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An expression for a dimensionless quantity $$P$$ is given by $$P = \frac{\alpha}{\beta} \log_e\left(\frac{kT}{\beta x}\right)$$; where $$\alpha$$ and $$\beta$$ are constants, $$x$$ is distance; $$k$$ is Boltzmann constant and $$T$$ is the temperature. Then the dimensions of $$\alpha$$ will be
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A person is standing in an elevator. In which situation, he experiences weight loss?
An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero?
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A ball is released from rest from point $$P$$ of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point $$Q$$ is $$A$$ while angular position of point $$Q$$ is $$\alpha$$ with respect to point $$P$$. Which of the following graphs represent the correct relation between $$A$$ and $$\alpha$$ when ball goes from $$Q$$ to $$R$$?
A thin circular ring of mass $$M$$ and radius $$R$$ is rotating with a constant angular velocity $$2$$ rad s$$^{-1}$$ in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass $$m$$ be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in rad s$$^{-1}$$).
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The variation of acceleration due to gravity $$(g)$$ with distance $$(r)$$ from the center of the earth is correctly represented by (Given $$R$$ = radius of earth)
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The efficiency of a Carnot's engine, working between steam point and ice point, will be
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A thermally insulated vessel contains an ideal gas of molecular mass $$M$$ and ratio of specific heats $$1.4$$. Vessel is moving with speed $$v$$ and is suddenly brought to rest. Assuming no heat is lost to the surrounding and vessel temperature of the gas increases by :
($$R$$ = universal gas constant)
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Time period of a simple pendulum in a stationary lift is $$T$$. If the lift accelerates with $$\frac{g}{6}$$ vertically upwards then the time period will be
(Where $$g$$ = acceleration due to gravity)
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Two capacitors having capacitance $$C_1$$ and $$C_2$$ respectively are connected as shown in figure. Initially, capacitor $$C_1$$ is charged to a potential difference $$V$$ volt by a battery. The battery is then removed and the charged capacitor $$C_1$$ is now connected to uncharged capacitor $$C_2$$ by closing the switch $$S$$. The amount of charge on the capacitor $$C_2$$, after equilibrium, is
An aluminium wire is stretched to make its length, $$0.4\%$$ larger. The percentage change in resistance is
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Given below two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A): Non-polar materials do not have any permanent dipole moment.
Reason (R): When a non-polar material is placed in an electric field, the centre of the positive charge distribution of it's individual atom or molecule coincides with the centre of the negative charge distribution.
In the light of above statements, choose the most appropriate answer from the options given below.
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A proton and an alpha particle of the same velocity enter in a uniform magnetic field which is acting perpendicular to their direction of motion. The ratio of the radii of the circular paths described by the alpha particle and proton is
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The magnetic flux through a coil perpendicular to its plane is varying according to the relation $$\phi = (5t^3 + 4t^2 + 2t - 5)$$ Weber. If the resistance of the coil is $$5$$ ohm, then the induced current through the coil at $$t = 2$$ s will be,
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If Electric field intensity of a uniform plane electro magnetic wave is given as
$$E = -301.6\sin(kz-\omega t)\hat{a}_x + 452.4\sin(kz-\omega t)\hat{a}_y$$ V m$$^{-1}$$. Then, magnetic intensity $$H$$ of this wave in A m$$^{-1}$$ will be [Given : Speed of light in vacuum $$c = 3 \times 10^8$$ m s$$^{-1}$$, Permeability of vacuum $$\mu_0 = 4\pi \times 10^{-7}$$ N A$$^{-2}$$]
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A wave of frequency $$= 3$$ GHz, strikes a particle of size $$\left(\frac{1}{100}\right)^{th}$$ of $$\lambda$$, then this phenomenon is called as
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An electron with speed $$v$$ and a photon with speed $$c$$ have the same de-Broglie wavelength. If the kinetic energy and momentum of electron are $$E_e$$ and $$P_e$$ and that of photon are $$E_{ph}$$ and $$P_{ph}$$ respectively. Which of the following is correct?
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How many alpha and beta particles are emitted when Uranium $$_{92}U^{238}$$ decays to lead $$_{82}Pb^{206}$$?
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The I - V characteristics of a $$p - n$$ junction diode in forward bias is shown in the figure. The ratio of dynamic resistance, corresponding to forward bias voltage of $$2$$ V and $$4$$ V respective is
Choose the correct statement for amplitude modulation
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A ball of mass $$0.5$$ kg is dropped from the height of $$10$$ m. The height, at which the magnitude of velocity becomes equal to the magnitude of acceleration due to gravity, is ______ m. [Use $$g = 10$$ m s$$^{-2}$$]
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A fighter jet is flying horizontally at a certain altitude with a speed of $$200$$ m s$$^{-1}$$. When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $$\theta$$ with the horizontal, to hit the jet. If the bullet speed is $$400$$ m s$$^{-1}$$, the value of $$\theta$$ will be ______ °.
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The elongation of a wire on the surface of the earth is $$10^{-4}$$ m. The same wire of same dimensions is elongated by $$6 \times 10^{-5}$$ m on another planet. The acceleration due to gravity on the planet will be ______ m s$$^{-2}$$. (Take acceleration due to gravity on the surface of earth $$= 10$$ m s$$^{-2}$$)
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The elastic behaviour of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of $$5 \times 10^{-4}$$ is ______ kJ m$$^{-3}$$. Assume that material is elastic upto the linear strain of $$5 \times 10^{-4}$$.
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An ideal fluid of density $$800$$ kg m$$^{-3}$$, flows smoothly through a bent pipe (as shown in the figure) that tapers in cross-sectional area from $$a$$ to $$\frac{a}{2}$$. The pressure difference between the wide and narrow sections of pipe is $$4100$$ Pa. At the wider section, the velocity of the fluid is $$\frac{\sqrt{x}}{6}$$ m s$$^{-1}$$ for $$x =$$ ______. (Given $$g = 10$$ m s$$^{-2}$$)
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A $$10$$ $$\Omega$$, $$20$$ mH coil carrying constant current is connected to a battery of $$20$$ V through a switch. Now after switch is opened current becomes zero in $$100$$ $$\mu$$s. The average e.m.f. induced in the coil is ______ V.
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A $$110$$ V, $$50$$ Hz, AC source is connected in the circuit (as shown in figure). The current through the resistance $$55$$ $$\Omega$$, at resonance in the circuit, will be ______ A.
A light ray is incident, at an incident angle $$\theta_1$$, on the system of two plane mirrors $$M_1$$ and $$M_2$$ having an inclination angle $$75°$$ between them (as shown in figure). After reflecting from mirror $$M_1$$ it gets reflected back by the mirror $$M_2$$ with an angle of reflection $$30°$$. The total deviation of the ray will be ______ degree.
As per the given circuit, the value of current through the battery will be ______ A.
In a vernier callipers, each cm on the main scale is divided into $$20$$ equal parts. If tenth vernier scale division coincides with nineth main scale division. Then the value of vernier constant will be ______ $$\times 10^{-2}$$ mm
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A commercially sold conc. HCl is $$35\%$$ HCl by mass. If the density of this commercial acid is $$1.46$$ g/mL, the molarity of this solution is : (Atomic mass : Cl $$= 35.5$$ amu, H $$= 1$$ amu)
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If the radius of the $$3^{rd}$$ Bohr's orbit of hydrogen atom is $$r_3$$ and the radius of $$4^{th}$$ Bohr's orbit is $$r_4$$. Then
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Consider the ions/molecule $$O_2^+, O_2, O_2^-, O_2^{2-}$$. For increasing bond order the correct option is
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An evacuated glass vessel weighs $$40.0$$ g when empty, $$135.0$$ g when filled with a liquid of density $$0.95$$ g mL$$^{-1}$$ and $$40.5$$ g when filled with an ideal gas at $$0.82$$ atm at $$250$$ K. The molar mass of the gas in g mol$$^{-1}$$ is : (Given : $$R = 0.082$$ L atm K$$^{-1}$$ mol$$^{-1}$$)
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The correct order of melting point is
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Given below are two statements :
Statement I : In 'Lassaigne's Test', when both nitrogen and sulphur are present in an organic compound, sodium thiocyanate is formed.
Statement II : If both nitrogen and sulphur are present in an organic compound, then the excess of sodium used in sodium fusion will decompose the sodium thiocyanate formed to give NaCN and $$Na_2S$$.
In the light of the above statements, choose the most appropriate answer from the options given below
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Which will have the highest enol content?
The $$\left(\frac{\partial E}{\partial T}\right)_P$$ of different types of half cells are as follows :
A: $$1 \times 10^{-4}$$, B: $$2 \times 10^{-4}$$, C: $$0.1 \times 10^{-4}$$, D: $$0.2 \times 10^{-4}$$
(Where E is the electromotive force). Which of the above half cells would be preferred to be used as reference electrode
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Given below are two statements :
Statement I : According to the Ellingham diagram, any metal oxide with higher $$\Delta G°$$ is more stable than the one with lower $$\Delta G°$$.
Statement II : The metal involved in the formation of oxide placed lower in the Ellingham diagram can reduce the oxide of a metal placed higher in the diagram.
In the light of the above statements, choose the most appropriate answer from the options given below
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Consider the following reaction :
$$2HSO_4^-(aq) \xrightarrow[(2) Hydrolysis]{(1) Electrolysis \atop } 2HSO_4^- + 2H^+ + A$$
The dihedral angle in product $$A$$ in its solid phase at $$110$$ K is
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The correct order of melting points of hydrides of group 16 elements is
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Consider the following reaction :
$$A + alkali \to B$$ (Major Product)
If B is an oxoacid of phosphorus with no P - H bond, then A is
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Polar stratospheric clouds facilitate the formation of
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Choose the correct stability order of group 13 elements in their $$+1$$ oxidation state.
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$$(C_7H_5O_2)_2 \xrightarrow{h\nu} [X] \to 2\dot{C}_6H_5 + 2CO_2$$
Consider the above reaction and identify the intermediate 'X'
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Consider the following reaction sequence and identify the product B.
Among the following structures, which will show the most stable enamine formation? (Where Me is $$-CH_3$$)
Which of the following sets are correct regarding polymer:
(A) Copolymer : Buna-S
(B) Condensation polymer : Nylon-6, 6
(C) Fibres : Nylon-6, 6
(D) Thermosetting polymer : Terylene
(E) Homopolymers : Buna-N
Choose the correct answer from given options below
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A chemical which stimulates the secretion of pepsin is
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Which statement is not true with respect to nitrate ion test?
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On complete combustion $$0.30$$ g of an organic compound gave $$0.20$$ g of carbon dioxide and $$0.10$$ g of water. The percentage of carbon in the given organic compound is ______ (Nearest Integer)
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For complete combustion of methanol
$$CH_3OH(l) + \frac{3}{2}O_2(g) \to CO_2(g) + 2H_2O(l)$$
the amount of heat produced as measured by bomb calorimeter is $$726$$ kJ mol$$^{-1}$$ at $$27°$$C. The enthalpy of combustion for the reaction is $$-x$$ kJ mol$$^{-1}$$, where $$x$$ is ______
(Given : $$R = 8.3$$ J K$$^{-1}$$ mol$$^{-1}$$)
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$$50$$ mL of $$0.1$$ M $$CH_3COOH$$ is being titrated against $$0.1$$ M NaOH. When $$25$$ mL of NaOH has been added, the pH of the solution will be ______ $$\times 10^{-2}$$. (Nearest integer)
(Given : $$pK_a(CH_3COOH) = 4.76$$)
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Compound 'P' on nitration with dil. $$HNO_3$$ yields two isomers (A) and (B). These isomers can be separated by steam distillations. Isomers (A) and (B) show the intramolecular and intermolecular hydrogen bonding respectively. Compound (P) on reaction with conc. $$HNO_3$$ yields a yellow compound 'C', a strong acid. The number of oxygen atoms is present in compound 'C'
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A $$0.5$$ percent solution of potassium chloride was found to freeze at $$-0.24°$$C. The percentage dissociation of potassium chloride is (Nearest integer)
(Molal depression constant for water is $$1.80$$ K kg mol$$^{-1}$$ and molar mass of KCl is $$74.6$$ g mol$$^{-1}$$)
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A flask is filled with equal moles of A and B. The half lives of A and B are $$100$$ s and $$50$$ s respectively and are independent of the initial concentration. The time required for the concentration of A to be four times that of B is ______ s.
(Given : $$\ln 2 = 0.693$$)
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$$2.0$$ g of $$H_2$$ gas is adsorbed on $$2.5$$ g of platinum powder at $$300$$ K and $$1$$ bar pressure. The volume of the gas adsorbed per gram of the adsorbent is ______ mL
(Given : $$R = 0.083$$ L bar K$$^{-1}$$ mol$$^{-1}$$)
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The spin-only magnetic moment value of the most basic oxide of vanadium among $$V_2O_3$$, $$V_2O_4$$ and $$V_2O_5$$ is ______ B.M. (Nearest integer)
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The spin-only magnetic moment value of an octahedral complex among $$CoCl_3 \cdot 4NH_3$$, $$NiCl_2 \cdot 6H_2O$$ and $$PtCl_4 \cdot 2HCl$$, which upon reaction with excess of $$AgNO_3$$ gives 2 moles of AgCl is ______ B.M. (Nearest Integer)
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The number of oxygen present in a nucleotide formed from a base, that is present only in RNA is ______
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Let $$A = \left\{z \in C : \left|\frac{z+1}{z-1}\right| < 1\right\}$$ and $$B = \left\{z \in C : \arg\left(\frac{z-1}{z+1}\right) = \frac{2\pi}{3}\right\}$$. Then $$A \cap B$$ is
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The remainder when $$(2021)^{2023}$$ is divided by $$7$$ is
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Let $$R$$ be the point $$(3, 7)$$ and let $$P$$ and $$Q$$ be two points on the line $$x + y = 5$$ such that $$PQR$$ is an equilateral triangle. Then the area of $$\triangle PQR$$ is
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Let $$C$$ be a circle passing through the points $$A(2,-1)$$ and $$B(3,4)$$. The line segment $$AB$$ is not a diameter of $$C$$. If $$r$$ is the radius of $$C$$ and its centre lies on the circle $$(x-5)^2 + (y-1)^2 = \frac{13}{2}$$, then $$r^2$$ is equal to
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Let the normal at the point $$P$$ on the parabola $$y^2 = 6x$$ pass through the point $$(5, -8)$$. If the tangent at $$P$$ to the parabola intersects its directrix at the point $$Q$$, then the ordinate of the point $$Q$$ is
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$$\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\sin(\cos^{-1} x) - x}{1 - \tan(\cos^{-1} x)}$$ is equal to
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Let $$\Delta, \nabla \in \{\wedge, \vee\}$$ be such that $$p\nabla q \to ((p\Delta q)\nabla r)$$ is a tautology. Then $$(p\nabla q) \Delta r$$ is logically equivalent to
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The mean of the numbers $$a, b, 8, 5, 10$$ is $$6$$ and their variance is $$6.8$$. If $$M$$ is the mean deviation of the numbers about the mean, then $$25M$$ is equal to
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Let $$A$$ be a $$3 \times 3$$ invertible matrix. If $$|\text{adj}(24A)| = |\text{adj}(3 \text{ adj}(2A))|$$, then $$|A|^2$$ is equal to
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The ordered pair $$(a, b)$$, for which the system of linear equations
$$3x - 2y + z = b$$
$$5x - 8y + 9z = 3$$
$$2x + y + az = -1$$
has no solution, is
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Let $$f(x) = \frac{x-1}{x+1}, x \in R - \{0, -1, 1\}$$. If $$f^{n+1}(x) = f(f^n(x))$$ for all $$n \in N$$, then $$f^6(6) + f^7(7)$$ is equal to
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$$f, g : R \to R$$ be two real valued functions defined as $$f(x) = \begin{cases} -|x+3| & x < 0 \\ e^x & x \geq 0 \end{cases}$$ and
$$g(x) = \begin{cases} x^2 + k_1 x & x < 0 \\ 4x + k_2 & x \geq 0 \end{cases}$$, where $$k_1$$ and $$k_2$$ are real constants. If $$gof$$ is differentiable at $$x = 0$$, then $$gof(-4) + gof(4)$$ is equal to
The sum of the absolute minimum and the absolute maximum values of the function
$$f(x) = |3x - x^2 + 2| - x$$ in the interval $$[-1, 2]$$ is
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Let $$S$$ be the set of all the natural numbers, for which the line $$\frac{x}{a} + \frac{y}{b} = 2$$ is a tangent to the curve $$\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$$ at the point $$(a, b), ab \neq 0$$. Then
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Let $$f(x) = 2\cos^{-1}x + 4\cot^{-1}x - 3x^2 - 2x + 10, x \in [-1, 1]$$. If $$[a, b]$$ is the range of the function, then $$4a - b$$ is equal to
The area bounded by the curve $$y = |x^2 - 9|$$ and the line $$y = 3$$ is
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If $$\vec{a} \cdot \vec{b} = 1, \vec{b} \cdot \vec{c} = 2$$ and $$\vec{c} \cdot \vec{a} = 3$$, then the value of $$\left[\vec{a} \times (\vec{b} \times \vec{c}), \vec{b} \times (\vec{c} \times \vec{a}), \vec{c} \times (\vec{b} \times \vec{a})\right]$$ is
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If the two lines $$l_1 : \frac{x-2}{3} = \frac{y+1}{-2}, z = 2$$ and $$l_2 : \frac{x-1}{1} = \frac{2y+3}{\alpha} = \frac{z+5}{2}$$ are perpendicular, then an angle between the lines $$l_2$$ and $$l_3 : \frac{1-x}{3} = \frac{2y-1}{-4} = \frac{z}{4}$$ is
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Let the plane $$2x + 3y + z + 20 = 0$$ be rotated through a right angle about its line of intersection with the plane $$x - 3y + 5z = 8$$. If the mirror image of the point $$(2, -\frac{1}{2}, 2)$$ in the rotated plane is $$B(a, b, c)$$, then
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Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is
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The sum of the cubes of all the roots of the equation $$x^4 - 3x^3 - 2x^2 + 3x + 1 = 0$$ is ______
There are ten boys $$B_1, B_2, \ldots, B_{10}$$ and five girls $$G_1, G_2, \ldots G_5$$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $$B_1$$ and $$B_2$$ together should not be the members of a group, is ______
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Let $$A = \sum_{i=1}^{10}\sum_{j=1}^{10} \min\{i, j\}$$ and $$B = \sum_{i=1}^{10}\sum_{j=1}^{10} \max\{i, j\}$$. Then $$A + B$$ is equal to ______
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If $$\sin^2(10°)\sin(20°)\sin(40°)\sin(50°)\sin(70°) = \alpha - \frac{1}{16}\sin(10°)$$, then $$16 + \alpha^{-1}$$ is equal to ______
Let the common tangents to the curves $$4(x^2 + y^2) = 9$$ and $$y^2 = 4x$$ intersect at the point $$Q$$. Let an ellipse, centered at the origin $$O$$, has lengths of semi-minor and semi-major axes equal to $$OQ$$ and $$6$$, respectively. If $$e$$ and $$l$$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $$\frac{l}{e^2}$$ is equal to ______
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Let $$A = \{n \in N : H.C.F.(n, 45) = 1\}$$ and let $$B = \{2k : k \in \{1, 2, \ldots, 100\}\}$$. Then the sum of all the elements of $$A \cap B$$ is ______
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Let $$f(x) = \max\{|x+1|, |x+2|, \ldots, |x+5|\}$$. Then $$\int_{-6}^{0} f(x)dx$$ is equal to ______
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The value of the integral $$\frac{48}{\pi^4}\int_0^{\pi}\left(\frac{3\pi x^2}{2} - x^3\right)\frac{\sin x}{1 + \cos^2 x}dx$$ is equal to ______
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Let the solution curve $$y = y(x)$$ of the differential equation $$(4 + x^2)dy - 2x(x^2 + 3y + 4)dx = 0$$ pass through the origin. Then $$y(2)$$ is equal to ______
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Let $$S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}$$. Let $$y = y(x), x \in S$$, be the solution curve of the differential equation $$\frac{dy}{dx} = \frac{1}{1 + \sin 2x}, y\left(\frac{\pi}{4}\right) = \frac{1}{2}$$. If the sum of abscissas of all the points of intersection of the curve $$y = y(x)$$ with the curve $$y = \sqrt{2}\sin x$$ is $$\frac{k\pi}{12}$$, then $$k$$ is equal to ______
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