**JEE Main Session 2 28 ^{th} July 2022 Shift 2**

**PHYSICS**

**Section-A**

**IMPORTANT INSTRUCTIONS:**

(1) The test is of **3 hours** duration:

(2) The Test Booklet consists of 90 questions. The maximum marks are 300.

(3) There are three parts in the question paper consisting of **Physics**, **Chemistry** and **Mathematics** having 30 questions in each part of equal weightage. Each part (subject) has two sections.

(i) **Section-A:** This section contains 20 multiple choice questions which have only one correct answer. each question carries **4 marks** for correct answer and **−1** mark for wrong answer.

(ii) **Section-B:** This section contains 10 questions. In Section-B, attempt any **five questions out of 10.** The answer to each of the questions is a numerical value. Each question carries 4 marks for correct answer and **−1 mark** for wrong answer. For Section-B, the answer should be rounded off to the nearest integer.

1. Consider the efficiency of Carnot engine is given by where α and β are constants. If T is temperature, k is Boltzmann constant, θ is angular displacement, and x has the dimensions of length. Then, choose the incorrect option

(A) Dimensions of βis same as that of force.

(B) Dimensions of α^{–1} x is same as that of energy.

(C) Dimensions of η^{–1}sinθ is same as that of αβ

(D) Dimensions of α is same as that of β

2. At time t = 0 a particle starts travelling from a height in a plane keeping z coordinate constant. At any instant of time it’s position along the directions are defined at 3t and 5t^{3} At t = 1 s acceleration of the particle will be

3. A pressure-pump has a horizontal tube of cross sectional area 10 cm^{2} for the outflow of water at a speed of 20 m/s. The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is

[given: density of water = 1000 kg/m^{3}]

(A) 300 N

(B) 500 N

(C) 250 N

(D) 400 N

4. A uniform metal chain of mass m and length ‘L’ passes over a massless and frictionless pully. It is released from rest with a part of its length ‘l’ is hanging on one side and rest of its length ‘L – l’ is hanging on the other side of the pully. At a certain point of time, when l = L/x, the acceleration of the chain is g/2. The value of x is ______.

(A) 6

(B) 2

(C) 1.5

(D) 4

5. A bullet of mass 200 g having initial kinetic energy 90 J is shot inside a long swimming pool as shown in the figure. If it’s kinetic energy reduces to 40 J within 1s, the minimum length of the pool, the bullet has a to travel so that it completely comes to rest is

(A) 45 m

(B) 90 m

(C) 125 m

(D) 25 m

6. Assume there are two identical simple pendulum Clocks-1 is placed on the earth and Clock-2 is placed on a space station located at a height h above the earth surface. Clock-1 and Clock-2 operate at time periods 4s and 6s respectively. Then the value of h is –

(consider radius of earth R_{E} = 6400 km and g on earth 10 m/s^{2})

(A) 1200 km

(B) 1600km

(C) 3200km

(D) 4800km

7. Consider a cylindrical tank of radius 1 m is filled with water. The top surface of water is at 15 m from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of 5 m from the bottom. A force of 5 × 10^{5} N is applied an the top surface of water using a piston. The speed of efflux from the hole will be:

(given atmospheric pressure P_{A} = 1.01 × 10^{5} Pa, density of water ρ_{w} = 1000 kg/m^{3} and gravitational acceleration g = 10 m/s^{2})

(A) 11.6 m/s

(B) 10.8m/s

(C) 17.8m/s

(D) 14.4m/s

8. A vessel contains 14 g of nitrogen gas at a temperature of 27°C. The amount of heat to be transferred to the gap to double the r.m.s. speed of its molecules will be : (Take R = 8.32 J mol^{–1}k^{–1})

(A) 2229 J

(B) 5616 J

(C) 9360 J

(D) 13,104 J

9. A slab of dielectric constant K has the same cross-sectional area as the plates of a parallel plate capacitor and thickness where d is the separation of the plates. The capacitance of the capacitor when the slab is inserted between the plates will be :

(Given C_{o} = capacitance of capacitor with air as medium between plates.)

10. A uniform electric field E = (8m/e) V/m is created between two parallel plates of length 1 m as shown in figure, (where m = mass of electron and e = charge of electron). An electron enters the field symmetrically between the plates with a speed of 2 m/s. The angle of the deviation (θ) of the path of the electron as it comes out of the field will be_______.

(A) tan^{−}^{1} (4)

(B) tan^{−}^{1} (2)

(C) tan^{−}^{1} (1/3)

(D) tan^{−}^{1} (3)

11. Given below are two statements :

**Statement I :** A uniform wire of resistance 80Ω is cut into four equal parts. These parts are now connected in parallel. The equivalent resistance of the combination will be 5Ω.

**Statement II :** Two resistance 2R and 3R are connected in parallel in a electric circuit. The value of thermal energy developed in 3R and 2R will be in the ratio 3 : 2.

In the light of the above statements, choose the most appropriate answer from the options given below

(A) Both statement I and statement II are correct

(B) Both statement I and statement II are incorrect

(C) Statement I is correct but statement II is incorrect

(D) Statement I is incorrect but statement II is correct

12. A triangular shaped wire carrying 10 A current is placed in a uniform magnetic field of 0.5 T, as shown in figure. The magnetic force on segment CD is (Given BC = CD = BD = 5 cm).

(A) 0.126 N

(B) 0.312 N

(C) 0.216 N

(D) 0.245 N

13. The magnetic field at the center of current carrying circular loop is B_{1}. The magnetic field at a distance of √3 times radius of the given circular loop from the center on its axis is B_{2}. The value of B_{1}/B_{2} will be

(A) 9 : 4

(B)12 : √5

(C) 8 : 1

(D) 5 :√3

14. A transformer operating at primary voltage 8 kV and secondary voltage 160 V serves a load of 80 kW. Assuming the transformer to be ideal with purely resistive load and working on unity power factor, the loads in the primary and secondary circuit would be

(A) 800 Ω and 1.06 Ω

(B) 10 Ω and 500 Ω

(C) 800 Ω and 0.32 Ω

(D) 1.0 Ω and 500 Ω

15. Sun light falls normally on a surface of area 36 cm2 and exerts an average force of 7.2 × 10^{–9} N within a time period of 20 minutes. Considering a case of complete absorption, the energy flux of incident light is

(A) 25.92 × 10^{2} W/cm^{2}

(B) 8.64 × 10^{−}^{6} W/cm^{2}

(C) 6.0 W/cm^{2}

(D) 0.06 W/cm^{2}

16. The power of a lens (biconvex) is 1.25 m–1 in particular medium. Refractive index of the lens is 1.5, and the radii of curvature are 20 cm and 40 cm, respectively. The refractive index of surrounding medium

(A) 1.0

(B) 9/7

(C) 3/2

(D) 4/3

17. Two streams of photons, possessing energies equal to five and ten times the work function of metal are incident on the metal surface successively. The ratio of maximum velocities of the photoelectron emitted, in the two cases respectively, will be

(A) 1 : 2

(B) 1 : 3

(C) 2 : 3

(D) 3 : 2

18. A radioactive sample decays 7/8 times its original quantity in 15 minutes. The half-life of the sample is

(A) 5 min

(B) 7.5min

(C) 15min

(D) 30min

19. An npn transistor with current gain β = 100 in common emitter configuration is shown in the figure. The output voltage of the amplifier will be

(A) 0.1 V

(B) 1.0 V

(C) 10 V

(D) 100 V

20. A FM Broad cast transmitter, using modulating signal of frequency 20 kHz has a deviation ratio of 10. The Bandwidth required for transmission is

(A) 220 kHz

(B) 180kHz

(C) 360kHz

(D) 440kHz

**SECTION-B**

21. A ball is thrown vertically upwards with a velocity of 19.6 ms^{–1} from the top of a tower. The ball strikes the ground after 6 s. The height from the ground up to which the ball can rise will be (k/5) m. The value of k is ______ (use g = 9.8 m/s^{2})

22. The distance of centre of mass from end A of a one dimensional rod (AB) having mass density and length L (in meter) is The value of α is ______. (where x is the distance from end A)

23. A string of area of cross-section 4mm^{2} and length 0.5 m is connected with a rigid body of mass 2 kg. The body is rotated in a vertical circular path of radius 0.5 m. The body acquires a speed of 5 m/s at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is ______×10^{–5}.

(use young’s modulus 10^{11} N/m^{2} and g = 10 m/s^{2})

24. At a certain temperature, the degrees of freedom per molecule for gas is 8. The gas performs 150 J of work when it expands under constant pressure. The amount of heat absorbed by the gas will be ______ J.

25. The potential energy of a particle of mass 4 kg in motion along the x-axis is given by U = 4 (1–cos 4x) J. The time period of the particle for small oscillation (sin θ≃θ) is The value of K is __________

26. An electrical bulb rated 220 V, 100 W, is connected in series with another bulb rated 220 V, 60 W. If the voltage across combination is 220 V, the power consumed by the 100 W bulb will be about ____________ W.

27. For the given circuit the current through battery of 6 V just after closing the switch ‘S’ will be __________ A.

28. An object ‘o’ is placed at a distance of 100 cm in front of a concave mirror of radius of curvature 200 cm as shown in the figure. The object starts moving towards the mirror at a speed 2 cm/s. The position of the image from the mirror after 10s will be at _______ cm.

29. In an experiment with a convex lens, the plot of the image distance (ν′) against the object distance (μ′) measured from the focus gives a curve ν′μ′ = 225. If all the distances are measured in cm. The magnitude of the focal length of the lens is _______ cm.

30. In an experiment to find acceleration due to gravity (g) using simple pendulum, time period of 0.5 s is measured from time of 100 oscillations with a watch of 1 s resolution. If measured value of length is 10 cm known to 1 mm accuracy, the accuracy in the determination of g is found to be x %. The value of x is _________.

**CHEMISTRY**

**SECTION-A**

1. Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R

**Assertion A :** Zero orbital overlap is an out of phase overlap.

**Reason :** It results due to different orientation/ direction of approach of orbitals.

In the light of the above statements. Choose the correct answer from the options given below

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is NOT the correct explanation of A

(C) A is true but R is false

(D) A is false but R is true

2. The correct decreasing order for metallic character is

(A) Na > Mg > Be > Si > P

(B) P > Si > Be > Mg > Na

(C) Si > P > Be > Na > Mg

(D) Be > Na > Mg > Si > P

3. Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R

**Assertion A :** The reduction of a metal oxide is easier if the metal formed is in liquid state than solid state.

**Reason R :** The value of ∆G^{⊝}becomes more on negative side as entropy is higher in liquid state than solid state.

In the light of the above statements. Choose the most appropriate answer from the options given below

(A) Both A and R are correct and R is the correct explanation of A

(B) Both A and R are correct but R is NOT the correct explanation of A

(C) A is correct but R is not correct

(D) A is not correct but R is correct

4. The products obtained during treatment of hard water using Clark’s method are:

(A) CaCO_{3} and MgCO_{3}

(B) Ca(OH)_{2} and Mg(OH)_{2}

(C) CaCO_{3} and Mg(OH)_{2}

(D) Ca(OH)_{2} and MgCO_{3}

5. **Statement I:** An alloy of lithium and magnesium is used to make aircraft plates.

**Statement II:** The magnesium ions are important for cell-membrane integrity.

In the light the above statements, choose the correct answer from the options given below

(A) Both Statement I and Statement II are true

(B) Both Statement I and Statement II are false

(C) Statement I is true but Statement II is false

(D) Statement I is false but Statement II is true

6. White phosphorus reacts with thionyl chloride to give

(A) PCl_{5}, SO_{2} and S_{2}Cl_{2}

(B) PCl_{3}, SO_{2} and S_{2}Cl_{2}

(C) PCl_{3}, SO_{2} and Cl_{2}

(D) PCl_{5}, SO_{2} and Cl_{2}

7. Concentrated HNO_{3} reacts with Iodine to give

(A) HI, NO_{2} and H_{2}O

(B) HIO_{2}, N_{2}O and H_{2}O

(C) HIO_{3}, NO_{2} and H_{2}O

(D) HIO_{4}, N_{2}O and H_{2}O

8. Which of the following pair is not isoelectronic species?

(At. no.Sm, 62; Er, 68: Yb, 70: Lu, 71; Eu, 63: Tb, 65; Tm, 69)

(A) Sm^{2+} and Er^{3+}

(B) Yb^{2+} and Lu^{3+}

(C) Eu^{2+} and Tb^{4+}

(D) Tb^{2+} and Tm^{4+}

9. Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R

**Assertion A:** Permanganate titrations are not performed in presence of hydrochloric acid.

**Reason R:** Chlorine is formed as a consequence of oxidation of hydrochloric acid.

In the light of the above statements, choose the correct answer from the options given below

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is NOT the correct explanation of A

(C) A is true but R is false

(D) A is false but R is true

10. Match List I with List II

Choose the correct answer from the options given below:

(A) A-IV, B-I, C-III, D-II

(B) A-I. B-IV, C-III, D-II

(C) A-I. B-IV, C-II, D-III

(D) A-IV, B-I, C-II. D-III

11. Dinitrogen and dioxygen. the main constituents of air do not react with each other in atmosphere to form oxides of nitrogen because

(A) N_{2} is unreactive in the condition of atmosphere.

(B) Oxides of nitrogen are unstable.

(C) Reaction between them can occur in the presence of a catalyst.

(D) The reaction is endothermic and require very high temperature.

12. The major product in the given reaction is

13. Arrange the following in increasing order of reactivity towards nitration

(A) p-xylene (B) bromobenzene

(C)mesitylene (D) nitrobenzene

(E)benzene

Choose the correct answer from the options given below

(A) C < D < E < A < B

(B) D < B < E < A < C

(C) D < C < E < A < B

(D) C < D < E < B < A

14. Compound I is heated with Conc. HI to give a hydroxy compound A which is further heated with Zn dust to give compound B. Identify A and B.

15. Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R

**Assertion A:** Aniline on nitration yields ortho, meta¶ nitro derivatives of aniline.

**Reason R:** Nitrating mixture is a strong acidic mixture.

In the light of the above statements, choose the correct answer from the options given below

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is NOT the correct explanation of A

(C) A is true but R is false

(D) A is false but R is true

16. Match List I with List II

Choose the correct answer from the options given below:

(A) A-II, B-III, C-IV, D-I

(B) A-III, B-II, C-IV, D-I

(C) A-III, B-I, C-IV, D-II

(D) A-I. B-III, C-IV, D-II

17. Two statements in respect of drug-enzyme interaction are given below

**Statement I:** Action of an enzyme can be blocked only when an inhibitor blocks the active site of the enzyme.

**Statement II:** An inhibitor can form a strong covalent bond with the enzyme.

In the light of the above statements. Choose the correct answer from the options given below

(A) Both Statement I and Statement II are true

(B) Both Statement I and Statement II are false

(C) Statement I is true but Statement II is false

(D) Statement I is false but Statement II is true

18. Given below are two statements : One is labelled as Assertion A and the other is labelled as Reason R

**Assertion A:** Thin layer chromatography is an adsorption chromatography.

**Reason:** A thin layer of silica gel is spread over a glass plate of suitable size in thin layer chromatography which acts as an adsorbent.

In the light of the above statements, choose the correct answer from the options given below

(A) Both A and R are true and R is the correct explanation of A

(B) Both A and R are true but R is NOT the correct explanation of A

(C) A is true but R is false

(D) A is false but R is true

19. The formulas of A and B for the following reaction sequence are

(A) A = C_{7}H_{14}O_{8}, B = C_{6}H_{14}

(B) A = C_{7}H_{13}O_{7}, B = C_{7}H_{14}O

(C) A = C_{7}H_{12}O_{8}, B = C_{6}H_{14}

(D) A = C_{7}H_{14}O_{8}, B = C_{6}H_{14}O_{6}

20.

Find out the major product for the above reaction.

**SECTION-B**

21. 2L of 0.2 M H_{2}SO_{4} is reacted with 2L of 0.1 M NaOH solution, the molarity of the resulting product Na_{2}SO_{4} in the solution is ____ millimolar. (Nearest integer).

22. Metal M crystallizes into a FCC lattice with the edge length of 4.0×10^{−8} The atomic mass of the metal is ______ g/mol.

(Nearest integer). (Use : NA = 6.02×10^{23} mol^{−1}, density of metal, M = 9.03 g cm^{−3})

23. If the wavelength for an electron emitted from Hatom is 3.3×10^{−}^{10} m, then energy absorbed by the electron in its ground state compared to minimum energy required for its escape from the atom, is _____times. (Nearest integer).

[Given : h = 6.626 ×10^{−}^{34}Js, Mass of electron = 9.1×10^{−}^{1}]

24. A gaseous mixture of two substances A and B, under a total pressure of 0.8 atm is in equilibrium with an ideal liquid solution. The mole fraction of substance A is 0.5 in the vapour phase and 0.2 in the liquid phase. The vapour pressure of pure liquid A is _______ atm. (Nearest integer)

25. At 600K, 2 mol of NO are mixed with 1 mol of O_{2}.

2NO_{(g)} + O_{2}(g) ⇄ 2NO_{2}(g)

The reaction occurring as above comes to equilibrium under a total pressure of 1 atom. Analysis of the system shows that 0.6 mol of oxygen are present at equilibrium. The equilibrium constant for the reaction is _______. (Nearest integer).

26. A sample of 0.125 g of an organic compound when analysed by Duma’s method yields 22.78 mL of nitrogen gas collected over KOH solution at 280K and 759 mm Hg. The percentage of nitrogen in the given organic compound is ____. (Nearest integer).

(a) The vapour pressure of water at 280 K is 14.2 mm Hg

(b) R = 0.082 L atm K^{–1}mol^{–1}

27. On reaction with stronger oxidizing agent like KIO_{4}, hydrogen peroxide oxidizes with the evolution of O_{2}. The oxidation number of I in KIO_{4}changes to ______.

28. For a reaction, given below is the graph of ln k vs 1/T. The activation energy for the reaction is equal to ________ cal mol^{−}^{1}. (Nearest integer).

(Given : R = 2 cal K^{−}^{1} mol^{−}^{1})

29. Among the following the number of curves not in accordance with Freundlich adsorption isotherm is ______.

30. Among the following the number of state variable is ______.

Internal energy (U)

Volume (V)

Heat (q)

Enthalpy (H)

**MATHEMATICS**

**SECTION-A**

1. Let and T = {x ∈Z : x^{2} – 7|x| + 9 ≤ 0}. Then the number of elements in S ∩ T is

(A) 7

(B) 5

(C) 4

(D) 3

2. Let α, β be the roots of the equation x^{2} – √2x + √6 = 0 and be the roots of the equation x^{2} + ax + b = 0 . Then the roots of the equation x^{2} – (a + b – 2)x + (a + b + 2) = 0 are

(A) Non-real complex number

(B) Real and both negative

(C) Real and both positive

(D) Real and exactly one of them is positive

3. Let A and B be any two 3 × 3 symmetric and skew symmetric matrices, respectively. Then which of the following is NOT true?

(A) A^{4} – B^{4} is a symmetric matrix

(B) AB – BA is a symmetric matrix

(C) B^{5} – A^{5} is a skew-symmetric matrix

(D) AB + BA is a skew-symmetric matrix

4. Let f(x) = ax^{2} + bx + c be such that f(1) = 3, f(-2) = λ and f(3) = 4. If f(0) + f(1) + f(-2) + f(3) = 14, then λ is equal to

(A) −4

(B) 13/2

(C) 23/2

(D) 4

5. The function f: ℝ → ℝ defined by is continuous for all x in

(A) ℝ − {−1}

(B) ℝ − {−1, 1}

(C) ℝ − {1}

(D) ℝ − {0}

6. The function f(x) = xe^{x(1}^{−}^{x)}, x ∈ ℝ is

(A) Increasing in (−1/2, 1)

(B) Decreasing in (1/2, 2)

(C) Increasing in (−1, −1/2)

(D) Decreasing in (−1/2, 1/2)

7. The sum of the absolute maximum and absolute minimum values of the function f(x) = tan^{−}^{1} (sin x – cos x) in the interval [0, π] is

(A) 0

(B)

(C)

(D) –π/12

8. Let and Then is equal to

(A) −2√2/3

(B) 2/3

(C) 1/3

(D) −2/3

9. Let n = 1, 2, 3, ….. Then

(A) 50I_{6} – 9I_{5} = xI′_{5}

(B) 50I_{6} – 11I_{5} = xI′_{5}

(C) 50I_{6} – 9I_{5} = I′_{5}

(D) 50I_{6} – 11I_{5}= I′_{5}

10. The area enclosed by the curves y = log_{e} (x + e^{2}), and x = log_{e}2, above the line y = 1 is

(A) 2 + e – log_{e} 2

(B) 1 + e – log_{e} 2

(C) e– log_{e} 2

(D) 1 + log_{e} 2

11. Let y = y(x) be the solution curve of the differential equation passing through the point Then √7 (8) is equal to

(A) 11 + 6log_{e} 3

(B) 19

(C) 12 – 2log_{e} 3

(D) 19 – 6log_{e} 3

12. The differential equation of the family of circles passing through the points (0, 2) and (0, –2) is

13. Let the tangents at two points A and B on the circle x^{2} + y^{2} – 4x + 3 = 0 meet at origin O(0, 0). Then the area of the triangle OAB is

(A) 3√3/2

(B) 3√3/4

(C) 3/2√3

(D) 3/4√3

14. Let the hyperbola pass through the point (2√2, −2√2). A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?

(A) (2√3, 3√2)

(B) (3√3, −6√2)

(C) (√3, −√6)

(D) (3√6, 6√2)

15. Let the lines and be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?

(A) (0, −2, −2)

(B) (−5, 0, −1)

(C) (3, −1, 0)

(D) (0, 4, 5)

16. A plane P is parallel to two lines whose direction rations are –2, 1, –3 and –1, 2, –2 and it contains the point (2, 2, –2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts α, β, γ. If V is the volume of the tetrahedron OABC, where O is the origin and p = α + β + γ, then the ordered pair (V, p) is equal to :

(A) (48, –13)

(B) (24, –13)

(C) (48, 11)

(D) (24, –5)

17. Let S be the set of all a∈ R for which the angle between the vectors and is acute. Then S is equal to

(A) (−∞, −4/3)

(B) Φ

(C) (−4/3, 0)

(D) (12/7, ∞)

18. A horizontal park is in the shape of a triangle OAB with AB = 16. A vertical lamp post OP is erected at the point O such that ∠PAO = ∠PBO = 15° and ∠PCO = 45°, where C is the midpoint of AB. Then (OP)^{2} is equal to

19. Let A and B be two events such that and Consider

(S1) P(A′ ∪ B) = 5/6,

(S2) P(A′ ∩ B′) = 1/18. Then

(A) Both (S1) and (S2) are true

(B) Both (S1) and (S2) are false

(C) Only (S1) is true

(D) Only (S2) is true

20. Let

p : Ramesh listens to music.

q :Ramesh is out of his village.

r : It is Sunday.

s : It is Saturday.

Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as

(A) ((~q) ∧ (r ∨ s)) ⇒ P

(B) (q∧ (r ∨ s)) ⇒ P

(C) p⇒ (q ∧ (r ∨ s))

(D) p⇒ ((~q)∧ (r ∨ s))

**SECTION-B**

21. Let the coefficients of the middle terms in the expansion of and respectively form the first three terms of an A.P. If d is the common difference of this A.P., then is equal to _______

22. A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then b + 3 g is equal to ______.

23. Let the tangents at the points P and Q on the ellipse meet at the point R(√2, 2√2 – 2). If S is the focus of the ellipse on its negative major axis, then SP^{2} + SQ^{2}is equal to ________.

24. If 1 + (2 + ^{49}C_{1} + ^{49}C_{2} + … ^{49}C_{49}) (^{50}C_{2} + ^{50}C_{4} + … ^{50}C_{50}) is equal to 2^{n}. m, where m is odd, then n + m is equal to ______.

25. Two tangent lines l_{1} and l_{2} are drawn from the point (2, 0) to the parabola 2y^{2} = – x. If the lines l_{1} and l_{2} are also tangent to the circle (x – 5)^{2} + y^{2} = r, then 17r is equal to _________.

26. If where m is odd, then m.n is equal to ______

27. Let Then the number of elements in the set

A = {θ∈S : tan θ(1 + √5 tan(2θ)) = √5 – tan(2θ)} is ______

28. Let z = a + ib, b≠ 0 be complex numbers satisfying Then the least value of n ∈ N, such that z^{n} = (z + 1)^{n}, is equal to _____.

29. A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If σ^{2} is the variance of X, then 100 σ^{2} is equal to ____.

30. The value of the integral is equal to _______

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