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The volumes of containers A and B, connected by a tube and a closed valve are V and 4 V, respectively. Both the containers A and B have the same ideal gas at pressures (temperatures) 5.0 ×10^{5} Pa(300 K) and 1.0 ×10^{5}Pa (400 K), respectively. The valve is opened to allow the pressure to equalise, but the temperature of each container is kept constant at its initial value. Find the common pressure in the containers.
∴ p_{1} > p_{2} so, the equalise pressure of the p is reduced from p_{1}
Given, T_{1} = 300
T_{2} = 400
V_{2} = 4V
p_{1} = 5 x 10^{5}
A block of mass 10 kg is suspended through two light spring balances as shown in figure
The FED of the spring balances and the block are as shown in figure.
T_{1}=10g T_{1}=T_{2} ⇒ T_{2}=10g where, T_{1} and T_{2 }are readings of spring balances as shown in figure.
If E = 100 sin (100t) volt and are the instantaneous values of voltage and current, then the R.M.S values of voltage and current are respectively.
The instantanous valur of voltae is
E= 100sin (100 t) V ...(1)
(compare it with
E = E_{0 }sin (ωt) V
we get,
E_{0 }= 100 V, ω = 100 rads^{1}
the r.m.s value of voltage is
The instantaneous value of current is
compare it with, I=I_{0}sin(ω t + Φ)
we get
I_{0} = 100 mA, ω = 100 rads^{1}
The r.m.s value of current is
Two spherical bodies of mass M and 5 M and radii R and 2 R, respectively are released in free space with initial separation between their centres equal to 12 R. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is
Initial separation between both centers = 12R
Final separation (when collision occurs) = 3R
Thus both the centers need to travel 9R combinely in such a way that their COM does not move as no external force is acting over the 2 mass system.
Thus let say if smaller mass travel x distance, bigger would eventually travel rest 9R  x
Thus we we write the equation of displacement of the COM, and taking direction of displacement of bigger mass as positive, we get
M (x) + 5M (9R  x) / (M + 5M) = 0
Thus we get, Mx + 45MR  5Mx = 0
We get 6x = 45R
Thus x = 7.5R
A charge moving along a circle is equivalent to a current carrying coil, but with respect to magnetic field on the axis of circle. It is equivalent only for the average value of the magnetic field and not for instantaneous values. While if two charge particles are moving symmetrically along a circle at diametrically opposite points then the average as well as instantaneous magnetic field on its axis is same as due to a current carrying coil.
The length of a given cylindrical wire is increased by 100%. Due to the consequent decrease in diameter the change in the resistance of the wire will be
Initial volume = final volume
Students I, II and III perform an experiment for measuring the acceleration due to gravity (g) using a simple pendulum. They use different lengths of the pendulum and /or record time for different number of oscillations. The observations are shown in the table.
Least count for length = 0.1 cm
Least count for time = 0.1 s
If E_{I}, E_{II} and E_{III}_{ }are the percentage errors in g, i.e.,
for students I, II and III, respectively, then
The period of oscillation (T) of a simple pendulum of length ℓ is given by
T = 2π√(ℓ/g)
Therefore, g = 4π2 ℓ/T2 so that the fractional error in g is given by
∆g/g = (∆ℓ/ℓ) + 2(∆T/T)
[The above expression is obtained by taking logarithm of both sides and then differentiating. Note that the sign of the second term on the RHS is changed from negative to positive since we have to consider the maximum possible error].
Here ∆ℓ = 0.1 cm and ∆T = 0.1 s
The percentage error is 100 times the fractional error so that
E_{I} = ∆g/g = [(0.1/64) + 2(0.1/128)]×100 = 5/16 %,
E_{II} = ∆g/g = [(0.1/64) + 2(0.1/64)]×100 = 15/32 % and
E_{III} = ∆g/g = [(0.1/20) + 2(0.1/36)]×100 = 19/18 %
Thus E_{I} is minimum.
Write down the expression for capacitance of a spherical capacitor whose conductors radii are R_{1 }and R_{2}(R_{2}>R1),when inner sphere is grounded.
A mass of M kg is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of 45° with the initial vertical direction is
Here, position 1 and position 2 can be calculated by using
work energy Theorem
Acceleration of each block is given as g/5√2. Find the magnitude and direction of force exerted by string on pulley. (μ = 0.4)
Let coefficient of friction be u. and and 3m block is moving down the incline, then Acceleration
Force exerted by string/on pulley is √2T as shown in figure
∴ F = 6mg/5
A pipe of length 85 cm is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below 1250 Hz. The velocity of sound in air is 340 m/s.
for closed pipes,
so in this case 6 possibilities
Three identical spheres, each of mass 1 kg are kept as shown in the figure below, touching each other, with their centres on a straight line. If their centres are marked P, Q, R respectively, the distance of centre of mass of the system from P is
Radius
r_{1}= 0
r_{2} = PQ
r_{3} = PR
the distance of centre of mass of the system from 'P' is given by
A resistor R and 2μF capacitor in series is connected through a switch to 200 V direct supply. Across the capacitor is a neon bulb that lights up at 120 V. Calculate the value of R to make the bulb light up 5 s after the switch has been closed (log_{10} 2.5 = 0.4)
Wen Neon bulb is fulled with gas.
No, current frows through it because the resistance is infine.
A α−particle passes through a potential difference of 2×10^{6}V and then it becomes incident on a silver foil. The charge number of silver is 47. The energy of incident particles will be: (in joule)
Energy of incident particles ie, alpha particles will be
Direction: Question is based on the following paragraph.
A wire of length L, mass m and carrying a current is suspended from point O as shown. An infinitely long wire carrying the same current I is at a distance L below the lower end of the wire. Given, I = 2A, L= 1m and m = 0.1 kg (ln 2 = 0.693)
What is angular acceleration of the wire just after it is released from the position shown?
The figure shows three circuits with identical batteries, inductors and resistances. Rank the circuits according to the currents through the battery just after the switch is closed, greatest first
In circuit ( 1 ), on closing the switch, the current in the inductor is zero due to self induction, ie, i_{1}=0.
In circuit (2), on closing the switch the current in the inductor is zero due to selfinduction.
Therefore,
i_{2 }= i′ = E / R1
In circuit (3), on closing the switch, the current in the inductor is again zero due to the same reason.
Therefore,
i_{2 }= i′ = E / (R1 + R2)
Thus, it is obvious that,
i2>i3>i1(=0)
A bullet when fired into a fixed target loses half of its velocity after penetrating 20 cm.How much further it will penetrate before coming to rest?
Directions: Question are Assertion  Reaction type each of these contains two statements: Statement I (Assertion), Statement II (Reason) Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes a, b, c and d given below:
Statement I : The isothermal curves intersect each other at a certain point
Statement II: The isothermal changes takes place slowly, so the isothermal curves have very little slope.
To carry out isothermal process, a perfect gas is compressed or allowed to expand very slowly. Isothermal curves never intersect each other as they have very little slope.
A parallel plate air capacitor has a capacitance C. When it is half filled with a dielectric of dielectric constant 5, the percentage increase in the capacitance will be
∴ increase in capiacitance
A ball is dropped from a bridge at a height of 176.4 m over a river. After 2s, a second ball is thrown straight downwards. What should be the initial velocity of the second ball so that both hit the water simultaneously? (Take g=10m/s^{2})
for second ball
Directions: The answer to this question is a singledigit integer, ranging from 0 to 9. Enter the correct digit in the box given below.
Q. A 40 cm diameter pipe branches into two pipes of diameters 10 cm and 20 cm each. If the average velocities of water that flows through 10 cm and 20 cm pipes are 6 m/s and 2 m/s, respectively, then the speed of the water that flows through the 40 cm pipe (in m/s) is
(Answer up to nearest integer)
Let x be the velocity of water flowing through 40 cm diameter branch
V(40): Volume of water flowing through 40cm diameter
V(20): Volume of water flowing through 20cm diameter
V(10): Volume of water flowing through 10cm diameter
V(40) = V(20) + V(10)    (1)
Volume of water flowing = (Crosssectional area) * (Velocity of water)
(1) → x*π (0.20)2 = 2*π (0.10)2 + 6*π (0.05)2
x*(0.04) = 2*(0.01) + 6*(0.0025)
x = (0.035) / (0.04) = 35/40 = 0.875
Nearest integral value is 1
Directions: The answer to this question is a singledigit integer, ranging from 0 to 9. Enter the correct digit in the box given below.
Q. Two point objects A and B are 40 cm apart. A convex lens L of focal length 15 cm should be placed in between them such that the images due to the two point objects coincide. If the least distance between L and A is x, then the value of x – 1 is
Let V_{A} be the image distance due to the object A. X is the distance between the object A and the lens.
image due to the second object:
As one of the images should be virtual in order to satisfy the given condition.
the lesser value is 10. Hence, the value of x1 = 9
Directions: The answer to this question is a singledigit integer, ranging from 0 to 9. Enter the correct digit in the box given below.
Q. An airfilled parallel plate capacitor having circular plates has a capacitance of 10 pF. When the radii of the plates are increased two times, the distance between them is halved and if a medium of dielectric constant k is introduced, the capacitance increases 16 times. The value of k is
Directions: The answer to this question is a singledigit integer, ranging from 0 to 9. Enter the correct digit in the box given below.
Q. Two parallel identical plates carry equal and opposite charges having a uniform charge of 88.9 C. Positive plate is fixed on the ceiling of a box and the negative plate has to be suspended. If the area of the plates is 6.35 sq. m and 'm' is the mass of the negative plate, then the value of [m] in kg, where [ ] stands for maximum integer value, is
force of attraction between the plates = weight of the negative plate for it to be suspended
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
A radioactive sample S_{1} having an activity of 5 μ Ci and half life of 20 years has twice the number of nuclei as another sample S_{2}, which has an activity of 10 μCi. The half lives (in years) of S_{2} is
Directions: Questions are based on the following paragraph.
When ammonium vanadate is heated with oxalic add solution, a compound Z is formed. A sample of Z was titrated with KMnO_{4} solution in hot acidic solution. The resulting liquid was reduced with SO_{2}, the excess SO_{2 }boiled off, and the liquid again titrated with KMnO_{4}. The ratio of the volumes of KMnO_{4} used in the two titrations was 5 : 1. KMnO_{4} oxidises all oxidation state of vanadium to Vanadium (+V) and SO_{2} reduces vanadium (+V) to vanadium (+IV). Read the above experiment and answer the following questions. If vanadium exists as , reduced species by SO_{2} would be
is reduced to +4 oxidation state which is
In a cubic dosed packed structure of mixed oxides, the lattice is made up of oxide ions, one eighth of tetrahedral/voids are occupied by divalent ions (A^{2}+), while one half of the octahedral voids are occupied by trivalent ions(B^{3+})What is the formula of the oxide ?
Let number of oxides = x
Number of octahedral void = x
Number of tetrahedral void = 2x
Number of
Hence, formula of oxide is AB_{2}O_{4}.
Which of the following sequence of reaction is the best means to furnish the conversion RCH_{2}OH→RCH_{2}NH_{2}
The standard heat of combustion of carbon(s), sulphur (s) and carbon disulphide (l) are 393.3, 293.72 and  1108.76 kJ/mol respectively. The standard heat of formation of carbon disulphide(l) is
on putting various enthalpy of formation in equation III
(reactants)  1108.76 = [393.3 + 2(293.72)] 
=1108.76 = 393.3 2 x 293.72 
The volume of 0.1 M oxalic acid that can be completely oxidised by 20 mL of 0.025 M KMnO_{4} solution is
Anomers are diastereo isomers of cyclic forms of sugars or similar molecules differing in the configuration at the anomeric carbon (C1 atom of aldose or the C2 atom of a 2ketose).The cyclic forms of carbohydrates can exist in two forms , α – and β based on the position of the substituent at the anomeric centre.
Match List I with List II and select the correct answer using the codes given below the lists:
Cyanide process is used for the extraction of Au, floatation process uses pine oil as a foaming agent, electrolytic reduction is used in the extraction of Al and zone refining process produces ultra pure Ge.
Which one of the following is the structure of polyacrylonitrile?
Polyacrylonitrile has molecular formula (C_{3}H_{3}N)_{n}.
Hence its structure is
In the hydrocarbon
The state of hybrization of carbons 1,3 and 5 are in the following sequence :
Among the four (I) and (II) have chirality . so, here, optical isomers are obtained.
for reaction
For the elementary reaction M → N, the rate of disappearance of M increases by factor of 8 upon doubling the concentration of M. The order of the reaction with respect to M is
On doubling cone of [M], Rate becomes 8 times
In which of the following case, increase in concentration of ion cause increase in E_{cell}?
Directions: Questions are based on the following paragraph.
When ammonium vanadate is heated with oxalic add solution, a compound Z is formed. A sample of Z was titrated with KMnO_{4} solution in hot acidic solution. The resulting liquid was reduced with SO_{2}, the excess SO_{2 }boiled off, and the liquid again titrated with KMnO_{4}. The ratio of the volumes of KMnO_{4 }used in the two titrations was 5 : 1. KMnO_{4} oxidises all oxidation state of vanadium to Vanadium (+V) and SO_{2} reduces vanadium (+V) to vanadium (+IV). Read the above experiment and answer the following questions.
Q. What is the oxidation state of vanadium in the compound Z?
vanadate ion is reduce to V^{x} (species Z) by
ion in acidic medium. V^{x }is oxidised by
∴ to which in turn isoxidised to
volumes of used in Eqs (1) , (2) are in ratio , of
∴
Direction: Question is assertion reason type. These question contains two statements Statement I (Assertion), Statement II (Reason). These question also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below:
Statement I : Detection of chlorine in 2, 4, 6 trinitrochlorobenzene can be done directly by addition of aq. AgNO_{3} solution.
Statement II: CCl bond is weakened by electron withdrawing  NO_{2} group
2,4, 6  trinitrocholorobenzene.The presence of electron withdrawing group (like NO_{2}) makes the nucleophilic aromatic substitution easier, as it decreases the strength of CCl bond. Thus,gives ppt. of AgCl with aq. AgNo_{3}
Silver (atomic weight = 108 g mol^{1} ) has a density of 10.5 g cm^{3}. The number of silver atoms on a surface of area 10^{12} m^{2 }can be expressed in scientific notation as y * 10^{x} . The value of x is
since the number of atom of silver
Names of some compounds are given. Which one is not correct in IUPAC system?
Correct IUPAC name is 4ethyl3methyl heptane.
Structure of 1 butene3yne we know that double bond contains one sigma and one pibond while a triple bond contains one sigma and two pibonds. So, total number of sigma and pibonds.
= (5+1+1) σ + (0+1+2) π i.e 7σ, 3π
1butene3yne contains 7 sigma and 3 Pibonds
consider the follwing reaction,
the value of x, y and z in the reaction respectively are
Given
after balancing the equation, we get
∴ The value of x, y and z in the given equation are 2,5 and 16 respectively
In the given structures A and B ring A is 6 membered ring containg one oxygen with α – glycosidic linkage, hence it is pyranose where as ring B is 5 membered ring containing one oxygen is furanos with β – glycosidic linkage.
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. X x 10^{2} moles of NaCl need to be added to precipitate out PbCl_{2} from 1 L solution of Pb(NO_{3})_{2} having concentration 0.01M. If K_{sp} of PbCl_{2} at 298K is 1.6 x 10^{5}, then the value of X will be
Hence [NaCl] required 4 x 10^{2}
Nimber of moles of NaCl to be added = 4 x 10^{2}
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. For a diatomic molecule having bond distance , the dipole moment is 1.2 D. What is the integral value obtained as a product of the fraction of charge separated and 16?
fraction of electric charge on ach atom
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. On complete combustion of a 9 L mixture of ethane and propane, 21 L of CO_{2} at STP is produced. The molar ratio of ethane and propane in the mixture is
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. How many open chain structures are possible for NMethyl butanamine (including it) that are referred to as metamers.
Molecular formula of Nmethyl butanamine is C_{5}H_{13}N. Primary, secondary and tertiary amines are different functional groups and metamerism cannot coexist with functional isomerism. Hence, we shall consider only those isomers of Nmethyl butanamine which are secondary (2^{o}) amines.
Every molecule of group A is a meramer of every molecule in group B
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. 100gm of is reduced to 3.125 g in 25 days. Half life of bismuth 210 (in days) is
A bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag replacing the ball each time after the draw till one of them draws a white ball and wins the game. A begins the game. If the probability of A winning the game is three times that of B, then the ratio a : b is
w → drawing white ball at any draw and B that for a black ball
then
Also P(B wins the game)
If Z is the set of integers. Then, the relation R= {(a, b) :1+ ab > 0} on Z is
so it is reflexive
so, it is symmetric
So, it is not transitive.
Direction: Question is AssertionReason type question. These question contains two statements: Statement I (Assertion) and Statement II (Reason). These question also has four alternative choices, only one of which is the correct answer. You have to select the correct choice in the cedes (a), (b), (c) and (d) in the given below:
Statement I:
Staement II: x is non differential at x = 0
Given
found, since condition on x is not given
also x is nondifferential at x = 0
For any two real numbers θ and ϕ, we define θRϕ , if and only if sec^{2} θ  tan^{2}ϕ =1. The relation R is
Direction: Question is AssertionReason type question. These question contains two statements: Statement I (Assertion) and Statement II (Reason). These question also has four alternative choices, only one of which is the correct answer. You have to select the correct choice in the cedes (a), (b), (c) and (d) in the given below:
Now statement II is valid only where
∴ Statement I is true and statement II is false
if n≥ 2 is an integer and I is the identity matrix of order 3. then
Given
similarly
Put
put
The number of real solutions of the equation 2^{x/2}+(√2+1)^{x}=(5+2√2)^{x/2 }is
Thus no. of real solution is 1.
Using Leibnitz's rule
on differentiating w.r.t θ we get
If m_{1}, m_{2}, m_{3} and m_{4} are, respectively the magnitudes of the vectors
then the corect order of m_{1}, m_{2}, m_{3} and m_{4} is
The area bounded by y = xsin x and X  axis between x = 0, x = 2π is
If θ is the angle between the tangents from (1, 0) to the circle x^{2} + y^{2}  5x + 4y  2 = 0, then θ is equal to
we have, angle between the two tangents from (x_{1},y_{1})
Let S_{k} be the sum of an infinite GP series whose first term k is k and common ratio is k/(k + 1) (k> 0). Then, the value of is equal to
Let z_{1} ≠ z_{2} and z_{1} = z_{2}. If z_{1} has positive real part and z_{2} has negative imaginary part. Then,
Now,
= a purely imaginary or 0 if (x_{1}/x_{2} )= (y_{1}/y_{2})
if (x_{1}/x_{2} )= (y_{1}/y_{2}) then
x_{1} + iy_{1} = k(x_{2}+iy_{2})
is purely imaginary.
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. if , then the number of solutions of
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Q. The points (1, 3) and (5, 1) are two opposites vertices of a rectangle and the other two vertices lie on the line y – 2x + C = 0. Then the value of C is
Since the diagonals of a rectangle bisect each other, so the point
lies on y = 2x  C, which gives C = 4
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
Given that α and γ are the roots of the equation Ax^{2} – 4x + 1 = 0, and β and δ are the roots of the equation Bx^{2} – 6x + 1 = 0. Find the value of B  A, such that α, β, γ, δ are in H.P.
As per the given condition
Let d be the common difference
Adding both of the equations, we get
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
if the greatest value of is
Directions: The answer to the following question is a single digit integer ranging from 0 to 9. Enter the correct digit in the box given below.
if range of the function f(x) = sin^{–1} x + 2 tan^{–1} x + x^{2} + 4x + 1 is [a, b], then the value of a + b is
therefore, f(x) is an increasing function. Hence, a is minimum value of f(x). therefore
And, b is maximum value of f(x) . Therefore
Therefore, the range of f(x) is . Therefore
Hence, it is required solution.
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