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QUESTION: 1

In the circuit shown C_{1} = 2C_{2}. Switch S is closed at time t = 0. Let i_{1} & i_{2 }be the current flowing through C_{1 }and C_{2} at any time t, then the ratio I_{1}/I_{2}

Solution:

B

QUESTION: 2

The capacitor of an oscillatory circuit of frequency 10^{4}Hz is enclosed in a container. When the container is evacuated, the frequency changes by 50 Hz, the dielectric constant of the gas is

Solution:

QUESTION: 3

A current carrying wire is placed in the grooves of an insulating semicircular disc of radius ‘R’ as shown. The current enters atpoint A and leaves point B. Determine the magnetic field at point D.

Solution:

B

QUESTION: 4

The organ pipes, both closed at one end, have lengths ℓ and ( ℓ + Δ ℓ) vibrates in fundamental mode. Neglect end correction. If the velocity of sound in air is v, then the number of beats/sec is (assume Δ ℓ << ℓ)

Solution:

C

QUESTION: 5

Light is incident normally on face AB of a prism as shown. Aliquid of refractive index μ is placed on face AC of the prism.The prism is made of glass of refractive index 3/2. The limit of μ for which total internal reflection takes place on face AC is

Solution:

*Multiple options can be correct

QUESTION: 6

Figure shows variation of magnification m(produced by a thin convexlens) and distance v of image from pole of the lens. Which of the following statements are correct?

Solution:

A, B, C

QUESTION: 7

A certain substance has a mass of 50g/mol. When 300J of heat is added to 25 gm of sample of this material, its temperature rises from 25 to 45^{o}C.

**Q.** The specific heat capacity of the substance is

Solution:

D

Specific heat capacity,

QUESTION: 8

A certain substance has a mass of 50g/mol. When 300J of heat is added to 25 gm of sample of thismaterial, its temperature rises from 25 to 45^{o}C.

**Q.** The molar heat capacity of the substance is

Solution:

A

Molar heat capacity

QUESTION: 9

Matrix – Match Type

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example:

If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s

and t; then the correct darkening of bubbles will look like the following:

**1. Match the following**

Solution:

**1.**

In wrist watch spring force is not dependent on gravity.

*Answer can only contain numeric values

QUESTION: 10

A projectile is projected with a speed 20 m/s from the floor of a 5 m high room as shown. The maximum horizontal range of the projectile is found to be m. Find n [Take g = 10 ms^{-2}].

Solution:

*Answer can only contain numeric values

QUESTION: 11

A solid cylinder is made of radius R and height 3R having mass density ρ. Now two half spheres of radius R are removed from both ends. The moment of intertia of remaining portion about axis ZZ’ can be calculated as Find K.

Solution:

*Answer can only contain numeric values

QUESTION: 12

Two particles each of mass m are connected by a string of length 1m. The particles are kept on a smooth horizontal plane. The initial separation between the particles is 0.5 m. One of the particles is given a velocity 18 m/s as shown. Find the magnitude of angular velocity of one particle with respect to the other particle just after the string becomes taut (in m/s)

Solution:

*Answer can only contain numeric values

QUESTION: 13

A 10 m long nichrome wire having 80Ω resistance has current carrying capacity of 5 A. This wire can be cut into equal parts and equal parts can be connected in series or parallel. What is the maximum power which can be obtained as heat by the wire from a 200 V mains supply (in KW).

Solution:

When the wire is divided into two equal parts and connected in parallel we get maximum power.

∴ Resistance of each part = 40Ω

*Answer can only contain numeric values

QUESTION: 14

In the given figure, a potential of +12 volt is given to point A, and point B is earthed. What is the potential at point P in volt?

Solution:

QUESTION: 15

Cyclopentadiene

Which one of the following in not true about D.

Solution:

C

Co – ordination no of Fe^{+2} in D is 6

(a sandwhich compound)

QUESTION: 16

Which one of the following reaction is not possible:-

Solution:

QUESTION: 17

(I) Co(CO)_{6}

(II) Co(CO)_{5}NH_{3}

(III) Co(CO)_{4}(NH_{3})_{2}

(IV) Co(CO)_{3}(NH_{3})_{3}

(V) Co(CO)_{2}(NH_{3})_{4}

(VI) Co(CO)(NH_{3})_{5}

Which statement is correct for these compounds?

Solution:

A

Synergic bonding.

*Multiple options can be correct

QUESTION: 18

Which among the following statements is/are correct?

Solution:

B, C

(b) conjugate base of

(c) pH of 0.1 M NaCl (aqueous solution) =

*Multiple options can be correct

QUESTION: 19

4.5 g of Al (atomic mass = 27) is deposited at cathode from Al^{+3} solution by certain quantity of charge. The correct statement/s is/are:

Solution:

A, C

Number of moles of Al produced

Number of moles of electrons used

Number of moles of H_{2} produced

Volume of H_{2 }produced

QUESTION: 20

On the basis of aromaticity, there are three types of compounds i.e. aromatic, non-aromatic and antiaromatic. The increasing order of stability of these compounds are is under: Anti-aromatic compound < non-aromatic compound < aromatic compounds. Compounds to be aromatic follow the following conditions (according to valence bond theory)

(i) The compounds must be be cyclic in structure having (4n + 2)π e^{–}, where n = Hückel’s number = 0, 1, 2, 3 et.c

(ii) The each atoms of the cyclic structure must have unhybridised p-orbital i.e. the atoms of the compounds have unhybridised p-orbital i.e. usually have sp^{2} hybrid or planar.

(iii) There must be a ring current of π electrons in the ring or cyclic structure i.e. cyclic structure must undergo resonance .

Compounds to be anti-aromatic, it must have 4nπe^{–} where n = 1, 2… and it must be planar and undergo resonance. Non-aromatic compounds the name itself spells that compounds must be non-planar irrespective of number of π electrons. Either it has 4nπe^{– }or (4n + 2) π electrons it does not matter. The rate of reaction of any aromatic compounds depends upon the following factors:

(i) Electron density

(ii) stability of carbocation produced

Higher the amount of electron density of the ring, higher will be its rate towards aromatic electrophilic substitution and vice-versa. Similarly, higher will be the stability of the produced carbocation after the attack of electrophile, higher will be its rate towards aromatic electrophilic substitution. There is a great effect of kinetic labelling on the rate of aromatic electrophilic substitution. As we known that higher the atomic weight or, molecualr weight, higher will be the van der Waal’s force of attraction or, bond energy. Since there will be effect of kienetic labelling if the 2^{nd} step of the reaction will be the slow step, (r.d.s.) otherwise there will be no effect of kinetic labelling.

**Q.** Which of the following is correct order of the rate of reaction of C_{6}H_{6}, C_{6}D_{6} and C_{6}T_{6} towards

sulphonation?

Solution:

C

2^{nd} step is r.d.s. in sulphonation reaction.

QUESTION: 21

On the basis of aromaticity, there are three types of compounds i.e. aromatic, non-aromatic and antiaromatic. The increasing order of stability of these compounds are is under:

Anti-aromatic compound < non-aromatic compound < aromatic compounds. Compounds to be aromatic follow the following conditions (according to valence bond theory)

(i) The compounds must be be cyclic in structure having (4n + 2)π e^{–}, where n = Hückel’s number = 0, 1,2, 3 et.c

(ii) The each atoms of the cyclic structure must have unhybridised p-orbital i.e. the atoms of the compounds have unhybridised p-orbital i.e. usually have sp^{2} hybrid or planar.

(iii) There must be a ring current of π electrons in the ring or cyclic structure i.e. cyclic structure must undergo resonance .

Compounds to be anti-aromatic, it must have 4nπe^{–} where n = 1, 2… and it must be planar and undergo resonance. Non-aromatic compounds the name itself spells that compounds must be non-planar irrespective of number of π electrons. Either it has 4nπe^{– }or (4n + 2) π electrons it does not matter.

The rate of reaction of any aromatic compounds depends upon the following factors:

(i) Electron density

(ii) stability of carbocation produced

Higher the amount of electron density of the ring, higher will be its rate towards aromatic electrophilic substitution and vice-versa. Similarly, higher will be the stability of the produced carbocation after the attack of electrophile, higher will be its rate towards aromatic electrophilic substitution. There is a great effect of kinetic labelling on the rate of aromatic electrophilic substitution. As we known that higher the atomic weight or, molecualr weight, higher will be the van der Waal’s force of attraction or, bond energy. Since there will be effect of kienetic labelling if the 2^{nd} step of the reaction will be the slow step, (r.d.s.) otherwise there will be no effect of kinetic labelling.

**Q.** Which of the following is correct order of the rate of reaction of C_{6}H_{6}, C_{6}D_{6} and C_{6}T_{6} towards

nitrations?

Solution:

B

2^{nd} step is fast step in nitration.

QUESTION: 22

The Bertholet’s equation of state for 1 mole real gas is given as under:

Where a and b are van der Waal’s constants. van der Waal’s constant “a” signifies, the force of attraction among the gas molecules. van der Waal’s constant “b” signifies incompressible volume i.e. volume having no effect of compression and expansion. It is also known as co-volume. Thereby the volume having effect of compression and expansion i.e. compressible volume = V – b (for 1 mole real gas). One of the form of van der Waal’s equation of state is virial equation. The virial equation for 1 mole real gas is as under:

where A, B, C are known as 2^{nd}, 3^{rd} and 4^{th} virial coefficients respectively. The temperature at which real gases obey ideal gas equation (PV = RT), is known as Boyles temperature i.e. T_{B}.

**Q. **If where Ti is the inversion temperature, then which of the following is the correct relation of the Boyle’s temperature T_{B} obtained by the Bertholet’s equation

Solution:

A

on converting into virial equation from, we get

to higher power of n …(i)

For Boyle’s temperature, 2^{nd} virial co-efficient becomes zero

QUESTION: 23

The Bertholet’s equation of state for 1 mole real gas is given as under:

Where a and b are van der Waal’s constants. van der Waal’s constant “a” signifies, the force of attraction among the gas molecules. van der Waal’s constant “b” signifies incompressible volume i.e. volume having no effect of compression and expansion. It is also known as co-volume. Thereby the volume having effect of compression and expansion i.e. compressible volume = V – b (for 1 mole real gas). One of the form of van der Waal’s equation of state is virial equation. The virial equation for 1 mole real gas is as under:

where A, B, C are known as 2^{nd}, 3^{rd} and 4^{th} virial coefficients respectively. The temperature at which real gases obey ideal gas equation (PV = RT), is known as Boyles temperature i.e. T_{B}.

**Q. **If the value of a and b increases by 8 times and 2 times respecively, then new Boyle’s temperature will increases by how many times and what will be effect on the ease of liquification ?

Solution:

A

Let Boyle’s temperature, 'T_{B}

QUESTION: 24

The Bertholet’s equation of state for 1 mole real gas is given as under:

Where a and b are van der Waal’s constants. van der Waal’s constant “a” signifies, the force of attraction among the gas molecules. van der Waal’s constant “b” signifies incompressible volume i.e. volume having no effect of compression and expansion. It is also known as co-volume. Thereby the volume having effect of compression and expansion i.e. compressible volume = V – b (for 1 mole real gas). One of the form of van der Waal’s equation of state is virial equation. The virial equation for 1 mole real gas is as under:

where A, B, C are known as 2^{nd}, 3^{rd} and 4^{th} virial coefficients respectively. The temperature at which real gases obey ideal gas equation (PV = RT), is known as Boyles temperature i.e. T_{B}.

**Q. **If the third virial coefficient of 4 moles of mono-atomic as is 0.04 (lit/mol)^{2} at NTP, then what will be the real gas at NTP?

Solution:

D

b^{2} = 0.04

b = 0.2 lit/mol

V_{real} = 4 x 22.4 + 4 x 0.2 = 90.4 lit

QUESTION: 25

(Matrix-Match Type)

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as

illustrated in the following example:

If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s and t; then the correct darkening of bubbles will look like the following:

1. Match the following.

Solution:

(A) Sic covalent carbide (Not hydrolysed)

QUESTION: 26

(Matrix-Match Type)

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as

illustrated in the following example:

If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s and t; then the correct darkening of bubbles will look like the following:

**2. **Match the solution given in column I with the properties given in column II.

Solution:

*Answer can only contain numeric values

QUESTION: 27

Examine the structural fromuals shown below and identify how many compounds will show coupling reaction with diazonium salt faster than anisole

Solution:

*Answer can only contain numeric values

QUESTION: 28

For the first order reaction The value of x will be

Solution:

*Answer can only contain numeric values

QUESTION: 29

Total no of molecules/ions which have non zero dipole moment is

Solution:

QUESTION: 30

Which one of following statement is true

Solution:

QUESTION: 31

If ‘P’ be a point on unit circle with centre at point ‘O’. If Q is another point inclined at angle to = r, then the interval in which ‘r’ lies, so that

Solution:

B

Where 'Q' is the angle between

QUESTION: 32

If a bag contains five balls. Two balls are drawn and are found to be white. What is the probabilitythat all the balls are white?

Solution:

C

Here two balls are drawn and the result is known that the two balls are white, therefore Bayes’s theorem should be used.

Required probability

QUESTION: 33

If f(x) and g(x) are defined and differentiable for x ≥ x_{0} andthen

Solution:

D

QUESTION: 34

If z_{1} and z_{2} are two variable complex numbers such that, . Then themaximum value of ,is

Solution:

QUESTION: 35

If .........................up to ‘n’ terms &

Solution:

QUESTION: 36

The perpendicular from origin to the tangent at any point on a curve is equal to the obscissa ofthe point of contact. If the curve passes through point (1, 2), then it also passes through point,

Solution:

A

Equation of tangent at 'P' is

putting y = vx

Solving Differential equation

It passes through (1,2) ⇒ C = 5

Equation of curve ≡

QUESTION: 37

For an ellipse with vertices A & A’, tangent drawn at the point ‘P’ in the first quadrantmeets the y – axis in Q and the chord A’P meets the y – axis in M. If ‘O’ is the centre of ellipse,then OQ^{2} – MQ2^{ is equal to,}

Solution:

= 4

*Multiple options can be correct

QUESTION: 38

If . Then f(x) = 0 has,

Solution:

A, B, D

QUESTION: 39

Which of the following is/are not true,

Solution:

*Multiple options can be correct

QUESTION: 40

If then

Solution:

B, C

QUESTION: 41

If α,β are the roots of equation, x^{2 }- 2x a^{2} + 1 = 0 and γ, δ are the roots of the equation, x^{2} - 2(a + 1)x + x(a - 1) = 0 such that α,β ∈ γ,δ then the interval in which ‘a’ lies,

Solution:

A, B

QUESTION: 42

Consider the curves,

c_{3} ≡ x = 0

Q. Maximum area bounded by all of these three curve is,

Solution:

C

The value of K_{1 }& K_{2} decide the opening of parabola

QUESTION: 43

Consider the curves,

c_{3} ≡ x= 0

Q. Minimum area bounded by all of these three curve is,

Solution:

D

QUESTION: 44

Triangle ABC has circumcentre ‘O’ and orthocentre ‘H’. Points E and F are chosen on sides AC and ABrespectively such that AE = AO and AF = AH. If ‘R’ be radius of circle circumscribing the triangle ABC.Then,

Q. The area of quadrilateral BFEC is equal to,

Solution:

D

QUESTION: 45

Triangle ABC has circumcentre ‘O’ and orthocentre ‘H’. Points E and F are chosen on sides AC and ABrespectively such that AE = AO and AF = AH. If ‘R’ be radius of circle circumscribing the triangle ABC.Then,

Q. The length of segment EF is equal to,

Solution:

AE = R,AF = 2RcosA

⇒ EF = R

QUESTION: 46

Triangle ABC has circumcentre ‘O’ and orthocentre ‘H’. Points E and F are chosen on sides AC and ABrespectively such that AE = AO and AF = AH. If ‘R’ be radius of circle circumscribing the triangle ABC.Then,

Q. Let , then sum of squares of three altitudes of triangle AFE is equal to,

Solution:

⇒ sum of square of altitudes

QUESTION: 47

(Matrix-Match Type)

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example:

If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s and t; then the correct darkening of bubbles will look like the following:

1. Match the following:

Solution:

(C) 6

QUESTION: 48

This section contains 2 questions. Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column II are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example:

If the correct matches are A – p, s and t; B – q and r; C – p and q; and D – s and t; then the correct darkening of bubbles will look like the following:

2. Match the following:

Solution:

*Answer can only contain numeric values

QUESTION: 49

A line makes an angle of α,β,γ,δ with the four diagonals of a cube then the value of

is, (where[.]denotes the GIF)...............

Solution:

D.C's of OP are,

Let D.C's of required line be (ℓ,m,n)

*Answer can only contain numeric values

QUESTION: 50

Tangents drawn from any point ‘P’ on the curve xy - 2x - y - 2 = 0 intersect the asymptotes at A & B. Then the area of triangle QAB, is 4K (where Q is centre of the curve). Then value of

K is………………

Solution:

2

The gives equation is (x - 1) (y - 2) = 4

XY = 4

Equationof tangent is,

*Answer can only contain numeric values

QUESTION: 51

Consider a triangle ABC whose sides length are AB = 4, BC = 2 & CA = 3. If the bisector of the angle ‘C’ cuts AB in D and the circumcircle in E, then the value of is…………(where, [.] denotes the GIF)

Solution:

*Answer can only contain numeric values

QUESTION: 52

If Then value of R is…………..

Solution:

*Answer can only contain numeric values

QUESTION: 53

If the number of ways in which four distinct balls can be put into two identical boxes so that no box remains empty is equal to K, then K =………….

Solution:

7

Every ball have two options

⇒ 4 balls can be put in 2^{4 }ways.

Arrangement among themselves in 2!

But, above count also includes the one case in which all the balls are put in one box,

⇒ Required number of ways = 2^{3} – 1 = 7.

*Answer can only contain numeric values

QUESTION: 54

The value of |K| for which the following system of equations is consistent,

Solution:

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