ and $$(\alpha^2, \beta)$$ are collinear, then $$\beta$$ is equal to
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A particle is moving in the $$xy$$-plane along a curve $$C$$ passing through the point $$(3, 3)$$. The tangent to the curve $$C$$ at the point $$P$$ meets the $$x$$-axis at $$Q$$. If the $$y$$-axis bisects the segment $$PQ$$, then $$C$$ is a parabola with
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Let the maximum area of the triangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{4} = 1, a > 2$$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $$y$$-axis, be $$6\sqrt{3}$$. Then the eccentricity of the ellipse is:
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Consider the following statements:
$$A$$: Rishi is a judge.
$$B$$: Rishi is honest.
$$C$$: Rishi is not arrogant.
The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is
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Let the system of linear equations
$$x + y + az = 2$$
$$3x + y + z = 4$$
$$x + 2z = 1$$
have a unique solution $$(x^*, y^*, z^*)$$. If $$((a, x^*), (y^*, \alpha)$$ and $$(x^*, -y^*)$$ are collinear points, then the sum of absolute values of all possible values of $$\alpha$$ is:
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Let $$x \times y = x^2 + y^3$$ and $$(x \times 1) \times 1 = x \times (1 \times 1)$$. Then a value of $$2\sin^{-1}\left(\frac{x^4 + x^2 - 2}{x^4 + x^2 + 2}\right)$$ is
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Let $$f(x) = \begin{cases} \frac{\sin(x-|x|)}{x-|x|}, & x \in (-2, -1) \\ \max(2x, 3[|x|]), & |x| < 1 \\ 1, & \text{otherwise} \end{cases}$$
where $$[t]$$ denotes greatest integer $$\leq t$$. If $$m$$ is the number of points where $$f$$ is not continuous and $$n$$ is the number of points where $$f$$ is not differentiable, the ordered pair $$(m, n)$$ is:
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If $$y = \tan^{-1}\left(\sec x^3 - \tan x^3\right), \frac{\pi}{2} < x^3 < \frac{3\pi}{2}$$, then
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The number of distinct real roots of the equation $$x^7 - 7x - 2 = 0$$ is
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Let $$\lambda^*$$ be the largest value of $$\lambda$$ for which the function $$f_\lambda(x) = 4\lambda x^3 - 36\lambda x^2 + 36x + 48$$ is increasing for all $$x \in \mathbb{R}$$. Then $$f_{\lambda^*}(1) + f_{\lambda^*}(-1)$$ is equal to:
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The value of the integral $$\int_{-\pi/2}^{\pi/2} \frac{dx}{(1+e^x)(\sin^6 x + \cos^6 x)}$$ is equal to
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$$\lim_{n \to \infty} \left(\frac{n^2}{(n^2+1)(n+1)} + \frac{n^2}{(n^2+4)(n+2)} + \frac{n^2}{(n^2+9)(n+3)} + \cdots + \frac{n^2}{(n^2+n^2)(n+n)}\right)$$ is equal to
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The slope of normal at any point $$(x, y), x > 0, y > 0$$ on the curve $$y = y(x)$$ is given by $$\frac{x^2}{xy - x^2y^2 - 1}$$. If the curve passes through the point $$(1, 1)$$, then $$e \cdot y(e)$$ is equal to
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Let $$\mathbf{a}$$ and $$\mathbf{b}$$ be two unit vectors such that $$|\mathbf{a} + \mathbf{b}| + 2|\mathbf{a} \times \mathbf{b}| = 2$$. If $$\theta \in (0, \pi)$$ is the angle between $$\hat{a}$$ and $$\hat{b}$$, then among the statements:
$$(S1) : 2|\hat{a} \times \hat{b}| = |\hat{a} - \hat{b}|$$
$$(S2)$$ : The projection of $$\hat{a}$$ on $$(\hat{a} + \hat{b})$$ is $$\frac{1}{2}$$
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If the shortest distance between the lines $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{\lambda}$$ and $$\frac{x-2}{1} = \frac{y-4}{4} = \frac{z-5}{\frac{1}{\sqrt{3}}}$$, then the sum of all possible values of $$\lambda$$ is:
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Let the points on the plane $$P$$ be equidistant from the points $$(-4, 2, 1)$$ and $$(2, -2, 3)$$. Then the acute angle between the plane $$P$$ and the plane $$2x + y + 3z = 1$$ is
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A random variable $$X$$ has the following probability distribution:
| $$X$$ | 0 | 1 | 2 | 3 | 4 |
| $$P(X)$$ | $$k$$ | $$2k$$ | $$4k$$ | $$6k$$ | $$8k$$ |
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Let $$S = \{z \in \mathbb{C} : |z - 3| \leq 1$$ and $$z(4 + 3i) + \bar{z}(4 - 3i) \leq 24\}$$. If $$\alpha + i\beta$$ is the point in $$S$$ which is closest to $$4i$$, then $$25(\alpha + \beta)$$ is equal to ______.
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The number of 7-digit numbers which are multiples of 11 and are formed using all the digits $$1, 2, 3, 4, 5, 7$$ and $$9$$ is ______.
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The remainder on dividing $$1 + 3 + 3^2 + 3^3 + \ldots + 3^{2021}$$ by $$50$$ is ______.
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Let a circle $$C : (x - h)^2 + (y - k)^2 = r^2, k > 0$$, touch the $$x$$-axis at $$(1, 0)$$. If the line $$x + y = 0$$ intersects the circle $$C$$ at $$P$$ and $$Q$$ such that the length of the chord $$PQ$$ is $$2$$, then the value of $$h + k + r$$ is equal to ______.
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Let $$P_1$$ be a parabola with vertex $$(3, 2)$$ and focus $$(4, 4)$$ and $$P_2$$ be its mirror image with respect to the line $$x + 2y = 6$$. Then the directrix of $$P_2$$ is $$x + 2y =$$ ______.
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Let the hyperbola $$H : \frac{x^2}{a^2} - y^2 = 1$$ and the ellipse $$E : 3x^2 + 4y^2 = 12$$ be such that the length of latus rectum of $$H$$ is equal to the length of latus rectum of $$E$$. If $$e_H$$ and $$e_E$$ are the eccentricities of $$H$$ and $$E$$ respectively, then the value of $$12(e_H^2 + e_E^2)$$ is equal to ______.
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The sum of all the elements of the set $$\{\alpha \in \{1, 2, \ldots, 100\} : HCF(\alpha, 24) = 1\}$$ is ______.
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Let $$S = \left\{\begin{pmatrix} -1 & a \\ 0 & b \end{pmatrix} ; a, b \in \{1, 2, 3, \ldots 100\}\right\}$$ and let $$T_n = \{A \in S : A^{n(n+1)} = I\}$$. Then the number of elements in $$\bigcap_{n=1}^{100} T_n$$ is ______.
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The area (in sq. units) of the region enclosed between the parabola $$y^2 = 2x$$ and the line $$x + y = 4$$ is ______.
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In an examination, there are $$10$$ true-false type questions. Out of $$10$$, a student can guess the answer of $$4$$ questions correctly with probability $$\frac{3}{4}$$ and the remaining $$6$$ questions correctly with probability $$\frac{1}{4}$$. If the probability that the student guesses the answers of exactly $$8$$ questions correctly out of $$10$$ is $$\frac{27k}{4^{10}}$$, then $$k$$ is equal to ______.
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