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Match List-I with List-II.
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A particle of mass $$m$$ projected with a velocity $$u$$ making an angle of $$30°$$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $$h$$ is :
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All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass $$2 \text{ kg}$$ is
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A particle is placed at the point $$A$$ of a frictionless track $$ABC$$ as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point $$B$$ is: (Take $$g = 10 \text{ m s}^{-2}$$).
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A spherical body of mass $$100 \text{ g}$$ is dropped from a height of $$10 \text{ m}$$ from the ground. After hitting the ground, the body rebounds to a height of $$5 \text{ m}$$. The impulse of force imparted by the ground to the body is given by: (given $$g = 9.8 \text{ m s}^{-2}$$)
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The gravitational potential at a point above the surface of earth is $$-5.12 \times 10^7 \text{ J kg}^{-1}$$ and the acceleration due to gravity at that point is $$6.4 \text{ m s}^{-2}$$. Assume that the mean radius of earth to be $$6400 \text{ km}$$. The height of this point above the earth's surface is:
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Young's modulus of material of a wire of length $$L$$ and cross-sectional area $$A$$ is $$Y$$. If the length of the wire is doubled and cross-sectional area is halved then Young's modulus will be:
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At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $$47°C$$?
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Two thermodynamical processes are shown in the figure. The molar heat capacity for process $$A$$ and $$B$$ are $$C_A$$ and $$C_B$$. The molar heat capacity at constant pressure and constant volume are represented by $$C_P$$ and $$C_V$$, respectively. Choose the correct statement.
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The electrostatic potential due to an electric dipole at a distance $$r$$ varies as :
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A potential divider circuit is shown in the figure below. The output voltage $$V_0$$ is
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An electric toaster has resistance of $$60 \text{ } \Omega$$ at room temperature $$(27°C)$$. The toaster is connected to a $$220 \text{ V}$$ supply. If the current flowing through it reaches $$2.75 \text{ A}$$, the temperature attained by toaster is around: (if $$\alpha = 2 \times 10^{-4} \text{ °C}^{-1}$$)
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Two insulated circular loop $$A$$ and $$B$$ radius $$a$$ carrying a current of $$I$$ in the anti clockwise direction as shown in figure. The magnitude of the magnetic induction at the centre will be:
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A series $$LR$$ circuit connected with an ac source $$E = (25 \sin 1000t) \text{ V}$$ has a power factor of $$\frac{1}{\sqrt{2}}$$. If the source of emf is changed to $$E = (20 \sin 2000t) \text{ V}$$, the new power factor of the circuit will be :
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Primary coil of a transformer is connected to $$200 \text{ V AC}$$. Primary and secondary turns of the transforms are $$100$$ and $$10$$ respectively. Secondary coil of transformer is connected to two series resistances as shown in figure. The output voltage $$(V_0)$$ is :
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The electric field of an electromagnetic wave in free space is represented as $$\vec{E} = E_0 \cos(\omega t - kz)\hat{i}$$. The corresponding magnetic induction vector will be :
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The diffraction pattern of a light of wavelength $$400 \text{ nm}$$ diffracting from a slit of width $$0.2 \text{ mm}$$ is focused on the focal plane of a convex lens of focal length $$100 \text{ cm}$$. The width of the $$1^{st}$$ secondary maxima will be :
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The work function of a substance is $$3.0 \text{ eV}$$. The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately:
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The ratio of the magnitude of the kinetic energy to the potential energy of an electron in the $$5^{th}$$ excited state of a hydrogen atom is :
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A Zener diode of breakdown voltage $$10 \text{ V}$$ is used as a voltage regulator as shown in the figure. The current through the Zener diode is
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The displacement and the increase in the velocity of a moving particle in the time interval of $$t$$ to $$(t + 1)$$ s are $$125 \text{ m}$$ and $$50 \text{ m s}^{-1}$$, respectively. The distance travelled by the particle in $$(t + 2)^{th}$$ s is __________ m.
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Consider a disc of mass $$5 \text{ kg}$$, radius $$2 \text{ m}$$, rotating with angular velocity of $$10 \text{ rad s}^{-1}$$ about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is _________ J.
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Each of the three blocks $$P$$, $$Q$$ and $$R$$ shown in the figure has a mass of $$3 \text{ kg}$$. Each of the wire $$A$$ and $$B$$ has a cross-sectional area $$0.005 \text{ cm}^2$$ and a Young's modulus $$2 \times 10^{11} \text{ N m}^{-2}$$. Neglecting friction, the longitudinal strain on wire $$B$$ is $$\_\_\_\_ \times 10^{-4}$$. (Take $$g = 10 \text{ m s}^{-2}$$)
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In a closed organ pipe, the frequency of fundamental note is $$30 \text{ Hz}$$. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to $$110 \text{ Hz}$$. If the organ pipe has a cross-sectional area of $$2 \text{ cm}^2$$, the amount of water poured in the organ tube is ________ g. (Take speed of sound in air is $$330 \text{ m s}^{-1}$$)
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A capacitor of capacitance $$C$$ and potential $$V$$ has energy $$E$$. It is connected to another capacitor of capacitance $$2C$$ and potential $$2V$$. Then the loss of energy is $$\frac{x}{3}E$$, where $$x$$ is ______.
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Two cells are connected in opposition as shown. Cell $$E_1$$ is of $$8 \text{ V}$$ emf and $$2 \text{ } \Omega$$ internal resistance; the cell $$E_2$$ is of $$2 \text{ V}$$ emf and $$4 \text{ } \Omega$$ internal resistance. The terminal potential difference of cell $$E_2$$ is ______ V.
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A ceiling fan having $$3$$ blades of length $$80 \text{ cm}$$ each is rotating with an angular velocity of $$1200 \text{ rpm}$$. The magnetic field of earth in that region is $$0.5 \text{ G}$$ and angle of dip is $$30°$$. The emf induced across the blades is $$N\pi \times 10^{-5} \text{ V}$$. The value of $$N$$ is ______.
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The horizontal component of earth's magnetic field at a place is $$3.5 \times 10^{-5} \text{ T}$$. A very long straight conductor carrying current of $$\sqrt{2} \text{ A}$$ in the direction from South east to North West is placed. The force per unit length experienced by the conductor is ________ $$\times 10^{-6} \text{ N m}^{-1}$$.
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The distance between object and its two times magnified real image as produced by a convex lens is $$45 \text{ cm}$$. The focal length of the lens used is ________ cm.
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An electron of hydrogen atom on an excited state is having energy $$E_n = -0.85 \text{ eV}$$. The maximum number of allowed transitions to lower energy level is _______.
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Given below are two statements:
Statement-I: The orbitals having same energy are called as degenerate orbitals.
Statement-II: In hydrogen atom, $$3p$$ and $$3d$$ orbitals are not degenerate orbitals.
In the light of the above statements, choose the most appropriate answer from the options given
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Given below are the two statements: one is labeled as Assertion (A) and the other is labeled as Reason (R).
Assertion (A): There is a considerable increase in covalent radius from $$N$$ to $$P$$. However from $$As$$ to $$Bi$$ only a small increase in covalent radius is observed.
Reason (R): covalent and ionic radii in a particular oxidation state increases down the group.
In the light of the above statement, choose the most appropriate answer from the options given below:
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Match List - I with List-II
Structure of $$4-\text{Methylpent}-2-\text{enal}$$ is:
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Example of vinylic halide is
Given below are two statements one is labeled as Assertion (A) and the other is labeled as Reason (R).
Assertion (A): $$CH_2=CH-CH_2-Cl$$ is an example of allyl halide
Reason (R): Allyl halides are the compounds in which the halogen atom is attached to $$sp^2$$ hybridised carbon atom.
In the light of the two above statements, choose the most appropriate answer from the options given below:
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Which of the following molecule/species is most stable?
Compound $$A$$ formed in the following reaction reacts with $$B$$ gives the product $$C$$. Find out $$A$$ and $$B$$.
In the given reactions identify the reagent $$A$$ and reagent $$B$$
What happens to freezing point of benzene when small quantity of naphthalene is added to benzene?
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Diamagnetic Lanthanoid ions are:
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Match List-I with List-II
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Choose the correct Statements from the following:
(A) Ethane-1,2-diamine is a chelating ligand.
(B) Metallic aluminium is produced by electrolysis of aluminium oxide in presence of cryolite.
(C) Cyanide ion is used as ligand for leaching of silver.
(D) Phosphine acts as a ligand in Wilkinson catalyst.
(E) The stability constants of $$Ca^{2+}$$ and $$Mg^{2+}$$ are similar with EDTA complexes.
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Aluminium chloride in acidified aqueous solution forms an ion having geometry
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This reduction reaction is known as:
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Following is a confirmatory test for aromatic primary amines. Identify reagent (A) and (B)
The final product $$A$$, formed in the following multistep reaction sequence is:
Given below are two statements:
Statement-I: The gas liberated on warming a salt with dil $$H_2SO_4$$, turns a piece of paper dipped in lead acetate into black, it is a confirmatory test for sulphide ion.
Statement-II: In statement-I the colour of paper turns black because of formation of lead sulphite.
In the light of the above statements, choose the most appropriate answer from the options given below:
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The Lassaigne's extract is boiled with dil $$HNO_3$$ before testing for halogens because,
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Sugar which does not give reddish brown precipitate with Fehling's reagent is:
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$$0.05 \text{ cm}$$ thick coating of silver is deposited on a plate of area $$0.05 \text{ m}^2$$. The number of silver atoms deposited on plate are _______ $$\times 10^{23}$$. (At mass $$Ag = 108$$, $$d = 7.9 \text{ g cm}^{-3}$$) Round off to the nearest integer.
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If IUPAC name of an element is "Unununnium" then the element belongs to $$n$$th group of periodic table. The value of $$n$$ is _______.
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The total number of molecular orbitals formed from $$2s$$ and $$2p$$ atomic orbitals of a diatomic molecule is _______.
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An ideal gas undergoes a cyclic transformation starting from the point $$A$$ and coming back to the same point by tracing the path $$A \rightarrow B \rightarrow C \rightarrow A$$ as shown in the diagram. The total work done in the process is _____ J.
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The pH at which $$Mg(OH)_2 [K_{sp} = 1 \times 10^{-11}]$$ begins to precipitate from a solution containing $$0.10 \text{ M } Mg^{2+}$$ ions is ______.
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$$2MnO_4^- + bI^- + cH_2O \rightarrow xI_2 + yMnO_2 + zOH^-$$
If the above equation is balanced with integer coefficients, the value of $$z$$ is ________.
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On a thin layer chromatographic plate, an organic compound moved by $$3.5 \text{ cm}$$, while the solvent moved by $$5 \text{ cm}$$. The retardation factor of the organic compound is ____________ $$\times 10^{-1}$$.
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The mass of sodium acetate $$(CH_3COONa)$$ required to prepare $$250 \text{ mL}$$ of $$0.35 \text{ M}$$ aqueous solution is _____ g. (Molar mass of $$CH_3COONa$$ is $$82.02 \text{ g mol}^{-1}$$) Round off to the nearest integer.
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The rate of first order reaction is $$0.04 \text{ mol L}^{-1} \text{s}^{-1}$$ at $$10$$ minutes and $$0.03 \text{ mol L}^{-1} \text{s}^{-1}$$ at $$20$$ minutes after initiation. Half life of the reaction is ______ minutes. (Given $$\log 2 = 0.3010$$, $$\log 3 = 0.4771$$)
Round off your answer to the nearest integer.
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The compound formed by the reaction of ethanal with semicarbazide contains _______ number of nitrogen atoms.
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If $$z = x + iy$$, $$xy \neq 0$$, satisfies the equation $$z^2 + i\bar{z} = 0$$, then $$|z^2|$$ is equal to :
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Let $$S_a$$ denote the sum of first $$n$$ terms an arithmetic progression. If $$S_{20} = 790$$ and $$S_{10} = 145$$, then $$S_{15} - S_5$$ is :
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If $$2\sin^3 x + \sin 2x \cos x + 4\sin x - 4 = 0$$ has exactly $$3$$ solutions in the interval $$\left[0, \frac{n\pi}{2}\right]$$, $$n \in \mathbb{N}$$, then the roots of the equation $$x^2 + nx + (n - 3) = 0$$ belong to :
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A line passing through the point $$A(9, 0)$$ makes an angle of $$30°$$ with the positive direction of $$x$$-axis. If this line is rotated about $$A$$ through an angle of $$15°$$ in the clockwise direction, then its equation in the new position is
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If the circles $$(x + 1)^2 + (y + 2)^2 = r^2$$ and $$x^2 + y^2 - 4x - 4y + 4 = 0$$ intersect at exactly two distinct points, then
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The maximum area of a triangle whose one vertex is at $$(0, 0)$$ and the other two vertices lie on the curve $$y = -2x^2 + 54$$ at points $$(x, y)$$ and $$(-x, y)$$ where $$y > 0$$ is :
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If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
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Let $$f : \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbb{R}$$ be a differentiable function such that $$f(0) = \frac{1}{2}$$. If $$\lim_{x \to 0} \frac{x \int_0^x f(t) dt}{e^{x^2} - 1} = \alpha$$, then $$8\alpha^2$$ is equal to :
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Let $$M$$ denote the median of the following frequency distribution.
Then $$20M$$ is equal to :
If $$f(x) = \begin{vmatrix} 2\cos^4 x & 2\sin^4 x & 3 + \sin^2 2x \\ 3 + 2\cos^4 x & 2\sin^4 x & \sin^2 2x \\ 2\cos^4 x & 3 + 2\sin^4 x & \sin^2 2x \end{vmatrix}$$ then $$\frac{1}{5}f'(0)$$ is equal to ________.
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Consider the system of linear equation $$x + y + z = 4\mu$$, $$x + 2y + 2\lambda z = 10\mu$$, $$x + 3y + 4\lambda^2 z = \mu^2 + 15$$, where $$\lambda, \mu \in \mathbb{R}$$. Which one of the following statements is NOT correct?
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If the domain of the function $$f(x) = \cos^{-1}\left(\frac{2 - |x|}{4}\right) + (\log_e(3 - x))^{-1}$$ is $$[-\alpha, \beta) - \{\gamma\}$$, then $$\alpha + \beta + \gamma$$ is equal to :
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Let $$g : \mathbb{R} \rightarrow \mathbb{R}$$ be a non constant twice differentiable such that $$g'\left(\frac{1}{2}\right) = g'\left(\frac{3}{2}\right)$$. If a real valued function $$f$$ is defined as $$f(x) = \frac{1}{2}[g(x) + g(2 - x)]$$, then
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The value of $$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^3}{(n^2 + k^2)(n^2 + 3k^2)}$$ is :
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The area (in square units) of the region bounded by the parabola $$y^2 = 4(x - 2)$$ and the line $$y = 2x - 8$$.
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Let $$y = y(x)$$ be the solution of the differential equation $$\sec x \, dy + \{2(1 - x)\tan x + x(2 - x)\}dx = 0$$ such that $$y(0) = 2$$. Then $$y(2)$$ is equal to :
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Let $$A(2, 3, 5)$$ and $$C(-3, 4, -2)$$ be opposite vertices of a parallelogram $$ABCD$$ if the diagonal $$\vec{BD} = \hat{i} + 2\hat{j} + 3\hat{k}$$ then the area of the parallelogram is equal to
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Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$ and $$\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$$ be two vectors such that $$|\vec{a}| = 1$$; $$\vec{a} \cdot \vec{b} = 2$$ and $$|\vec{b}| = 4$$. If $$\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$$, then the angle between $$\vec{b}$$ and $$\vec{c}$$ is equal to :
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Let $$(\alpha, \beta, \gamma)$$ be the foot of perpendicular from the point $$(1, 2, 3)$$ on the line $$\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. then $$19(\alpha + \beta + \gamma)$$ is equal to :
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Two integers $$x$$ and $$y$$ are chosen with replacement from the set $$\{0, 1, 2, 3, \ldots, 10\}$$. Then the probability that $$|x - y| > 5$$ is :
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Let $$\alpha, \beta \in \mathbb{R}$$ be roots of equation $$x^2 - 70x + \lambda = 0$$, where $$\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{Z}$$. If $$\lambda$$ assumes the minimum possible value, then $$\frac{(\sqrt{\alpha - 1} + \sqrt{\beta - 1})(\lambda + 35)}{|\alpha - \beta|}$$ is equal to :
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Let $$\alpha = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + \ldots$$ upto $$10$$ terms and $$\beta = \sum_{n=1}^{10} n^4$$. If $$4\alpha - \beta = 55k + 40$$, then $$k$$ is equal to _______.
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Number of integral terms in the expansion of $$\left\{7^{(1/2)} + 11^{(1/6)}\right\}^{824}$$ is equal to ______.
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Let the latus rectum of the hyperbola $$\frac{x^2}{9} - \frac{y^2}{b^2} = 1$$ subtend an angle of $$\frac{\pi}{3}$$ at the centre of the hyperbola. If $$b^2$$ is equal to $$\frac{l}{m}(1 + \sqrt{n})$$, where $$l$$ and $$m$$ are co-prime numbers, then $$l^2 + m^2 + n^2$$ is equal to __________.
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A group of $$40$$ students appeared in an examination of $$3$$ subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, $$20$$ students passed in Mathematics, $$25$$ students passed in Physics, $$16$$ students passed in Chemistry, at most $$11$$ students passed in both Mathematics and Physics, at most $$15$$ students passed in both Physics and Chemistry, at most $$15$$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.
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Let $$A = \{1, 2, 3, \ldots, 7\}$$ and let $$P(A)$$ denote the power set of $$A$$. If the number of functions $$f : A \rightarrow P(A)$$ such that $$a \in f(a), \forall a \in A$$ is $$m^n$$, $$m$$ and $$n \in \mathbb{N}$$ and $$m$$ is least, then $$m + n$$ is equal to ______.
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If the function $$f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases}$$ is differentiable on $$\mathbb{R}$$, then $$48(a + b)$$ is equal to _______.
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The value $$9\int_0^9 \left[\sqrt{\frac{10x}{x+1}}\right] dx$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is _____.
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Let $$y = y(x)$$ be the solution of the differential equation $$(1 - x^2)dy = \left[xy + (x^3 + 2)\sqrt{3(1 - x^2)}\right]dx$$, $$-1 < x < 1$$, $$y(0) = 0$$. If $$y\left(\frac{1}{2}\right) = \frac{m}{n}$$, $$m$$ and $$n$$ are coprime numbers, then $$m + n$$ is equal to __________.
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If $$d_1$$ is the shortest distance between the lines $$x + 1 = 2y = -12z$$, $$x = y + 2 = 6z - 6$$ and $$d_2$$ is the shortest distance between the lines $$\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$$, $$\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$$, then the value of $$\frac{32\sqrt{3} \, d_1}{d_2}$$ is :
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