The integral \[\int\limits_{1}^{e}{\left\{ {{\left( \frac{x}{e} \right)}^{2x}}-{{\left( \frac{e}{x} \right)}^{x}} \right\}}{{\log }_{e}}\] is equal to:
Let \[\vec{a},\vec{b},\]be three unit vectors, out of which vectors \[\vec{b}\] and \[\vec{c}\] are non-parallel If \[\alpha \] and \[\beta \] are the angles which vector \[\vec{a}\]makes with vectors \[\vec{b}\] and \[\vec{c}\] respectively and \[\vec{a}\times \left( \vec{b}\times \vec{c} \right)=\frac{1}{2}\vec{b},\] then \[\left| a-\beta \right|\] is Equal to:
If a curve passes through the point \[(1,-2)\] and has slope of the tangent at any point \[(x,y)\] on it as \[\frac{{{x}^{2}}-2y}{x},\] then the curve also passes through the point:
If an angle between the line, \[\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}\]and the plane, \[x-2y-kz=3\]is \[{{\cos }^{-1}}\left( \frac{2\sqrt{2}}{3} \right),\] then a value of k is:
\[\underset{x+\infty}{\mathop{\lim}}\,\left( \frac{n}{{{n}^{2}}+{{1}^{2}}}+\frac{n}{{{n}^{2}}+{{n}^{2}}}+\frac{n}{{{n}^{2}}+{{3}^{2}}}+...\frac{1}{5n} \right)\] is equal to:
In a game, a man wins Rs.100 if he gets 5 or 6 on a throw of a fair die and loses Rs.50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is:
If a straight line passing through the point \[P(-3,4)\] is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is:
There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is:
Let S be the set of all real values of \[\lambda \] such that a plane passing through the points \[(-{{\lambda }^{2}},1,1),\] \[(1,-{{\lambda }^{2}},1)\] also passes through the point \[(-1,-1,1).\]Then S is equal to:
Let \[|{{z}_{1}}|\]and\[|{{z}_{2}}|\]be two complex numbers satisfying \[\left| {{z}_{1}} \right|=\,9\]and \[\left| {{z}_{2}}-3-4i \right|=4.\] Then the minimum value of \[\left| {{z}_{2}}-{{z}_{2}} \right|\]is:
In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. of these students is selected at random, the probability that the student selected has opted for NCC nor for NSS is:
The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3, 4 and 4; then the absolute value of the difference of the other two observations, is:
Inner surface of a cone is coated with a reflecting layer forms a conical mirror. A small point source S is placed over the axis of this cone. Minimum angle a of the cone for which rays emitted by source S will be reflected from conical surface only once is
A system consists of three coins that can come up either head or tail. Coins are tossed and that results in two tails and one head. Entropy of three coins system will be
Two sound waves A and B have intensities of \[10\frac{\mu W}{c{{m}^{2}}}\] and \[500\frac{\mu W}{c{{m}^{2}}}.\] Difference in their intensity levels is
Electric field in a region is \[E=\frac{a\,(x\hat{i}+y\hat{j}+z\hat{k})}{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\] where, a is a positive constant. Charge enclosed in a sphere of radius R and centre at origin is
A positively charged ion is accelerated by using a voltage V and then it is allowed to enter a region of perpendicular magnetic field. Frequency of rotation of ion is x If accelerating voltage is increased four times, then frequency becomes
An alloy weighing 1.05 g of \[\operatorname{Pb}-Ag\] was dissolved in desired amount of \[{{\operatorname{HNO}}_{3}}\] and the volume was made 350 mL. An Ag electrode was dipped in solution and \[{{E}_{cell}}\] of the cell \[Pt,\,\underset{1\,atm}{\mathop{{{H}_{2}}}}\,\underset{1M}{\mathop{\left| {{H}^{+}} \right|}}\,A{{g}^{+}}|Ag\] was 0.503 V at 298 K. The percentage of lead in alloy is: [Given\[E_{Ag+/Ag}^{o}~= 0.80 V\]]
In nitroprusside ion the iron and NO exist as \[{{\operatorname{Fe}}^{\operatorname{II}}}\] and \[{{\operatorname{NO}}^{+}}\] rather than \[{{\operatorname{Fe}}^{III}}\] and \[\operatorname{NO}\]. These forms can be differentiated by
The heats of atomization of \[P{{H}_{3}}(g)\] and \[{{P}_{2}}{{H}_{4}}(g)\] are \[954 kJ mo{{l}^{-1}}\] and \[1485 kJ mo{{l}^{-1}}\] respectively. The P-P bond energy in \[\operatorname{kJ} mo{{l}^{-1}}\]is
The edge length of unit cell of a metal having molecular weight \[75 g mo{{l}^{-1}}\] is \[5\,\overset{\text{o}}{\mathop{\text{A}}}\,\] which crystallizes in cubic lattice. If the density is 2g/cc then find the radius of metal atom. \[({{N}_{A}}=6\times 1{{0}^{23}}).\] Give the answer in pm.
Resistance of a conductivity cell filled with a solution of an electrolyte of concentration 0.1M is \[100\Omega \]. The conductivity of this solution is\[1.29 S {{m}^{-1}}\]. Resistance of the same cell when filled with 0.02 M of the same solution is \[520\Omega \]. The molar conductivity of 0.02 M solution of electrolyte will be
The ratio of the frequency corresponding to the third line in Lyman series of hydrogen atomic spectrum to that of the first line in Balmer series of \[L{{i}^{2+}}\]spectrum is
A complex cation is formed by Pt (in some oxidation state) with ligands (in proper number so that coordination number of Pt becomes six). Which of the following can be its correct IUPAC name?
A)
Diammineethylenediaminedithiocyanato-S-Platinum (II)
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B)
Diammineethylenediaminedithiocyanato- S-Palatinate (IV) ion
doneclear
C)
Diammineethylenediaminedithiocyanato-S-Platinum (IV) ion
doneclear
D)
Diamminebis (ethyienediamnie) dithiocyanato-S-platinum (IV) ion
The intermediate lobe of the pituitary gland produces a secretion which causes a dramatic darkening of the skin of many fishes, amphibians and reptiles. It is -
If a circle of radius R passes through the origin\[O\] and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from 0 on AB is:
If the function/given by \[f(x)={{x}^{3}}-3\,(a-2){{x}^{2}}\]\[+3ax+7,\] for some \[a\in R\] is increasing in \[(0,1]\] and decreasing in \[[1,5),\] then a root of the equation,\[\frac{f(4)-14}{{{(x-1)}^{2}}}=0(x\ne 1)is:\]
Let Z be the set of integers. If\[A-\left\{ x\in Z:{{2}^{(x+2)({{x}^{2}}-5x+6)}}=1 \right\}\] and \[B=\left\{ x\in Z:-3<2x-1<9 \right\}\] then the number of subsets of the set \[A\times B,\]is:
Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If \[\Delta S'BS\]is a right angled triangle with right angle at B and \[(\Delta S'BS)=8sq.\]units, then the length of a latus rectum of the ellipse is:
If the angle of elevation of a cloud from which is 25 m above a lake be \[35{}^\circ \] and the depression of reflection of the cloud in the P be \[60{}^\circ ,\] then the height of the cloud (in the surface of the lake is:
If the sum of the first 15 terms of the series \[{{\left( \frac{3}{4} \right)}^{3}}+{{\left( 1\frac{1}{2} \right)}^{3}}+{{\left( 2\frac{1}{4} \right)}^{3}}+{{3}^{3}}+{{\left( 3\frac{3}{4} \right)}^{3}}+...\]is equivalent to : 225 k, then k is equal to:
There is a hole in centre of a thin circular biconvex lens of focal length 4 cm. Diameter of hole is half of diameter of the lens. A point source of light is placed 9 cm from the wall and lens is placed in between, such that a single circular illuminated spot with sharp bright edges is formed on the wall. Distance of lens and source must be
A tumor on a person's leg has a mass of 3g. What is the minimum activity a radiation source can have, if it is to deliver a dose of 10 Gy (Gray) to the tumor in 14 min. Assume each disintegration provides an energy of \[0.7\text{ }MeV\]to the tumor.
An electron typically spends about \[{{10}^{-8}}\] in an excited state before it drops to a lower state by emitting a photon. Number of revolutions made by an electron in \[n=2,\] Bohr's orbit in \[1.00\times {{10}^{-\,8}}s\] is
Let R = radius of a spherical refracting surface and l= least distance between conjugate focii. Then, refractive index of medium of refracting surface is
For an ideal gas, let \[{{C}_{P}}=\alpha +cT\] and \[{{C}_{V}}=b+cT.\] where, a, b and c are constants. T-V relation that holds in adiabatic expansion is
A driver is caught crossing a red light. The driver claims to the judge the colour she actually saw was green \[({{f}_{G}}=5.6\times {{10}^{14}}Hz)\]and not red\[({{f}_{R}}=4.80\times {{10}^{14}}Hz)\]because of the Doppler effect.
The judge accepts this explaination and instead fines her for speeding at the rate of 1Rs. for each kilometer per hour she exceeded the speed limit erf 100 km/h.
The time taken for \[75%\] reaction of P is twice the time taken for \[50%\] reaction of P. The concentration of Q varies with reaction time as shown in the figure. The overall order of the reaction is
The degree of dissociation of HI at a particular temperature is 0.8. The volume of 2 M \[N{{a}_{2}}{{S}_{2}}{{O}_{3}}\] solution required to neutralise the iodine present in an equilibrium mixture of a reaction when 2 mole each of \[{{H}_{2}}\] and \[{{I}_{2}}\], are heated in a closed vessel of 2 litre capacity is
Among the following, the number of compounds than can react with \[PC{{l}_{5}}\] to give \[POC{{l}_{3}}\]is \[{{O}_{2}},S{{O}_{2}},{{H}_{2}}O,{{H}_{2}}S{{O}_{4}},{{P}_{4}}{{O}_{10}}\]
If we apply potential difference so that an electron is accelerated continuously in a vaccum tube such that a decrease of 10% occurs in its de Broglie wave length, hi such a case the change observed in kinetic energy of electron will be approximately
The standard enthalpies of formation of \[{{\operatorname{CO}}_{2}}\](g), \[{{\operatorname{H}}_{2}}O\left( l \right)\] and glucose(s) at \[25{}^\circ C\] are \[-400 kJ/mol,\] \[-300 kJ/mol\] and\[-1300 kJ/mol,\]respectively. The standard enthalpy of combustion per gram of glucose at \[25{}^\circ C\] is
Match the revolutions given under Column I with their respective fields given under Column-II; choose the answer which gives correct combination of alphabets of two columns -