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Water is filled up to a height 'h' in a cylindrical vessel. Now, a hole of area 'A' is made at bottom of the vessel. Water drains out of the hole in time 't' seconds. If we repeat the above process with a height of water as '4h', then how much time it require for water to drain out of the cylinder?
[Assume A << A_{0 } (area of tank)]
Time required to empty the tank
Which of the following graphs correctly represent the variation of β = −[(dV/dp)/V] with p for an ideal gas at constant temperature?
PV = constant, according to Boyle’s law
∴ PV = constant
∴ PdV + VdP = 0
The equation between β and P is of the form xy = constant which represents a rectangular hyperbola.
∴ Graph between β and P will be a rectangular hyperbola represented by graph (a)
What is the potential difference between A and B?
(where A is the centre of left ring and B is the centre of right ring)
Halflife of a radioactive substance A is 4 days. The probability of a nucleus that, from the given sample that it will decay in two halflives is
After two halflives 1/4 th fraction of nuclei will remain undecayed. Or, 3/4 th fraction will decay. Hence, the probability that a nucleus decays in two half lives is 3/4
Which of the following statement is false for electromagnetic waves ?
Electromagnetic waves can propagate in material as well as vacuum.
The voltage of clouds is 4 × 10^{6} V with respect to ground. In a lightning strike lasting 100 ms, a charge of 4 C is delivered to the ground. The average power of lightning strike is (assume complete discharges)
Energy stored between cloud and earth, assuming the capacitor model is
The power of lightning strike is
A tuning fork vibrates with a frequency of 256 Hz. Taking the speed of sound to be 345.6 m s^{1} in the air, find the wavelength and the distance, which the sound travels during the time, fork makes 60 vibrations.
In one cycle, sound advances by distance λ
Therefore, the required distance travelled by sound,
S = 60. λ = 81 m
If the radius of the earth were to shrink by one percent and its mass remains the same, the acceleration due to gravity on the earth's surface would
g = GM/R^{2}
or g∝1/R^{2}
g will increase if R decreases.
Three capacitors each of capacity 4 μF are to be connected in such a way that the effective capacitance is 6 μF. This can be done by
To get equivalent capacitance 6 μF Out of the 4 μF capacitance, two are connected in series and third one is connected in parallel.
Six wires of current I_{1} = 1A, I_{2} = 2A, I_{3} = 3A, I_{4} = 1A, I_{5} = 5A and I_{6} = 4A cut the page perpendicularly at the points 1,2,3,4,5 and 6 respectively as shown in the figure. Find the value of integral ∮B→.dℓ around the circular path c.
A conducting circular loop is placed in a uniform magnetic field, B = 0.025 T with its plane perpendicular to the direction of magnetic field. The radius of the loop is made to shrink at a constant rate of 1 mm s^{1}. Find the induced emf in the loop when it's radius is 2 cm.
Here;
Magnetic field. B = 0.025 T
Radius of the loop, r = 2 cm = 2 x 10^{2} m
Constant rate at which radius of the loop shrinks,
Magnetic flux linked with the loop is
From Faraday's law, the magnitude of the induced emf is
A body executing S.H.M. has amplitude 5 cm and frequency 5 vibrations per second. Calculate the displacement of particle from mean position after 0.32 s.
Here, A = 5 cm ; f = 5 Hz ; t = 0.32 s
Now,y = A sin ωt = A sin 2π f t = 5 sin (576°) = –2.94 cm
For network shown in figure, determine V_{0 }and I_{D}. (Where, I_{D} = Current flowing in the diodes)
Direction of current l_{D} is the same direction as arrow in the diode symbol. Therefore, both the diodes are in forward Bias; hence voltage drops across their terminals are V_{Si} and V_{Ge} are 0.7 V and 0.3 V respectively.
A particle is projected vertically upwards with a velocity u, from a point O. When it returns to the point of projection, which of the following is incorrect ?
Total displacement is 0 as the particle returns to the original position, thus the average velocity zero.
Total distance travelled is 2s and total time taken is 2t
0^{2} = u^{2}  2gs
Hence option (Its average speed is u) is correct.
Hence only option 3 has incorrect option.
Two blocks each of mass m, lie on a smooth table. They are attached to two other masses as shown in figure. The pulleys and strings are light. An object O is kept at rest on the table. The sides AB and CD of the two blocks are made reflecting. The acceleration of two images formed in those two reflecting surfaces with respect to each other is
In a plane mirror the image moves twice as faster as mirror thus
acceleration of image in mirror AB = a → AB
acceleration of image in mirror CD= a → cd
acceleration of images w.r.t. each other is
An electron is in an excited state in a hydrogen like atom. It has a total energy of 3.4 eV. The kinetic energy is E and its de Broglie wavelength is λλ. Then
So, K E = 3.14 ev
Let p = momentum and m = mass of the electron.
de Broglie wavelength,
On substituting the values, we get
Two cylindrical rods of uniform crosssectional area A and 2A, having free electrons per unit volume 2n and n respectively are joined in series. A constant current I flows through them in steady state. The ratio of drift velocity of free electrons in the left rod to that of the right rod is (VL/VR) is:
Since current I = neAV_{d} through both rods is same
2(n)e AV_{L }= ne (2A) V_{R}
or V_{L}/V_{R }=1
Uncharged capacitor of capacitance 4 μF4 and a resistance of 2.5 M ΩM Ω are connected in series with 12 V battery at t=0. Find the time after which the potential difference across the capacitor is 3 times the potential difference across the resistor. [Given, ln (2) = 0.693]
Given : V_{C} = 3VR = 3(V  V_{C})
Here: V is the applied potential.
⇒ t = 2 x 10 x 0.693 = 13.86 sec
A refrigerator is to maintain the eatables, kept inside it, at 9^{o}C. The coefficient of performance of the refrigerator, if the room temperature is 36^{o}C, is
Here, T_{1} = 36°C = 36 4  273 = 309 K
T_{2} = 9°C = 9 + 273 = 282 K
The coefficient of performance of a refrigerator is given by
A ball is projected up an incline of 30o with a velocity of 30 ms^{−1} at an angle of 30^{o} with reference to the inclined plane from the bottom of the inclined plane. If g = 10 ms^{−2}, then the range on the inclined plane is
Three balls A, B and C whose masses are m, km and 4m respectively kept at rest on the horizontal smooth surface, as shown in the figure. The ball A is given velocity v_{0}, rightward and it collides with the ball B elastically. Then ball B collides elastically third ball C. For what value of k, does the third ball C receive the maximum speed?
Solving elastic collisions we get
maximizing v_{c} we get k=2
ABCDECA is a planar body of mass m of uniform thickness and same material. The dimensions are as shown in the figure. The moment of inertia of the body about an axis passing through point A and perpendicular to planar body is 1_{1} and that of about an axis passing through C and perpendicular to planer body is l_{2}. If 1_{1}/1_{2} is k. Find the l value of k.
Form a symmetric structure by completing squares as shown
using superposition and parallel axis theorem we can find l_{1} = 2ma^{2} Rotate the parts about to form a square as
A body of mass 6.25 kg is travelling in a horizontal straight line with a velocity of 3 m/sec when a horizontal force P is applied to it at right angle to the initial direction of motion. If P varies according to the accompanying graph, remains constant in direction and is the only force acting on the body in its plane of motion, find the magnitude of the velocity of the body when t = 2 sec.
Momentum agained in direction of force = Area
If a car is moving rightward with acceleration rightward as shown in the figure. Find the value of k so that, rod maintains its orientation as shown in the figure. Neglect the friction and mass of the small rollers at A and B.
The rod will align effective gravity in frame of cart
For identical rods, each of mass m are welded at their ends to form a square, and the corners are then welded to a light metal hoop of radius r. If the rigid assembly of rods and hoop is allowed to roll down the inclined rough surface. If the minimum value of the coefficient of static friction which will prevent slipping is k/10
Find the value of k.
If the nitrogen atom had electronic configuration 1s^{7}, it would have energy lower than that of the normal ground state configuration 1s^{2} 2s^{2 }2p^{3}, because the electrons would be closer to the nucleus, yet 1s^{7} is not observed because it violates
1s^{7 }violate Pauli exclusion principle, according to which an orbital cannot have more than two electrons.
Lone pair is not involved in resonance, most basic. In all other cases, lonepair of nitrogen is involved in resonance, less basic.
How many EDTA (ethylenediaminetetraacetic acid) molecules are required to make an octahedral complex with a Ca^{2+}
EDTA, Which has four lone pair donor oxygen atoms and two lone pair donor nitrogen atoms in each molecule forms complex with Ca^{2+} ion. EDTA is hexa due to need 1 EDTA to form octahedral completes with ca^{+2}
The first orbital of H is represented by where a_{0} is Bohr's radius The probability of finding the electron at a distance r, from the nucleus in the region dV is
P(r) = ψ^{2}4πr^{2}dr
4πr^{2}dr = dv = vol of thin spherical shell around nucleus at distance (r)
20% of N_{2}O_{4} molecules are dissociated in a sample of gas at 27^{∘}C and 760 torr. Mixture has the density at equilibrium equal to:
Total Moles 1 + α at equilibrium observed Molar Mass at equilibrium
Apart from this option, the other compounds don’t show geometrical isomerism.
Nickel (Z = 28) combines with a uninegative mono dentate ligand X to form a paramagnetic complex [NiX_{4}]^{2}.The number of unpaired electron(s) in the nickel and geometry of this complex ion are, respectively
Number of unpaired electrons = 2
Geometry = tetrahedral.
The correct stability order of the following resonance structures is?
No. of π π bonds ∝∝ Resonance Energy ∝∝ Stability
+ve charge on electro positive element
ve charge on electro negative element
unlike charges on adjacent atoms like charges to separate out far.
(i) Two π π bonds unlike charges on adjacent atoms
(iii) Two π π Bonds unlike charges on adjacent undesired atoms
(ii) One π π bond unlike charges separated but on desired atoms.
(iv) One π π bond unlike charges separated but on undesired atoms
(i) > (iii) > (ii) > (iv)
In presence of H_{2}O_{2}, HBr adds in antiMarkownikoffs way (peroxide effect).
Due to presence of two lone pairs of electrons, it is square planar in shape. Lone pairs are arranged at pyramidal position as per Bent's Rule.
(i) Tensile strength of vulcanised rubber is almost ten times more than raw rubber.
(ii) Elasticity of raw rubber is very high.
Choose the correct option.
Natural rubber is soft and sticky and after vulcanisation its strength increases due to cross linked formed between different
The correct order of reducing ability for the four successive elements Cr, Mn, Fe and Co is: Their Eored values are given below.
The values of E^{o}_{red }gives us an idea of the ease with which metals get reduced. A higher negative value implies that it has greater tendency to get oxidised.
For a metal to act as a reducing agent, it should itself get oxidised. Hence, the order of reducing ability is proportional to ease of oxidation which is inversly proportional to E^{o}_{red}.
Hence, the answer will be:
Mn > Cr > Fe > Co
An unknown alkyl halide (A) reacts with alcoholic KOH to produce a hydrocarbon (C_{4}H_{8}) as the major product. Ozonolysis of the hydrocarbon affords one mole of propanaldehyde and one mole of formaldehyde. Suggest which organic compound among the following has the correct structure of the above alkyl halide (A)?
Since the product on ozonolysis is propanaldehyde and formaldehyde, it implies that the the double bond is shared between C_{1} and C_{2}.
If the compound were option 1, the result of such elimination would have been 2 butene whose ozonolysis would give ethanals
1 butene on ozonolysis gives propanaldehyde and formaldehyde thus the hydrocarbon is
CH_{3}CH_{2}CH = CH_{2}
(C_{4}H_{8}) 1butene
1 butene can be obtained by
Which one of the following reactions of Xenon compounds is not feasible?
The reaction XeO_{3}+6HF⟶XeF_{6}+3H_{2}O is not feasible because XeF_{6 }formed will further produce XeO_{3} by getting hydrolysed.
XeF_{6} + H_{2}O → XeOF_{4} + 2HF
XeOF_{4} + H_{2}O → XeO_{2}F_{2} + 2HF
XeO_{2}F_{2 }+ H_{2}O → XeO_{3} + 2HF
One mole of a nonideal gas undergoes a change of state(2.0 atm, 3.0 L, 95 K) ⟶ (4.0 atm, 5.0 L, 245 K) with a change in internal energy, ΔE = 30.0 Latm. The change in enthalpy (ΔH) of the process in Latm
H = E+PVH
ΔH = ΔE+Δ(PV)
= ΔE+(P_{2}V_{2}−P_{1}V_{1})
= 30.0+(4×5−2×3)
= 30.0+14.0
= 44 L atm.
How many moles of KMnO_{4} are needed to oxidise a mixture of 1 mole each of FeSO_{4}, FeC_{2}O_{4} and Fe_{2}(C_{2}O_{4})_{3} completely in acidic medium :
Equialents of {FeSO_{4} + Fe(C_{2}O_{4}) + Fe_{2} (C_{2}O_{4})_{3}}
= {1 x1 + 1 x 3 + 1 x 6} = 10 Equialents
Equialents of KMnO_{4} = 10 Equialents
Moles of KMnO_{4} = 10/5 = 2 Moles
The catalyst used in the manufacture of polyethylene by Ziegler method is
Ziegler  Natta catalyst is used in polymerisation of ethene : (C_{2}H_{5})_{3} Al + TiCl_{4}.
The reagent with which both acetaldehyde and acetone react easily is
Grignard’s reagent reacts with both aldehydes and ketones while other three reagents reacts only with aldehydes, not with ketones.
Which of the following facts about the complex [Cr(NH_{3})_{6}]Cl_{3} is wrong?
The complex [Cr(NH_{3})_{6}]Cl_{3} involves d^{2}sp^{3} hybridization as it involves (n  1)d orbitals for hybridization. It is an inner orbital complex.
Consider the reaction 2NO(g)+O_{2}(g) ⟶ 2NO_{2}(g) Predict whether the reaction is spontaneous at 298 K. Δ_{f}G(NO) = 86.69kJ/mol,Δ_{f}G(NO_{2}) = 51.84kJ/mol
2NO(g)+O_{2}(g) ⟶ 2NO_{2}(g)
ΔG = 2Δ_{f}G(NO_{2})−[2Δ_{f}G(NO)+Δ_{f}G(O_{2})]
= (2×51.84)−(2×86.69+0)
= 103.68−173.38
= −69.7kJ/mol ; Δ G is negative
So the reaction is spontaneous
The pH range of a basic indicator (InOH, pK_{ln}.= 4) is 3.4 to 4.6. For what minimum ratio of does the solution appear in the colour of In+
x moles of phosgene gas is allowed to attain equilibrium with its gaseous decomposition products in a 1 litre vessel. For what value of x; half the chlorine atoms in the equilibrium mixture remain with phosgene. (KC for phosgene decomposition = 3)
If the number of 109°28' angles in the structure of TMS (tetra methyl silane) is x, find x/6.
How many moles of water vapour is evolved when 1 mole of hydrated aluminium chloride dimer (Al_{2}Cl_{6}.12H_{2}O) is strongly heated.
Calculate the energy required (in Joules) to convert all atoms of Mg to Mg^{2+} ions present in 48 mg of Mg vapours. IE_{1} and IE_{2} of Mg are 740 and 1450 kJ mol^{–1} respectively.
Mg_{(g)} ➔ Mg^{+}_{(g)} + e–; IE_{1} = 740 kJ mol^{–1}
Mg^{+}_{(g)} ➔ Mg^{2+}_{(g) }+ e^{–}; IE_{2} = 1450 kJ mol^{–1}
∴ Total energy required to convert 1 mol of Mg_{(g)} into
Mg^{2+}_{(g)} ion = IE_{1} + IE_{2}
= (740 + 1450) kJ mol^{–1}
= 2190 kJ mol^{–1}
^{}
= 2 × 10^{3} mol
∴ Energy required to ionize 2 × 10^{3} mol of Mg_{(g)}
= 2190 × 2 × 10^{3}
= 4.380 kJ
= 4380 J
[Cancelling x as x ≠ 0]
Alternate Solution
Use L’Hospital Rule,
The equation of a circle C_{1} is x^{2 }+ y^{2 }− 4x − 2y − 11 = 0. A circle C_{2} of radius 1 unit rolls on the outside of the circle C_{1} touching it externally. The locus of the centre of C_{2} has the equation
The centre of C_{1} = (2,1) and the radius
Two vertices of a triangle are (3, 2) and (2, 3) and its orthocentre is ( 6, 1). The coordinates of its third vertex are
Solving Eqs. (i) and (ii), we get
α =  1 , β = 6
∴ Third vertex is (1,6).
The equation of the common tangent to the equal parabolas y^{2} = 4ax and x^{2} = 4ay is
Any tangent to y^{2} = 4ax is It touches x^{2} = 4ay if
has equal roots.
So m^{3} + 1 = 0, i e , m = 1 Hence, the common tangent is y =  x  a
A line makes angles of 45° and 60° with the positive directions of x and y axes respectively. An angle, which the line can make with the positive direction of zaxis is:
Let α , β , γ be the angles, which the line makes with the positive directions of xaxis, yaxis and zaxis respectively
Hence the line makes an angle of 60° with the positive direction of zaxis
If the fourth term of is equal to 200 and x > 1, then x is equal to
∴ x = 10^{4}, 10^{1}
If the line 3x + 2y = 13 divides the area enclosed by the curve, 9x^{2 }+ 4y^{2 }− 18x − 16y − 11 = 0 into two parts then the ratio of the larger area to the smaller area is
E quation o f given curve is
It can be reduced to which is the equation of ellipse having it’s principal axis parallel to coordinate axes
⇒ lt's area = πab = π (2) ( 3) = 6π
From the figure we can see that points of intersection of and line 3x + 2y = 13, have x = 1, 3
⇒ Smaller area between ellipse and line
The third vertex of the triangle whose centroid is (7,−2, 5) and whose other two vertices are (2, 6, −4) and (4,−2, 3) is
Let A (2, 6,  4) and B (4,  2, 3) be the two given vertices of the triangle. Let C (α , β, γ) be the third vertex
Since G (7,  2, 5) is the centroid of ΔABC,
Hence C (15,  10,16) is the third vertex
⇒ m is a multiple of 4.
Hence the smallest positive value of m is 4.
Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r is equal to
From figure it is clear that ΔPRQ and ΔRSP are similar
If a^{3} + b^{6} = 2, then the maximum value of the term independent of x in the expansion of (ax^{1/3} + bx^{1/6})^{9} (a > 0, b > 0) is
Let (r + 1) th term be independent of x, then
As T_{r + 1} is independent of x
The coordinates of the point which divides the line segment joining the points (5,4, 2) and (−1,−2, 4) in the ratio 2 : 3 externally is
Let A (5 , 4 , 2) and B ( 1,  2, 4) be the given points.
Let P (x , y, z) be the point, which divides the line segment [AB] in the ratio  2 : 3
The area of an expanding rectangle is increasing at the rate of 48 cm^{2}/sec. The length of the rectangle is always equal to the square of the breadth. At the instant when the breadth is 4.5 cm, The length is increasing at the rate of
If ^{n}P_{r }= 1680 and ^{n}C_{r }= 70, then 69 n + r! is equal to
From equation (i) and (ii), we get
On putting the value of r in equation (i), we get
Consider the straight line ax + by = c, where a, b, c ∈ R^{+ }this line meets the coordinate axes at A and B respectively. If the area of the ΔOAB, O being origin, is always a constant equal to half, then
Given line is ax + by = c
⇒ This line cut the coordinate axes as points
Equating this area to half, we get,
c^{2} = ab
Therefore, a, c, b are in GP
cot A⋅cot B⋅cot C>0⇒cot A>0, cot B>0, cot C > 0
Because two or more of cot A, cot B, cot C cannot be negative at the same time in a triangle, as no two angle could be more than 90 degree.
The equation of the smallest circle passing through the intersection of the line x + y = 1 and the circle x^{2 }+ y^{2} = 9 is
Any circle passing through the points of intersection of the given line and circle has the equation x^{2} + y^{2}  9 + λ (x + y  1) = 0. Its centre
The circle is the smallest if center
Putting this value for λ, the equation of the smallest circle is x^{2} + y^{2}  9  (x + y  1) = 0
⇒ x^{2} + y^{2}  x  y  8 = 0
So given integration
The probability of a bomb hitting a bridge is 1/2 and two direct hits are needed to destroy it. Find the least number of bombs required so that the probability of the bridge being destroyed is greater than 0.9.
Let n ne the number of bombs required and X the number of bombs that hit the bridge.
Then X follows a bionomial distribution with parameters n and p=1/2. Now
If sin 3αα= 4 sin αα sin (x+ a) sin (x  αα), then 864 sin^{2} x+ 3620 cos^{2} x is equal to
we have
and f(1) = 2,
∴ f(101) = f(1) + 100 x 1/2 = 2 + 50
∴ f(101) = 52
If points (0,0,9), (1,1,8), (1,0,7) and (2,2, λ) are coplanar, then λ =
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