A cylindrical wire of mass $$(0.4 \pm 0.01)$$ g has length $$(8 \pm 0.04)$$ cm and radius $$(6 \pm 0.03)$$ mm. The maximum error in its density will be
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A cylindrical wire of mass $$(0.4 \pm 0.01)$$ g has length $$(8 \pm 0.04)$$ cm and radius $$(6 \pm 0.03)$$ mm. The maximum error in its density will be
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Two forces having magnitude A and $$\dfrac{A}{2}$$ are perpendicular to each other. The magnitude of their resultant is:
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Dimension of $$\dfrac{1}{\mu_0 \varepsilon_0}$$ should be equal to
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Two projectiles A and B are thrown with initial velocities of 40 m s$$^{-1}$$ and 60 m s$$^{-1}$$ at angles 30° and 60° with the horizontal respectively. The ratio of their ranges respectively is $$(g = 10$$ m s$$^{-2}$$)
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At any instant the velocity of a particle of mass 500 g is $$(2t\hat{i} + 3t^2\hat{j})$$ m s$$^{-1}$$. If the force acting on the particle at $$t = 1$$ s is $$(\hat{i} + x\hat{j})$$ N. Then the value of $$x$$ will be:
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Given below are two statements:
Statement I: If $$E$$ be the total energy of a satellite moving around the earth, then its potential energy will be $$\dfrac{E}{2}$$.
Statement II: The kinetic energy of a satellite revolving in an orbit is equal to the half the magnitude of total energy $$E$$.
In the light of the above statements, choose the most appropriate answer from the options given below.
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The weight of a body on the earth is 400 N. Then weight of the body when taken to a depth half of the radius of the earth will be:
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An aluminium rod with Young's modulus $$Y = 7.0 \times 10^{10}$$ N m$$^{-2}$$ undergoes elastic strain of 0.04%. The energy per unit volume stored in the rod in SI unit
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An air bubble of volume 1 cm$$^3$$ rises from the bottom of a lake 40 m deep to the surface at a temperature of 12°C. The atmospheric pressure is $$1 \times 10^5$$ Pa, the density of water is 1000 kg m$$^{-3}$$ and $$g = 10$$ m s$$^{-2}$$. There is no difference of the temperature of water at the depth of 40 m and on the surface. The volume of air bubble when it reaches the surface will be
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Given below are two statements:
Statement I: If heat is added to a system, its temperature must increase.
Statement II: If positive work is done by a system in a thermodynamic process, its volume must increase.
In the light of the above statements, choose the correct answer from the options given below
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The engine of a train moving with speed 10 m s$$^{-1}$$ towards a platform sounds a whistle at frequency 400 Hz. The frequency heard by a passenger inside the train is: (Neglect air speed. Speed of sound in air = 330 m s$$^{-1}$$)
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Graphical variation of electric field due to a uniformly charged insulating solid sphere of radius R, with distance r from the centre O is represented by:
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In this figure the resistance of the coil of galvanometer G is 2 $$\Omega$$. The emf of the cell is 4 V. The ratio of potential difference across C$$_1$$ and C$$_2$$ is
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Certain galvanometers have a fixed core made of non magnetic metallic material. The function of this metallic material is
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A charge particle moving in magnetic field B, has the components of velocity along B as well as perpendicular to B. The path of the charge particle will be
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In a reflecting telescope, a secondary mirror is used to:
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Proton (P) and electron (e) will have same de-Broglie wavelength when the ratio of their momentum is (assume, $$m_p = 1849 m_e$$)
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For a nucleus $$^A_Z X$$ having mass number A and atomic number Z
A. The surface energy per nucleon $$(b_s) = -a_1 A^{2/3}$$.
B. The Coulomb contribution to the binding energy $$b_c = -a_2 \dfrac{Z(Z-1)}{A^{1/3}}$$.
C. The volume energy $$b_v = a_3 A$$
D. Decrease in the binding energy is proportional to surface area.
E. While estimating the surface energy, it is assumed that each nucleon interacts with 12 nucleons. ($$a_1, a_2$$ and $$a_3$$ are constants)
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For the logic circuit shown, the output waveform at Y is
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A TV transmitting antenna is 98 m high and the receiving antenna is at the ground level. If the radius of the earth is 6400 km, the surface area covered by the transmitting antenna is approximately:
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The momentum of a body is increased by 50%. The percentage increase in the kinetic energy of the body is ______%.
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The moment of inertia of a semicircular ring about an axis, passing through the center and perpendicular to the plane of ring, is $$\dfrac{1}{x}$$ MR$$^2$$, where R is the radius and M is the mass of the semicircular ring. The value of x will be ______.
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An air bubble of diameter 6 mm rises steadily through a solution of density 1750 kg m$$^{-3}$$ at the rate of 0.35 cm s$$^{-1}$$. The co-efficient of viscosity of the solution (neglect density of air) is ________ Pas (given, $$g = 10$$ m s$$^{-2}$$).
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An organ pipe 40 cm long is open at both ends. The speed of sound in air is 360 m s$$^{-1}$$. The frequency of the second harmonic is ________ Hz.
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An electric dipole of dipole moment is $$6.0 \times 10^{-6}$$ C m placed in a uniform electric field of $$1.5 \times 10^3$$ N C$$^{-1}$$ in such a way that dipole moment is along electric field. The work done in rotating dipole by 180° in this field will be ______ mJ.
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A current of 2 A flows through a wire of cross-sectional area 25.0 mm$$^2$$. The number of free electrons in a cubic meter are $$2.0 \times 10^{28}$$. The drift velocity of the electrons is ______ $$\times 10^{-6}$$ ms$$^{-1}$$
(given, charge on electron = $$1.6 \times 10^{-19}$$ C).
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The magnetic intensity at the centre of a long current carrying solenoid is found to be $$1.6 \times 10^3$$ A m$$^{-1}$$. If the number of turns is 8 per cm, then the current flowing through the solenoid is ______ A.
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An oscillating LC circuit consists of a 75 mH inductor and a 1.2 $$\mu$$F capacitor. If the maximum charge to the capacitor is 2.7 $$\mu$$C. The maximum current in the circuit will be ______ mA.
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Two vertical parallel mirrors A and B are separated by 10 cm. A point object O is placed at a distance of 2 cm from mirror A. The distance of the second nearest image behind mirror A from the mirror A is ______ cm.
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A nucleus with mass number 242 and binding energy per nucleon as 7.6 MeV breaks into two fragment each with mass number 121. If each fragment nucleus has binding energy per nucleon as 8.1 MeV, the total gain in binding energy is ______ MeV.
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The correct order of electronegativity for given elements 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: Butan-1-ol has higher boiling point than ethoxyethane.
Reason R: Extensive hydrogen bonding leads to stronger association of molecules.
In the light of the above statements, choose the correct answer from the options given below :
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$$2IO_3^- + xI^- + 12H^+ \rightarrow 6I_2 + 6H_2$$ What is the value of x?
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Given below are two statements:
Statement I: Lithium and Magnesium do not form superoxide
Statement II: The ionic radius of Li$$^+$$ is larger than ionic radius of Mg$$^{2+}$$
In the light of the above statements, choose the most appropriate answer from the questions given below:
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What is the purpose of adding gypsum to cement?
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Match List I with List II
List-I (Species) List-2 (Maximum allowed concentration in ppm in drinking water)
A. F$$^-$$ I. < 50 ppm
B. SO$$_4^{2-}$$ II. < 5 ppm
C. NO$$_3^-$$ III. < 2 ppm
D. Zn IV. < 500 ppm
Choose the correct answer from the options given below.
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The reaction occurs in which of the given galvanic cell?
$$\dfrac{1}{2}H_2(g) + AgCl(s) \rightleftharpoons H^+(aq) + Cl^-(aq) + Ag(s)$$
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The water gas on reacting with cobalt as a catalyst forms
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Which of the following represents the Freundlich adsorption isotherms?
(A)
(B)
(C)
(D)
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Which of the following metals can be extracted through alkali leaching technique?
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In chromyl chloride, the number of d-electrons present on chromium is same as in (Given at no. of Ti: 22, V: 23, Cr: 24, Mn: 25, Fe: 26)
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Which halogen is known to cause the reaction given below?
Which of the following complex is octahedral, diamagnetic and the most stable?
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The correct order of spin only magnetic moments for the following complex ions is
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Choose the halogen which is most reactive towards SN1 reaction in the given compounds (A, B, C & D)
A.
B.
C.
D.
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The major product formed in the following reaction is
CO$$_2$$H + CO$$_2$$Et compound treated with (i) LiBH$$_4$$/EtOH (ii) H$$_3$$O$$^+$$
Match List I with List II:
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Match List I with List II:
is reacted with reagents in List I to form products in List II
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Match List I with List II
List I List II
A. Saccharin I. High potency sweetener
B. Aspartame II. First artificial sweetening agent
C. Alitame III. Stable at cooking temperature
D. Sucralose IV. Unstable at cooking temperature
Choose the correct answer from the options given below :
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Sulphur (S) containing amino acids from the following are:
(a) isoleucine
(b) cysteine
(c) lysine
(d) methionine
(e) glutamic acid
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0.5 g of an organic compound (X) with 60% carbon will produce ______ $$\times 10^{-1}$$ g of CO$$_2$$ on complete combustion.
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The number of following statement/s which is/are incorrect is ______
A) Line emission spectra are used to study the electronic structure
B) The emission spectra of atoms in the gas phase show a continuous spread of wavelength from red to violet.
C) An absorption spectrum is like the photographic negative of an emission spectrum
D) The element helium was discovered in the sun by spectroscopic method
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The number of following factors which affect the percent covalent character of the ionic bond is______
A) Polarising power of cation
B) Extent of distortion of anion
C) Polarisability of the anion
D) Polarising power of anion
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Three bulbs are filled with CH$$_4$$, CO$$_2$$ and Ne as shown in the picture. The bulbs are connected through pipes of zero volume. When the stopcocks are opened and the temperature is kept constant throughout, the pressure of the system is found to be ______ atm. (Nearest integer).
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When a 60 W electric heater is immersed in a gas for 100 s in a constant volume container with adiabatic walls, the temperature of the gas rises by 5°C. The heat capacity of the given gas is JK$$^{-1}$$ (Nearest integer)
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The titration curve of weak acid vs. strong base with phenolphthalein as indicator is shown below. The $$K_{phenolphthalein} = 4 × 10^{−10}$$
Given: $$log 2 = 0.3$$
The number of following statement/s which is/are correct about phenolphthalein is ______
A. It can be used as an indicator for the titration of weak acid with weak base.
B. It begins to change colour at pH = 8.4
C. It is a weak organic base
D. It is colourless in acidic medium
Molar mass of the hydrocarbon (X) which on ozonolysis consumes one mole of O$$_3$$ per mole of (X) and gives one mole each of ethanal and propanone is ______ g mol$$^{-1}$$ (Molar mass of C: 12 g mol$$^{-1}$$, H: 1 g mol$$^{-1}$$)
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The vapour pressure vs. temperature curve for a solution solvent system is shown below.
The boiling point of the solvent is ______°C.
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The number of given statement/s which is/are correct is______
(A) The stronger the temperature dependence of the rate constant, the higher is the activation energy.
(B) If a reaction has zero activation energy, its rate is independent of temperature.
(C) The stronger the temperature dependence of the rate constant, the smaller is the activation energy.
(D) If there is no correlation between the temperature and the rate constant then it means that the reaction has negative activation energy.
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XeF$$_4$$ reacts with SbF$$_5$$ to form [XeF$$_m$$]$$^{n+}$$[SbF$$_y$$]$$^{2-}$$. $$m + n + y + z$$ = ?
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Let $$\alpha, \beta, \gamma$$ be the three roots of the equation $$x^3 + bx + c = 0$$ if $$\beta\gamma = 1 = -\alpha$$ then $$b^3 + 2c^3 - 3\alpha^3 - 6\beta^3 - 8\gamma^3$$ is equal to
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If for $$z = \alpha + i\beta$$, $$|z + 2| = z + 4(1+i)$$, then $$\alpha + \beta$$ and $$\alpha\beta$$ are the roots of the equation
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The number of arrangements of the letters of the word 'INDEPENDENCE' in which all the vowels always occur together is
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The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is
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Let $$S_K = \dfrac{1+2+\ldots+K}{K}$$ and $$\sum_{j=1}^n S_j^2 = \dfrac{n}{A}(Bn^2 + Cn + D)$$ where $$A, B, C, D \in N$$ and $$A$$ has least value, then
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If the coefficients of three consecutive terms in the expansion of $$(1+x)^n$$ are in the ratio 1:5:20 then the coefficient of the fourth term is
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Let $$C(\alpha, \beta)$$ be the circumcentre of the triangle formed by the lines $$4x + 3y = 69$$, $$4y - 3x = 17$$, and $$x + 7y = 61$$. Then $$(\alpha - \beta)^2 + \alpha + \beta$$ is equal to
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Let $$R$$ be the focus of the parabola $$y^2 = 20x$$ and the line $$y = mx + c$$ intersect the parabola at two points P and Q. Let the points G(10, 10) be the centroid of the triangle PQR. If $$c - m = 6$$, then $$PQ^2$$ is
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$$\lim_{x \to 0} \left(\left(\dfrac{1-\cos^2(3x)}{\cos^3(4x)}\right)\left(\dfrac{\sin^3(4x)}{(\log_e(2x+1))^5}\right)\right)$$ is equal to
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Negation of $$(p \to q) \to (q \to p)$$ is
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Let the number of elements in sets $$A$$ and $$B$$ be five and two respectively. Then the number of subsets of $$A \times B$$ each having at least 3 and at most 6 elements is
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Let $$\begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$. If $$|adj(adj(adj(2A)))| = (16)^n$$, then $$n$$ is equal to
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Let $$P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix}$$, $$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ and $$Q = PAP^T$$. If $$P^TQ^{2007}P = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ then $$2a + b - 3c - 4d$$ is equal to
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Let $$f(x) = \dfrac{\sin x + \cos x - \sqrt{2}}{\sin x - \cos x}$$, $$x \in [0, \pi] - \{\dfrac{\pi}{4}\}$$, then $$f\left(\dfrac{7\pi}{12}\right) f''\left(\dfrac{7\pi}{12}\right)$$ is equal to
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Let $$I(x) = \int \dfrac{x+1}{x(1+xe^x)^2} dx$$, $$x > 0$$. If $$\lim_{x \to \infty} I(x) = 0$$ then $$I(1)$$ is equal to
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The area of the region $$\{(x,y): x^2 \le y \le 8-x^2, y \le 7\}$$ is
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If the points with position vectors $$\alpha\hat{i} + 10\hat{j} + 13\hat{k}$$, $$6\hat{i} + 11\hat{j} + 11\hat{k}$$, $$\dfrac{9}{2}\hat{i} + \beta\hat{j} - 8\hat{k}$$ are collinear, then $$(19\alpha - 6\beta)^2$$ is equal to
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The shortest distance between the lines $$\dfrac{x-4}{4} = \dfrac{y+2}{5} = \dfrac{z+3}{3}$$ and $$\dfrac{x-1}{3} = \dfrac{y-3}{4} = \dfrac{z-4}{2}$$ is
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If the equation of the plane containing the line $$x + 2y + 3z - 4 = 0 = 2x + y - z + 5$$ and perpendicular to the plane $$\vec{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})$$ is $$ax + by + cz = 4$$ then $$(a - b + c)$$ is equal to
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In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found defective then the probability that it is manufactured by the machine C is
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The largest natural number $$n$$ such that $$3n$$ divides 66! is ______.
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Let $$[t]$$ denote the greatest integer $$\le t$$. If the constant term in the expansion of $$\left(3x^2 - \dfrac{1}{2x^5}\right)^7$$ is $$\alpha$$ then $$[\alpha]$$ is equal to ______.
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Consider a circle $$C_1: x^2 + y^2 - 4x - 2y = \alpha - 5$$. Let its mirror image in the line $$y = 2x + 1$$ be another circle $$C_2: 5x^2 + 5y^2 - 10fx - 10gy + 36 = 0$$. Let $$r$$ be the radius of $$C_2$$. Then $$\alpha + r$$ is equal to ______.
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Let the mean and variance of 8 numbers x, y, 10, 12, 6, 12, 4, 8 be 9 and 9.25 respectively. If $$x > y$$, then $$3x - 2y$$ is equal to ______.
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Let $$A = \{0, 3, 4, 6, 7, 8, 9, 10\}$$ and $$R$$ be the relation defined on $$A$$ such that $$R\{(x,y) \in A \times A: x-y$$ is odd positive integer or $$x-y = 2\}$$. The minimum number of elements that must be added to the relation $$R$$, so that it is a symmetric relation, is equal to ______.
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If $$a_\alpha$$ is the greatest term in the sequence $$a_n = \dfrac{n^3}{n^4 + 147}$$, $$n = 1, 2, 3, \ldots$$, then $$\alpha$$ is equal to ______.
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Let $$[t]$$ denote the greatest integer $$\le t$$. Then $$\dfrac{2}{\pi} \int_{\pi/6}^{5\pi/6} (8[\csc x] - 5[\cot x]) dx$$ is equal to ______.
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If the solution curve of the differential equation $$(y-2\log_e x)dx + (x\log_e x^2)dy = 0$$, $$x \gt 1$$ passes through the points $$(e, \dfrac{4}{3})$$ and $$(e^4, \alpha)$$, then $$\alpha$$ is equal to ______.
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Let $$\vec{a} = 6\hat{i} + 9\hat{j} + 12\hat{k}$$, $$\vec{b} = \alpha\hat{i} + 11\hat{j} - 2\hat{k}$$ and $$\vec{c}$$ be vectors such that $$\vec{a} \times \vec{c} = \vec{a} \times \vec{b}$$. If $$\vec{a} \cdot \vec{c} = -12$$, and $$\vec{c} \cdot (\hat{i} - 2\hat{j} + \hat{k}) = 5$$ then $$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k})$$ is equal to ______.
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Let $$\lambda_1, \lambda_2$$ be the values of $$\lambda$$ for which the points $$\left(\dfrac{5}{2}, 1, \lambda\right)$$ and $$(-2, 0, 1)$$ are at equal distance from the plane $$2x + 3y - 6z + 7$$. If $$\lambda_1 > \lambda_2$$ then the distance of the point $$(\lambda_1 - \lambda_2, \lambda_2, \lambda_1)$$ from the line $$\dfrac{x-5}{1} = \dfrac{y-1}{2} = \dfrac{z+7}{2}$$ is ______.
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