Practice Questions

Q14.Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: (1) 10 (2) 7 (3) 8 (4) 9

Q14.Let Tr be the rth term of an A.P. If for some m, Tm = 251 , T25 = 201 , and 20 ∑25r=1 Tr = 13, then 5 m ∑2r=mm Tr is equal to (1) 98 (2) 126 (3) 142 (4) 112

Q14.The perpendicular distance, of the line x−1 2 = −1 = z+32 from the point P(2, −10, 1), is : (1) 6 (2) 5√2 (3) 4√3 (4) 3√5

Q14.If the domain of the function log5 (18x −x2 −77) is (α, β) and the domain of the function is (γ, δ), then α2 + β2 + γ 2 is equal to : log(x−1) ( 2x2+3x−2x2−3x−4 ) (1) 195 (2) 179 (3) 186 (4) 174

Q14.The number of complex numbers z , satisfying |z| = 1 and z¯z + ¯zz = 1, is : (1) 4 (2) 8 (3) 10 (4) 6 Q15. ⎡ 0 ⎤ ⎡ 0 ⎤ ⎡4⎤ ⎡0⎤ ⎡2 ⎤ ⎡1 ⎤ Let A = [aij] be 3 × 3 matrix such that A 1 = 0 , A 1 = 1 and A 1 = 0 , then a23 equals : ⎣ 0 ⎦ ⎣ 1 ⎦ ⎣3⎦ ⎣0⎦ ⎣2 ⎦ ⎣0 ⎦ (1) -1 (2) 2 (3) 1 (4) 0 2 x sin 2 dx equals : 3 3

Q15. In an arithmetic progression, if S40 = 1030 and S12 = 57, then S30 −S10 is equal to : (1) 525 (2) 510 (3) 515 (4) 505

Q15.Let ABC be a triangle formed by the lines 7x −6y + 3 = 0, x + 2y −31 = 0 and 9x −2y −19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y −53 = 0. Then h2 + k2 + hk is equal to: (1) 47 (2) 37 (3) 36 (4) 40 is:

Q15.If ∑nr=1 Tr = (2n−1)(2n+1)(2n+3)(2n+5)64 , then limn→∞∑nr=1 ( Tr1 ) (1) 0 (2) 23 (3) 1 (4) 13

Q15. x + y + 2z = 6 If the system of linear equations : 2x + 3y + az = a + 1 where a, b ∈R, has infinitely many solutions, then −x −3y + bz = 2 b 7a + 3b is equal to : (1) 16 (2) 12 (3) 22 (4) 9 = 0, y ∈(−π2 , π2 ) with

Q15.Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C , whose mid-point is (1, 2), is : (1) √3 (2) 2√2 (3) 2√3 (4) 4√2

Q15. A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8 , and B wins if he throws a sum of 8 before A throws a sum of 5 . The probability, that A wins if A makes the first throw, is (1) 8 (2) 9 17 19 (3) 9 (4) 8 17 19

Q15.Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If x denote the number of defective oranges, then the variance of x is (1) 28/75 (2) 18/25 (3) 26/75 (4) 14/25 x > 0 and f(2) = 3. Then f(6) is equal to

Q15.If f(x) = ∫ 1 dx, f(0) = −6, then f(1) is equal to : x1/4(1+x1/4) (1) 4 (loge 2 −2) (2) 2 −loge2 2 (3) loge 2 + 2 (4) 4 (loge 2 + 2)

Q16.The value of ∫e4e2 x ( e((loge x)2+1)−1 +e((6−loge x)2+1)−1 )dx (1) 2 (2) loge 2 (3) 1 (4) e2 2025 (23 Jan Shift 1) JEE Main Previous Year Paper

Q16.Suppose A and B are the coefficients of 30th and 12th terms respectively in the binomial expansion of (1 + x)2n−1 . If 2 A = 5 B , then n is equal to : (1) 22 (2) 20 (3) 21 (4) 19

Q16.If x = f(y) is the solution of the differential equation (1 + y2) + (x −2etan−1 y) dydx is equal to : f(0) = 1, then f ( √31 ) (1) eπ/12 (2) eπ/4 (3) eπ/3 (4) eπ/6

Q16.Let for some function y = f(x), ∫x0 tf(t)dt = x2f(x), (1) 1 (2) 3 (3) 6 (4) 2 π dx = π (απ2 + β), α, β ∈Z , then (α + β)2 equals

Q16.If I = ∫ 0π 3 dx, then ∫210 sin4x sinx+cos4x cos xx 2 2 x sin x+cos (1) π2 (2) π2 12 4 (3) π2 (4) π2 16 8 ∣∣ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2025 (23 Jan Shift 2) JEE Main Previous Year Paper

Q16.A coin is tossed three times. Let X denote the number of times a tail follows a head. If μ and σ2 denote the mean and variance of X , then the value of 64 (μ + σ2) is : (1) 51 (2) 64 (3) 32 (4) 48

Q16.The area of the region bounded by the curves x (1 + y2) = 1 and y2 = 2x is: (1) 2 ( π2 −13 ) (2) π2 −13 (3) π 4 −13 (4) 12 ( π2 −13 )

Q16.Let a straight line L pass through the point P(2, −1, 3) and be perpendicular to the lines x−12 = y+11 = z−3−2 and x−3 1 = y−23 = z+24 . If the line L intersects the yz -plane at the point Q , then the distance between the points P and Q is : (1) √10 (2) 2√3 (3) 2 (4) 3

Q16.Let f(x) = 2x+2+16 . Then the value of 8 (f ( 151 ) + f ( 152 ) + … + f ( 5915 )) is equal to 22x+1+2x+4+32 (1) 92 (2) 118 (3) 102 (4) 108 + + (1 + x2)dy = 0, y(0) = 0.

Q16.The value of limn→∞(∑nk=1 k3+6k2+11k+5(k+3)! ) (1) 4/3 (2) 2 (3) 7/3 (4) 5/3

Q17.If ∫ −π2 2 96x2(1+ex)cos2 x (1) 64 (2) 196 (3) 144 (4) 100

Q17.Let y = y(x) be the solution of the differential equation (xy −5x2√1 x2)dx Then y(√3) is equal to (1) √152 (2) 5√32 (3) 2√2 (4) √143 is: