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A solid ball rolls down a parabolic path ABC from a height h as shown in figure. Portion AB of the path is rough while BC is smooth. How high will the ball climb in BC ?
At B , total kinetic energy = mgh
Here,
m = mass of ball
The ratio of rotational to kinetic energy would be , Kr/Kt = 2/5
where, Kr = 2/7mgh and Kt = 5/7 mgh
In portion BC, friction is absent . Therefore, rotational K.E will remain constant and Translational K.E will convert into potential energy.
Hence, if H be the height to which ball climbs in BC, then
mgH = Kt
mgH = 5/7mgh
H = 5/7h
A heat engine has an efficiency η. Temperatures of source and sink are each decreased by 100 K. Then, the efficiency of the engine.
where T_{1} and T_{2} are the temperatures of a source and sink respectively.
When T_{1} and T_{2} both are decreased by 100 K each, (T_{1}  T_{2}) stays constant. T_{1} decreases.
∴ η increases.
A particle is moving in a uniform circular motion on a horizontal surface. Particle position and velocity at time t = 0 are shown in the figure in the coordinate system. Which of the indicated variable on the vertical axis is incorrectly matched by the graph shown alongside for particle's motion
In Coolidge tube experiment, if applied voltage is increased to three times, the short wavelength limit of continuous X  ray spectrum shifts by 20 pm. What is the initial voltage applied to the tube ?
Cutoff wavelength for continuous xray is given as :
∴ hc eV_{0} = λ & hc 3 eV_{0} = λ  Δλ
⇒ V_{0} = 2 hc 3 e Δ λ = 41 kV
A washer is made of metal having resistivity 10^{–7} Ωm. The washer has inner radius 1 cm, outer radius 3 cm and thickness 1 mm. A magnetic field, oriented normal to the plane of the washer, has the time dependent magnitude B = (2t) tesla/sec. Find the current (in ampere) around the washer
Electric field at a general radial distance is E
E = rN/c
J = σE
Current in circular element di = j(dr)t
So net current in washer i =
Ends of two wires A and B having resistivity and of same cross section area joined in series together to from a single wire. If the resistance of the joined wire does not change with temperature, then find the ratio of their lengths given that temperature coefficient of resistivity of wire A and B is
α_{A}= 4 ×10^{–5}/ °C and α_{B} = –6 × 10^{–6}/°C. Assume that mechanical dimensions do not change with temperature
As net resistance does not changes on change temp. so
A cone of radius = height = r is under a liquid of density d. Its base is parallel to the free surface of the liquid at a depth H from it as shown in the figure. What is the net force due to liquid on its curved surface? (neglect atmospheric pressure)
Force due to liquid on curved surface = F
F = weight of liquid above the cone
A spherical black body has a radius R and steady surface temperature T, heat sources ensure the heat evolution at a constant rate and distributed uniformly over its volume. What would be the new steady surface temperature of the object if the radius is decreased by half? Assume surrounding to be at absolute zero and heat evolution rate through unit volume remain same.
Net heat getting generated in complete volume of sphere = rate of heat radiated by its surface
The smallest length scale known in physics is the Planck length. It is an important ingredient in some current cosmological theories. Which of the following expressions could represent this Planck length? (Symbols has usual meaning)
Dimensional formula of only option D matches with length.
In the figure shown two motors P_{1} & P_{2} fixed on a plank which is hanging with light string passing over fixed Pulley P. If the motors start winding the thread with angular velocity ω1 & ω2 then velocity of plank V is (here R_{1} & R_{2} are the radii of motor rotor respectively) [Given: ω1 = 2 rad/s, R_{1} = 2m, ω2 = 2 rad/s, R_{2} = 3m]
Five identical balls each of mass m and radius r are strung like beads at random and at rest along a smooth, rigid horizontal thin road of length L, mounted between immovable supports. Assume 10r < L and that the collision between balls or between balls and supports are elastic. If one ball is struck horizontally so as to acquire a speed v, the average force felt by the support is
A double star is a system of two stars rotating about their centre of mass only under their mutual gravitational attraction. Let the stars have masses m and 2m and let their separation be l. Their time period of rotation about their centre of mass will be proportional to
If white light is used in a Young’s double – slit experiment. Point C represents centre of a screen
Consider a thermodynamic cycle in a PV diagram shown in the figure performed by one mole of a monoatomic gas. The temperature at A is T_{0} and volume at A and B are related as V_{B} = V_{C} = 2V_{A}. Choose the correct option(s) form the following
temprature at state 'B' is maximum
Net work done by gas in cyclic process
Heat capacity for process
If the plank’s constant would be double the present value, in the Bohr’s model for hydrogen atom
In 1906, Robert Millikan devised an experiment that allowed him to determine the charge of an electron. A schematic of Millikan’s set – up is shown below:
Two metal plates are connected by a series of batteries to form a capacitor. There is an electric field between the plates. The metal plates are inside an insulated cylindrical container.
Oil drops are introduced into the container through a small hole in the top. The oil drops acquire a negative charge as they pass through the nozzle of the oil can. Some of the drops fall through a hole in the upper plate. By adjusting the voltage between the plates, certain drops can be suspended between them. The relationship between the electric field between the plates and the voltage across the plates is ∆V = EL
Where E is the electric field and L is the plate separation.
Millikan chose oil because of its relatively low vapour pressure and high charge holding ability. (To answer the following question assume oil drops as to be nonconducting tiny spheres)
In order for an oil drop of mass m, radius r and volume charge density ρ, to be suspended between the plates, the magnitude and direction of the electric field must be:
In 1906, Robert Millikan devised an experiment that allowed him to determine the charge of an electron. A schematic of Millikan’s set – up is shown below:
Two metal plates are connected by a series of batteries to form a capacitor. There is an electric field between the plates. The metal plates are inside an insulated cylindrical container.
Oil drops are introduced into the container through a small hole in the top. The oil drops acquire a negative charge as they pass through the nozzle of the oil can. Some of the drops fall through a hole in the upper plate. By adjusting the voltage between the plates, certain drops can be suspended between them. The relationship between the electric field between the plates and the voltage across the plates is ∆V = EL
Where E is the electric field and L is the plate separation.
Millikan chose oil because of its relatively low vapour pressure and high charge holding ability. (To answer the following question assume oil drops as to be nonconducting tiny spheres)
Suppose the original oil droplet were replaced with a positively charged one that had twice the charge and three times the mass of the original droplet, how would the magnitude of the electric field have to be changed in order for the drop to remain suspended?
In a hypothetical atom, a negatively charged particle having a charge of magnitude 3e and mass 3m revolves around a proton. Here, e is the electronic charge and m is the electronic mass. Mass of proton may be assumed to be much larger than that of the negatively charged particle, thus the proton is at rest. This “atom” obeys Bohr’s postulate of quantization of angular momentum, that is It is given that for the first Bohr orbit of hydrogen atom: radius of orbit is r_{0} speed of electron is V_{0}, and total energy is –E_{0}_{.} Now answer the following questions.
Speed of the revolving particle is, in the first Bohr orbit.
If mass of revolving particle is m, and change q. change at nucleus Q
Energy of n^{th} orbit
Radius of first orbit of this atom
In a hypothetical atom, a negatively charged particle having a charge of magnitude 3e and mass 3m revolves around a proton. Here, e is the electronic charge and m is the electronic mass. Mass of proton may be assumed to be much larger than that of the negatively charged particle, thus the proton is at rest. This “atom” obeys Bohr’s postulate of quantization of angular momentum, that is It is given that for the first Bohr orbit of hydrogen atom: radius of orbit is r_{0} speed of electron is V_{0}, and total energy is –E_{0}_{.} Now answer the following questions.
Radius of hypothetical atom is
If mass of revolving particle is m, and change q. change at nucleus Q
Energy of n^{th} orbit
Radius of first orbit of this atom
In a hypothetical atom, a negatively charged particle having a charge of magnitude 3e and mass 3m revolves around a proton. Here, e is the electronic charge and m is the electronic mass. Mass of proton may be assumed to be much larger than that of the negatively charged particle, thus the proton is at rest. This “atom” obeys Bohr’s postulate of quantization of angular momentum, that is It is given that for the first Bohr orbit of hydrogen atom: radius of orbit is r_{0} speed of electron is V_{0}, and total energy is –E_{0}_{.} Now answer the following questions.
The momentum of an emitted photon when it makes a transition from the second excited state to ground state, is
If mass of revolving particle is m, and change q. change at nucleus Q
Energy of n^{th} orbit
Radius of first orbit of this atom
The given figure shows a plot of the time dependent force x F acting on a particle in motion along the xaxis. What is the total impulse (in kgm/s) delivered by this force to the particle from time t = 0 to t = 2 second?
In the figure shown a small block B of mass m is released from the top of a smooth movable wedge A of the same mass m. The height of wedge A shown in figure is h = 16 cm. B ascends another movable smooth wedge C of the same mass. Neglecting friction anywhere find the maximum height (in cm) attained by block B on wedge C.
Let u and v be the speed of wedge A and block B at just after the Block B get off the wedge A. Applying conservation of momentum
in horizontal direction, we get
mu = mv ...(1)
Applying conservation of energy between initial and final state as shown in figure (1), we get
At the instant block B reaches amximum height h' on the wedge C (figure 2) the speed of block B and wedge C are v'
Applying conservation of momentum in horizontal direction, we get
mv=(m+m)v' ...(4)
Applying conservation of energy between initial and final state
Mass 2m is kept on a smooth circular track (R = 9 meters) of mass m which is kept on a smooth horizontal surface. The circular track is given a horizontal velocity towards left and released. Find the maximum height reached by 2m in meters.
Let v be the final speed of block when it is at maximum height. At that instant the speed of circular track shall also be v
From conservation of momentum
Two blocks of masses m_{1} and m_{2} are connected by spring of constant K such that m_{2}/m_{1}=9. The spring is initially compressed and the system is released from rest at t = 0 second. The work done by spring on the blocks m_{1} and m_{2} be ω_{1} and ω_{2} respectively by time t. The speeds of both the blocks at time ‘t’ are nonzero. Then find the value of ω_{1}/ω_{2}.
Here in this question when we released the system this will start moving
So ω_{1} = work done by spring on the block 1 = change in kinetic energy of block 1
ω_{2} = work done by spring on the block 2 = change in kinetic energy of block 2
So only spring will do the work
So
and
so
And we know there is no external force on this system so momentum will be conserved
So m_{1}v_{1} = m_{2}v_{2}
so v_{1}/v_{2} = m_{1}/m_{2}
So now put this in the ratio
We get ω_{1}/ω_{2} = m_{2}/m_{1} = 9
Two simple pendulums of length 5 m and 20 m respectively are given small linear displacement in one direction at the same time. They will again be in the phase when the pendulum of shorter length has completed .... oscillations.
If t is the time taken by pendulums to come in same phase again first time after t=0.
and N_{S}= Number of oscillations made by shorter length pendulum with time period T_{S}.
N_{L}= Number of oscillations made by longer length pendulum with time period T_{L}
Then t=N_{S}T_{S}=N_{L}T_{L}
_{}
_{}
⇒ N_{S}=2N_{L} i.e. if N_{L}=1
⇒ N_{S}=2
Copper reduces into NO and NO_{2} depending upon concentration of HNO_{3} in solution. Assuming [Cu^{2+}] = 0.1M, and P_{NO} = P_{NO2} = 10^{–3} bar. At which concentration of HNO_{3}, thermodynamic tendency for reduction of into NO and NO_{2} by copper is same?
A radioactive material (t_{1/2} = 30 days) gets spilled over the floor of a room. If initial activity is ten times the permissible value, after how many days will it be safe to enter the room
It will be safe to enter the room after the nuclei decreases to 1/10 of its initial amount.
→ Half life = 30 days
→ Using integration law,
∴ After 100 days, it will be safe to enter the room.
Trinitro benzene diagenium ionis strong electrophile type and show coupling even with mesitylene
A mixture of all possible stereoisomers from the above structure is subjected to fractional distillation, which of the following statements is correct
Which of the following about SF_{4}, SOF_{4} and COF_{2} molecules is correct?
Equatorial FSF bond angle is less in SF_{4 }than in SOF_{4} since lone pair repulsion is more than two electron pairs in double bond
In both SF_{4} and SOF_{4} the hybridisation state of S is same sp^{3}d
The OCF bond angle on COF_{2} is more than 120° since two electron pairs in double bond repel more than one electron pair in CF bonds.
Due to repulsion by lone pair the axial FSF bond angle is less than 180°
B_{2}O_{3} substitutes nonmetal oxides from several metal salts because
Less volatile with more melting point (B_{2}O_{3}) acidic oxide can substitute more volatile acidic oxides.
NCl_{3}, NBr_{3} and NI_{3} are explosive.
The correct statement regarding various types of molecular speeds are
The compounds that should be used to prepare glycine and β – alanine by Gabriel phthalimide synthesis are
In which of these compounds, Nitrogen can be estimated by Duma’s method?
A black mineral (A) in solid state is fused with KOH and KNO_{3} and the mixture extracted with water to get a green coloured solution (B). On passing CO_{2} gas through the solution the colour changes to pink with a black residue (C). Which of the following is/are correct
Formation of “B” through the attack of first two reagents involve respectively
A white substance (A) reacts with dilute H_{2}SO_{4} to produce a colourless gas (B) and a colourless solution (C). The reaction between (B) and acidified K_{2}Cr_{2}O_{7} solution produces a green solution and a slightly coloured precipitate (D). The substance (D) burns in air to produce a gas (E), which reacts with (B) to yield (D) and a colourless liquid. Anhydrous copper sulphate is turned blue on addition of this colourless liquid. Addition of aqueous NH_{3} or NaOH to (C) produces first a precipitate which dissolves in the excess of the respective reagent to produce a clear solution in each case
(B) and (D) are respectively
A white substance (A) reacts with dilute H_{2}SO_{4} to produce a colourless gas (B) and a colourless solution (C). The reaction between (B) and acidified K_{2}Cr_{2}O_{7} solution produces a green solution and a slightly coloured precipitate (D). The substance (D) burns in air to produce a gas (E), which reacts with (B) to yield (D) and a colourless liquid. Anhydrous copper sulphate is turned blue on addition of this colourless liquid. Addition of aqueous NH_{3} or NaOH to (C) produces first a precipitate which dissolves in the excess of the respective reagent to produce a clear solution in each case
The precipitate obtained by addition of aqueous NH_{3} or NaOH to (C) initially is _____ which dissolves in excess reagent to produce ________
Which of the following is correct regarding solutions of sodium metal in liquid ammonia.
All the statements are true regarding solutions of alkali metals in liquid NH_{3}.
For the given reaction the correct statement is
Choose the correct option
Free radical obtained from x is stabilised due to resonance with one phenyl group and free radical from y is stabilised by resonance with two phelnyl rings. Also free radical from z is stabilised by resonance with one phenyl group and hyperconjugation with CH,Igroup But free radical from w is not stabilised by any effect. So ease of abstraction or order of reactivity of different Hatoms is y > z > x > w. Abstraction from x, y, z and w give 1, 2 (enantiomers), 2 (enantiomers) and 1 product respectively so total no. of products are 6. But both enantiomers appear in one fraction so total no. of fractions is 4.
For the cell (at 1 bar H_{2} pressure) Pt/H_{2}(g) H X (m_{1}), NaX(m_{2}), NaCl(m_{3})/AgCl/Ag/Pt it is found that the value of E
approaches 0.2490 in the limit of zero concentration. Calculate for the acid HX at 27°C. (R = 8.3 Jmole^{–1}K^{–1}, F = 96500C)
During the titration of 100 ml of a weak monobasic acid solution using 0.1 M NaOH, the solution became neutral at 40 mL addition of NaOH and equivalence point was obtained at 50 mL NaOH addition. The K_{a}_{ }of the acid is (log 2 = 0.3)
The sum of no of cyclic transition states and intermediates in the above reaction during the formation of product is/are....
How many carbon atoms (In A, B and (c) changed their hybridization till the formation of D? (Consider each reaction and do not consider stereoisomerism)
Let a, b and c be positive constants. The value of ‘a’ in terms of ‘c’ if the value of integral is independent of ‘b’ equals
Let are three vectors along the adjacent edges of a tetrahedron, if and then volume of tetrahedron is
Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r_{1}, r_{2} and r_{3} are the numbers obtained on the die, then the probability that
A dice is thrown thrice n(s)=6 x 6 x 6
Favorable events, ω^{r1} + ω^{r2} + ω^{r3}
(r_{1},r_{2},r_{3}) are ordered 3 triples which can take values.
(1,2,3)(1,5,3)(4,2,3)(4,5,3)
(1,2,6)(1,5,6)(4,2,6)(4,5,6)
i.e, 8 ordered pairs and each can be arranged in 3! ways.
3!=6
A ray of light travels along a line y = 4 and strikes the surface of a curves y^{2} = 4 (x + y), then equations of the line along which reflected ray travel is
Focus is (0,2)
Point os interflection property of parabola, reflected ray passes through the focus.
X = 0 is required line.
A triangle ABC having vertices A (5, 1), B (–1, –7) and C (1, 4) respectively. If L1 be the line mirror passing through C and parallel to AB, a light ray emanating from point A and goes along the direction of internal bisector of the angle A, which meets the mirror at E and BC at D. Then sum of the area of ΔACE and ΔABC is
The number of different ways in which the persons A, B, C having 6 one rupee coins, 7 one rupee coins, 8 one rupee coins respectively donate 10 one rupee coins collectively is equal
Let a = cos^{–1} (cos 20), b = cos^{–1} (cos 30) and c = sin^{–1} sin (a + b) then maximum value of sin (2 (a+b+c) x) + cos^{2} ((a+b+c) x) is
If ‘A_{1}’ is the area bounded by x – aiai + y = bibi, i ∈ N, where and then
A rectangle ABCD of dimensions r and 2r is folded along diagonal BD such that planes ABD and CBD are perpendicular to each other. Let the position of the vertex A remains unchanged and C_{1} is the new position of C.
The distance of C_{1} from A is equal to
Let the rectangle ABCD initially lies in xy plane with B King at origin BC' along x  axis and BA
Along yaxis.
Equation of BD in xyplane is y = 2x
So. the coordinates of foot N of C on BD are
Clearly. CN = C_{1}N
Hence, the coordinates of various points in 3D are
A(0, r, 0), C(r, 0 , 0) , D (r, 2r, 0),
A rectangle ABCD of dimensions r and 2r is folded along diagonal BD such that planes ABD and CBD are perpendicular to each other. Let the position of the vertex A remains unchanged and C_{1} is the new position of C.
If ∠∠ABC_{1} = θ, then cos θ is equal to
Let the rectangle ABCD initially lies in xy plane with B King at origin BC' along x  axis and BA
Along yaxis.
Equation of BD in xyplane is y = 2x
So. the coordinates of foot N of C on BD are
Clearly. CN = C_{1}N
Hence, the coordinates of various points in 3D are
A(0, r, 0), C(r, 0 , 0) , D (r, 2r, 0),
A square ABCD of diagonal 2a is folded along the diagonal AC so that the planes DAC and BAC are at right angle. The shortest distance between DC and AB is
When folded coordinates will be D(0,0,a);C(a,0,0);A(−a,0,0);B(0,−a,0)
Equation of DC is,
Equation of ABAB is,
∴ Shortest distance = 2a/√3
Family of curves which makes an angle of π/4 with the family of hyperbola xy = a, is (a > 0, and a is a parameter)
xy=a
Take log on both sides
log(xy)=loga
⇒logx+logy=loga
Differentiate with respect to x
which is the differential equation governing the given family of rectangular hyperbola Replacing y' by
In the Ordinary differential equation(ODE), we obtain the ODE govern in the required isogonal trajectories as
which simplifies as
Differentiate with respect to x
As derivative of denominator is present in numerator in L.H.S., hencewe can directly integrate to get
ln(v^{2}+2v−1) =−2lnx + lnA'
⇒ ln(v^{2}+2v−1) + 2lnx = lnA'
⇒ ln(v^{2}+2v−1) + lnx^{2} = lnA'
⇒ ln[x^{2}(v^{2}+2v−1)] = lnA'
⇒ x^{2}(v^{2}+2v−1)=A'
Now y=vx
Let position vector of the orthocenter of ΔABC be Then, which of the following statement(s) is/are correct (Given position vector of points A, B, C are
Let f (x) = x^{3} + 2x^{2} – x + 1, then which of the following statement(s) is/are correct
f (–2) = 3 and f (–3) is negative
So, equation has one real root in between (–3, –2)
The equations of AB, BC, AC, the three sides of ΔABC are –x + y –1 = 0, x + y –1 = 0 and x = –4. If (α, 0) and (0, β) lie inside the triangle where α, β ∈ Z, then
a, b ∈ I satisfies equation a(b – l) = 3 + b – b^{2}, then a + b is equal to
Let be a sequence of sets defined by a_{n} = (n^{2}  1)/n. Then,
A right angle triangle ABC, right angle at A is inscribed in hyperbola xy = c^{2} (c > 0) such that slope of BC is 2. If distance of point A from centre of xy = c^{2} is √10, then which of the following is/are correct for xy = c^{2}
Let the coordinates of point A are (ct. c/t)
So. the slope of normal at A will be t^{2}.
And normal will be parallel to BC
Let P(a, b) be a variable point satisfying Let R be the complete region represented in xy plane in which P can lie.
Area of region R is
Let z_{1}, z_{2}, z_{3} be complex numbers such that z_{1} = z_{2} = z_{3} = 3&(z_{1} ≠ z_{3}). Then find the value of
[min{az_{2},+ (1  a)z_{3 } z_{1}} = height, from z_{1} to the line, joining z_{2} , z_{3}
Let z represents a variable point in complex plane such that z − z_{1} is real, where z_{1} is a fixed point in the same plane? Let “m” be the number of “z” values such that z = λ, where λ >Im(z_{1}) & let “n” be the number of “z” values such that z = λ, where λ = Im(z_{1}) then value of m + n =__________
The least possible degree of a polynomial equation, with real coefficients having 2ω^{2}, 3 + 4ω, 3 + 4ω^{2}, 5 − ω − ω^{2} as roots is ________
(ω, ω^{2} are nonreal cube roots of 1).
A curve is defined as Two spiders, one male and other female were moving together along the curve. The female spider suddenly realizes that the male spider is a rogue spider and immediately tries to get away as far as possible from it. Hence it moved onto the another point on the curve. The maximum distance between two final points when both spiders try out all possibilities, is k. Then the value of __________[.] Greatest integer function
If α,β,γ are the roots of x^{3} − 3x^{2} + 3x + 7 = 0 and w is a nonreal cube root of 1, and the value of then number of ordered pairs (p, q) such that p + q = 15 is _______
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