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In the figure shown a parallel plate capacitor has a dielectric of width d/2 and dielectric constant K = 2. The other dimensions of the dielectric are same as that of the plates. The plates P_{1} and P_{2} of the capacitor have area 'A' each. The energy of the capacitor is :
A rod of length l having uniformly distributed charge Q is rotated about one end with constant frequency ' f '. Its magnetic moment.
Charge on the differential element dx,
equivalent current di = f dq
∴ magnetic moment of this element
dμ = (di) NA (N = 1)
⇒
Two identical spheres of same mass and specific gravity (which is the ratio of density of a substance and density of water) 2.4 have different charges of Q and – 3Q. They are suspended from two strings of same length l fixed to points at the same horizontal level, but distant l from each other. When the entire set up is transferred inside a liquid of specific gravity 0.8, it is observed that the inclination of each string in equilibrium remains unchanged. Then the dielectric constant of the liquid is
⇒
Two infinitely long parallel wires are a distance d apart and carry equal parallel currents I in the same direction as shown in the figure. If the wires are located on y axis (normal to xy plane) at y = d/2 and y = d/2, then the magnitude of xcoordinate of the point on xaxis where the magnitude of magnetic field is maximum is (Consider points on xaxis only)
The magnetic field at point P is where are magnetic field at P due to wire 1 and 2.
where r^{2} = x^{2} + (d/2)2
∵ field is along +y direction at point
P and its magnitude is
Figure shows a uniformly charged hemispherical shell. The direction of electric field at point p, that is offcentre (but in the plane of the largest circle of the hemisphere), will be along
Let electric field at point. 'p' has both x and y component.
So similar electric field will be, for other hemisphere (upper half).
Now lets overlap both.
(E_{net})_{p} = 2 E_{x} and it should be zero (as E inside a full shell = 0).
So E_{x} = 0, So electric field at 'p' is purely in y direction.
A wooden stick of length 3l is rotated about an end with constant angular velocity ω in a uniform magnetic field B perpendicular to the plane of motion. If the upper one third of its length in coated with copper, the potential difference across the copper coating of the stick is
When the rod rotates, there will be an induced current in the rod. The given situation can be treated as if a rod 'A' of length '3l' rotating in the clockwise direction, while an other rod 'B' of length '2l' rotating in the anticlockwise direction with same angular speed 'ω'.
For ‘A’ :
Resultant induced emf will be :
⇒
⇒
The resistance of each straight section is r. Find the equivalent resistance between A and B.
From symmetry, the current distribution in branches LP, MP, NP and OP are as shown in figure 1. Therefore junction at P can be broken as shown in figure 2
⇒
Hence equivalent resistance is 3.5 r.
PQ is an infinite current carrying conductor. AB and CD are smooth conducting rods on which a conductor EF moves with constant velocity V as shown. The force needed to maintain constant speed of EF is.
⇒ ⇒
⇒
⇒
Two capacitors C_{1} & C_{2} are charged to same potential V, but with opposite polarity as shown in fig. The switch S_{1} & S_{2} are then closed.
Net charge on both the capacitors is = C_{1}V  C_{2}V
The effective capacitance of system is C_{1} + C_{2} because both are in parallel.
Therefore p.d a cro ss the system is
Therefore ratio of final to initial energy is
In the figure shown ‘R’ is a fixed conducting ring of negligible resistance and radius ‘a’. PQ is a uniform rod of resistance r. It is hinged at the centre of the ring and rotated about this point in clockwise direction with a uniform angular velocity ω. There is a uniform magnetic field of strength ‘B’ pointing inwards. ‘r’ is a stationary resistance, then choose correct statements
Equivalent circuit :
(∵ Radius = a)
By nodal equation
5X = 4e
⇒
also direction of current in ‘r’ will be towards negative terminal i.e. from rim to origin
Alternately; by equivalent of cells (figure (ii)) :
In the circuit shown in figure, E_{1} and E_{2} are two ideal sources of unknown emfs. Some currents are shown. Potential difference appearing across 6? resistance is V_{A} – V_{B} = 10V.
after redrawing the circuit
(a) I_{4} = 5A ,
(b) , (c) From loop (1)  8(3) + E_{1}  4(3) = 0
⇒ E_{1} = 36 volt
from loop (2) + 4(5) + 5(2)  E_{2} + 8(3) = 0
E_{2} = 54 volt
(d) from loop (3)  2R  E_{1} + E_{2} = 0
A proton of charge 'e' and mass 'm' enters a uniform and constant magnetic field with an initial velocity Which of the following will be correct at any later time 't' :
The xcomponent of velocity, being parallel to magnetic field, shall remain unchanged.The component of velocity perpendicular to xaxis will always have magnitude Voy,
and at any time t it shall make an angle θ =ωt with yaxis as shown. so ycomponent of velocity is Voy cosωt and zcomponent of velocity along negative zdirection at any time t is Voy sinωt. Where ω = qB/m
Satement1 : Two cells of unequal emf E_{1} and E_{2} having internal resistances r_{1} and r_{2} are connected as shown in figure. Then the potential difference across any cell cannot be zero.
Satement2 : If two cells having nonzero internal resistance and unequal emf are connected across each other as shown, then the current in the
Let E_{1} < E_{2} and a current i flows through the circuit Then the potential difference across cell of emf E_{1} is E_{1} + ir_{1} Which is positive, hence potential difference across this cell cannot be zero Hence statement 1 is correct For current in the circuit to be zero, emf of both the cells should he equal But E_{1} ≠ E_{2} , Hence statement 2 is correct but it is not a correct explanation of statement 1,
Satement1 : A pendulum made of an insulated rigid massless rod of length l is attached to a small sphere of mass m and charge q. The pendulum is undergoing oscillations of small amplitude having time period T. Now a uniform horizontal magnetic field out of plane of page is switched on. As a result of this change, the time period of oscillations does not change.
Satement2 : A force acting along the string on the bob of a simple pendulum (such that tension in string is never zero) does not produce any restoring torque on the bob about the hinge.
The magnetic force on bob does not produce any restoring torque on bob about the hinge. Hence this force has no effect on time period of oscillation. Therefore both statements are correct and statement2 is the correct
In the shown circuit involving a resistor of resistance R W, capacitor of capacitance C farad and an ideal cell of emf E volts, the capacitor is initially uncharged and the key is in position 1. At t = 0 second the key is pushed to position 2 for t0 = RC seconds and then key is pushed back to position 1 for t0 = RC seconds. This process is repeated again and again. Assume the time taken to push key from position 1 to 2 and vice versa to be negligible.
Q. The charge on capacitor at t = 2RC second is
For t = 0 to t_{0} = RC seconds, the circuit is of charging type. The charging equation for this time is
Therefore the charge on capacitor at time t_{0} = RC is q_{0}
For t = RC to t = 2RC seconds, the circuit is of discharging type. The charge and current equation for this time are
and
Hence charge at t = 2 RC and current at t = 1.5 RC are
and
Since the capacitor gets more charged up from t = 2RC to t = 3RC than in the interval t = 0 to t = RC, the graph representing the charge variation is as shown in figure
In the shown circuit involving a resistor of resistance R W, capacitor of capacitance C farad and an ideal cell of emf E volts, the capacitor is initially uncharged and the key is in position 1. At t = 0 second the key is pushed to position 2 for t0 = RC seconds and then key is pushed back to position 1 for t0 = RC seconds. This process is repeated again and again. Assume the time taken to push key from position 1 to 2 and vice versa to be negligible.
Q. The current through the resistance at t = 1.5 RC seconds is
For t = 0 to t_{0} = RC seconds, the circuit is of charging type. The charging equation for this time is
Therefore the charge on capacitor at time t_{0} = RC is q_{0}
For t = RC to t = 2RC seconds, the circuit is of discharging type. The charge and current equation for this time are
and
Hence charge at t = 2 RC and current at t = 1.5 RC are
and
Since the capacitor gets more charged up from t = 2RC to t = 3RC than in the interval t = 0 to t = RC, the graph representing the charge variation is as shown in figure
In the shown circuit involving a resistor of resistance R W, capacitor of capacitance C farad and an ideal cell of emf E volts, the capacitor is initially uncharged and the key is in position 1. At t = 0 second the key is pushed to position 2 for t0 = RC seconds and then key is pushed back to position 1 for t0 = RC seconds. This process is repeated again and again. Assume the time taken to push key from position 1 to 2 and vice versa to be negligible.
Q. Then the variation of charge on capacitor with time is best represented by
For t = 0 to t_{0} = RC seconds, the circuit is of charging type. The charging equation for this time is
Therefore the charge on capacitor at time t_{0} = RC is q_{0}
For t = RC to t = 2RC seconds, the circuit is of discharging type. The charge and current equation for this time are
and
Hence charge at t = 2 RC and current at t = 1.5 RC are
and
Since the capacitor gets more charged up from t = 2RC to t = 3RC than in the interval t = 0 to t = RC, the graph representing the charge variation is as shown in figure
Matrix Match
A charged particle having non zero velocity is subjected to certain conditions given in Column I . Column II gives possible trajectories of the particle. Match the conditions in column I with the results in Column II
(A) Uniform electric field exerts constant force on the charged particle, hence the particle may move in straight line or a parabolic path.
(B) Under action of uniform magnetic field, the charged particle may move in straight line when projected along or opposite to direction of magnetic field. The charged particle moves in circle when it is projected perpendicular to the magnetic field. If the initial velocity of the charged particle makes an angle between 0° and 180° (except 90°) with magnetic field, the particle moves along a helical path of uniform pitch.
(C) If charged particle is shot parallel to both fields it moves along a straight line. If the charged particle is shot at any angle with both the field (except 0° and 180°) , the particle moves along a helix with nonuniform pitch.
(D) from results of A and B all the given paths are possible.
A uniformly charged ring of radius 0.1 m rotates at a frequency of 10^{4} rps about its axis. The ratio of energy density of electric field to the energy density of the magnetic field at a point on the axis at distance 0.2 m from the centre is in form X × 10^{9}. Find the value of X. (Use speed of light c = 3 × 108 m/s, π^{2} = 10)
Electric field at P is
Magnetic field at P is
f = frequency of revolution.
Electric energy density Magnetic energy density
In the circuit shown S_{1} and S_{2} are switches. S_{2} remains closed for a long time and S_{1} open. Now S_{1} is also closed. It is given that R = 10Ω , L = 1 mH and ε = 3V. Just after S_{1} is closed, the magnitude of rate of change of current (in ampere/sec.) that is , in the inductor L is x × 10^{2} A/s find x
∴
⇒
∴ Potential difference
And
The equivalent capacitance between terminals ‘A’ and ‘B’ is Find x. The letters have their usual meaning.
The current density inside a long, solid, cylindrical wire of radius a = 12 mm is in the direction of the central axis and its magnitude varies linearly with radial distance r from the axis according to where A/m^{2}. Find the magnitude of the magnetic field at r = a/2 in μT.
Current in the element = J (2πr . dr)
Current enclosed by Amperian loop of radius a/2
Applying Ampere's law
On putting values
B = 10 μT
(halflife = 15 hrs.) is known to contain some radioactive impurity (halflife = 3 hrs.) in a sample. This sample has an initial activity of 1000 counts per minute, and after 30 hrs it shows an activity of 200 counts per minute. What percent of the initial activity was due to the impurity ?
Let the activity due to impurity be ‘a’ cpm.
∴ due to Na it is (1000  a) cpm.
After 30 hrs ‘a’ would be reduced to (1/2)^{10} a cpm and (1000  a) would be reduced to cpm
∴ total activity after 30 hrs would be
solving we get
∴ ⇒ a = 200
Hence 20% activity was due to impurity.
For the cell (at 298 K)
Ag(s)  AgCl(s)  Cl– (aq)  AgNO_{3} (aq)  Ag(s)
Which of following is correct :
It [Ag^{+}]_{a} = [Ag^{+}]c then both the electrodes have same potential.
[Ag+] will increase in anodic compartment.
AgCI(s) precipitate in anodic compartment will increase.
At 298K the standard free energy of formation of H_{2}O(l) is –257.20 kJ/mole while that of its ionisation into H^{+} ions and OH^{–} ions is 80.35 kJ/mole, then the emf of the following cell at 298 K will be (Take F = 96500 C] :
H_{2}(g,1 bar)  H^{+} (1M)  OH¯ (1M)  O_{2} (g, 1bar)
Cell reaction
Cathode :
Anode :
________________________________________________
Also we have
Hence for cell reaction
Consider the reaction, NH_{2}NO_{2} (aq) ———? N_{2}O(g) + H_{2}O(l)
The concentration of nitramide as a function of time is shown below for a particular run.
Which line represents the correct tangent to the graph at the origin (t = 0) ?
In a hypothetical reaction
A(aq) 2B(aq) + C(aq) (1st order decomposition)
'A' is optically active (dextrorototory) while 'B' and 'C' are optically inactive but 'B' takes part in a titration reaction (fast reaction) with H_{2}O_{2}. Hence the progress of reaction can be monitored by measuring rotation of plane of plane polarised light or by measuring volume of H_{2}O_{2} consumed in titration.
In an experiment the optical rotation was found to be θ = 40° at t = 20 min and θ = 10° at t = 50 min. from start of the reaction. If the progress would have been monitored by titration method, volume of H_{2}O_{2} consumed at t = 15 min. (from start) is 40 ml then volume of H_{2}O_{2} consumed at t = 60 min will be:
As only A is optically active. So concentration of A at t = 20 min ∝ 40°
While concentration of A at t = 50 min ∝ 10°, so t_{1/2} = 15 min.
So volume consumed of H_{2}O_{2} at t = 15 min = t_{1/2} , is according to 50% production of B. at t = 60 min. production of B = 94.75% (four half lives)
So volume consumed = 75 ml Ans.
How many m.moles of sucrose should be dissolved in 500 grams of water so as to get a solution which has a difference of 103.57°C between boiling point and freezing point ?
(K_{f} = 1.86 K Kg mol^{–1}, K_{b} = 0.52 K Kg mol^{–1})
Boiling point of solution = boiling point + ΔT_{b} = 100 + ΔT_{b}
Freezing point of solution = freezing point  ΔT_{f} = 0  ΔT_{f}
Difference in temperature (given) = 100 + ΔT_{b}  ( ΔT_{f})
103.57 = 100 + AT_{b} + ΔT_{f} = 100 + molality x K_{b} + molality x K_{f}
= 100 + molality (0.52 + 1.86)
= 750mmoles
When a graph is plotted between log x/m and log p, it is straight line with an angle 45° and intercept 0.6020 on yaxis. If initial pressure is 0.3 atm, what will be the amount of gas adsorbed per gram of adsorbent :
Diamond has facecentred cubic lattice. There are two atoms per lattice point, with the atoms at (000) and coordinates. The ratio of the carboncarbon bond distance to the edge of the unit cell is:
Carbon atoms are at corners and are at alternate corners. So from geometry.
So required ratio
(A) ΔG = ΔH  TΔS < O as ΔS < O so ΔH has to be negative
(B) micelles formation will take place above T_{k} and above CMC
(C) this solution will be negatively charged.
(D) Fe^{3+} ions will have greater flocculatibility power so smaller flocculating value.
The polarimeter readings in an experiment to measure the rate of inversion of cane suger (1st order reaction) were as follows :
Identify the true statement (s) log 2 = 0.3, log 3 = 0.48, log 7 = 0.84, log_{e} 10 = 2.3
⇒ ⇒
⇒ x = 22.5  15 = 7.5°
For chloroform and acetone or for a solution of chloroform and acetone if p_{s} (observed (actual)) is compared with p_{s} (Theoretical (Raoult)) then which of the following is /are true ?
Chloroform form hydrogen bond with acetone
Due to hydrogen bond formation vapour pressure of the solution become less then expected results.
The standard reduction potentials of some half cell reactions are given below :
Pick out the correct statement :
On basis of given SRP values
Statement1 : The ratio of specific conductivity to the observed conductance does not depend upon the concentration of the solution taken in the conductivity cell.
Statement2 : Specific conductivity decreases with dilution where as observed conductance increases with dilution.
Statement1 : The ratio is cell constant
Statement2 : Number of ions in a unit volume decreases. Hence specific conductivity decreases.
Conductance increases as number of ions increases α increases).
Statement1 : When AgNO_{3} is treated with excess of KI, colloidal particles gets attracted towards anode.
Statement2 : Colloidal particles adsorb common ions and thus become charged.
in excess of kl common ion is adsorbed preferentially on precipitate of AgI and becomes negatively charged colloidal particle AgI/I^{} .
Ideal Solution at Fixed Temperature
Consider two liquids 'B' and 'C' that form an ideal solution. We hold the temperature fixed at some value T that
is above the freezing points of 'B' and 'C'. We shall plot the system's pressure P against X_{B}, the overall mole
fraction of B in the system :
Where are the number of moles of B in the liquid and vapor phases, respectively. For a closed system X_{B} is fixed, although may vary.
Let the system be enclosed in a cylinder fitted with a piston and immersed in a constanttemperature bath. To see what the Pversus–X_{B} phase diagram looks like, let us initially set the external pressure on the piston high enough for the system to be entirely liquid (point A in figure) As the pressure is lowered below that at A, the system eventually reaches a pressure where the liquid just begins to vaporizes (point D). At point D, the liquid has composition at D is equal to the overall mole fraction X_{B} since only an infinitesimal amount of liquid has vapourized. What is the composition of the first vapour that comes off ? Raoult's law, relates the vapourphase mole fractions to the liquid composition as follows :
............(1)
Where P_{B}^{0} and P_{C}^{0} are the vapour pressures of pure 'B' and pure 'C' at T, where the system's pressure P equals the sum P_{B }+ P_{C} of the partial pressures, where , and the vapour is assumed ideal.
............(2)
Let B be the more volatile component, meaning that . Above equation then shows that The vapour above an ideal solution is richer than the liquid in the more volatile component. Equations (1) and (2) apply at any pressure where liquid –vapour equilibrium exists, not just at point D.
Now let us isothermally lower the pressure below point D, causing more liquid to vaporize. Eventually, we
reach point F in figure , where the last drop of liquid vaporizes. Below F, we have only vapour. For any point
on the line between D and F liquid and vapour phases coexist in equilibrium.
Q. If the above process is repeated for all other compositions of mixture of C and B. If all the points where vapours start converting into liquid are connected and all the points where vapours get completely converted into liquid are connected then obtained graph will look like :
When liquid just starts forming vapours we have Roult’s law valid with X_{b} and X_{c} as mole fraction in liquid state so equation of curve obtained by collecting such points will be
The second curve will not be a straight line having equation
Ideal Solution at Fixed Temperature
Consider two liquids 'B' and 'C' that form an ideal solution. We hold the temperature fixed at some value T that
is above the freezing points of 'B' and 'C'. We shall plot the system's pressure P against X_{B}, the overall mole
fraction of B in the system :
Where are the number of moles of B in the liquid and vapor phases, respectively. For a closed system X_{B} is fixed, although may vary.
Let the system be enclosed in a cylinder fitted with a piston and immersed in a constanttemperature bath. To see what the Pversus–X_{B} phase diagram looks like, let us initially set the external pressure on the piston high enough for the system to be entirely liquid (point A in figure) As the pressure is lowered below that at A, the system eventually reaches a pressure where the liquid just begins to vaporizes (point D). At point D, the liquid has composition at D is equal to the overall mole fraction X_{B} since only an infinitesimal amount of liquid has vapourized. What is the composition of the first vapour that comes off ? Raoult's law, relates the vapourphase mole fractions to the liquid composition as follows :
............(1)
Where P_{B}^{0} and P_{C}^{0} are the vapour pressures of pure 'B' and pure 'C' at T, where the system's pressure P equals the sum P_{B }+ P_{C} of the partial pressures, where , and the vapour is assumed ideal.
............(2)
Let B be the more volatile component, meaning that . Above equation then shows that The vapour above an ideal solution is richer than the liquid in the more volatile component. Equations (1) and (2) apply at any pressure where liquid –vapour equilibrium exists, not just at point D.
Now let us isothermally lower the pressure below point D, causing more liquid to vaporize. Eventually, we
reach point F in figure , where the last drop of liquid vaporizes. Below F, we have only vapour. For any point
on the line between D and F liquid and vapour phases coexist in equilibrium.
Q. The equation of the curve obtained by connecting all those points where the vapors of above mixture (all mixtures of different composition are taken) just start forming will be :
When liquid just starts forming vapours we have Roult’s law valid with X_{b} and X_{c} as mole fraction in liquid state so equation of curve obtained by collecting such points will be
Ideal Solution at Fixed Temperature
Consider two liquids 'B' and 'C' that form an ideal solution. We hold the temperature fixed at some value T that
is above the freezing points of 'B' and 'C'. We shall plot the system's pressure P against X_{B}, the overall mole
fraction of B in the system :
Where are the number of moles of B in the liquid and vapor phases, respectively. For a closed system X_{B} is fixed, although may vary.
Let the system be enclosed in a cylinder fitted with a piston and immersed in a constanttemperature bath. To see what the Pversus–X_{B} phase diagram looks like, let us initially set the external pressure on the piston high enough for the system to be entirely liquid (point A in figure) As the pressure is lowered below that at A, the system eventually reaches a pressure where the liquid just begins to vaporizes (point D). At point D, the liquid has composition at D is equal to the overall mole fraction X_{B} since only an infinitesimal amount of liquid has vapourized. What is the composition of the first vapour that comes off ? Raoult's law, relates the vapourphase mole fractions to the liquid composition as follows :
............(1)
Where P_{B}^{0} and P_{C}^{0} are the vapour pressures of pure 'B' and pure 'C' at T, where the system's pressure P equals the sum P_{B }+ P_{C} of the partial pressures, where , and the vapour is assumed ideal.
............(2)
Let B be the more volatile component, meaning that . Above equation then shows that The vapour above an ideal solution is richer than the liquid in the more volatile component. Equations (1) and (2) apply at any pressure where liquid –vapour equilibrium exists, not just at point D.
Now let us isothermally lower the pressure below point D, causing more liquid to vaporize. Eventually, we
reach point F in figure , where the last drop of liquid vaporizes. Below F, we have only vapour. For any point
on the line between D and F liquid and vapour phases coexist in equilibrium.
Q. Two liquids A and B have the same molecular weight and form an ideal solution. The solution has a vapour pressure of 700 Torrs at 80ºC. It is distilled till 2/3^{rd} of the solution (2/3^{rd} moles out of total moles) is collected as condensate. The composition of the condensate is x'_{A} = 0.75 and that of the residue is x''_{A}= 0.30. If the vapour pressure of the residue at 80ºC is 600 Torrs, find the original composition of the liquid ?
If initially is liquid there are x moles of A and y moles of B then
we have
........(1)
.......(2)
and
(A)
For concentration cell E°_{cell }= 0
(B) H_{2}SO_{4} compartment acts as anode For concentration cell E°_{cell} = 0
(C) E = +ve, ΔG = ve
(D) Ag+ (0.01 M) →Ag+ (K_{sp}/0.1);E = +ve.
For concentration cell E°_{cell} = 0
A certain reactant XO_{3}^{–} is getting converted to X_{2}O_{7} in solution. The rate constant of this reaction is measured by titrating a volume of the solution with a reducing agent which reacts only with XO_{3}^{–} and X_{2}O_{7}. In this process of reduction both the compounds converted to X^{–}. At t = 0, the volume of the reagent consumed is 30mL and at t = 9.212 min. the volume used up is 36 mL. Find the rate constant(in hr^{–1}) of the conversion of XO_{3}^{–} to X_{2}O_{7} ? Asuming reaction is of Ist order. (Given that ln 10 = 2.303, log 2 = 0.30).
⇒ x = 3
The following two cells with initial concentration as given are connected with each other.
(1) Fe(s)  Fe(NO_{3})_{2}(aq.) (1M)  SnCl_{2}(aq.) (1M)  Sn(s)
(2) Zn(s)  ZnSO_{4}(aq.) (1M)  Fe(NO_{3})_{2}(aq.) (1M)  Fe(s)
After sufficient time equilibrium is established in the circuit. What will be the concentrations (in mmoles/L) of Fe^{2+} ions in first and second cells respectively ?
(use only 3 significant figures to fill your answer. For example if [Fe^{2+}]_{1} = 0.225M and [Fe^{2+}]_{2} = 0.425M then your answer should be 225425)
= 0.44 V, 2.3 x RT = 6433, log2 = 0.3]
If above cells are connected in parallel then first cell will get charged up and second cell will get discharged so net cell reaction will be.
Fe_{2}^{2+}(aq.) + Fe_{1}^{2+} (aq.) Zn^{2+} (aq.) + Sn^{2+}(aq.) E°_{net} = 0.02
So,  2.30 RT log k_{eq} = 2 x 96500 x 0.02
⇒ x = 1/3
So, concentration of Fe^{2+} ions in the first cell
= concentration of Fe^{2+} ions in the second cell = 2/3 M = 667 mmoles/L.
A solid cube of edge length = 25.32 mm of an ionic compound which has NaCl type lattice is added to 1kg of water. The boiling point of this solution is found to be 100.52°C (assume 100% ionisation of ionic compound). If radius of anion of ionic solid is 200 pm then calculate radius of cation of solid in pm (picometer) ?
(k_{b} of water = 0.52 K kg mole^{1}, Avogadro number, N_{A} = 6 x 10^{23}, = 4.22)
Effective molality of solution = 1
Hence, no. of moles of ionic solid in given cube = 0.5
so, no. of formula units in given cube
no. of unit cells
no. of unit cells alongone edge of cube
If edge length of unit cell = 600 pm
for NaCI type unit cell, a = 2 (r_{+} + r_)
So r_{+} = 100 pm.
Calculate the pH at which the following conversion(reaction) will be at equilibrium in basic medium ?
When the equilibrium concentrations at 300 K are [I^{}] = 0.10 M and [IO_{3}^{}] = 0.10 M
(state) = 0, ΔG_{r}°(reaction) = ∑v_{p} ΔG_{f}°(products)  ∑v_{r} ΔG°(reactants), where v_{p} and v_{r} are the stochiometric coefficients in the balanced chemical equation.}
{Given that ΔG_{f}°(I^{},aq) =  50 kJ/mole , Δ G_{f}^{0}(IO_{3}^{},a q) =  123.5 kJ/mole , ΔG_{f}^{0}(H_{2}O, ℓ) =  233 kJ/mole ,
ΔG_{f}^{0}(HO^{} ,aq) =  150 kJ/mole, Ideal gas constant = R = 25/3 Jmole^{1}K^{1}, log e = 2.3, ΔG_{f}°{element, standard}
Balanced equation will be
so [OH^{}] = 106. and therefore [H^{+}] = 10^{8} so, pH = 8 Ans.
If f(x) = ; then [where [.] and {.} represents greatest integer part and fractional part respectively.]
Consider the function f(x) in the interval (0, 2)
continuous at x = 1
∴ f(x) is not differentiable at x = 1
Since period of the function is 2,
∴ the range is {  1, 0, 1}
10,0000 characters of information are held on a magnetic tape in batches of x characters each, the batch processing time being 1600 + 16x^{2} seconds. The value of ‘x’ for the fastest processing is
Number of batches =
∴
If f(1) = 3, f´(1) = 2, f´´(1) = 4, then (f ^{–1})´´ (3) =
We know
If the slope of tangent to the curve y = f(x) is then f(x) is periodic function with principal period
∴
∴ f(x) is periodic with period π.
If f(x) = where [.] denotes the greatest integer function. Then the number of points of discontinuity of f(x) is :
clearly the number of points where f(x) is discontinuous is 3.
If f(x) = px + q, p < 0 is onto when defined from [–1, 1] to [0, 2] then is equal to
f(x) = px + q
f(X) = p < 0
∴ f(x) is decreasing
since, f(x) is onto
∴ f(1) = 2 and f(1) = 0
⇒ ⇒ p = 1, q = 1
i . e. f(0) = 1
∴ f(x) = X + 1
Now,
=  tan^{1} (tan 2)  sin^{1} (sin 3)
=  (2  π)  (π  3) = 1
f(x) = 1 ⇒  x + 1 = 1 ⇒ x = 0
Clearly f(x) is periodic with period 1, it is sufficient to consider f(x) in (0, 1)
Also f(x) hence least value of M = 1
From graph it is clear that b < n + 1  n sin 1 and a > n + 1  n sin 1
Clearly f(x) is periodic with period 1 and g(1)
⇒ g(x) is periodic with period 1
h(x) will be periodic with period 1 if h(1) = 0
∵ f(x) is not conitnuous at x = integer
⇒ g(x) is not differentiable at x = integer
∵ g(x) is continuous
⇒ h(x) is always differentiable
A nursery sells plants after 6 year of growth. Two seedlings A and B are planted each of height 5 inches whose growth rates are where heights h_{A} and h_{B} are in cms and t is the time in years. Then
(∵ h = 5 when t = 0)
plants are at equal height, when t = 0, 4
height of plants A after 6 years = 26 cms
height of plants B after 6 years = 29 cms
Let f(x) = minimum {1, cosx, 1  sinx},  π < x < π, then
f (x) = min {1, cos x, 1  sin x}
= min {cos x, 1,  sin x}
Clearly f (x) is not differentiable at x = 0
Statement1 :
Statement2 : The left hand limit as x → 0 does not exist because the function involves
is defined only for x > 0
Statement1 : Normal drawn at a fixed point P(t_{1}), t_{1} 0, on the parabola y^{2} = 4ax again intersects the parabola at point t_{2} for all nonzero real values of t_{2}.
Statement2 : Normal drawn at a point P(t_{1}), t_{1} 0, on the parabola y_{2} = 4ax again intersects the parabola at the point t_{2}, where t_{2}= –t_{1}–2/t_{1}
Statement1 if normal at t. intersects the parabola again at t_{2}.
thus t_{2} has particular value
∴ the statement is false
Read the passage answer the following :
In a problem of differentiation of , one student write the derivative of as and he find the correct result if g(x) = x^{2} and f(x) = 4. A circle 'C' of minimum radius is drawn which intersect both the curves y = f(x) & y = g(x) at two points at which they intersect. Let 'P' be a point on y = g(x).
Q.
Consider
by wrong calculations .
∴
∴
∴
∴
∴
Read the passage answer the following :
In a problem of differentiation of (f(x) over g(x)) , one student write the derivative of (f(x) over g(x)) as and he find the correct result if g(x) = x^{2} and f(x) = 4. A circle 'C' of minimum radius is drawn which intersect both the curves y = f(x) & y = g(x) at two points at which they intersect. Let 'P' be a point on y = g(x).
Q.
Coordinate of 'P' at which tangent to y = g(x) is parallel to common chord of y = f(x) & y = g(x) is
Points of intersection of y = x^{2} and
i.e. x^{2}  4x = 0 i.e. x = 0 x = 4
i.e. ( 0 , 0 ) (4, 16)
slope of the common chord = 4
g'(x) = 2x = 4 x = 2
∴ the point is (2, 4)
(C)
(D)
If f : [1, 1] → be a continuous function satisfying f(2x^{2}  1) = (x^{3} + x) f(x), then find
f(2x^{2}  1) = (x^{3} + x) f(x) ......(i)
replacing x by  x
f(2x^{2}  1) =  (x^{3} + x) f(x) ......(ii)
from (i) and (ii), we get
f(x) =  f(x) hence f(x) is an odd function and as it is continuous,
⇒ f(0) = 0
(f(x) is continuous at x = 0) ⇒
⇒ ⇒
Let f : ( ∞, a] → R defined by f(x) = x(x  2).If the set of all real values of a for which f(x) is manyone is (ℓ, ∞), then fined the value of ℓ
Value of f(x) is least at x = 1
∴ f(x) is many one if a > 1
∴ a ∈ (1, ∞)
∴ ℓ = 1
Given two curves, y = f(x) passing through (0, 1) and other passing through . If tangents at points with equal Abscissa on the two curves intersect on xaxis, then find the value of f(ln2).
consider y_{1} =
.......(i)
Equation of tangent is
similarly for other curve
⇒ but x intercept for both the tangents is same
∴
⇒
⇒ y_{2} = ky_{1} ....... (ii)
differentiating equation (ii)
⇒ ⇒
⇒ ⇒
Now , f(x) passes through (0, 1) ⇒ 1 = c
⇒
Now, y_{2} passes through
Hence y_{1} = f(x) = e^{2x}
If f(x) = 2e^{x}  ae^{x} + (2a+1)x  3 is monotonically increasing for all x € R and the range of values of ‘a’ are
a € [λ, ∞), then find the value of λ.
f ’(x) must be positive for all x e r
∴ 2e^{x} + ae^{x} + (2a + 1) > 0
⇒ e^{x} (2(e^{x})2 + (2a + 1)e^{x} + a} > 0
⇒ {2t^{2} + (2a + 1)t + a} > 0 where t = e^{x}, Possible graphs of lines are
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