If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
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If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
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One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is
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Let the position vectors of the vertices A, B and C of a tetrahedron ABCD be $$\widehat{i}+2\widehat{j}+\widehat{k},\widehat{i}+3\widehat{j}-2\widehat{k}$$ and $$2\widehat{i}+\widehat{j}-\widehat{k}$$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through of the triangle ABC at the point . If the length of AD is $$\frac{\sqrt{110}}{3}$$ and the volume of the tetrahedron is $$\frac{\sqrt{805}}{6\sqrt{2}}$$, then the position vector of is
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If A, B and $$(adj (A^{-1})+adj(B^{-1}))$$ are non-singular matrices of same order, then the inverse of $$A(adj(A^{-1}+adj(B^{-1}))^{-1}B$$, is equal to
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Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is
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Let a curve y=f(x) pass through the points (0,5) and $$(\log_{e}2,k)$$ . If the curve satisfies the differential equation $$2(3+y)e^{2x}dx-(7+e^{2x})dy=0$$ , then k is equal to
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If the function $$f(x)=\begin{cases}\frac{2}{x}\{\sin((k_1+1)x)+\sin(k_2-1)x\}, & x<0 \\4, & x=0 \\\frac{2}{x}\log_e\left(\frac{2+k_1x}{2+k_2x}\right), & x>0\end{cases}$$ is continuous at x=0, then $$k_1^2+k_2^2$$ is equal to
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If the line 3x-2y+12=0 intersects the parabola $$4y=3x^{2}$$ At the points A and B , then at the vertex of the parabola, the line segment AB subtends an angle equal to
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Let P be the foot of the perpendicular from the point Q(10,-3,-1) on the line $$\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$$. Then the area of the right angled triangle PQR , where R is the point (3,-2,1),is
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Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC, divides the arc AC such that $$\frac{\text{lenght of arc AB}}{\text{lenght of arc BC}}=\frac{1}{5}$$,and $$\overrightarrow{OC}=\alpha\overrightarrow{OA}+\beta\overrightarrow{OB}$$, then $$\alpha +\sqrt{2}(\sqrt{3}-1)\beta$$ is equal to
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Let $$f(x)=\log_{e}x$$ and $$g(x)=\frac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$$. Then the domain of $$f \circ g$$ is
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If the system of equations $$(\lambda-1)x+(\lambda-4)y+\lambda z=5 \\\lambda x+(\lambda-1)y+(\lambda-4)z=7 \\ (\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$$ has infinitely many solutions, then $$\lambda^{2}+\lambda$$ is equal to
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The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
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Let $$R=\left\{(1,2),(2,3),(3,3)\right\}$$ be a relation defined on the set $$\left\{1,2,3,4\right\}$$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:
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Let the area of a $$\triangle PQR$$ with vertices P(5,4), Q(-2,4) and R(a,b) be 35 square units. If its orthocenter and centroid are $$O(2,\frac{14}{5})$$ and C(c,d) respectively, then c+2d is equal to
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The value of $$\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{((log_{e}x)^{2}+1)^{-1}}}{e^{((log_{e}x)^{2}+1)^{-1}}+e^{((6-\log_{e}x)^{2}+1)^{-1}}}\right)dx$$ is
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Let $$\mid\frac{\overline{z}-i}{2\overline{z}+i}\mid=\frac{1}{3}, z\in C$$, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and $$(\alpha,0)$$ is 11 square units, then $$\alpha^{2}$$ equals:
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The value of $$\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ}\cot 70^{\circ}-1\right)$$ is
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Let $$I(x)=\int_{}^{} \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$$. If $$I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}}\right),b,c\in N$$, then 3(b+c) is equal to
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If $$\frac{\pi}{2}\leq x\leq \frac{3\pi}{4}$$, then $$\cos^{-1}\left(\frac{12}{13}\cos x+\frac{5}{13}\sin x\right)$$ is equal to
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Let the circle touch the line x-y+1=0, have the centre on the positive x -axis, and cut off a chord of length $$\frac{4}{\sqrt{13}}$$ along the line -3x+2y=1. Let H be the hyperbola $$\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$$ , whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then $$2\alpha^{2}+3\beta^{2}$$ is equal to ______
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If the equation $$a(b-c)x^{2}+b(c-a)x+c(a-b)=0$$ has equal roots, where a+c=15 and $$b=\frac{36}{5}$$, then $$a^{2}+c^{2}$$ is equal to
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If the set of all values of a, for which the equation $$5x^{3}-15x-a=0$$ has three distinct real roots, is the interval $$(\alpha, \beta)$$, then $$\beta-2\alpha $$ is equal to ______
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The sum of all rational terms in the expansion of $$(1+2^{1/2}+3^{1/2})^{6}$$ is equal to
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If the area of the larger portion bounded between the curves $$x^{2}+y^{2}=25$$ and y=|x-1| is $$\frac{1}{4}(b\pi+c),b,c\in N$$, then b+c is equal to
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A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r . The dipoles are made of opposite charge q separated by a distance 2a. For the charge particle at P not to experience any net
force, which of the following correctly describes the situation?
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A spherical surface of radius of curvature R, separates air from glass (refractive index = 1.5). The centre of curvature is in the glass medium. A point object $$'O'$$ placed in air on the optic axis of the surface, so that its real image is formed at $$'I'$$ inside glass. The line $$OI$$ intersects the spherical surface at $$P$$ and $$PO=PI$$ . The distance $$PO$$ equals to
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The position of a particle moving on x-axis is given by $$x(t)=A\sin t+B\cos^{2}t+Ct^{2}+D$$, where is time. The dimension of $$\frac{ABC}{D}$$ is
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Given a thin convex lens (refractive index $$\mu_{2}$$), kept in a liquid (refractive index $$\mu_{1},\mu_{1}<\mu_{2}$$) having radii of curvatures $$|R_{1}|$$ and $$|R_{2}|$$ . Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?
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Refer to the circuit diagram given in the figure. which of the following observations are correct?
A. Total resistance of circuit is $$6\Omega$$
B. Current in Ammeter is 1 A
C. Potential across AB is 4 Volts.
D. Potential across CD is 4 Volts
E. Total resistance of the circuit is $$8\Omega$$ .
Choose the correct answer from the options given below:
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Given below are two statements:
Statement I: The hot water flows faster than cold water
Statement II: Soap water has higher surface tension as compared to fresh water. In the light above statements, choose the correct answer from the options given below
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Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be
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The electric flux is $$\phi=\alpha \sigma+\beta\lambda$$ where $$\lambda$$ and $$\sigma$$ are linear and surface charge density, respectively. $$\left(\frac{\alpha}{\beta}\right)$$ represents
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A sub-atomic particle of mass $$10^{-30}$$kg is moving with a velocity $$2.21\times10^{6}$$ m/s . Under the matter wave consideration, the particle will behave closely like
$$\left(h=6.63\times10^{-34}J.s\right)$$
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Consider a moving coil galvanomenter (MCG):
A. The torsional constant in moving coil galvanometer has dimentions $$[ML^{2}T^{-2}]$$
B. Increasing the current sensitivity may not necessarily increase the voltage sensitivity.
C. If we increase number of turns (N) to its double (2N), then the voltage sensitivity doubles.
D. MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer.
E. Current sensitivity of MCG depends inversely on number of turns of coil. Choose the correct answer from the options given below:
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The electric field of an electromagnetic wave in free space is
$$\vec{E} = 57 \cos \left[ 7.5 \times 10^{6} t - 5 \times 10^{-3} (3x + 4y) \right] (4\hat{i} - 3\hat{j}) N / C$$. The associated magnetic field in Tesla is
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A gun fires a lead bullet of temperature 300 K into a wooden block. The bullet having melting temperature of 600 K penetrates into the block and melts down. If the total heat required for the process is 625 J , then the mass of the bullet is grams. (Latent heat of fusion of lead $$=2.5\times10^{4}JKg^{-1}$$ and specific heat capacity $$=125JKg^{-1}K^{-1}$$ of lead
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What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness 'h' in terms of the angle of incidence 'i' and angle of refraction 'r', if the glass slab is placed in air medium?
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A radioactive nucleus $$n_{2}$$ has 3 times the decay constant as compared to the decay constant of another radioactive nucleus $$n_{1}$$ . If initial number of both nuclei are the same, what is the ratio of number of nuclei of $$n_{2}$$ to the number of nuclei of $$n_{1}$$ , after one half-life of $$n_{1}$$ ?
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A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $$y\pi\times10^{-2}s$$, where the value of y is (Acceleration due to gravity, $$g=10 m/s^{2}$$, density of water $$=10^{3}kg/m^{3}$$
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Regarding self-inductance:
A. The self-inductance of the coil depends on its geometry.
B. Self-inductance does not depend on the permeability of the medium.
C. Self-induced e.m.f. opposes any change in the current in a circuit.
D. Self-inductance is electromagnetic analogue of mass in mechanics.
E. Work needs to be done against self-induced e.m.f. in establishing the current.
Choose the correct answer from the options given below:
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The motion of an airplane is represented by velocity-time graph as shown below. The distance covered by
airplane in the first 30.5 second is _______ km .
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Identify the valid statements relevant to the given circuit at the instant when the key is closed.
A. There will be no current through resistor R.
B. There will be maximum current in the connecting wires.
C. Potential difference between the capacitor plates A and B is minimum.
D. Charge on the capacitor plates is minimum.
Choose the correct answer from the options given below:
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A solid sphere of mass 'm' and radius 'r' is allowed to roll without slipping from the highest point of an inclined plane of length 'L' and makes an angle $$30^{\circ}$$ with the horizontal. The speed of the particle at the bottom of the plane is $$\upsilon_{1}$$ . If the angle of inclination is increased $$45^{\circ}$$ to while keeping L constant. Then the new speed of the sphere at the bottom of the plane is $$\upsilon_{2}$$ . The ratio $$\upsilon_1^2:\upsilon_2^2$$ is
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A positive ion A and a negative ion B has charges $$6.67\times10^{-19}C$$ and $$9.6\times10^{-10}C$$, and masses $$19.2\times10^{-27}Kg$$ and $$9\times10^{-27}Kg$$ respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is $$P\times10^{45}$$ , where the value of 10P is (Take $$\frac{1}{4\pi\epsilon_{0}}=9\times10^{9}NM^{2}C^{-1}$$ and universal gravitational constant as $$6.67\times10^{-11}NM^{2}Kg^{-2}$$)
Assume that charge may not be an integral multiple of electrons.
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In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of 8 A/s. At an instant when R is $$12\Omega$$ , the value of the current in the circuit will be ______ A.
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Two particles are located at equal distance from origin. The position vectors of those are represented by $$\overline{A}=2\widehat{i}+3n\widehat{j}+2\widehat{k}$$ and $$\overline{B}=2\widehat{i}-2\widehat{j}+4p\widehat{k}$$, respectively. If both the vectors are at right angle to each other, the value of $$n^{-1}$$ is _____ .
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An ideal gas initially at $$0^{\circ}$$C temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is 3/2, the change in temperature due to the thermodynamic process is _____ K.
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A force $$f=x^{2}y\widehat{i}+y^{2}\widehat{j}$$ acts on a particle in a plane x+y=10. The work done by this force during a displacement from (0,0) to (4m,2m) is Joule
(round off to the nearest integer)
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Given below are two statements:
Statement I: Fructose does not contain an aldehydic group but still reduces Tollen's reagent
Statement II: In the presence of base, fructose undergoes rearrangement to give glucose. In the light of the above statements, choose the correct answer from the options given below
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The complex that shows Facial - Meridional isomerism is:
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$$FeO_{4}^{2-}\xrightarrow{+2.0_{V}} Fe^{3+}\xrightarrow{0.8_{V}} Fe^{2+}\xrightarrow{-0.5_{V}}Fe^{0}$$ In the above diagram, the standard electrode potentials are given in volts (over the arrow). The value of $$E_{FeO_{4}^{2-}/Fe^{2+}}^{0}$$ is
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The element that does not belong to the same period of the remaining elements (modern periodic table) is:
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Match the LIST-I with LIST-II
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What amount of bromine will be required to convert 2 g of phenol into 2,4,6-tribromophenol? (Given molar mass in $$gmol^{-1}$$ of C, H, O, Br are 12, 1, 16, 80 respectively)
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Which among the following react with Hinsberg's reagent?
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The correct set of ions (aqueous solution) with same colour from the following is:
Given below are two statements:
Statement I: In Lassaigne's test, the covalent organic molecules are transformed into ionic compounds.
Statement II: The sodium fusion extract of an organic compound having N and S gives prussian blue colour with $$FeSO_{4}$$ and $$Na_{4}[Fe(CN)_{6}]$$ In the light of the above statements, choose
the correct answer from the options given below.
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Propane molecule on chlorination under photochemical condition gives two di-chloro products, " x " and " y ". Amongst " x " and " y ", " x " is an optically active molecule. How many tri-chloro products (consider only structural isomers) will be obtained from " x " when it is further treated with chlorine under the photochemical condition?
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$$CrCl_{3}\cdot xNH_{3}$$ can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of $$0.558^{\circ}C$$. Assuming 100% ionisation of this complex and coordination number of Cr is 6 , the complex will be (Given $$K_{f}$$ = 1.86 K kg $$mol^{-1}$$)
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Which of the following happens when $$NH_{4}OH$$ is added gradually to the solution containing 1 M $$A^{2+}$$ and $$1MB^{3+}$$ ions? Given : $$K_{sp}[A(OH)_{2}]= 9 \times 10^{-10}$$ and $$K_{sp}[B(OH)_{3}]= 27 \times 10^{-18}$$ at 298 K .
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The major product of the following reaction is:
Ice at $$-5^{\circ}C$$ is heated to become vapor with temperature of $$110^{\circ}C$$ at atmospheric pressure. The entropy change associated with this process can be obtained from
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The incorrect statement among the following is options .
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$$2.8 \times 10^{-3}$$ mol of $$CO_{2}$$ is left after removing $$10^{21}$$ molecules from its 'x' mg sample. The mass of $$CO_{2}$$ taken initially is Given: $$N_{A} = 6.02 \times 10^{23} mol^{-1}$$
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Match the LIST-I with LIST-II
Choose the correct answer from the options given below:
Heat treatment of muscular pain involves radiation of wavelength of about 900 nm . Which spectral line of H atom is suitable for this? Given : Rydberg constant $$R_{H}=10^{5} cm^{-1}$$, $$h=6.6 \times 10^{-34}$$ Js,$$c= 3\times 10^{8}$$ m/s
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The d- electronic configuration of an octahedral Co(II) complex having magnetic moment of 3.95 BM is:
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The correct stability order of the following species/molecules is:
The standard enthalpy and standard entropy of decomposition of $$N_{2}O_{4}$$ to $$NO_{2}$$ are 55.0 kJ $$mol^{-1}$$ and 175.0 J/K/mol respectively. The standard free energy change for this reaction at $$25^{\circ}C$$ in J $$mol^{-1}$$ is ______ (Nearest integer)
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For the thermal decomposition of $$N_{2}O_{5}(g)$$ at constant volume, the following table can be formed, for the reaction mentioned below. $$2 N_{2}O_{5}(g)\rightarrow 2 N_{2}O_{4}(g)+O_{2}(g)$$
$$x= .... \times 10^{-3}$$ atm [nearest integer] Given : Rate constant for the reaction is $$4.606 \times 10^{-2} s^{-1}.$$
During " S " estimation, 160 mg of an organic compound gives 466 mg of barium sulphate. The percentage of Sulphur in the given compound is _______ %. (Given molar mass in $$gmol^{-1}$$ of Ba : 137, S : 32, O : 16)
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If 1 mM solution of ethylamine produces pH=9, then the ionization constant $$(K_{b})$$ of ethylamine is $$10^{-x}$$. The value of is ______ (nearest integer). [The degree of ionization of ethylamine can be neglected with respect to unity.]
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Consider the following sequence of reactions to produce major product (A)
Molar mass of product (A) is $$gmol^{-1}$$. (Given molar mass in $$gmol^{-1}$$ of C : 12, H : 1, O : 16, Br : 80, N : 14, P : 31)
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