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*Multiple options can be correct

QUESTION: 1

The potential energy of a particle of mass m at a distance r from a fixed point O is given by (r)=kr^{2}/2, where k is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius R about the point O. If v is the speed of the particle and L is the magnitude of its angular momentum about O, which of the following statements is (are) true?

Solution:

The expression for the force acting on the particle is,

The negative sign indicates that the direction of force is towards the centre.

At r =R , F = −kR

This force must be equal to centripetal force for the particle to move in circular orbit.

The expression for the angular momentum of the particle is,

*Multiple options can be correct

QUESTION: 2

Consider a body of mass 1.0 ka at rest at the origin at time t =0. A force is applied on the body, where α= 1.0 Ns^{-1} and β=1.0 N. The torque acting on the body about the origin at time t = 1.0 s is . Which of the following statements is (are) true?

Solution:

Write the expression for the force on the body.

For, m = 1.0 kg , α = 1.0 N/s and β = 1.0 N,

The expression of velocity at time t = 1s is,

At t = 1s , the expression of the distance of the body from the origin is,

At time t = 1s ,

The torque acting on the particle at time is,

*Multiple options can be correct

QUESTION: 3

A uniform capillary tube of inner radius r is dipped vertically into a beaker filled with water. The water rises to a height h in the capillary tube above the water surface in the beaker. The surface tension of water is σ. The angle of contact between water and the wall of the capillary tube is θ. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

Solution:

Write the expression for the rise in the height of the water in the capillary tube dipped vertically.

The expression for the effective acceleration is,

g_{eff} = g + a

From the above equation, it is clear that

*Multiple options can be correct

QUESTION: 4

In the figure below, the switches S_{1} and S_{2} are closed simultaneously at t = 0 and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current I in the middle wire reaches its maximum magnitude I_{max} at time t =τ. Which of the following statements is (are) true?

Solution:

The below figure represents the circuit diagram:

The expression for the net current flow in the middle wire is,

Condition for the current to be maximum is,

At maximum current, t = τ thus,

So, the value of maximum current is,

*Multiple options can be correct

QUESTION: 5

Two infinitely long straight wires lie in the xy-plane along the lines x = ±R. The wire located at x = +R carries a constant current I_{1} and the wire located at x = -R carries a constant current I_{2}. A circular loop of radius R is suspended with its centre at (0, 0,√3R) and in a plane parallel to the xy-plane. This loop carries a constant current I in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the direction. Which of the following statements regarding the magnetic field is (are) true?

Solution:

The given condition is represented by the figure as shown below:

(A) Due to two straight wires, the magnetic field at origin is zero because both the wires are located at same distance from the origin; thus, they cancel out magnetic field of each other. Therefore, only the magnetic field due to the circular loop act on the origin and its direction is in −k . So, the total magnetic field could not be zero at origin.

(B) When I_{1} > 0 and I_{2} < 0 , the magnetic field due to the circular loop acting on the origin and its direction is in −k and the net magnetic field at origin due to both the straight wires is in + k direction. Thus it may be possible that they cancel out and make zero magnetic field at origin.

(C) When I_{1} < 0 and I_{2} > 0 , the magnetic field due to the circular loop acting on the origin and its direction is in −k and the net magnetic field at origin due to both the straight wires is in −k direction. So, the total magnetic field cannot be zero at origin.

(D) If I_{1} =I_{2} then B_{1} =B_{2} , then they cancel out each other at the center of the loop. So, only magnetic field exists due to the circular loop.

*Multiple options can be correct

QUESTION: 6

One mole of a monatomic ideal gas undergoes a cyclic process as shown in the figure (where V is the volume and T is the temperature). Which of the statements below is (are) true?

Solution:

(A). For process I, it is not an isochoric process because the volume is decreasing.

(B) For process II, there will be no change in the internal energy of the system because the temperature is constant. As the volume is increasing, the work done is positive ΔW> 0 .

From, first law of thermodynamics,

So, in the process II, heat is absorbed.

(C) For process IV, there will be no change in the internal energy of the system because the temperature is constant. As the volume is decreasing, the work done is negative ∆W < 0 .

From, first law of thermodynamics,

So, in process IV, heat is released.

(D) There is a linear relation between T and V in an isobaric process, that is

T ∝ V

So, the T −V must be linear. Thus, processes I and III are not isobaric processes.

*Answer can only contain numeric values

QUESTION: 7

Two vectors and are defined as and , where a is a constant and

ω = π/6 rad s^{-1}. If at time t = τ for the first time, the value of τ, in seconds, is _________ .

Solution:

Simplify the above equation,

Put n = 0 for the first time,

= 2s

*Answer can only contain numeric values

QUESTION: 8

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed 1.0 ms^{-1} and the man behind walks at a speed 2.0 ms^{-1}. A third man is standing at a height 12m above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency 1430 Hz. The speed of sound in air is 330 ms^{-1}. At the instant, when the moving men are 10 m apart, the stationary man is equidistant from them. The frequency of beats in Hz, heard by the stationary man at this instant, is_________.

Solution:

The figure as shown below represents the given condition:

The expression of the frequency at A is,

The expression of the frequency at B is,

The expression of the net frequency heard by the stationary man is,

Here,

*Answer can only contain numeric values

QUESTION: 9

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is then the height of the top of the inclined plane, in meters, is __________. Take g = 10m s^{-2}.

Solution:

The below figure represents the given condition:

At any instant, say O point on the incline plane.

The expression of the moment of inertia of the disk is,

The expression of the moment of inertia of the ring is,

I_{ring} = 2mR^{2}

So,

Similarly,

From the Newton’s equation,

As per the condition given,

Simplify the above equation,

*Answer can only contain numeric values

QUESTION: 10

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is 2.0 Nm^{-1} and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition.

Another block of mass 1.0 kg moving with a speed of 2.0 ms^{-1 }collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ____________ .

Solution:

Apply conservation of momentum,

Initial momentum = final momentum

1 x u_{2} = 2v_{2} + v_{1}

1 x 2 = 2v_{2} + v_{1}

2v_{2} + v_{1} = 2 ...(1)

The coefficient of restitution must be 1, since the collision is perfectly elastic.

v_{2} - v_{1} = 2 ....(2)

Solve equation (1) and equation (2),

The time period of the oscillation is,

For a half time period,

The distance of the block after collision is given as,

*Answer can only contain numeric values

QUESTION: 11

Three identical capacitors C_{1}, C_{2} and C_{3} have a capacitance of 1.0 μF each and they are uncharged initially. They are connected in a circuit as shown in the figure and C_{1} is then filled completely with a dielectric material of relative permittivity ∈_{r}. The cell electromotive force (emf) V_{o} = 8 V. First the switch S_{1} is closed while the switch S_{2} is kept open. When the capacitor C_{3} is fully charged, S_{1} is opened and S_{2} is closed simultaneously. When all the capacitors reach equilibrium, the charge on C_{3} is found to be 5 μC. The value of ∈_{r} = ______________.

Solution:

The capacitor C_{3} charges to 8 μC as the switch S_{1} is closed.

But when switch S_{1} is opened and S_{2} is closed, then capacitor C_{3} is charged to 5 μC , thus the net charge 8μC − 5μC= 3μC resides on C_{1} and C_{2} , as they are connected in series.

So, Apply the Kirchhoff loop,

*Answer can only contain numeric values

QUESTION: 12

In the xy-plane, the region y > 0 has a uniform magnetic field and the region y < 0 has another uniform magnetic field A positively charged particle is projected from the origin along the positive y-axis with speed V_{o}=πms^{-1} at t = 0, as shown in the figure. Neglect gravity in this problem. Let t = T be the time when the particle crosses the x-axis from below for the first time. If B_{2} = 4B_{1}, the average speed of the particle, in ms^{-1}, along the x-axis in the time interval T is .

Solution:

The expression of the distance traveled by the positively charged particle in uniform magnetic field B_{1} is,

The expression of the distance traveled by the positively charged particle in uniform magnetic field B_{2} is,

Write the expression of the total distance along x-axis.

Write the expression of the total time of the charged particle.

The average velocity of the charged particle is given as,

*Answer can only contain numeric values

QUESTION: 13

Sunlight of intensity 1.3 kW m^{-2} is incident normally on a thin convex lens of focal length 20 cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, in kW m^{-2}, at a distance 22 cm from the lens on the other side is__________.

Solution:

The below diagram shows falling of the sunlight parallel to a thin convex lens:

The ratio of area is calculated as,

The final intensity of the sunlight is calculated as,

*Answer can only contain numeric values

QUESTION: 14

Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures T_{1} = 300 K and T_{2} = 100 K, as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are K_{1} and K_{2} respectively. If the temperature at the junction of the two cylinders in the steady state is 200 K, then K_{1}/K_{2 = ________________.
}

Solution:

The rate of heat flow is same at steady state,

The expression of the thermal resistance of the cylinder is,

QUESTION: 15

PARAGRAPH “X”

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E] and [B] stand for dimensions of electric and magnetic fields respectively, while [∈O] and [μO] stand for dimensions of the permittivity and permeability of free space respectively. [L] and [T] are dimensions of length and time respectively. All the quantities are given in SI units.

Q. The relation between [E] and [B] is

Solution:

The electrostatic force is, F_{e} = qE

The magneto static force is,

F_{m} = qvB

Since, the dimension of force is same, qE = qvB

E = vB

In terms of dimension,

QUESTION: 16

PARAGRAPH “X”

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, [E] and [B] stand for dimensions of electric and magnetic fields respectively, while [∈_{O}] and [μ_{O}] stand for dimensions of the permittivity and permeability of free space respectively. [L] and [T] are dimensions of length and time respectively. All the quantities are given in SI units.

Q. The relation between [∈_{O}] and [μ_{O}] is

Solution:

Write the expression for the relation between μ_{0} and ∈_{0} .

In terms of dimension,

QUESTION: 17

PARAGRAPH “A”

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are Δx, Δy and Δz, respectively, then

. The series expansion for to first power in Δy/y is The relative errors in independent variables are always added. So the error in z will be

The above derivation makes the assumption that Δx/x << 1, Δy/y << 1, Therefore, the higher powers of these quantities are neglected.

Q. Consider the ratio to be determined by measuring a dimensionless quantity a. If the error in the measurement of a is then what is the error Δr in determining r?

Solution:

The ratio is,

For small error,

The expression for the magnitude of the error in r is,

QUESTION: 18

PARAGRAPH “A”

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation z = x/y. If the errors in x, y and z are Δx, Δy and Δz, respectively, then

The series expansion for to first power in Δy/y, is The relative errors in independent variables are always added. So the error in z will be

The above derivation makes the assumption that Δx/x <<1, Δy/y << 1. Therefore, the higher powers of these quantities are neglected.

Q. In an experiment the initial number of radioactive nuclei is 3000. It is found that 1000 ± 40 nuclei decayed in the first 1.0 s. For up to first power in x. The error Δλ, in the determination of the decay constant λ, in s^{-1}, is

Solution:

The expression of the number of nuclei left after radioactive decay is,

Simplify above equation,

QUESTION: 19

The compound(s) which generate(s) N_{2} gas upon thermal decomposition below 300 °C is (are)

Solution:

The decomposition process of given species below 300 °C is,

The compound Mg_{3}N_{2} does not undergo decomposition.

*Multiple options can be correct

QUESTION: 20

The correct statement(s) regarding the binary transition metal carbonyl compounds is (are) (Atomic numbers: Fe = 26, Ni = 28 )

Solution:

(A) The total number of valence electrons in Fe(CO)_{5} and (Ni(CO)_{4} is 18.

(B) Carbonyl group is a strong field ligand. Hence, the complexes are low spin in nature.

(C) For the lower oxidation state of metal, the electron density of the metal is high. Therefore, it leads to an increased back bonding. Therefore, the strength of M −C bond increases.

(D) Increased oxidation state of metal decreases back bonding and therefore it leads to an increased strength of C −O bond.

*Multiple options can be correct

QUESTION: 21

Based on the compounds of group 15 elements, the correct statement(s) is (are)

Solution:

(A) The compound Bi_{2}O_{5} is metallic oxide, hence it is basic while the compound N_{2}O_{5} is non-metallic oxide and thus it is acidic.

(B) All the elements in NF_{3} are non-metals but in BiF_{3} , bismuth is metallic. So, the covalent character of NF_{3} is greater than BiF_{3}.

(C) NH_{3} has higher boiling point than PH_{3} due to the presence of hydrogen bonding in NH_{3}.

(D) N −N bond is weak because the lone pair-lone pair repulsion is more in dinitrogen due to its small size.

QUESTION: 22

In the following reaction sequence, the correct structure(s) of X is (are)

Solution:

The correct sequence of reaction is,

*Multiple options can be correct

QUESTION: 23

The reaction(s) leading to the formation of 1,3,5-trimethylbenzene is (are)

Solution:

(A)

(B)

(C)

(D)

*Multiple options can be correct

QUESTION: 24

A reversible cyclic process for an ideal gas is shown below. Here, P, V, and T are pressure, volume, and temperature, respectively. The thermodynamic parameters q, w, H, and U are heat, work, enthalpy and internal energy, respectively.

The correct option(s) is (are)

Solution:

(A) Process A to C is isochoric. So,

The work done is given as

Thus, A is incorrect,

Therefore, it is correct.

The final solution for each is negative. So,

Thus, it is correct.

Thus, it is incorrect.

*Answer can only contain numeric values

QUESTION: 25

Among the species given below, the total number of diamagnetic species is ______. H atom, NO_{2} monomer, O^{2-} (superoxide), dimeric sulphur in vapour phase, Mn_{3}O_{4}, (NH_{4})_{2}[FeCl_{4}], (NH_{4})_{2}[NiCl_{4}], K_{2}MnO_{4}, K_{2}CrO_{4}

Solution:

Hydrogen atom is paramagnetic because it has one electron.

The given species NO_{2} monomer is paramagnetic because it has an odd electron.

The given species O_{2}^{−} superoxide is paramagnetic because it has one unpaired electron in its π^{*}.

The given species S_{2} is paramagnetic because it has two unpaired electrons in its π*.

The given species Mn_{3}O_{4} is paramagnetic because it exists as

In the given species (NH_{4})_{2} [FeCl_{4}], iron exists in +2

oxidation state. Thus, it is paramagnetic.

In the given species (NH_{4})_{2} [NiCl_{4}], nickel exists in +2

oxidation state. Thus, it is paramagnetic.

In the given species K_{2}MnO_{4}, manganese exists in +6 oxidation state. Thus, it is paramagnetic.

In the given species K_{2}CrO_{4}, manganese exists in +6

oxidation state. Thus, it is diamagnetic.

*Answer can only contain numeric values

QUESTION: 26

The ammonia prepared by treating ammonium sulphate with calcium hydroxide is completely used by NiCl_{2}.6H_{2}O to form a stable coordination compound. Assume that both the reactions are 100% complete. If 1584 g of ammonium sulphate and 952 g of NiCl_{2}.6H_{2}O are used in the preparation, the combined weight (in grams) of gypsum and the nickel-ammonia coordination compound thus produced is______.

(Atomic weights in g mol^{-1}: H = 1, N = 14, O = 16, S = 32, Cl = 35.5, Ca = 40, Ni = 59)

Solution:

The given reactions are,

Mass of gypsum 12x 172 = 2064 g

Number of moles of NH_{3} that is released = 24 mol

Number of moles of [Ni( NH_{3})_{6}] Cl_{2} =4 mol

Mass of [Ni( NH_{3})_{6}] = 4 x 232 = 928 g

Total Mass = ( 2064 +928) g

= 2992g

*Answer can only contain numeric values

QUESTION: 27

Consider an ionic solid MX with NaCl structure. Construct a new structure (Z) whose unit cell is constructed from the unit cell of MX following the sequential instructions given below. Neglect the charge balance.

(i) Remove all the anions (X) except the central one

(ii) Replace all the face centered cations (M) by anions (X)

(iii) Remove all the corner cations (M)

(iv) Replace the central anion (X) with cation (M)

Solution:

(i) Number of cations is 4 and number of anions is 1.

(ii) Number of cations is 1 and number of anions is 4.

(iii) Number of cations is 1 and number of anions is 4.

(iv) Number of cations is 1 and number of anions is 3.

*Answer can only contain numeric values

QUESTION: 28

For the electrochemical cell,

the standard emf of the cell is 2.70 V at 300 K. When the concentration of Mg^{2+} is changed to x M, the cell potential changes to 2.67 V at 300 K. The value of x is ____. (given, where F is the Faraday constant and R is the gas constant, ln(10) = 2.30)

Solution:

The reaction at anode is,

The reaction at cathode is,

The final equation is shown below.

The Nernst equation with substituted values is shown below.

*Answer can only contain numeric values

QUESTION: 29

A closed tank has two compartments A and B, both filled with oxygen (assumed to be ideal gas). The partition separating the two compartments is fixed and is a perfect heat insulator (Figure 1). If the old partition is replaced by a new partition which can slide and conduct heat but does NOT allow the gas to leak across (Figure 2), the volume (in m^{3}) of the compartment A after the system attains equilibrium is ____.

Solution:

Number of moles in system A

Number of moles in system B

Write the conditions after attaining the equilibrium.

Volume of system A is,

*Answer can only contain numeric values

QUESTION: 30

Liquids A and B form ideal solution over the entire range of composition. At temperature T, equimolar binary solution of liquids A and B has vapour pressure 45 Torr. At the same temperature, a new solution of A and B having mole fractions x_{a} and x_{b}, respectively, has vapour pressure of 22.5 Torr. The value of x_{a}/x_{b} in the new solution is______. (given that the vapour pressure of pure liquid A is 20 Torr at temperature T)

Solution:

The total vapor pressure is,

The new total vapor pressure is,

As it is known that x_{A} + x_{B}= 1, therefore, x_{B} = 0.05 .

So

,

*Answer can only contain numeric values

QUESTION: 31

The solubility of a salt of weak acid (AB) at pH 3 is Yx10^{-3} mol L^{-1}. The value of Y is________. (Given that the value of solubility product of AB (K_{sp}) = 2x10^{-10} and the value of ionization constant of HB (K_{a}) = 1X10^{-8})

Solution:

The expression of the dissociation of salt AB is,

The expression of the formation of HB is,

Further equation is solved as,

Now, the equation is solved as shown below.

*Answer can only contain numeric values

QUESTION: 32

The plot given below shows P-T curves (where P is the pressure and T is the temperature) for two solvents X and Y and isomolal solutions of NaCl in these solvents. NaCl completely dissociates in both the solvents.

On addition of equal number of moles of a non-volatile solute S in equal amount (in kg) of these solvents, the elevation of boiling point of solvent X is three times that of solvent Y. Solute S is known to undergo dimerization in these solvents. If the degree of dimerization is 0.7 in solvent Y, the degree of dimerization in solvent X is ______.

Treatment of benzene with CO/HCl in the presence of anhydrous AlCl_{3}/CuCl followed by reaction with Ac_{2}O/NaOAc gives compound X as the major product. Compound X upon reaction with Br_{2}/Na_{2}CO_{3}, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H_{2}/Pd-C, followed by H_{3}PO_{4} treatment gives Z as the major product.

Solution:

The solution for solvent X is done as,

The solution for solvent Y is done as,

Divide the above two equations,

The van’t Hoff factor (i) = 1- α/2

The value of van’t Hoff factor ( i ) for

= 0.65

According to the given information,

QUESTION: 33

PARAGRAPH “X”

Treatment of benzene with CO/HCl in the presence of anhydrous AlCl_{3}/CuCl followed by reaction with Ac_{2}O/NaOAc gives compound X as the major product. Compound X upon reaction with Br_{2}/Na_{2}CO_{3}, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H_{2}/Pd-C, followed by H_{3}PO_{4} treatment gives Z as the major product.

Q. The compound Y is

Solution:

The correct sequence for the formation of Y is,

QUESTION: 34

PARAGRAPH “X”

Treatment of benzene with CO/HCl in the presence of anhydrous AlCl_{3}/CuCl followed by reaction with Ac_{2}O/NaOAc gives compound X as the major product. Compound X upon reaction with Br_{2}/Na_{2}CO_{3}, followed by heating at 473 K with moist KOH furnishes Y as the major product. Reaction of X with H_{2}/Pd-C, followed by H_{3}PO_{4} treatment gives Z as the major product.

Q The compound Z is

Solution:

The correct sequence for the formation of Z is,

QUESTION: 35

PARAGRAPH “A”

An organic acid P (C_{11}H_{12}O_{2}) can easily be oxidized to a dibasic acid which reacts with ethyleneglycol to produce a polymer dacron. Upon ozonolysis, P gives an aliphatic ketone as one of the products. P undergoes the following reaction sequences to furnish R via Q. The compound P also undergoes another set of reactions to produce S.

Q. The compound R is

Solution:

The correct sequence for the formation of R is,

QUESTION: 36

PARAGRAPH “A”

An organic acid P (C_{11}H_{12}O_{2}) can easily be oxidized to a dibasic acid which reacts with ethyleneglycol to produce a polymer dacron. Upon ozonolysis, P gives an aliphatic ketone as one of the products. P undergoes the following reaction sequences to furnish R via Q. The compound P also undergoes another set of reactions to produce S.

Q. The compound S is

Solution:

The correct sequence for the formation of S is,

*Multiple options can be correct

QUESTION: 37

For a non-zero complex number z, let arg(z) denote the principal argument with - π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE?

Solution:

(A) The argument of the complex number is calculated as follows.

Hence, option (A) is false.

(B) The definition of the function f(t) = arg ( −1 + it ) is,

It is clear from the definition of the function that the function is discontinuous at t =0 .

Hence, option (B) is false.

(C) The value of

is calculated as follows..

Hence, option (C) is true.

(D) If,

Then,

is a real number

Hence are concyclic. That is,

Hence, option (D) is false.

*Multiple options can be correct

QUESTION: 38

In a triangle PQR, let ∠PQR = 30° and the sides PQ and QR have lengths 10√3 and 10, respectively. Then, which of the following statement(s) is (are) TRUE?

Solution:

The required triangle is as follows,

(A) Apply cosine angle formula,

Since, two sides are equal. Hence, the triangle is isosceles.

Using the property of triangle,

Hence, option (A) is false.

(B) The area of ΔPQR is calculated as follows,

Hence, option (B) is true.

(C) The radius of ΔPQR is calculated as follows,

Rationalize the above expression,

Hence, option (C) is correct.

(D) The radius of the circumcircle is,

The area of the circle is,

A = πR^{2}

=100 π

Hence, option (D) is correct.

*Multiple options can be correct

QUESTION: 39

Let P_{1}: 2x + y - z = 3 and P_{2}: x + 2y + z = 2 be two planes. Then, which of the following statement(s) is (are) TRUE?

Solution:

The direction ratios of line of intersection is,

Hence, option (A) is false.

(B) The equation of line is,

The lines are parallel.

Hence, option (B) is false.

(C) The angle between P_{1} and P_{2} is,

Hence, option (C) is true.

(D) The equation of line passing through intersection of P_{1} and P_{2} is,

If the plane passes through this line, then (4, 2, -2) must satisfy the equation.

The equation of plane is,

x − y +z = 0

The perpendicular distance of point (2, 1, 1) from the plane is,

Hence, option (D) is true.

*Multiple options can be correct

QUESTION: 40

For every twice differentiable function with which of the following statement(s) is (are) TRUE?

Solution:

(A) It is clear from the definition of the function *f*(x) that the

function is a continuous function. For every continuous function, there exists an open interval (r, s) where, (r < s)

such that function f is a one- one function on the open interval.

Hence, option (A) is true.

(B) The first order derivative of function is,

The limits of the function are as follows,

Combine the above limits of the function,

(C) The limit of the function f (x ) such that x→ ∞ is,

.....(1)

It is clear from equation (1) that does not exist..

Hence, option (C) is false.

(D) Consider the function,

From option (B), there exist some real number such that

for some x in the range (-4, 0).

Assume an element p such that P∈ (-4, 0) for which h ( p ) =5.

Assume a small element q such that q ∈ ( 0, 4 ) for which h (q) =5 .

Hence, by Rolle’s theorem is (p,q).

Hence, h (x ) > 5 as we move from x = p to x = q and

Thus,

Hence, option (D) is true.

*Multiple options can be correct

QUESTION: 41

Let be two non-constant be two non-constant. if for all x ∈ R, and *f*(1) = g(2) = 1, then which of the following statement(s) is (are) True?

Solution:

The first order derivative of the function is,

On further expansion of the expression,

The conclusion from the above expression is,

*Multiple options can be correct

QUESTION: 42

Let be a continuous function such that Then, which of the following statement(s) is (are) TRUE?

Solution:

The modified form of the given function is,

Differentiate the above expression with respect to x ,

The integrating factor of the function is e^{-2x}

The function for the area is,

....(1)

Substitute the values in equation (1).

1 = 1+c

c = 0

The equation of the line is,

y= 1 - x

The curve passes through the point (2,-1) but does not pass through (1,2).

The graph shows the function,

The required area of the curve is,

*Answer can only contain numeric values

QUESTION: 43

The value of is ________.

Solution:

The required value is,

*Answer can only contain numeric values

QUESTION: 44

The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is _____ .

Solution:

In order to obtain a 5 digit number which are divisible by 4, with the digits from the set, the options for the last two digit of the numbers are as (12), (24), (32), (44) and (52).

The total number of five digit number that can be formed is,

*Answer can only contain numeric values

QUESTION: 45

Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11,… , and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23,…. Then, the number of elements in the set X ∪ Y is _____.

Solution:

The given sets are,

The intersection of set X and set Y is,

The number of elements in the set X∩Y is,

The number of elements in the set X∪Y is,

*Answer can only contain numeric values

QUESTION: 46

The number of real solutions of the equation

lying in the interval is _____. (Here, the inverse trigonometric functions sin^{-1}x and cos ^{-1}x assume values in respectively.)

Solution:

The number of real solution of the equation is,

The root of the equation is,

It is clear from the above calculation that one root lies between 0 and 1/2.

So, the equation has 2 roots.

*Answer can only contain numeric values

QUESTION: 47

For each positive integer n, let For x ∈ R , let [x] be the greatest integer less than or equal to x. if then the value of [L] is _________.

Solution:

The value of [L] is

On further expansion,

*Answer can only contain numeric values

QUESTION: 48

Let be two unit vectors such that . For some and the vector is inclined at the same angle α to both then the value of 8cos^{2} α is _____ .

Solution:

The dot product of is

The cross product of is

*Answer can only contain numeric values

QUESTION: 49

Let a, b, c be three non-zero real numbers such that the equation has two distinct real roots α and β with Then, the value of b/a is _________.

Solution:

The value of b/a is ,

Subtract the above equations,

*Answer can only contain numeric values

QUESTION: 50

A farmer F_{1} has a land in the shape of a triangle with vertices at (0, 0), (1, 1) and R(2, 0). From this land, a neighbouring farmer F_{2} takes away the region which lies between the side PQ and a curve of the form y = x^{n} (n > 1). If the area of the region taken away by the farmer F_{2} is exactly 30% of the area of ΔPQR, then the value of n is________.

Solution:

The figure of the farmers land is as follows,

The area of land taken by another farmer is, Area under curve is,

Substitute the values,

QUESTION: 51

PARAGRAPH “X”

Let S be the circle in the xy-plane defined by the equation x^{2} + y^{2} = 4.

Let E_{1}E_{2} and F_{1}F_{2} be the chords of S passing through the point P_{0}(1, 1) and parallel to the x-axis and the y-axis, respectively. Let G_{1}G_{2} be the chord of S passing through P_{0} and having slope -1. Let the tangents to S at E_{1} and E_{2} meet at E_{3}, the tangents to S at F_{1} and F_{2} meet at F_{3}, and the tangents to S at G_{1} and G_{2} meet at G_{3}. Then, the points E_{3}, F_{3}, and G_{3} lie on the curve

Solution:

The figure below shows the chords E_{1}E_{2} and F_{1}F_{2} of the circle.

The equation of circle is,

x^{2} +y^{2}= 4 .. (A)

The equation of chord E_{1}E_{2} is,

y = 1 …… (B)

The equation of chord F_{1}F_{2} is,

x = 1 …… (C)

The coordinates of E_{1} and E_{2 }are calculated by solving equation (A) and (B).

The coordinates of F_{1} and F_{2} are calculated by solving equation (A) and (C).

The equation of tangent at E_{1} is as follows,

(D)

The equation of tangent at E_{2} is as follows,

The coordinates of point E_{3} are calculated by solving equation (D) and (E).

E_{3} = ( 0, 4 )

The equation of tangent at F_{1} is as follows,

The equation of tangent at F_{2} is as follows,

The coordinates of point F_{3} are calculated by solving equation (F) and (G).

F_{3} = ( 4, 0 )

Similarly, the coordinates of point G_{3} are (2,2).

The point (0,4), (4,0) and (2,2) lies on the equation of line, x +y= 4

QUESTION: 52

PARAGRAPH “X”

Let S be the circle in the xy-plane defined by the equation x^{2} + y^{2} = 4.

Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point of the line segment MN must lie on the curve

Solution:

The assumed coordinates of point P are ( 2 cosθ , 2 sinθ ) .

The expression for the tangent on the circle is, x cosθ +y sinθ= 2

The coordinate of point M and N are as follows,

The coordinate of the mid-point of the segment MN are,

The required equation is,

QUESTION: 53

PARAGRAPH “A”

There are five students S_{1}, S_{2}, S_{3}, S_{4} and S_{5} in a music class and for them there are five seats R_{1}, R_{2},R_{3},R_{4} and R_{5} arranged in a row, where initially the seat R_{i} is allotted to the student S_{i}, i=1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

Q.

The probability that, on the examination day, the student S_{1} gets the previously allotted seat R_{1}, and NONE of the remaining students gets the seat previously allotted to him/her is

Solution:

The event happening is,

The event already occured is,

The probability that the student S_{1} gets the previously allotted seat R_{1} is,

QUESTION: 54

PARAGRAPH “A”

There are five students S_{1}, S_{2}, S_{3}, S_{4} and S_{5} in a music class and for them there are five seats R_{1}, R_{2},R_{3},R_{4} and R_{5} arranged in a row, where initially the seat R_{i} is allotted to the student S_{i}, i=1, 2, 3, 4, 5. But, on the examination day, the five students are randomly allotted the five seats.

Q. For i = 1, 2, 3, 4, let T_{i} denote the event that the students S_{i} and S_{i} + 1 do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event T_{1} ∩ T_{2} ∩ T_{3} ∩ T_{4} is

Solution:

The total number of ways in which seat can be allotted are,

The number of ways in which student S_{1}, S_{2}, S_{3} and S_{4} do not sit together is,

The probability that the students S_{i} and student S_{i} +1 do not sit together is,

### JEE Advanced 2018 Question Paper paper1

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### JEE (Advanced) 2018 Solved Paper - 2

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### JEE (Advanced) 2018 Solved Paper - 1

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