JEE Main 2024 Past Paper (Mathematics) - 09 April 2024 - PDF

JEE | NEET | Class 8 - 10 Download eSaral App JEE(Main)-2024 | 09 April 2024 (Shift-1 Morning) | Question Paper | Memory Based MATHEMATICS 2 − tan x 1....

JEE | NEET | Class 8 - 10 Download eSaral App JEE(Main)-2024 | 09 April 2024 (Shift-1 Morning) | Question Paper | Memory Based MATHEMATICS 2 − tan x 1. ∫ 3 + tan x = αx + βn (3cos x + sin x ) + γ Where γ is constant of integration. Find α + β. 2. f(x) = 3ax3 + bx2 + cx + 41 f(1) = 41, f'(1) = 2, f"(1) = 4. Find a2 + b2 + c2. 2024 3. The remainder when 428 ( ) is divided by 21 is  x−1  4. ( ) If the domain of function f x = sin−1   is R – (α, β). Then find 12αβ.  2x + 3   tan 8x   8  tan 7x π ,x <    2  7 π 5. f (x) = a−8 ,x=  b 2  ( )  1 + cot x a tan x ,x> π 2  π f(x) is continuous at x =. Find a2 + b2. 2 1 1 1 1 6. + + +.... + 5. Find the value of 50d. = 1 + d ( 1 + d)( 1 + 2d) ( 1 + 2d)( 1 + 3d) ( 1 + 9d)( 1 + 10d) 1 7. cos θ cos ( 60° − θ ) cos ( 60° + θ ) ≤ , θ ∈ 0,2π . Find the sum of value of θ for which 8 cos3θ is maximum. 8. A variable line passing through (3, 5) cut positive x & y axis. Find minimum area made between axis and line. 9. If the roots of equation x2 + 2 2x − 1 =0 are α and β. Find the equation whose roots are α4 + β4 and 1 10 ( α6 + β6. ) 10. Given system as 3x + 4y + λz = 4 5x + 7y + 2z = 8 97x + 197y + 83z = µ. Find λ + 3µ if the system has infinite solutions. JEE-Main 09-04-2024 Morning Shift www.esaral.com 1  JEE | NEET | Class 8 - 10 Download eSaral App       11. A triangle ABC is made of three vectors a,b and c. a , b and c are   ( )( ) 2 αˆi + 5ˆj + 4kˆ , 3iˆ + 5ˆj + 4kˆ and a − b respectively. Area of ∆ABC is given as 5 6. Find c. 12. A circle with centre (α, β) passes through point (0, 0) and (0, 1) and touches the circle x2 + y2 = 9 for all possible values of (α, β). Find value of 4 α 4 + β4 ? ( ) 98 97 54 13. ( x2 1 + x ) ( + x3 1 + x ) ( +... + x46 1 + x ) x70. If the coefficient of = 99 Cp − 54Cq , find p + q. x x2 y 2 14. Given ( ) f x= x2 + 9 and ( ) g x = x−9. And a curve a + b 1, = where =a fog = ( ) ( ) 10 , b gof 3. Then find 8e2 + 2 , (Where e = eccentricity,  = latus rectum length) 15. A circle x2 + y2 = 5 and a parabola y2 = 4x intersecting each other. Then find the area of smallest intersecting region. 16. A tetrahedral dice written 1, 2, 3, 4 on their faces is thrown, find the probability such that quadratic equation ax2 + bx + c = 0 has real roots. 2022 ∑ f ( λ + k ) ≤ (2022) 2 17. ( ) ( ) () If f m + n= f m + f n and f 1 = 1. (). Find maximum value of λ. k=1 18. The solution of the differential equation x2 + y2 dx − 5xy dy = ( 0. y 1 = 0 ) ()        19. For a quadrilateral OABC, given that OA = 6a + 2b and OC = 3b. It is also given that = 2a,OB area of parallelogram with adjacent sides OA and OC is 15. Then find the area of quadrilateral OABC.         5 20. If 2 a − b = a + b , a = n b and angle between a and b is cos−1 = then find n = ? 9 21. If set = A {z : z − 1 ≤ 1} and set = B {z : z − 5i ≤ z − 5 } , if z = a + ib, where a, b ∈ Ι. The sum of modulus squares of A ∩ B is 22. A ray of light passing through (1, 2) after reflecting on x-axis at point Q passes through R(4, 3). If S(h, k) is such that PQRS is a parallelogram then find (h, k) 23. If A is 3 × 3 matrix, det(3adj(2adjA)) = 2-13. 3-10 and det(3adj(2A)) = 2-m. 2-n then 2m + 2n is equals to JEE-Main 09-04-2024 Morning Shift www.esaral.com 2  JEE | NEET | Class 8 - 10 Download eSaral App PHYSICS 1. Find the work done by one mole of monoatomic gas undergoing adiabatic expansion such that the volume changes from V to 2V. 5 2. Angle between two vectors is cos-1   , if A + B = 2 A − B and A = nB find the value of n. 9 3. Find the dimensional formula of latent heat. 4. Find the energy equivalent (in MeV) for 1 gm mass of substance? 5. A vehicle travels half of the distance with speed 3 m/s and other half of distance in two equal time intervals with speed 6 m/s and 9 m/s. The average speed of vehicle is: 6. What will be the order of de-Broglie wavelength of  particle, proton, electron if their kinetic energies are same? 7. In an atwood machine two masses m1 and m2 are suspended and the magnitude of g acceleration of the masses is. Find the ratio of masses. 8 8. Statement-1: Concave lens always forms erect and virtual images Statement-2: If a object is placed at one centre of curvature of concave lens then image forms at centre of curvature of other side. (1) Only statement -1 is correct. (2) Only statement -2 is correct. (3) Both of the statements are correct. (4) None of the statements is correct. 9. Find the ratio of initial to final pressure for a gas compressed adiabatically from 5 litres to 3 4 litres. (Given  = ). 2 10. If particle A is on earth surface and other particle B is revolving around the earth R/20 above earth’s surface. Then the difference in mechanical energies of A and B will be: (Radius of Earth is R = 6570 km) 11. If a particle performing SHM has x = 4 m, v = 2 m/s, a = 16 m/s 2, then what will be its amplitude? 12. If a rod of weight w is resting on head of a man at an angle  as shown in figure, find the load on man’s head. JEE-Main 09-04-2024 Morning Shift www.esaral.com 3  JEE | NEET | Class 8 - 10 Download eSaral App 13. Find R equivalent. A 8  B 8 7 7 9 8 6 14. Find the wavelength of light emitted by the bulb which uses the LED having the band gap of 1.42 eV. 15. If the velocity of particle of mass m is given by v =  x. Find the work done by the particle to go from x = 0 to x = d? 16. Find the current passing through 1  resistance. 4 4 10 V 1 2 2 5V 17. An inductor when connected to a 20V DC battery gives current of 5A and when connected to a (20V, 50Hz) AC supply the current through the inductor is 4A. Find the inductance of the loop ( = 3). 18. A square loop of side 2 m carrying current i is placed in a magnetic field B = (1 + 4x) K̂. Find the net force acting on the loop. y (0,0) x 19. A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. The width of each stair is a and the height is d. Find the capacitance of assembly. A/3 A/3 d a a A/3 d d a JEE-Main 09-04-2024 Morning Shift www.esaral.com 4  JEE | NEET | Class 8 - 10 Download eSaral App 20. In a Young’s double slit experiment, the slits are 1 mm apart and are illuminated by a light of  = 600nm. What should be the minimum distance from central maximum where intensity 1 of light is of maximum intensity on a screen placed at 1 m distance from the plane of slits. 4 21. If the young’s modulus of the rod (shown in the figure) is Y = 1011 N/m2. Then find elongation in the rod (l). 22. EM wave travelling in x direction with E = 60 v/m ˆj. Find magnetic field B. 23. A body of moment of inertia I = 0.4 kg-m2 and radius r = 10 cm is given as shown in the figure. If a force of F = 40N is applied on the periphery of the body for 10 seconds, then angular velocity attained will be? 24. A half ring of R = 10 cm and linear density is 4 nC/m. Find the potential at center of the ring. JEE-Main 09-04-2024 Morning Shift www.esaral.com 5  JEE | NEET | Class 8 - 10 Download eSaral App CHEMISTRY 1. Which of the following are paramagnetic? O2 , O22 , C2 , CN , N2 , O2 2. What will be the increasing order of energy? (i) n = 4,  = 0 (ii) n = 4,  = 2 (iii) n = 4,  = 1 (iv) n = 3,  = 2 (v) n = 3,  = 1 (1) v < i < iv < iii < ii (2) v < ii < iv < iii < i (3) iv < i < v < iii < ii (4) v < i < iii < iv < ii 3. Which of the following is ambident ligand NO2 , CN , SCN , H2O , NH3, C2O42 4. PbS when react with dil. HNO3 then which of the following will not formed (1) S (2) N2O (3) NO (4) Pb(NO3)2 5. Statement I: [Co(en)2Cl2]+1 have 3 G.I. Statement II: [Co(en)2Cl2]+1 have octahedral geometry In the light of the above statements, choose the most appropriate answer from the options given below: (1) Both Statement-I and Statement-II are false (2) Statement-I is false but Statement-II is true (3) Both Statement I and Statement-II are true (4) Statement-I is true but Statement-II is false 6. Which of the following are colourless La3+, Eu3+, Sm3+, Nd2+, Lu3+ 7. Correct pair of sp2 hybridised molecule (1) BF3, NO2 (2) BF3, NH2 (3) BF3, H2O (4) H2O, NO2 8. Statement I: Sulphur exist as monoclinic and Rhombic form having S 8 unit but O2 do not have. Statement II: Because p p bond is present in O2 but not present in sulphur In the light of the above statements, choose the most appropriate answer from the options given below: (1) Both Statement-I and Statement-II are false (2) Statement-I is false but Statement-II is true (3) Both Statement I and Statement-II are true (4) Statement-I is true but Statement-II is false JEE-Main 09-04-2024 Morning Shift www.esaral.com 6  JEE | NEET | Class 8 - 10 Download eSaral App 9. Statement I: r = k[A]2[B] on doubling concentration rate becomes x times. Conc. Statement II: order = y Time Find the value of x + y. 10. Equal volume of 1 M HCl and 1 M H2SO4 neutralized by dil. NaOH and heat released is x and y kcal respectively, then which is correct? (1) x = y (2) x = 0.5 y (3) x = 0.4 y (4) x = 2y 11. What is the decreasing order of PKa value for given phenolic compounds. OH OH OH OH NO2 NO2 OCH3 I II III IV (1) I > II > III > IV (2) IV > III > II > I (3) IV > III > I > II (4) I > II > IV > III 12. How many of following are essential  Amino acids? (1) Histidine (2) Arginine (3) Lysine (4) Valine (5) Proline (6) Glutamic acid (7) Phenyl alanine (8) Tryptophan (9) Glycine 13. Chemical formula of compound present in in tooth enamel? (1) Ca10(PO4)6F2 (2) Ca8(PO4)4F2 (3) Ca6(PO4)2F2 (4) Ca8(PO4)6F2 14. Consider the following electronic configuration: Cu2+ = [Ar] 3d4 4s0 Cu+ = [Ar] 3d10 4s0 Which option is correct? (1) Cu2+ is more stable in aqueous solution (2) Cu+ is more stable in aqueous solution (3) Cu+ and Cu2+ are equally stable in aqueous solution (4) Depends upon copper salt 15. Predict the basic strength order N (i) (ii) (iii) N N | H (1) iii > i > ii (2) ii > iii > i (3) iii > ii > i (4) ii > i > iii JEE-Main 09-04-2024 Morning Shift www.esaral.com 7  JEE | NEET | Class 8 - 10 Download eSaral App 16. Heat of solution of CuSO4(s) & CuSO4·5H2O(s) is 72 and 12 KJ/mol respectively. Mole of heat of hydration of anhyd. CuSO4 is___ 17. Given E0 = 1.33 V For Cr2O72 + 14H+ + 6e → 2Cr3+ + 7H2O How many of following will oxidise. Fe, Ni, Cr, Cu, Ag, Au (CO+HCl) 18. ⎯⎯⎯⎯⎯⎯ AlCl → Product is? 3 CHO COOH CH3 CH2 Cl (1) (2) (3) (4) 19. Identify the correct product for below given reaction AlCl Zn−Hg Conc.H SO + ⎯⎯⎯⎯ 3 →P ⎯⎯⎯⎯⎯⎯ Conc.HCl/  → Q ⎯⎯⎯⎯⎯⎯  2 4 →R O (1) (2) (3) (4) m  I 20. II c I II (1) Strong electrolyte Weak electrolyte (2) Weak electrolyte Strong electrolyte (3) Strong electrolyte Strong electrolyte (4) Weak electrolyte Weak electrolyte 21. Method of purification of compounds depends on (1) Impurity (2) Nature of compound and impurity (3) Nature of compound (4) Does not depend on impurity JEE-Main 09-04-2024 Morning Shift www.esaral.com 8  JEE | NEET | Class 8 - 10 Download eSaral App P 22. CH3 CH=CH2 Q Product P and Q will be P Q OH OH (1) OH OH (2) OH OH (3) OH OH (4) JEE-Main 09-04-2024 Morning Shift www.esaral.com 9