JEE Advanced 2023 Paper 2

Let the position vectors of the points ,,PQR and S be ˆ ˆˆ25 ai j k, ˆ ˆˆ36 3bi j k , 17 16ˆ ˆˆ 755ci jk  and ˆ ˆˆ2di j k  , respectively. Then which of t he following statements is true?

1. The points ,,PQR and S are NOT coplanar
2. 2 3bd is the position vector o f a point which divides PR internally in the ratio 5:4
3. 2 3bd is the position vector of a point which divides PR externally in the ratio 5:4
4. The square of the magnitude of the vector bd is 95

Let () , , { 1 , 2 , 3 } ,ij Mai j be the 33 matrix such that 1ija if 1j is divisible by i, otherwise 0ija. Then which of the following statements is(are) true?

1. M is invertible
2. There exists a nonzero column matrix 1 2 3a a a   such that 1 2 3a Ma a  = 1 2 3a a a 
3. The set 3{: } { }XM X  00  , where 0 = 0 0 0  
4. The matrix (2 )M I is invertible, where I is the 33 identity matrix  For example, in a question, if (A),
5. and
6. are the ONLY th ree options corresponding to correct answers, then choosing ONLY (A),
7. and
8. will get ൅4 marks; choosing ONLY (A) and
9. will get ൅2 marks; choosing ONLY (A) and
10. will get ൅2marks; choosing ONLY
11. and
12. will get ൅2 marks; choosing ONLY (A) will get ൅1 mark; choosing ONLY
13. will get ൅1 mark; choosing ONLY
14. will get ൅1 mark; choosing no option(s) (i.e. the question is unanswered) will ge t 0 marks and choosing any other option(s) will get െ2 marks.

Let :( 0 ,1 )f be the function defined as 211() [ 4] ,42fx x x x    where []x denotes the greatest integer less than or equal to x. Then which of the following statements is(are) true?

1. The function f is discontinuous exactly at one point in (0,1)
2. There is exactly one point in (0,1) at which the function f is continuous but NOT differentiable
3. The function fis NOT differentiable at more than three points in (0,1)
4. The minimum value of the function fis 1 512

Let S be the set of all twice differentiable functions f from  to  such that 2 2() 0dfxdx for all (1 , 1 ) .x For fS, let fX be the number of points (1 , 1 )x for which () .fxx Then which of the following sta tements is(are) true?

1. There exists a function fS such that 0fX
2. For every function fS, we have 2fX
3. There exists a function fS such that 2fX
4. There does NOT exist any function f in S such that 1fX

For x, let 1tan ( ) ,22()x . Then the minimum value of the function :f defined by 1 (c o s ) 2023 0()1xt a n x ttefxd tt  is

For , x let ()yx be a solution of the differential equation 22 2(5 ) 2 2 (5 )dyxx y x xdx   such that (2) 7.y Then the maximum value of the function ()yx is

Let X be the set of all five digit numb ers formed using 1,2,2,2,4,4, 0. For example, 22240 is in X while 02244 and 44422 are not in .X Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability tha t an element chosen at rando m is a multiple of 20 given that it is a multiple of 5. Then the value of 38p is equal to

Let 123 8,,,,AAA A be the vertices of a regular octagon that lie on a circle of r adius 2. Let P be a point on the circle and let iPA denote the distance between the points Pand iA for 1, 2, , 8.i If P varies over the circle, then the maximum value of the product 12 8PAP A P A , is

Let 3 2 : , , , { 0, 3, 5, 7,11,13,17,19 } 050ab Rc d a b c d . Then the number of invertible matrices in R is

Let 1C be the circle of radius 1 with center at the origin. Let 2C be the circle of radius r with center at the point (4,1)A , where 13 .r Two distinct common tangents PQ and STof 1C and 2C are drawn. The tangent PQ touches 1C at P and 2C at Q. The tangent ST touches 1C at S and 2C at .T Mid points of the line segments PQand STare joined to form a line which meets the x-axis at a point B. If 5 AB , then the value of 2r is PARAGRAPH “I” Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is 2 and whose sides are in arithmet ic progression. Suppose that th e vertices of this triangle lie on a circle of radius 1. (There are two questions based on PARAGRAPH “I”, the question given below is one of them)

Let a be the area of the triangle . ABC Then the value of 264a is PARAGRAPH “I” Consider an obtuse angled triangle ABC in which the difference between the largest and the smallest angle is 2 and whose sides are in arithmet ic progression. Suppose that th e vertices of this triangle lie on a circle of radius 1. (There are two questions based on PARAGRAPH “I”, the question given below is one of them)

Then the inradius of the triangle ABC is PA RAGRAPH “II” Consider the 66square in the figure. Let 12 4 9,,,AAA be the points of intersections (dots in the picture ) in some order. We say that iA and jA are friends if they are adjacent along a row or along a column. Assume that each point iA has an equal chance of being chosen. (There are two questions based on PARAGRAPH “II”, the question given below is one of them)

Let ip be the probability that a randomly chosen point has i many friends, 0,1, 2,3, 4i . Let X be a random variable such that for 0,1, 2,3, 4i , the probability ()i PXi p . Then the value of 7( )EX is PA RAGRAPH “II” Consider the 66square in the figure. Let 12 4 9,,,AAA be the points of intersections (dots in the picture ) in some order. We say that iA and jA are friends if they are adjacent along a row or along a column. Assume that each point iA has an equal chance of being chosen. (There are two questions based o n PARAGRAPH “II” , the question given below is one of them)

Two distinct points are chosen randomly out of the points 12 4 9,,,AAA . Let p be the probability that they are fri ends. Then the value of 7p is

An electric dipole is formed by two charges +𝑞 and −𝑞 located in 𝑥𝑦-plane at (0,2) mm and (0,−2) mm, respectively, as shown in the figure . The electric potential at point P (100,100) mm due to the dipole is 𝑉0. The charges +𝑞 and −𝑞 are then moved to the points (−1,2) mm and (1,−2) mm, respectively. What is t he value of electric potential at P due to the new dipole ?

1. 𝑉0/4
2. 𝑉0/2
3. 𝑉0/√2
4. 3𝑉0/4

Young’s modulus of elasticity 𝑌 is expressed in terms of three derived quantities, namely , the gravitational constant 𝐺, Planck’s constant ℎ and the speed of light 𝑐, as 𝑌=𝑐𝛼ℎ𝛽𝐺𝛾. Which of the following is the correct option ?

1. 𝛼=7,𝛽=−1,𝛾=−2
2. 𝛼=−7,𝛽=−1,𝛾=−2
3. 𝛼=7,𝛽=−1,𝛾=2
4. 𝛼=−7,𝛽=1,𝛾=−2

A particle of mass 𝑚 is moving in the 𝑥𝑦-plane such that i ts velocity at a point (𝑥,𝑦) is given as v⃗ =𝛼(𝑦𝑥̂+2𝑥𝑦̂), where 𝛼 is a non -zero constant . What is the force 𝐹 acting on the particle ?

1. 𝐹 =2𝑚𝛼2(𝑥𝑥̂+𝑦𝑦̂ )
2. 𝐹 =𝑚𝛼2(𝑦𝑥̂+2𝑥𝑦̂ )
3. 𝐹 =2𝑚𝛼2(𝑦𝑥̂+𝑥𝑦̂ )
4. 𝐹 =𝑚𝛼2(𝑥𝑥̂+2𝑦𝑦̂ ) Physics

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is 𝑛. The internal energy of one mole of the gas is 𝑈𝑛 and the speed of sound in the gas is v𝑛. At a fixed temperature and pressure, which of the following is the correct option?

1. v3<v6 and 𝑈3>𝑈6
2. v5>v3 and 𝑈3>𝑈5
3. v5>v7 and 𝑈5<𝑈7
4. v6<v7 and 𝑈6<𝑈7 • For example, in a question, if (A), (B ) and
5. are the ONLY three options corresponding to correct answers , then choosing ONLY (A),
6. and
7. will get +4 marks; choosing ONLY (A) and ( B) will get +2 marks; choosing ONLY (A) and ( D) will get +2marks; choosing ONLY (B ) and ( D) will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY
8. will get +1 mark; choosing ONLY
9. will get +1 mark; choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other op tion(s) will get −2 marks.

A monochromatic light wave is incident normally on a glass slab of thickness 𝑑, as shown in the figure. The refractive index of the slab increases linearly from 𝑛1 to 𝑛2 over the height ℎ. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?

1. It will deflect up by an angle tan−1[(𝑛22−𝑛12)𝑑 2ℎ].
2. It will deflect up by an angle tan−1[(𝑛2−𝑛1)𝑑 ℎ].
3. It will not deflect .
4. The deflection angle depends only on (𝑛2−𝑛1) and not on the individual values of 𝑛1 and 𝑛2.

An annular disk of mass 𝑀, inner radius 𝑎 and outer radius 𝑏 is placed on a horizontal surface with coefficient of friction 𝜇, as shown in the figure . At some time, an impulse ℐ0𝑥̂ is applied at a height ℎ above the center of the disk. If ℎ=ℎ𝑚 then the disk rolls without slipping along the 𝑥-axis. Which of the following statement(s) is(are) correct?

1. For 𝜇≠0 and 𝑎→0, ℎ𝑚=𝑏/2.
2. For 𝜇≠0 and 𝑎→𝑏, ℎ𝑚=𝑏.
3. For ℎ=ℎ𝑚, the initial angular velocity does not depend on the inner radius 𝑎.
4. For 𝜇=0 and ℎ=0, the wheel always slides without rolling.

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by 𝐸⃗ =30(2𝑥̂+𝑦̂)sin[2𝜋(5×1014𝑡−107 3𝑧)] V m−1. Which of the following option (s) is(are) correct ? [Given: The speed of light in vacuum , 𝑐=3×108 m s⁻¹]

1. 𝐵𝑥=−2×10⁻⁷ sin[2𝜋(5×1014 𝑡−107 3𝑧)] Wb m−2.
2. 𝐵𝑦=2×10⁻⁷ sin[2𝜋(5×1014 𝑡−107 3𝑧)] Wb m−2.
3. The wave is polarized in the 𝑥𝑦-plane with polarization angle 30° with respect to the 𝑥-axis.
4. The refractive index of the medium is 2.

A thin circular coin of mass 5 gm and radius 4/3 cm is initially in a horizontal 𝑥𝑦-plane . The coin is tossed vertically up (+𝑧 direction ) by applying an impulse of √𝜋 2×10−2 N-s at a distance 2/3 cm from its center. The coin spins about its diameter and moves along the +𝑧 direction. By the time the coin reaches back to its initial position, it completes 𝑛 rotations. The value of 𝑛 is ____. [Given: The acceleration due to gravity 𝑔=10 m s−2]

Numerical answer type

A rectangular conducting loop of length 4 cm and width 2 cm is in the 𝑥𝑦-plane , as shown in the figure . It is being moved away from a thin and long conducting wire along the direction √3 2 𝑥̂+1 2 𝑦̂ with a constant speed v. The wire is carrying a steady current 𝐼=10 A in the positive 𝑥-direction . A current of 10 𝜇A flows through the loop when it is at a distance 𝑑=4 cm from the wire . If the resistance of the loop is 0.1 Ω, then the value of v is _______ m s⁻¹. [Given: The permeability of free space 𝜇0=4𝜋×10⁻⁷ N A−2]

Numerical answer type

A string of length 1 m and mass 2×10−5 kg is under tension 𝑇. When the string vibrates, two successive harmonics are found to occur at frequencies 750 Hz and 1000 Hz. The value of tension 𝑇 is ____ _ Newton .

Numerical answer type

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the airtight piston hole , as shown in the figure. The air in the container is isothermally compressed from i ts original volume 𝑉0 to 100 101𝑉0 with the movable piston. Consider ing air as an ideal gas , the height (ℎ) of the liquid column in the capillary above the liquid level i n cm is_______. [Given: Surface tension of the liquid is 0.075 N m−1, atmospheric pressure is 105 N m−2, acceleration due to gravity (𝑔) is 10 m s−2, density of the liquid is 103 kg m−3 and contact angle of capillary surface with the liquid is zero ]

Numerical answer type

In a radioactive decay process, the activity is defined as 𝐴=−𝑑𝑁 𝑑𝑡, where 𝑁(𝑡) is the number of radioactive nuclei at time 𝑡. Two r adioactive sources , 𝑆1 and 𝑆2 have same activity at time 𝑡=0. At a later time, the activit ies of 𝑆1 and 𝑆2 are 𝐴1 and 𝐴2, respectively . When 𝑆1 and 𝑆2 have just completed their 3rd and 7th half-lives, respectively, the ratio 𝐴1/𝐴2 is __________.

Numerical answer type

One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the 𝑃-𝑉 diagram s below . In cycle I, processes 𝑎,𝑏,𝑐 and 𝑑 are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes 𝑎′,𝑏′,𝑐′ and 𝑑′ are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is 𝑊𝐼 and that during cycle II is 𝑊𝐼𝐼. The ratio 𝑊𝐼/𝑊𝐼𝐼 is _______. PARAGRAPH I 𝑆1 and 𝑆2 are two identical sound source s of frequency 656 Hz. The source 𝑆1 is located at 𝑂 and 𝑆2 moves anti-clockwise with a uniform speed 4√2 m s⁻¹ on a circular path around 𝑂, as shown in the figure . There are three points 𝑃,𝑄 and 𝑅 on this path such that 𝑃 and 𝑅 are diametrically opposite while 𝑄 is equidistant from them. A sound detector is placed at point 𝑃. The source 𝑆1 can move along direction 𝑂𝑃. [Given: The speed of sound in air is 324 m s⁻¹]

Numerical answer type

When only 𝑆2 is emitting sound and it is at 𝑄, the frequency of s ound measured by the detector in Hz is _________. PARAGRAPH I 𝑆1 and 𝑆2 are two identical sound sources of frequency 656 Hz. The source 𝑆1 is located at 𝑂 and 𝑆2 moves anti -clockwise with a uniform speed 4√2 m s⁻¹ on a circular path around 𝑂, as shown in the figure. There are three points 𝑃,𝑄 and 𝑅 on this path such that 𝑃 and 𝑅 are diametrically opposite while 𝑄 is equidistant from them. A sound detector is placed at point 𝑃. The source 𝑆1 can move along direction 𝑂𝑃. [Given: The speed of sound in air is 324 m s⁻¹]

Numerical answer type

Consider both sources emitting sound . When 𝑆2 is at 𝑅 and 𝑆1 approaches the detector with a speed 4 m s⁻¹, the beat frequency measured by the detector is _______ Hz. PARAGRAPH II A cylindrical furnace has height (𝐻) and diamete r (𝐷) both 1 m. It is maintained at temperature 360 K. The air gets heated inside the furnace at constant pressure 𝑃𝑎 and its temperature becomes 𝑇=360 𝐾. The hot air with density 𝜌 rises up a vertical chimney of diameter 𝑑=0.1 m and height ℎ=9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density 𝜌𝑎= 1.2 kg m−3, pressu re 𝑃𝑎 and temperature 𝑇𝑎=300 K enters the furnace . Assume air as an ideal gas, neglect the variations in 𝜌 and 𝑇 inside the chimney and the furnace . Also ignore the viscous effects. [Given: The acceleration due to gravity 𝑔=10 m s−2 and 𝜋=3.14]

Numerical answer type

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _______ gm s⁻¹. PARAGRAPH II A cylindrical furnace has height ( 𝐻) and diameter ( 𝐷) both 1 m. It is maintained at temperature 360 K. The air gets heated inside the furnace at constant pressure 𝑃𝑎 and its temperature becomes 𝑇=360 𝐾. The hot air with density 𝜌 rises up a vertical chimney of diameter 𝑑=0.1 m and height ℎ=9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density 𝜌𝑎= 1.2 kg m−3, pressure 𝑃𝑎 and temperature 𝑇𝑎=300 K enters the furnace. Assume air as an ideal gas, neglect the variations in 𝜌 and 𝑇 inside the chimney and the furnace . Also ignore the viscous effects. [Given: The acceleration due to gravity 𝑔=10 m s−2 and 𝜋=3.14]

Numerical answer type

When t he chimney is c losed using a cap at the top , a pressure difference Δ𝑃 develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow , are negligible then the value of Δ𝑃 is ______ N m−2.

Numerical answer type

The correct molecular orbital diagram for F 2 molecule in the ground state is

1. Chemistry

Consider the following statements related to colloids. (I) Lyophobic colloids are not formed by simple mixing of dispersed phase and dispersion medium. (II) For emulsions , both the dispersed phase and the dispersion medium are liquid. (III) Micelles are produced by dissolving a surfactant in any solvent at any temperature. (IV) Tyndall effect can be observed from a colloidal solution with dispersed phase having the same refractive index as that of the dispersion medium. The option with the correct set of statements is

1. (I) and (II)
2. (II) and (III)
3. (III) and (IV)
4. (II) and (IV)

In the following reactions, P, Q, R, and S are the major products . The correct statement about P, Q, R, and S is

1. P is a primary alcohol with four carbons.
2. Q undergo es Kolbe’s electrolysis to give an eight -carbon product.
3. R has six carbons and it undergo es Cannizzaro reaction.
4. S is a primary amine with six carbons .

A disaccharide X cannot be oxidi sed by bromine water. The acid hydrolysis of X leads to a laevorotatory solution. The disaccharide X is

The complex(es) , which can exhibit the type of isomerism shown by [Pt(NH 3)2Br2], is(are) [en = H 2NCH 2CH 2NH 2]

1. [Pt(en)(SCN) 2]
2. [Zn(NH 3)2Cl₂]
3. [Pt(NH 3)2Cl4]
4. [Cr(en) 2(H₂O)(SO 4)]+  For example, in a question, if (A),
5. and
6. are the ONLY three options corresponding to correct answers, then choo sing ONLY (A),
7. and
8. will get +4 marks; choosing ONLY (A) and
9. will get +2 marks; choosing ONLY (A) and
10. will get +2marks; choosing ONLY
11. and
12. will get +2 marks; choosing ONLY (A) will get +1 mark; choosing ONLY
13. will get +1 mark; choosing ONLY
14. will get +1 mark; choosing no option(s) (i.e. the question is unanswered) will get 0 marks and choosing any other option(s) will get −2 marks.

Atoms of metals x, y, and z form face -centred cubic (fcc) unit cell of edge length Lx, body -centred cubic (bcc) unit cell of edge length Ly, and simple cubic unit cell of edge length Lz, respectively. If rz= √3 2 ry ; ry= 8 √3 rx ; Mz= 3 2 My and MZ=3M x, then the correct statement(s) is(are) [Given : Mx, My, and Mz are molar masses of metals x, y, and z, respectively. rx, ry, and rz are atomic radii of metal s x, y, and z, respectively.]

1. Packing efficiency of unit cell of x  Packing efficiency of unit cell of y  Packing efficiency of unit cell of z
2. Ly  Lz
3. Lx  Ly
4. Density of x  Density of y

In the following reactions, P, Q, R, and S are the major products . The correct statement(s) about P, Q, R, and S is(are)

1. P and Q are monomers of polymer s dacron and glyptal , respectively .
2. P, Q, and R are dicarboxylic acids.
3. Compounds Q and R are the same .
4. R does not undergo aldol condensation and S does not undergo Cannizzaro reaction .

H₂S (5 moles) reacts completely with acidified aqueous potassium permanganate solution. In this reaction , the number of moles of water produced is x, and the number of moles of electrons involved is y. The value of (x + y) is ____.

Numerical answer type

Among [I 3]+, [SiO 4]4−, SO 2Cl₂, XeF 2, SF 4, ClF3, Ni(CO) 4, XeO 2F2, [PtCl 4]2−, XeF 4, and SOCl 2, the total number of species having sp3 hybridi sed central atom is ______.

Numerical answer type

Consider the following molecules: Br3O8, F2O, H 2S4O6, H 2S5O6, and C 3O₂. Count the number of atoms existing in their zero oxidation state in each molecule . Their sum is ____.

Numerical answer type

For He+, a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm . The wavelength (in nm) of the emitted photon during the transition is ___. [Use: Bohr radius, a = 52.9 pm Rydberg constant, 𝑅H = 2.2 ×10−18 J Planck’s constant, h = 6.6 ×10⁻³4 J s Speed of light, c = 3 ×108 m s⁻¹]

Numerical answer type

50 mL of 0.2 molal urea solution (density = 1.012 g mL⁻¹ at 300 K ) is mixed with 250 mL of a solution containing 0.06 g of urea . Both the solutions were prepared in the same solvent. The osmotic pressure (in Torr) of the resulting solution at 300 K is ___. [Use: Molar mass of urea = 60 g mol⁻¹; gas constant, R = 62 L Torr K⁻¹ mol⁻¹ ; Assume , mixH = 0, mixV = 0]

Numerical answer type

The reaction of 4 -methyloct -1-ene (P, 2.52 g) with HBr in the presence of (C 6H5CO) 2O₂ gives two isomeric bromides in a 9 : 1 ratio, with a combined yield of 50%. Of these, t he entire amount of the primary alkyl bromide was reacted with an appropriate amount of diethylamine followed by treatment with aq. K 2CO 3 to give a non -ionic product S in 100% yield. The mass (in mg) of S obtained is ___. [Use molar mass (in g mol–1): H = 1, C = 12, N = 14, Br = 80] “PARAGRAPH I ” The entropy versus temperature plot for phases α and β at 1 bar pressure is given. 𝑆T and 𝑆0 are entropies of the phases at temperatures T and 0 K, respectively. The transition temperature for α to β phase change is 600 K and 𝐶p,β− 𝐶p,α = 1 J mol⁻¹ K⁻¹. Assume (𝐶p,β− 𝐶p,α) is independent of temperature in the range of 200 to 700 K. 𝐶p,α and 𝐶p,β are heat capacities of α and β phases, respectively.

Numerical answer type

The value of entropy change, Sβ−Sα (in J mol⁻¹ K⁻¹), at 300 K is ___. [Use: ln 2 = 0.69 Given: Sβ−Sα=0 at 0 K] “PARAGRAPH I” The entropy versus temperature plot for phases α and β at 1 bar pressure is given. 𝑆T and 𝑆0 are entropies of the phases at temperatures T and 0 K, respectively. The transition temperature for α to β phase change is 600 K and 𝐶p,β− 𝐶p,α = 1 J mol⁻¹ K⁻¹. Assume (𝐶p,β− 𝐶p,α) is independent of temperature in the range of 200 to 700 K. 𝐶p,α and 𝐶p,β are heat capacities of α and β phases, respectively.

Numerical answer type

The value of enthalpy change, Hβ−Hα (in J mol⁻¹), at 300 K is ___. “PARAGRAPH II ” A trinitro compound, 1, 3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of Sn/HCl gives a major product, which on treatment with an excess of NaNO 2/HCl at 0 oC provides P as the product. P, upon treatment with excess of H₂O at room temperature , gives the product Q. Bromination of Q in aqueous medium furnishes the product R. The compound P upon treatment with an excess of phenol under basic conditions gives the product S. The molar mass difference between compounds Q and R is 474 g mol–1 and between compounds P and S is 172.5 g mol–1.

Numerical answer type

The number of heteroatoms present in one molecule of R is ______ . [Use: Molar mass (in g mol–1): H = 1, C = 12, N = 14, O = 16, Br = 80 , Cl = 35 .5 Atoms other than C and H are considered as heteroatoms] “PARAGRAPH I I” A trinitro compound, 1, 3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of Sn/HCl gives a major product, which on treatment with an excess of NaNO 2/HCl at 0 oC provides P as the product. P, upon treatment with excess of H₂O at room temperature , gives the product Q. Bromination of Q in aqueous medium furnishes the product R. The comp ound P upon treatment with an excess of phenol under basic conditions gives the product S. The molar mass difference between compounds Q and R is 474 g mol–1 and between compounds P and S is 172.5 g mol–1.

Numerical answer type

The total number of carbon atoms and heteroatoms present in one molecule of S is ______ . [Use: Molar mass (in g mol–1): H = 1, C = 12, N = 14, O = 16, Br = 80 , Cl = 35.5 Atoms other than C and H are considered as heteroatoms]

Numerical answer type