If two vectors $$\vec{P} = \hat{i} + 2m\hat{j} + m\hat{k}$$ and $$\vec{Q} = 4\hat{i} - 2\hat{j} + m\hat{k}$$ are perpendicular to each other. Then, the value of $$m$$ will be
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If two vectors $$\vec{P} = \hat{i} + 2m\hat{j} + m\hat{k}$$ and $$\vec{Q} = 4\hat{i} - 2\hat{j} + m\hat{k}$$ are perpendicular to each other. Then, the value of $$m$$ will be
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The frequency ($$\nu$$) of an oscillating liquid drop may depend upon radius ($$r$$) of the drop, density ($$\rho$$) of liquid and the surface tension ($$s$$) of the liquid as: $$\nu = r^a \rho^b s^c$$. The values of $$a$$, $$b$$ and $$c$$ respectively are
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The velocity-time graph of a body moving in a straight line is shown in figure.
The ratio of displacement and distance travelled by the body in time 0 to 10 s is
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A body of mass 200 g is tied to a spring of spring constant 12.5 N m$$^{-1}$$, while the other end of spring is fixed at point $$O$$. If the body moves about $$O$$ in a circular path on a smooth horizontal surface with constant angular speed 5 rad s$$^{-1}$$, then the ratio of extension in the spring to its natural length will be:
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Given below are two statements:
Statement I: Acceleration due to earth's gravity decreases as you go 'up' or 'down' from earth's surface.
Statement II: Acceleration due to earth's gravity is same at a height 'h' and depth 'd' from earth's surface, if $$h = d$$.
In the light of above statements, choose the most appropriate answer form 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 pendulum clock when taken to Mount Everest becomes fast.
Reason $$R$$: The value of g (acceleration due to gravity) is less at Mount Everest than its value on the surface of earth.
In the light of the above statements, choose the most appropriate answer from the options given below
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If the distance of the earth from Sun is $$1.5 \times 10^6$$ km, then the distance of an imaginary planet from Sun, if its period of revolution is 2.83 years 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): Steel is used in the construction of buildings and bridges.
Reason (R): Steel is more elastic and its elastic limit is high.
In the light of above statements, choose the most appropriate answer from the options given below
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In an Isothermal change, the change in pressure and volume of a gas can be represented for three different temperature; $$T_3 > T_2 > T_1$$ as:
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Let $$\gamma_1$$ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and $$\gamma_2$$ be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio $$\frac{\gamma_1}{\gamma_2}$$ is:
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The electric potential at the centre of two concentric half rings of radii $$R_1$$ and $$R_2$$, having same linear charge density $$\lambda$$ is
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A cell of emf 90 V is connected across series combination of two resistors each of 100 $$\Omega$$ resistance. A voltmeter of resistance 400 $$\Omega$$ is used to measure the potential difference across each resistor. The reading of the voltmeter will be:
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A long solenoid is formed by winding 70 turns cm$$^{-1}$$. If 2.0 A current flows, then the magnetic field produced inside the solenoid is _____.
($$\mu_0 = 4\pi \times 10^{-7}$$ T m A$$^{-1}$$)
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A metallic rod of length $$L$$ is rotated with an angular speed of $$\omega$$ normal to a uniform magnetic field $$B$$ about an axis passing through one end of rod as shown in the figure. The induced emf will be:
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The electric field and magnetic field components of an electromagnetic wave going through vacuum is described by $$E_x = E_0 \sin(kz - \omega t)$$ and $$B_y = B_0 \sin(kz - \omega t)$$. Then the correct relation between $$E_0$$ and $$B_0$$ is given by
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When a beam of white light is allowed to pass through convex lens parallel to principal axis, the different colours of light converge at different point on the principle axis after refraction. This is called:
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An $$\alpha$$ particle, a proton and an electron have the same kinetic energy. Which one of the following is correct in case of their de-Broglie wavelength:
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A photon is emitted in transition from $$n = 4$$ to $$n = 1$$ level in hydrogen atom. The corresponding wavelength for this transition is (given, $$h = 4 \times 10^{-15}$$ eV s)
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The logic gate equivalent to the given circuit diagram is:
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Match List I with List II
LIST I LIST II
A. AM Broadcast I. 88 - 108 MHz
B. FM Broadcast II. 540 - 1600 kHz
C. Television III. 3.7 - 4.2 GHz
D. Satellite Communication IV. 54 MHz - 590 MHz
Choose the correct answer from the options given below:
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A body of mass 1 kg begins to move under the action of a time dependent force $$\vec{F} = (t\hat{i} + 3t^2\hat{j})$$ N, where $$\hat{i}$$ and $$\hat{j}$$ are the unit vectors along $$x$$ and $$y$$ axis. The power developed by above force, at the time $$t = 2$$ s, will be _____ W.
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A uniform solid cylinder with radius $$R$$ and length $$L$$ has moment of inertia $$I_1$$, about the axis of cylinder. A concentric solid cylinder of radius $$R' = \frac{R}{2}$$ and length $$L' = \frac{L}{2}$$ is carved out of the original cylinder. If $$I_2$$ is the moment of inertia of the carved out portion of the cylinder then $$\frac{I_1}{I_2} =$$ _____.
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A spherical ball of radius 1 mm and density 10.5 g cc$$^{-1}$$ is dropped in glycerine of coefficient of viscosity 9.8 poise and density 1.5 g cc$$^{-1}$$. Viscous force on the ball when it attains constant velocity is $$3696 \times 10^{-x}$$ N. The value of $$x$$ is
(Given, $$g = 9.8$$ m s$$^{-2}$$ and $$\pi = \frac{22}{7}$$)
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A mass $$m$$, attached to free end of a spring executes SHM with a period of 1 s. If the mass is increased by 3 kg, the period of oscillation increases by one second. The value of mass $$m$$ is _____ kg.
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A parallel plate capacitor with air between the plate has a capacitance of 15 pF. The separation between the plate becomes twice and the space between them is filled with a medium of dielectric constant 3.5. Then the capacitance becomes $$\frac{x}{4}$$ pF. The value of $$x$$ is _____.
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If a copper wire is stretched to increase its length by 20%. The percentage increase in resistance of the wire is _____%.
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A single turn current loop in the shape of a right angle triangle with sides 5 cm, 12 cm, 13 cm is carrying a current of 2 A. The loop is in a uniform magnetic field of magnitude 0.75 T whose direction is parallel to the current in the 13 cm side of the loop. The magnitude of the magnetic force on the 5 cm side will be $$\frac{x}{130}$$ N. The value of $$x$$ is _____.
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Three identical resistors with resistance $$R = 12$$ $$\Omega$$ and two identical inductors with self inductance $$L = 5$$ mH are connected to an ideal battery with emf of 12 V as shown in figure. The current through the battery long after the switch has been closed will be _____ A.
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A convex lens of refractive index 1.5 and focal length 18 cm in air, is immersed in water. The change in the focal length of the lens will be _____ cm.
(Given refractive index of water = $$\frac{4}{3}$$)
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The energy released per fission of nucleus of $$^{240}$$X is 200 MeV. The energy released if all the atoms in 120 g of pure $$^{240}$$X undergo fission is _____ $$\times 10^{25}$$ MeV.
(Given $$N_A = 6 \times 10^{23}$$)
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What is the number of unpaired electron(s) in the highest occupied molecular orbital of the following species: N$$_2$$, N$$_2^+$$, O$$_2$$, O$$_2^+$$?
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Choose the correct representation of conductometric titration of benzoic acid vs sodium hydroxide.
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In which of the following reactions the hydrogen peroxide acts as a reducing agent?
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Identify the correct statements about alkali metals.
A. The order of standard reduction potential (M$$^+$$ | M) for alkali metal ions is Na > Rb > Li.
B. CsI is highly soluble in water.
C. Lithium carbonate is highly stable to heat.
D. Potassium dissolved in concentrated liquid ammonia is blue and paramagnetic.
E. All alkali metal hydrides are ionic solids.
Choose the correct 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: Beryllium has less negative value of reduction potential compared to the other alkaline earth metals.
Reason R: Beryllium has large hydration energy due to small size of Be$$^{2+}$$ but relatively large value of atomisation enthalpy.
In the light of the above statements, choose the most appropriate answer from the options given below.
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The number of s-electrons present in an ion with 55 protons in its unipositive state is
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Which will undergo deprotonation most readily in basic medium?
Given below are two statements:
Statement I:
under Clemmensen reduction conditions will give
Statement II:
under Wolff-Kishner reduction condition will give
In the light of the above statements, choose the correct answer from the options given below:
Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Benzene is more stable than hypothetical cyclohexatriene.
Reason R: The delocalized $$\pi$$ electron cloud is attracted more strongly by nuclei of carbon atoms.
In the light of the above statements, choose the correct answer from the options given below:
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The metal which is extracted by oxidation and subsequent reduction from its ore is:
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Which one amongst the following are good oxidising agents?
(a) Sm$$^{2+}$$
(b) Ce$$^{2+}$$
(c) Ce$$^{4+}$$
(d) Tb$$^{4+}$$
Choose the most appropriate answer from the options below:
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K$$_2$$Cr$$_2$$O$$_7$$ paper acidified with dilute H$$_2$$SO$$_4$$ turns green when exposed to
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Which of the following cannot be explained by crystal field theory?
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The hybridization and magnetic behaviour of cobalt ion in [Co(NH$$_3$$)$$_6$$]$$^{3+}$$ complex, respectively is
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Find out the major products from the following reactions.
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Given below are two statements:
Statement I: Pure Aniline and other arylamines are usually colourless.
Statement II: Arylamines get coloured on storage due to atmospheric reduction.
In the light of the above statements, choose the most appropriate answer from the options given below:
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Choose the correct colour of the product for the following reaction.
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Correct statement is:
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Match List I with List II
LIST I (Type) LIST II (Name)
A. Antifertility drug I. Norethindrone
B. Tranquilizer II. Meprobromate
C. Antihistamine III. Seldane
D. Antibiotic IV. Ampicillin
Choose the correct answer from the options given below:
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Following figure shows spectrum of an ideal black body at four different temperatures. The number of correct statement/s from the following is _____.
A. $$T_4 > T_3 > T_2 > T_1$$
B. The black body consists of particles performing simple harmonic motion.
C. The peak of the spectrum shifts to shorter wavelength as temperature increases.
D. $$\frac{T_1}{\nu_1} = \frac{T_2}{\nu_2} = \frac{T_3}{\nu_3}$$
E. The given spectrum could be explained using quantisation of energy
Sum of $$\pi$$-bonds present in peroxodisulphuric acid and pyrosulphuric acid is
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The number of statement's, which are correct with respect to the compression of carbon dioxide from point (a) in the Andrews isotherm from the following is _____.
A. Carbon dioxide remains as a gas upto point (b)
B. Liquid carbon dioxide appears at point (c)
C. Liquid and gaseous carbon dioxide coexist between points (b) and (c)
D. As the volume decreases from (b) to (c), the amount of liquid decreases
One mole of an ideal monoatomic gas is subjected to changes as shown in the graph. The magnitude of the work done (by the system or on the system) is _____ J (nearest integer)
Given: log 2 = 0.3, ln 10 = 2.3
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If the pKa of lactic acid is 5, then the pH of 0.005 M calcium lactate solution at 25°C is _____ $$\times 10^{-1}$$ (Nearest integer)
The Total pressure observed by mixing two liquid A and B is 350 mm Hg when their mole fractions are 0.7 and 0.3 respectively. The Total pressure becomes 410 mm Hg if the mole fractions are changed to 0.2 and 0.8 respectively for A and B. The vapour pressure of pure A is _____ mm Hg. (Nearest integer) Consider the liquids and solutions behave ideally
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The number of units, which are used to express concentration of solutions from the following is _____.
(Mass percent, Mole, Mole fraction, Molarity, ppm, Molality.)
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A student has studied the decomposition of a gas $$AB_3$$ at 25°C. He obtained the following data
| p (mm Hg) | 50 | 100 | 200 | 400 |
|---|---|---|---|---|
| Relative $$t_{1/2}$$ (s) | 4 | 2 | 1 | 0.5 |
The order of the reaction is
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The number of statement/s which are the characteristics of physisorption is _____.
A. It is highly specific in nature
B. Enthalpy of adsorption is high
C. It decreases with increase in temperature
D. It results into unimolecular layer
E. No activation energy is needed
Maximum number of isomeric monochloro derivatives which can be obtained from 2,2,5,5-tetramethylhexane by chlorination is _____
Total number of tripeptides possible by mixing of valine and proline is _____.
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The number of real solutions of the equation $$3\left(x^2 + \frac{1}{x^2}\right) - 2\left(x + \frac{1}{x}\right) + 5 = 0$$, is
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The value of $$\left(\frac{1 + \sin\frac{2\pi}{9} + i\cos\frac{2\pi}{9}}{1 + \sin\frac{2\pi}{9} - i\cos\frac{2\pi}{9}}\right)^3$$ is
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The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is
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If $$\frac{1^3 + 2^3 + 3^3 + \ldots \text{upto n terms}}{1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots \text{upto n terms}} = \frac{9}{5}$$ then the value of $$n$$ is
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If $$\binom{30}{1}^2 + 2\binom{30}{2}^2 + 3\binom{30}{3}^2 + \ldots + 30\binom{30}{30}^2 = \frac{\alpha \cdot 60!}{(30!)^2}$$, then $$\alpha$$ is equal to
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The locus of the middle points of the chords of the circle $$C_1: (x-4)^2 + (y-5)^2 = 4$$ which subtend an angle $$\theta_i$$ at the centre of the circle $$C_i$$, is a circle of radius $$r_i$$. If $$\theta_1 = \frac{\pi}{3}$$, $$\theta_3 = \frac{2\pi}{3}$$ and $$r_1^2 = r_2^2 + r_3^2$$, then $$\theta_2$$ is equal to
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The equations of sides $$AB$$ and $$AC$$ of a triangle $$ABC$$ are $$(\lambda + 1)x + \lambda y = 4$$ and $$\lambda x + (1 - \lambda)y + \lambda = 0$$ respectively. Its vertex $$A$$ is on the $$y$$-axis and its orthocentre is $$(1, 2)$$. The length of the tangent from the point $$C$$ to the part of the parabola $$y^2 = 6x$$ in the first quadrant is
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The set of values of $$a$$ for which $$\lim_{x \to a} ([x-5] - [2x+2]) = 0$$, where $$[\zeta]$$ denotes the greatest integer less than or equal to $$\zeta$$ is equal to
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Let $$p$$ and $$q$$ be two statements. Then $$\sim(p \wedge (p \to \sim q))$$ is equivalent to
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Let the six numbers $$a_1, a_2, \ldots, a_6$$ be in A.P. and $$a_1 + a_3 = 10$$. If the mean of these six numbers is $$\frac{19}{2}$$ and their variance is $$\sigma^2$$, then $$8\sigma^2$$ is equal to
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The number of square matrices of order 5 with entries from the set $$\{0, 1\}$$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is
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Let $$A$$ be a $$3 \times 3$$ matrix such that $$|adj(adj(adj \cdot A))| = 12^4$$. Then $$|A^{-1} adj A|$$ is equal to
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If the system of equations $$x + 2y + 3z = 3$$, $$4x + 3y - 4z = 4$$ and $$8x + 4y - \lambda z = 9 + \mu$$ has infinitely many solutions, then the ordered pair $$(\lambda, \mu)$$ is equal to
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If $$f(x) = \frac{2^{2x}}{2^{2x}+2}$$, $$x \in \mathbb{R}$$, then $$f\left(\frac{1}{2023}\right) + f\left(\frac{2}{2023}\right) + f\left(\frac{3}{2023}\right) + \ldots + f\left(\frac{2022}{2023}\right)$$ is equal to
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Let $$f(x)$$ be a function such that $$f(x + y) = f(x) \cdot f(y)$$ for all $$x, y \in \mathbb{N}$$. If $$f(1) = 3$$ and $$\sum_{k=1}^{n} f(k) = 3279$$, then the value of $$n$$ is
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If $$f(x) = x^3 - x^2f'(1) + xf''(2) - f'''(3)$$, $$x \in \mathbb{R}$$, then
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Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 - 3y^2)dx + 3xy$$ dy = 0, $$y(1) = 1$$. Then $$6y^2(e)$$ is equal to
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Let the sum of the coefficient of first three terms in the expansion of $$\left(x - \frac{3}{x^2}\right)^n$$; $$x \neq 0$$, $$n \in \mathbb{N}$$ be 376. Then, the coefficient of $$x^4$$ is equal to:
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Let $$S = \{\theta \in [0, 2\pi) : \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$$, then $$\sum_{\theta \in S} \sin^2\left(\theta + \frac{\pi}{4}\right)$$ is equal to
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The equations of the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ are $$2x + y = 0$$, $$x + py = 21a$$ ($$a \neq 0$$) and $$x - y = 3$$ respectively. Let $$P(2, a)$$ be the centroid of the triangle $$ABC$$, then $$(BC)^2$$ is equal to
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The minimum number of elements that must be added to relation $$R = \{(a,b), (b,c), (b,d)\}$$ on the set $$\{a, b, c, d\}$$, so that it is an equivalence relation is
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$$\displaystyle\int_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}} \frac{48}{\sqrt{9-4z^2}} dz$$ is equal to
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Let $$f$$ be a differentiable function defined on $$\left[0, \frac{\pi}{2}\right]$$ such that $$f(x) > 0$$ and $$f(x) + \int_0^x f(t)\sqrt{1 - (\log_e(f(t)))^2} dt = e$$ $$\forall x \in \left[0, \frac{\pi}{2}\right]$$, then $$\left\{6\log_e\left(f\left(\frac{\pi}{6}\right)\right)\right\}^2$$ is equal to
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If the area of the region bounded by the curves $$y^2 - 2y = -x$$ and $$x + y = 0$$ is $$A$$, then $$8A =$$
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Let $$\vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k}$$ and $$\vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k}$$. Let $$\vec{\beta_1}$$ be parallel to $$\vec{\alpha}$$ and $$\vec{\beta_2}$$ be perpendicular to $$\vec{\alpha}$$. If $$\vec{\beta} = \vec{\beta_1} + \vec{\beta_2}$$, then the value of $$5\vec{\beta_2} \cdot (\hat{i} + \hat{j} + \hat{k})$$ is
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Let $$\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}$$, $$\vec{b} = 3\hat{i} - 5\hat{j} - \lambda\hat{k}$$, $$\vec{a} \cdot \vec{c} = 7$$, $$2(\vec{b} \cdot \vec{c}) + 43 = 0$$, $$\vec{a} \times \vec{c} = \vec{b} \times \vec{c}$$, then $$|\vec{a} \cdot \vec{b}|$$ is equal to
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Let the plane containing the line of intersection of the planes $$P_1: x + (\lambda + 4)y + z = 1$$ and $$P_2: 2x + y + z = 2$$ pass through the points $$(0, 1, 0)$$ and $$(1, 0, 1)$$. Then the distance of the point $$(2\lambda, \lambda, -\lambda)$$ from the plane $$P_2$$ is
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If the foot of the perpendicular drawn from $$(1, 9, 7)$$ to the line passing through the point $$(3, 2, 1)$$ and parallel to the planes $$x + 2y + z = 0$$ and $$3y - z = 3$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to
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If the shortest distance between the lines $$\frac{x+\sqrt{6}}{2} = \frac{y-\sqrt{6}}{3} = \frac{z-\sqrt{6}}{4}$$ and $$\frac{x-\lambda}{3} = \frac{y-2\sqrt{6}}{4} = \frac{z+2\sqrt{6}}{5}$$ is 6, then square of sum of all possible values of $$\lambda$$ is
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The urns $$A$$, $$B$$ and $$C$$ contains 4 red, 6 black; 5 red, 5 black and $$\lambda$$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $$C$$ is 0.4, then the square of length of the side of largest equilateral triangle, inscribed in the parabola $$y^2 = \lambda x$$ with one vertex at vertex of parabola is
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