Practice Questions

Q67. lim 𝑒2sin𝑥- 2sin𝑥- 1 𝑥→0 𝑥2 (1) is equal to -1 (2) does not exist (3) is equal to 1 (4) is equal to 2

Q67.If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is : (1) √5 (2) √3 3 2 (3) 1 (4) 2 √3 √5 π 1 x ∫x0 f(t)dt lim = α, then 8α2 is equal

Q69.Let M denote the median of the following frequency distribution. Class 0 −4 4 −8 8 −12 12 −16 16 −20 Frequency 3 9 10 8 6 Then 20M is equal to : (1) 416 (2) 104 (3) 52 (4) 208 Q70. 2 cos4 x 2 sin4 x 3 + sin2 2x If f(x) = 3 + 2 cos4 x 2 sin4 x sin2 2x then 15 f ′(0) is equal to ________. 2 cos4 x 3 + 2 sin4 x sin2 2x JEE Main 2024 (30 Jan Shift 1) JEE Main Previous Year Paper (1) 0 (2) 1 (3) 2 (4) 6

Q70.Considering only the principal values of inverse trigonometric functions, the number of positive real values of 𝑥 satisfying tan-1 (x) + tan-1 (2x) = π is : 4 (1) More than 2 (2) 1 (3) 2 (4) 0

Q73.Let 𝑓𝑥= 2𝑥2 + 5𝑥- 3, 𝑥∈𝑅. If 𝑚 and 𝑛 denote the number of points where 𝑓 is not continuous and not differentiable respectively, then 𝑚+ 𝑛 is equal to: (1) 5 (2) 2 (3) 0 (4) 3

Q78.Let →𝑎 and →𝑏 be two vectors such that | →𝑏| = 1 and | →𝑏× →𝑎| = 2 Then |( →𝑏× →𝑎) - →𝑏| (1) 3 (2) 5 (3) 1 (4) 4

Q79.Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is 2 4 (1) (2) 25 25 (3) 2 (4) 4 3 75

Q79.Let P(3, 2, 3), Q(4, 6, 2) and R(7, 3, 2) be the vertices of Δ PQR. Then, the angle ∠QPR is (1) π 6 (2) cos−1( 187 ) (3) cos−1( 181 ) (4) π3

Q80.Let Ajay will not appear in JEE exam with probability 𝑝= 2 while both Ajay and Vijay will appear in the 7, exam with probability 𝑞= 15. Then the probability, that Ajay will appear in the exam and Vijay will not appear is: 9 18 (1) (2) 35 35 (3) 24 (4) 3 35 35

Q80.A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is- 2 1 (1) (2) 9 9 (3) 2 (4) 1 27 27

Q1. A particle starts with an initial velocity of 10. 0 ms−1 along x-direction and accelerates uniformly at the rate of 2. 0 m s−2 . The time taken by the particle to reach the velocity of 60. 0 m s−1 is _____. (1) 25 s (2) 3 s (3) 6 s (4) 30 s

Q1. Match List I with List II : List-I (Physical Quantity) List-II (Dimensional Formula) A Pressure gradient I [M0L2T−2] B Energy density II [M1L−1T−2] C Electric Field III [M1L−2T−2] D Latent heat IV [M1L1T−3A−1] Choose the correct answer from the options given below: (1) A-III, B-II, C-I, D-IV (2) A-II, B-III, C-IV, D-I (3) A-III, B-II, C-IV, D-I (4) A-II, B-III, C-I, D-IV

Q1. A person travels 𝑥 distance with velocity 𝑣1 and then 𝑥 distance with velocity 𝑣2 in the same direction. The average velocity of the person is 𝑣, then the relation between 𝑣, 𝑣1 and 𝑣2 will be 𝑣1 + 𝑣2 1 1 1 (1) 𝑣= (2) 𝑣= 𝑣1 + 𝑣2 2 2 1 1 (3) 𝑣= 𝑣1 + 𝑣2 (4) 𝑣= 𝑣1 + 𝑣2

Q1. A vector in x −y plane makes an angle of 30o with y -axis. The magnitude of y -component of vector is 2√3. The magnitude of x-component of the vector will be : (1) 1 (2) 6 √3 (3) 2 (4) √3

Q1. When vector A = 2ˆi + 3ˆj + 2ˆk is subtracted from vector B, it gives a vector equal to 2ˆj. Then the magnitude of → vector B will be: (1) √5 (2) 3 (3) √6 (4) √33

Q1. If two vectors P = ˆi + 2mˆj + mˆk and Q = 4ˆi −2ˆj + mˆk are perpendicular to each other. Then, the value of m will be (1) −1 (2) 2 (3) 3 (4) 1

Q1. Electric field in a certain region is given by →𝐸= 𝐴 ^i + 𝐵 ^j. The SI unit of 𝐴 and 𝐵 are : 𝑥2 𝑦3 (1) N m3 C-1; N m2 C-1 (2) N m2 C-1; N m3 C-1 (3) N m3 C; N m2 C (4) N m2 C; N m3 C

Q1. Match List I with List II List I List II A Torque I kg m–1 s–2 B Energy density II kg m s–1 C Pressure gradient III kg m–2 s–2 D Impulse IV kg m2 s–2 Choose the correct answer from the options given below : (1) A-IV, B-III, C-I, D-II (2) A-I, B-IV, C-III, D-II (3) A-IV, B-I, C-II, D-III (4) A-IV, B-I, C-III, D-II

Q1. If 𝑅, 𝑋𝐿 and 𝑋𝐶 represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless: 𝑅 (1) 𝑅 𝑋𝐿 𝑋𝐶 (2) √𝑋𝐿𝑋𝑐 𝑅 𝑋𝐿 (3) (4) 𝑅 𝑋𝐿𝑋𝑐 𝑋𝑐

Q1. Match List I with List II List List II A Young's Modulus (Y ) I [ML–1 T –1] B Co-efficient of Viscosity (η) II [ML2 T –1] C Planck's Constant (h) III [ML–1 T –2] D Work Function (ϕ) IV [ML2 T–2] Choose the correct answer from the options given below: (1) A-II, B-III, C-IV, D-I (2) A-III, B-I, C-II, D-IV (3) A-I, B-III, C-IV, D-II (4) A-I, B-II, C-III, D-IV

Q2. Match List I with List II List - I List - II A Surface tension I. kg m−1 s−1 B Pressure II. kg m s−1 C Viscosity III. kg m−1 s−2 D Impulse IV. kg s−2 Choose the correct answer from the options given below: (1) A-IV, B-III, C-II, D-I (2) A-IV, B-III, C-I, D-II (3) A-III, B-IV, C-I, D-II (4) A-II, B-I, C-III, D-IV

Q2. A vehicle travels 4 km with speed of 3 km h−1 and another 4 km with speed of 5 km h−1 , then its average speed is : (1) 4. 25 km h−1 (2) 3. 50 km h−1 (3) 4. 00 km h−1 (4) 3. 75 km h−1

Q2. Match List I with List II List I List II A Angular momentum I [ML2 T−2] B Torque II [ML−2 T−2] C Stress III [ML2 T−1] D Pressure gradient IV [ML−1 T−2] Choose the correct answer from the options given below : (1) A-I, B-IV, C-III, D-II (2) A-III, B-I, C-IV, D-II (3) A-II, B-III, C-IV, D-I (4) A-IV, B-II, C-I, D-III

Q2. The equation of a circle is given by x2 + y2 = a2 , where a is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation : (x −At)2 + (y − Bt ) 2 = a2 The dimensions of t is given as [T−1] (1) A = [L–1T], B = [LT–1] (2) A = [LT], B = [L–1T –1] (3) A = [L–1T –1 ], B = [LT–1] (4) A = [L–1T –1], B = [LT]

Q2. The distance travelled by a particle is related to time t as x = 4t2 . The velocity of the particle at t = 5 s is (1) 40 m s–1 (2) 25 m s–1 (3) 20 m s–1 (4) 8 m s–1