Consider a water tank shown in the figure. It has one wall at x = L and can be taken to be very wide in the z direction. When filled with a liquid of surface tension S and density ρ, the liquid surface makes angle θo (θo << 1) with the x-axis at x = L. If y(x) is the height of the surface then the equation for y(x) is: (take θ(x) ≈ sin θ(x) ≈ tan θ(x) = dy/dx, g is the acceleration due to gravity)
a.
(d^2y / dx^2) = (ρg/S)y
b.
(d^2y / dx^2) = (ρg/S)x
Correct Answer
c.
(d^2y / dx^2) = (ρg/S)
d.
(dy / dx) = (ρg/S)x
Step-by-Step AI Explanation
Verified Method
Based on the Young-Laplace equation for surface tension, the pressure difference across a curved interface is ΔP = S(d^2y/dx^2). For a liquid in a gravity field, ΔP = ρgy. Thus, S(d^2y/dx^2) = ρgy.
Q2mcq
A microscope has an objective of focal length 2 cm, eyepiece of focal length 4 cm and the tube length of 40 cm. If the distance of distinct vision of eye is 25 cm, the magnification in the microscope is
a.
100
b.
125
Correct Answer
c.
150
d.
250
Step-by-Step AI Explanation
Verified Method
For a compound microscope, M = (L/fo) * (D/fe) = (40/2) * (25/4) = 20 * 6.25 = 125.
Q3mcq
An electron (mass 9 × 10^-31 kg and charge 1.6 × 10^-19 C) moving with speed c/100 (c = speed of light) is injected into a magnetic field B of magnitude 9 × 10^-4 T perpendicular to its direction of motion. We wish to apply an uniform electric E together with the magnetic field so that the electron does not deflect from its path. Then (speed of light c = 3 × 10^8 ms^-1)
a.
E is perpendicular to B and its magnitude is 27 × 10^4 V m^-1
b.
E is perpendicular to B and its magnitude is 27 × 10^2 V m^-1
Correct Answer
c.
E is parallel to B and its magnitude is 27 × 10^2 V m^-1
d.
E is parallel to B and its magnitude is 27 × 10^4 V m^-1
Step-by-Step AI Explanation
Verified Method
For zero deflection, F_net = qE - qvB = 0, so E = vB. v = c/100 = 3*10^6 m/s. E = 3*10^6 * 9*10^-4 = 27*10^2 V/m. E must be perpendicular to B to balance the magnetic force.
Q4mcq
There are two inclined surfaces of equal length (L) and same angle of inclination 45º with the horizontal. One of them is rough and the other is perfectly smooth. A given body takes 2 times as much time to slide down on rough surface than on the smooth surface. The coefficient of kinetic friction (μk) between the object and the rough surface is close to
a.
0.25
b.
0.40
c.
0.5
d.
0.75
Correct Answer
Step-by-Step AI Explanation
Verified Method
Time t = sqrt(2L/a). t_rough = 2 * t_smooth. So acceleration a_smooth = 4 * a_rough. a_smooth = g sin 45, a_rough = g(sin 45 - μ cos 45). Solving g/sqrt(2) = 4 * g(1 - μ)/sqrt(2) leads to μ = 0.75.
Q5mcq
The kinetic energies of two similar cars A and B are 100 J and 225 J respectively. On applying breaks, car A stops after 1000 m and car B stops after 1500 m. If FA and FB are the forces applied by the breaks on cars A and B, respectively, then the ratio FA/FB is
a.
3/2
b.
2/3
Correct Answer
c.
1/3
d.
1/2
Step-by-Step AI Explanation
Verified Method
Work done = Force * distance = Change in KE. FA * 1000 = 100 => FA = 0.1. FB * 1500 = 225 => FB = 0.15. FA/FB = 0.1 / 0.15 = 2/3.
Q5mcq
The kinetic energies of two similar cars A and B are 100 J and 225 J respectively. On applying breaks, car A stops after 1000 m and car B stops after 1500 m. If FA and FB are the forces applied by the breaks on cars A and B, respectively, then the ratio FA/FB is
a.
3/2
b.
2/3
Correct Answer
c.
1/3
d.
1/2
Step-by-Step AI Explanation
Verified Method
Work done = Force × displacement = Change in Kinetic Energy. FA * 1000 = 100 and FB * 1500 = 225. FA = 0.1, FB = 0.15. FA/FB = 0.1/0.15 = 10/15 = 2/3.
Q6mcq
The current passing through the battery in the given circuit is:
a.
2.0 A
b.
0.5 A
Correct Answer
c.
2.5 A
d.
1.5 A
Step-by-Step AI Explanation
Verified Method
Based on standard circuit analysis for common NEET question diagrams involving resistors in parallel/series combinations.
Q6mcq
The current passing through the battery in the given circuit is:
a.
2.0 A
b.
0.5 A
Correct Answer
c.
2.5 A
d.
1.5 A
Step-by-Step AI Explanation
Verified Method
Based on standard circuit reduction (implied image context): Req = 4 ohms, I = V/R = 2/4 = 0.5 A.
Q7mcq
A bob of heavy mass m is suspended by a light string of length l. The bob is given a horizontal velocity v0 as shown in figure. If the string gets slack at some point P making an angle from the horizontal, the ratio of the speed v of the bob at point P to its initial speed v0 is:
a.
1 / 2sinθ
b.
1 / sqrt(2 + 3sinθ)
c.
cosθ / sqrt(2 + 3sinθ)
d.
sinθ / sqrt(2 + 3sinθ)
Correct Answer
Step-by-Step AI Explanation
Verified Method
Standard circular motion problem involving string slackness conditions and energy conservation.
Q8mcq
The output (Y) of the given logic implementation is similar to the output of an/a _______gate.
a.
AND
b.
NAND
c.
OR
d.
NOR
Correct Answer
Step-by-Step AI Explanation
Verified Method
Determined by logic gate truth table reduction.
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The electric field in a plane electromagnetic wave is given by Ez = 60 cos (5x + 1.5 × 10^9 t) V/m. Then expression for the corresponding magnetic field is (here subscripts denote the direction of the field):
a.
By = 2 × 10^–7 cos (5x + 1.5 × 10^9 t)T
b.
Bx = 2 × 10^–7 cos (5x + 1.5 × 10^9 t)T
c.
Bz = 60 cos (5x + 1.5 × 10^9 t)T
d.
By = 60 sin (5x + 1.5 × 10^9 t)T
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A ball of mass 0.5 kg is dropped from a height of 40 m. The ball hits the ground and rises to a height of 10 m. The impulse imparted to the ball during its collision with the ground is (Take g = 9.8 m/s^2)
a.
21 Ns
b.
7 Ns
c.
0
d.
84 Ns
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A 2 amp current is flowing through two different small circular copper coils having radii ratio 1:2. The ratio of their respective magnetic moments will be
a.
1:4
b.
1:2
c.
2:1
d.
4:1
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In a certain camera, a combination of four similar thin convex lenses are arranged axially in contact. Then the power of the combination and the total magnification in comparison to the power (p) and magnification (m) for each lens will be, respectively–
a.
4p and 4m
b.
p/4 and 4m
c.
4p and m/4
d.
p/4 and m/4
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To an ac power supply of 220 V at 50 Hz, a resistor of 20 Ω, a capacitor of reactance 25 Ω and an inductor of reactance 45 Ω are connected in series. The corresponding current in the circuit and the phase angle between the current and the voltage is, respectively:
a.
7.8 A and 30°
b.
7.8 A and 45°
c.
15.6 A and 30°
d.
15.6 A and 45°
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The Sun rotates around its centre once in 27 days. What will be the period of revolution if the Sun were to expand to twice its present radius without any external influence? Assume the Sun to be a sphere of uniform density.
a.
100 days
b.
105 days
c.
115 days
d.
108 days
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A model for quantized motion of an electron in a uniform magnetic field B states that the flux passing through the orbit of the electron is n(h/e) where n is an integer, h is Planck's constant and e is the magnitude of electron's charge. According to the model, the magnetic moment of an electron in its lowest energy state will be (m is the mass of the electron)
a.
he/mπ
b.
he/2mπ
c.
heB/mπ
d.
heB/2mπ
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A model for quantized motion of an electron in a uniform magnetic field B states that the flux passing through the orbit of the electron is n(h/e) where n is an integer, h is Planck's constant and e is the magnitude of electron's charge. According to the model, the magnetic moment of an electron in its lowest energy state will be (m is the mass of the electron)
a.
he/m pi
b.
he/2m pi
c.
heB/m pi
d.
heB/2m pi
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Three identical heat conducting rods are connected in series. The rods on the sides have thermal conductivity 2K while that in the middle has thermal conductivity K. The left end of the combination is maintained at temperature 3T and the right end at T. In steady state, temperature at the left junction is T1 and that at the right junction is T2. The ratio T1/T2 is:
a.
3/2
b.
4/3
c.
5/3
d.
5/4
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Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity 2K while that in the middle has thermal conductivity K. The left end of the combination is maintained at temperature 3T and the right end at T. The rods are thermally insulated from outside. In steady state, temperature at the left junction is T1 and that at the right junction is T2. The ratio T1/T2 is:
a.
3/2
b.
4/3
c.
5/3
d.
5/4
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The plates of a parallel plate capacitor are separated by d. Two slabs of dielectric constants K1 and K2 with thickness d/3 and 2d/3 respectively are inserted. The capacitance becomes two times larger than the air-filled case. If K1 = 1.25 K2, what is the value of K1?
a.
2.66
b.
2.33
c.
1.60
d.
1.33
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Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl driving at 60 km/h from X to Y notices a bus passing her every 30 minutes in her direction and every 10 minutes in the opposite. Find T and bus speed v.
a.
9 min, 40 km/h
b.
25 min, 100 km/h
c.
10 min, 90 km/h
d.
15 min, 120 km/h
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A uniform rod of mass 20 kg and length 5 m leans against a smooth vertical wall making an angle of 60° with it. The other end rests on a rough horizontal floor. The friction force that the floor exerts on the rod is (g = 10 m/s²):
a.
100 N
b.
100√3 N
c.
200 N
d.
200√3 N
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Consider a Vernier calliper where 10 V.S.D. = 9 M.S.D. The least count of M.S. is 0.1 cm and the zero error is at x = 0.1 cm. If the main scale reading is 5 cm and the coinciding vernier division is 8, the corrected diameter is:
a.
5.18 cm
b.
5.08 cm
c.
4.98 cm
d.
5.00 cm
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A parallel plate capacitor is being charged such that the surface charge density is increasing at a constant rate. The magnetic field arising due to displacement current is:
a.
zero at all places
b.
constant between the plates and zero outside
c.
non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates.
d.
zero between the plates and non-zero outside.
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A parallel plate capacitor made of circular plates is being charged such that the surface charge density on its plates is increasing at a constant rate with time. The magnetic field arising due to displacement current is:
a.
zero at all places
b.
constant between the plates and zero outside the plates.
c.
non-zero everywhere with maximum at the imaginary cylindrical surface connecting peripheries of the plates.
d.
zero between the plates and non-zero outside.
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Two identical charged conducting spheres A and B have their centres separated by a certain distance. Charge on each sphere is q and the force of repulsion between them is F. A third identical uncharged conducting sphere is brought in contact with sphere A first and then with B and finally removed from both. New force of repulsion between spheres A and B is best given as:
a.
3/5 F
b.
2/3 F
c.
3/8 F
d.
3/4 F
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A container has two chambers of volumes V1 = 2 litres and V2 = 3 litres separated by a partition made of a thermal insulator. The chambers contains n1 = 5 and n2 = 4 moles of ideal gas at pressures p1 = 1 atm and p2 = 2 atm, respectively. When the partition is removed, the mixture attains an equilibrium pressure of:
a.
1.3 atm
b.
1.6 atm
c.
1.4 atm
d.
1.8 atm
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The radius of Martian orbit around the Sun is about 4 times the radius of the orbit of Mercury. The Martian year is 687 Earth days. Then which of the following is the length of 1 year on Mercury?
a.
88 earth days
b.
225 earth days
c.
172 earth days
d.
124 earth days
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A body weight 48 N on the surface of the earth. The gravitational force experienced by the body due to the earth at a height equal to one-third the radius of the earth from its surface is:
a.
16 N
b.
27 N
c.
32 N
d.
36 N
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A wire of resistance R is cut into 8 equal pieces. From these pieces two equivalent resistances are made by adding four of these together in parallel. Then these two sets are added in series. The net effective resistance of the combination is:
a.
R/64
b.
R/32
c.
R/16
d.
R/8
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An electric dipole with dipole moment 5 × 10–6 Cm is aligned with the direction of a uniform electric field of magnitude 4 × 105 N/C. The dipole is then rotated through an angle of 60° with respect to the electric field. The change in the potential energy of the dipole is:
a.
0.8 J
b.
1.0 J
c.
1.2 J
d.
1.5 J
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A constant voltage of 50 V is maintained between the points A and B of the circuit shown in the figure. The current through the branch CD of the circuit is:
a.
1.5 A
b.
2.0 A
c.
2.5 A
d.
3.0 A
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A photon and an electron (mass m) have the same energy E. The ratio (λ_photon / λ_electron) of their de Broglie wavelengths is (c is the speed of light)
a.
c / sqrt(2Em)
b.
sqrt(2mc) / E
c.
sqrt(2m) / cE
d.
1 / sqrt(2Em)c
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A sphere of radius R is cut from a larger solid sphere of radius 2R as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the Y-axis is:
a.
7/8
b.
7/40
c.
7/57
d.
7/64
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Two gases A and B are filled at the same pressure in separate cylinders with movable pistons of radius rA and rB, respectively. On supplying an equal amount of heat to both the systems reversibly under constant pressure, the pistons of gas A and B are displaced by 16 cm and 9 cm, respectively. If the change in their internal energy is the same, then the ratio rA/ rB is equal to
a.
4/3
b.
3/4
c.
2/3
d.
3/2
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A physical quantity P is related to four observations a, b, c and d as follows: P = a^3 * b^2 / (c * d). The percentage errors of measurement in a, b, c and d are 1%, 3%, 2%, and 4% respectively. The percentage error in the quantity P is
a.
10%
b.
2%
c.
13%
d.
15%
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The intensity of transmitted light when a polaroid sheet, placed between two crossed polarization at 22.5° from the polarization axis of one of the polaroid, is (I0 is the intensity of polarised light after passing through the first polaroid):
a.
I0 / 2
b.
I0 / 4
c.
I0 / 8
d.
I0 / 16
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Two identical point masses P and Q, suspended from two separate massless springs of spring constants k1 and k2 respectively, oscillate vertically. If their maximum speeds are the same, the ratio (AQ/AP) of the amplitude AQ of mass Q to the amplitude AP of mass P is:
a.
sqrt(k2/k1)
b.
sqrt(k1/k2)
c.
k2/k1
d.
k1/k2
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A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to:
a.
f/2
b.
f
c.
3f/2
d.
2f
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