Chemistry
1. Given are 4 statements related to the chemical properties of Glucose.
Identify the two incorrect statements from the fol2. At $$700 \mathrm{~K}$$, the Equilibrium constant value for the formation of $$\mathrm{HI}$$ from $$\mathrm{H}_2$$ and $$3. A compound having molecular formula $$\mathrm{C}_4 \mathrm{H}_{11} \mathrm{N}$$, reacts with $$\mathrm{CHCl}_3$$ in alco4. A Hydrocarbon [A] (molecular formula $$\mathrm{C}_3 \mathrm{H}_6$$) on reaction with $$\mathrm{Br}_2 / \mathrm{CCl}_4$$ 5. Identify the end-product [D] formed when solution salicylate undergoes the following series of reactions
6. For a given reaction, ,$$\mathrm{X}(\mathrm{g})+\mathrm{Y}(\mathrm{g} \rightarrow \mathrm{Z}(\mathrm{g})$$, the order of7. Study the graph between partial pressure and mole fraction of some gases and arrange the gases P, Q, R and S dissolved i8. Identify the final product formed when Toluene undergoes a series of reactions with reagents given in the order:
(i) $$\9. 4 statements are given below. Identify the incorrect statement
A. Phenol has lower $$\mathrm{pK}_{\mathrm{a}}$$ value th10. One of the reactions A, B, C, D given below yields a product which will not answer Hinsberg's test when reacted with Ben11. Identify the correct statement from the following.12. Identify the compound which is non-aromatic in nature.
13. Given that the freezing point of benzene is $$5.48^{\circ} \mathrm{C}$$ and its $$\mathrm{K}_{\mathrm{f}}$$ value is $$514. Identify the final product formed when Benzamide undergoes the following reactions:
15. Identify the correct statement regarding corrosion of iron rod left exposed to atmosphere.16. Two statements, Assertion and Reason are given below. Choose the correct option.
Assertion: n-propyl tert-butyl ether ca17. Two statements, Assertion and Reason are given. Choose the correct option from the following.
Assertion: In the secondar18. An inorganic compound $$\mathrm{W}$$ undergoes the following reactions:
$$
\begin{gathered}
W+\frac{\mathrm{Na}_2 \mathr19. Given are the names of 4 compounds. Two of these compounds will not undergo Cannizzaro's reaction. Identify the two.
A. 20. Identify the product $$[\mathrm{C}]$$ formed at the end of the reaction below
1,1,2,2- Tetrabromopropane + 2 $$\mathrm{Z21. What is/are the product/s formed when Benzaldehyde and Ethanal react in presence of dil. $$\mathrm{NaOH}$$ followed by h22. In the estimation of element $$\mathrm{X}$$ in an organic compound, $$0.8 \mathrm{~g}$$ of the compound containing $$\ma23. Identify the pair of molecules in which one of them is a molecule with an odd electron and the other has an expanded oct24. Arrange the following compounds in the increasing order of their reactivity when each of them is reacted with chloroetha25. What mass of Silver chloride, in grams, gets precipitated when $$150 \mathrm{~ml}$$ of $$32 \%$$ solution of Silver nitr26. Which of the following 2 compounds exhibit both Geometrical and Structural isomerism?
$$\begin{aligned}
& \mathrm{A}=\le27. What is the wave number (Units $$\mathrm{cm}^{-1}$$) of the longest wave length transition in the Balmer series of Hydro28. Given below are 2 statements: Assertion and Reason. Choose the correct option.
Assertion: When Molar conductivity for a 29. Based on Valence Bond Theory, match the complexes listed in Column I with the number of unpaired electrons on the centra30. What are the Principal & Azimuthal quantum number values of the valence electrons in tripositive Lutetium?31. What is the quantity of charge, in Faraday units, required for the reduction of 3.5 moles of $$\mathrm{Cr}_2 \mathrm{O} 32. A dry cell consists of a moist paste of $$\mathrm{NH}_4 \mathrm{Cl}$$ and $$\mathrm{ZnCl}_2$$ contained in a $$\mathrm{Z33. Compounds $$\mathrm{A}$$ and $$\mathrm{B}$$, having the same molecular formula $$(\mathrm{C}_4 \mathrm{H}_8 \mathrm{O})$34. Match the names of reactions given in Column I with the appropriate reactions given in Column II.
.tg {border-collapse35. With reference to Pauling's Electronegativity scale, which one of the following options shows the correct order of elect36. What volume of $$0.2 \mathrm{~M}$$ Acetic acid is to be added to $$100 \mathrm{ml}$$ of $$0.4 \mathrm{M}$$ Sodium acetat37. Arrange the compounds $$\mathrm{A}, \mathrm{B}, \mathrm{C}$$ and $$\mathrm{D}$$ in the increasing order of their reactiv38. For a reaction $$5 X+Y \rightarrow 3 Z$$, the rate of formation of $$Z$$ is $$2.4 \times 10^{-5} \mathrm{~mol} \mathrm{~39. In the redox reaction between $$\mathrm{Cr}_2 \mathrm{O}_7^{2-} / \mathrm{H}^{+}$$ and sulphite ion, what is the number 40. Which of the following two molecular species are Diamagnetic in nature?
$$[\mathrm{A}]=\mathrm{O}_2^{-} \quad[\mathrm{B}41. A solute $$\mathrm{X}$$ is found to exist as a dimer in water. A 4 molal solution of $$\mathrm{X}$$ shows a boiling poin42. The Lanthanoid ion which would form coloured compounds is -------------.
Atomic numbers: $$\mathrm{Yb}=70, \quad \mathrm43. Given below are 4 statements. Two of these are correct statements. Identify them.
A. $$\mathrm{Co}^{2+}$$ is easily oxid44. The Enthalpy of combustion of $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{COOH}(\mathrm{s})$$ at $$25^{\circ} \mathrm{c}$$ and 145. Sulphuric acid used in Lead Storage battery has a concentration of $$4.5 \mathrm{~M}$$ and a density of $$1.28 \mathrm{~46. Given :
$$
\begin{gathered}
\Delta \mathrm{H}^0 \mathrm{f}_{\text {of }} \mathrm{CO}_2(\mathrm{~g})=-393.5 \mathrm{~kJ} 47. A current of 3.0A is passed through 750 ml of 0.45 M solution of CuSO$$_4$$ for 2 hours with a current efficiency of 90%48. Match the Vitamins given in Column I with the diseases caused by their deficiency as given in Column II.
.tg {border-c49. Which one of the following is the correct order of reagents to be used to convent [A] to [X] ?
$$
\text { Reagents: } \50. Arrange the following redox couples in the increasing order of their reducing strength:
$$\begin{array}{ll}
{[\mathrm{A}51. In the presence of a catalyst at a given temperature of $$27^{\circ} \mathrm{C}$$, the Activation energy of a specific r52. 5.0 moles of an Ideal gas at 3.0 atm pressure and $$27^{\circ} \mathrm{C}$$ is compressed isothermally to half its volum53. Identify the products C, D and F formed in the following sets of reactions.
54. Given below are 4 reactions. Two of these reactions will give product which is an equimolar mixture of the d and 1 forms55. $$200 \mathrm{ml}$$ of an aqueous solution contains $$3.6 \mathrm{~g}$$ of Glucose and $$1.2 \mathrm{~g}$$ of Urea maint56. The following data was recorded for the decomposition of XY compound at 750K
.tg {border-collapse:collapse;border-spac57. Identify the final product $$[\mathrm{D}]$$ formed when Benzyl alcohol undergoes the following series of reactions
$$\ma58. Choose the incorrect statement from the following:
A. Isoelectronic molecules/ions have the same bond order.
B. Dipole m59. Based on Crystal Field theory, match the Complex ions listed in Column I with the electronic configuration in the d orbi60. The Activation energy for the reaction $$A \rightarrow B+C$$, at a temperature $$\mathrm{TK}$$ was $$0.04606 \mathrm{~RT
Mathematics
1. $$
\text { The rate of change of the volume of a sphere with respect to its surface area } \mathrm{S} \text { is }
$$2. While shuffling a pack of cards, 3 cards were accidently dropped, then find the probability that the missing cards belon3. The area of the triangle whose vertices are $$(-2, a)(2,-6)$$ and $$(5,4)$$ is 35 sq units then the value of '$$\mathrm{4. $$
\text { If } 3 A+4 B^t=\left(\begin{array}{ccc}
7 & -10 & 17 \\
0 & 6 & 31
\end{array}\right) \text { and } 2 B-3 A^t5. The domain of the function $$y=\frac{1}{\log _{10}(3-x)}+\sqrt{x+7}$$ is6. Consider an infinite geometric series with first term '$$a$$' and common ratio '$$r$$'.
If the sum of infinite geometric7. Let $$\alpha$$ and $$\beta$$ be the distinct roots of $$a x^2+b x+c=0$$, then $$\lim _\limits{x \rightarrow \alpha} \fra8. If two positive numbers are in the ratio $$3+2 \sqrt{2}: 3-2 \sqrt{2}$$, then the ratio between their A.M (arithmetic me9. $$
\text { The value of the integral } \int_\limits{\frac{1}{3}}^1 \frac{\left(x-x^3\right)^{\frac{1}{3}}}{x^4} d x \tex10. $$
\text { The area of the region enclosed by the curve }\left\{(x, y): 4 x^2+25 y^2=100\right\} \text { is }
$$11. The mean of five observations is 4 and their variance is 5.2 . If three of these observations are 1, 2 and 6, then the o12. The line joining two points $$A(2,0) B(3,1)$$ is rotated about $$A$$ in anticlockwise direction through an angle of $$1513. $$
\text { The value of } \lim _\limits{x \rightarrow 1} \frac{x^{15}-1}{x^{10}-1}=
$$14. Let $$\mathrm{A}$$ and $${B}$$ be two events such that $$P(A / B)=\frac{1}{2}$$ and $$P(B / A)=\frac{1}{3}$$ and $$P(A \15. $$
\text { If } \frac{\cos x}{\cos (x-2 y)}=\lambda \text { then } \tan (x-y) \tan y=
$$16. $$
\text { If } y=f(x), \quad p=\frac{d y}{d x} ; q=\frac{d^2 y}{d x^2} \text { then } \frac{d^2 x}{d y^2} \text { is eq17. $$
\text { If } \hat{\imath}+\hat{\jmath}-\hat{k} \quad \&~ 2 \hat{\imath}-3 \hat{\jmath}+\hat{k} \text { are adjacent s18. Which of the following relations on the set of real numbers $$\mathrm{R}$$ is an equivalence relation?19. A number consists of three digits in geometric progression. The sum of the right hand and left hand digits exceeds twice20. $$
\text { If } y=\sqrt{\sin x+y} \text { then find } \frac{d y}{d x} \text { at } x=0, \quad y=1
$$21. $$
\text { If } A=\left[\begin{array}{cc}
5 a & -b \\
3 & 2
\end{array}\right] \text { and } A \operatorname{adj} A=A A^22. $$
\text { If } f(x)=\left\{\begin{array}{cc}
x & , \quad 0 \leq x \leq 1 \\
2 x-1 & , \quad x>1
\end{array}\right. \tex23. The measure of the angle between the lines $$x=k+1, \quad y=2 k-1, \quad z=2 k+3, \quad k \in R \quad$$ and $$\quad \fra24. $$
\text { Evaluate: } \cot ^{-1}\left(-\frac{3}{\sqrt{3}}\right)-\sec ^{-1}\left(-\frac{2}{\sqrt{2}}\right)-\operatorna25. The shaded region in the Venn diagram represents
26. $$
\text { The solution set for the inequality } 13 x-527. The general solution of the differential equation $$x \frac{d y}{d x}=y+x \tan \left(\frac{y}{x}\right)$$ is28. $$
\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 8 \theta}}} \text { where } \theta \in\left[-\frac{\pi}{8}, \frac{\pi}{8}\right] \text29. The turning point of the function $$y=\frac{a x-b}{(x-1)(x-4)}$$ at the point $$P(2,-1)$$ is 30. A coin is tossed until a head appears or until the coin has been tossed three times. Given that 'head' does not appear o31. $$
\text { The value of } \int \frac{d x}{\sqrt{2 x-x^2}} \text { is }
$$32. The equation of the circle which touches the $$x$$-axis, passes through the point $$(1,1)$$ and whose centre lies on the33. If the matrix $A$ is such that $$A\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)=\left(\begin{array}{cc}-4 & 134. In the parabola $$y^2=4 a x$$ the length of the latus rectum is 6 units and there is a chord passing through its vertex 35. The points on the $$x$$-axis whose perpendicular distance from the line $$\frac{x}{3}+\frac{y}{4}=1$$ is 4 units are36. The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio37. Suppose we have three cards identical in form except that both sides of the first card are coloured red, both sides of t38. $$
\text { The function } f(x)=\tan ^{-1}(\sin x+\cos x) \text { is an increasing function in }
$$39. For an examination a candidate has to select 7 questions from three different groups $$\mathrm{A}, \mathrm{B}$$ and C. T40. The letters of the word "COCHIN" are permuted and all the permutations are arranged in alphabetical order as in an Engli41. The co-ordinate of the foot of the perpendicular from $$P(1,8,4)$$ on the line joining $$R(0,-1,3)$$ and $$Q(2,-3,-1)$$ 42. $$
\text { The general solution of the differential equation }(1+\tan y)(d x-d y)+2 x d y=0 \text { is }
$$43. If $$a$$ is a real number such that $$\int_\limits0^a x d x \leq a+4$$ then44. Find the value of '$$b$$' such that the scalar product of the vector $$\hat{\imath}+\hat{\jmath}+\hat{k}$$ with the unit45. $$
\text { Value of } \cos 105^{\circ} \text { is }
$$46. The area bounded by the curve $$y=\cos x, x=0$$ and $$x=\pi$$ is47. $$
\text { If } I_n=\int_\limits0^{\frac{\pi}{4}} \tan ^n x d x \text {, for } n \geq 2 \text {, then } I_n+I_{n-2}=
$$48. In the expansion $$\left(\frac{1}{x}+x \sin x\right)^{10}, \quad$$ the co - efficient of $$6^{\text {th }}$$ term is equ49. $$
\text { If }(1-4 i)^3=a+i b \text { then the value of } \mathrm{a} \text { and } \mathrm{b} \text { is }
$$50. Two finite sets have '$$m$$' and '$$n$$' number of elements respectively. The total number of subsets of the first set i51. The sum of the order and degree of the differential equation $$\left(\frac{d^2 y}{d x^2}\right)^5+\frac{4\left(\frac{d^252. $$
\int e^x\left[\frac{x^2+1}{(x+1)^2}\right] d x \quad \text { is equal to }
$$53. $$
\text { Evaluate: } \cos ^{-1}\left(\cos \frac{35 \pi}{18}\right)-\sin ^{-1}\left(\sin \frac{35 \pi}{18}\right)
$$54. The maximum value of $$P=500 x+400 y$$ for the given constraints $$x+y \leq 200, \quad x \geq 20, \quad y \geq 4 x, \qua55. $$
\text { If } y=\sin ^{-1}\left(\frac{5 x+12 \sqrt{1-x^2}}{13}\right) \text { then } \frac{d y}{d x} \text { equals }
56. If the straight lines $$\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-t}$$ and $$\frac{x-1}{t}=\frac{y-4}{2}=\frac{z-5}{1}$$ a57. $$
\text { If } \frac{d y}{d x}=y+3>0 \text { and } y(0)=2 \text { then } y(\log 2) \text { is equal to }
$$58. What is the probability of a randomly chosen 2 digit number being divisible by 3 ?59. If $$A=\left[\begin{array}{ccc}0 & x & 16 \\ x & 5 & 7 \\ 0 & 9 & x\end{array}\right]$$ is a singular matrix then $$x$$ 60. What is the nature of the function $$f(x)=x^3-3 x^2+4 x$$ on real numbers?
Physics
1. The binding energy per nucleon for $$\mathrm{C}^{12}$$ is $$7.68 \mathrm{~MeV}$$ and that for $$\mathrm{C}^{13}$$ is $$72. The distance of closest approach when an alpha particle of kinetic energy $$6.5 \mathrm{~MeV}$$ strikes a nucleus of ato3. A scooter moves with a speed of $$7 \mathrm{~ms}^{-1}$$, on a straight road and is stopped by applying the brakes. Befor4. In Young's double slit experiment light of wavelength $$500 \mathrm{~nm}$$ is used to form interference pattern. A unifo5. Figure below shows a network of resistors, cells, and a capacitor at steady state.
What is the current through the resi6. A cricketer of height $$2.5 \mathrm{~m}$$ throws a ball at an angle of $$30^{\circ}$$ with the horizontal such that it i7. If an electron in a hydrogen atom jumps from the third orbit to the second orbit, it emits a photon of wavelength $$\lam8. A solid cylinder of mass $$2 \mathrm{~kg}$$ and radius $$0.2 \mathrm{~m}$$ is rotating about its own axis without fricti9. A neutral water molecule is placed in an electric field $$E=2.5 \times 10^4 \mathrm{NC}^{-1}$$. The work done to rotate 10. A telescope has an objective of focal length $$60 \mathrm{~cm}$$ and eyepiece of focal length $$5 \mathrm{~cm}$$. The te11. A parallel plate capacitor having a dielectric constant 5 and dielectric strength $$10^6 \mathrm{~V} \mathrm{~m}^{-1}$$ 12. The coefficient of volume expansion of glycerine is $$49 \times 10^{-5} \mathrm{~K}^{-1}$$. The percentage change in its13. A current I flows in an infinitely long wire with cross section in the form of semi-circular ring of radius $$1 \mathrm{14. Three bulbs of $$40 \mathrm{~W}, 60 \mathrm{~W}$$, and $$100 \mathrm{~W}$$ are arranged in series with a $$220 \mathrm{~15. A photon emitted during the de-excitation of electron from a state $$\mathrm{n}$$ to the second excited state in a hydro16. The acceleration due to gravity at pole and equator can be related as17. A bar magnet is held perpendicular to a uniform field. If the couple acting on the magnet is to be halved, by rotating i18. A negative charge particle is moving upward in a magnetic field which is towards north. The particle is deflected toward19. Two point charges $$\mathrm{M}$$ and $$\mathrm{N}$$ having charges $$+q$$ and $$-q$$ respectively are placed at a distan20. A conducting circular loop is placed in a uniform magnetic field $$\mathrm{B}=0.125 \mathrm{~T}$$ with its plane perpend21. Figure below shows a lens of refractive index, $$\mu=1.4$$. $$C_1$$ and $$C_2$$ are the centres of curvature of the two 22. The temperature of a wire is doubled. The Young's modulus of elasticity23. A wire of length $$2 \mathrm{~m}$$ carries a current of $$1 \mathrm{~A}$$ along the $$\mathrm{x}$$ axis. A magnetic fiel24. The dimension $$[\mathrm{ML}^{-1} \mathrm{~T}^{-2}]$$ is the physical quantity of25. The resistance of a $$10 \mathrm{~m}$$ long wire is $$10 \Omega$$. Its length is increased by $$25 \%$$ by stretching th26. A body is executing SHM. When its displacements from the mean position are $$4 \mathrm{~cm}$$ and $$5 \mathrm{~cm}$$ it 27. When water falls from a height of $$80 \mathrm{~m}$$ at the rate of $$20 \mathrm{~kg} \mathrm{~s}^{-1}$$ to operate a tu28. An electron has a mass of $$9.1 \times 10^{-31} \mathrm{~kg}$$. It revolves round the nucleus in a circular orbit of rad29. $$
\text { If the nuclear radius of }{ }^{27} \mathrm{Al} \text { is } 3.6 \text { fermi, the nuclear radius of }{ }^{1230. When an A.C. source is connected to a inductive circuit,31. A satellite is revolving around the earth in a circular orbit with kinetic energy of $$1.69 \times 10^{10} \mathrm{~J}$$32. A ball is moving in a circular path of radius $$5 \mathrm{~m}$$. If tangential acceleration at any instant is $$10 \math33. If 216 drops of the same size are charged at $$200 \mathrm{~V}$$ each and they combine to form a bigger drop, the potent34. The conductivity of a semiconductor increases with increase in temperature because
A) number density of free current car35. Four resistors, each of resistance R, are connected as shown in the figure below.
36. In the $$\mathrm{A} . \mathrm{C}$$. circuit given below, voltmeters $$\mathrm{V}_1$$ and $$\mathrm{V}_2$$ read $$100 \ma37. Joule second is the unit of38. When a biconvex lens of glass of refractive index 1.5 is dipped in a liquid, it acts like a plane sheet of paper.
This m39. The latent heat of vaporisation of water is $$2240 \mathrm{~J}$$. If the work done in the process of vaporisation of $$140. A particle of mass $$2 \mathrm{mg}$$ has the same wavelength as a neutron moving with a velocity of $$3 \times 10^5 \mat41. Two narrow parallel slits illuminated by a coherent monochromatic light produces an interference pattern on a screen
pla42. Three point charges are located on a circular arc at $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ as shown in the figur43. A voltmeter of resistance $$1000 \Omega .0 .5 \mathrm{~V} /$$ div is to be converted into a voltmeter to make it to read44. Column - I lists the waves of the electromagnetic spectrum. Column - II gives approximate frequency range of these waves45. A monochromatic light of wavelength $$800 \mathrm{~nm}$$ is incident normally on a single slit of width $$0.020 \mathrm{46. Internal energy of $$\mathrm{n}_1$$ moles of hydrogen at temperature T is equal to internal energy of $$\mathrm{m}_2$$ m47. A body initially at rest undergoes rectilinear motion. The forcetime (F-t) graph for the motion of the body is given bel48. Incident light of wavelength $$\lambda=800 \mathrm{~nm}$$ produces a diffraction pattern on a screen $$1.5 \mathrm{~m}$$49. A transformer of $$100 \%$$ efficiency has 200 turns in the primary and 40000 turns in the secondary. It is connected to50. Action and reaction can never balance out because51. The current in a coil changes steadily from $$3 \mathrm{~A}$$ to $$5 \mathrm{~A}$$ in $$0.2 \mathrm{~s}$$ when an emf of52.
The output of the given circuit is
A. Negatively rectified half wave
B. Positively rectified half wave
C. Negatively re53. For a paramagnetic material, the dependence of the magnetic susceptibility $$\chi$$ on the absolute temperature is given54. The magnetic flux linked with a coil is given by the equation: $$\phi=8 t^2+t+10$$. The e.m.f. induced in the coil in th55. A cylinder of fixed capacity 44.81 contains hydrogen gas at STP. What is the amount of heat needed to raise the temperat56. A hollow prism is filled with water and placed in air. It will deviate the incident rays57. The number of possible natural oscillations of air column in a pipe closed at one end of length $$85 \mathrm{~cm}$$ whos58. The figure shows a network of five capacitors connected to a 20 V battery. Calculate the charge acquired by each 10 $$\m59. What is the relation obeyed by the angles of contact $$\theta_1, \theta_2$$ and $$\theta_3$$ of 3 liquids of different d60. The following are the graphs of potential barrier versus width of the depletion region for a p-n junction diode.
Which
1
COMEDK 2024 Evening Shift
MCQ (Single Correct Answer)
+1
-0
Action and reaction can never balance out because
A
They are unequal in magnitudes
B
Though they are equal in magnitude and opposite in direction they act on different bodies.
C
They are equal in magnitude but not opposite always.
D
They are unequal in magnitudes even though opposite in direction
2
COMEDK 2024 Evening Shift
MCQ (Single Correct Answer)
+1
-0
The current in a coil changes steadily from $$3 \mathrm{~A}$$ to $$5 \mathrm{~A}$$ in $$0.2 \mathrm{~s}$$ when an emf of $$2 \mu \mathrm{V}$$ is induced in it. The self-inductance of the coil is
A
0.2 mH
B
20 $$\mu H$$
C
2 $$\mu H$$
D
0.2 $$\mu H$$
3
COMEDK 2024 Evening Shift
MCQ (Single Correct Answer)
+1
-0
The output of the given circuit is
A. Negatively rectified half wave
B. Positively rectified half wave
C. Negatively rectified full wave
D. Zero all times
A
B
B
D
C
A
D
C
4
COMEDK 2024 Evening Shift
MCQ (Single Correct Answer)
+1
-0
For a paramagnetic material, the dependence of the magnetic susceptibility $$\chi$$ on the absolute temperature is given as
A
Independent of $$\mathrm{T}$$
B
$$
X \propto \frac{1}{T^2}
$$
C
$$
X \propto T
$$
D
$$
X \propto\frac{1}{T}
$$
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Mathematics
60
Physics
60
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