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. IfI(m, n) = ∫10 xm−1(1 −x)n−1dx, m, (1) I(19, 27) (2) I(9, 1) (3) I(1, 13) (4) I(9, 13)

Q14.Let M and m respectively be the maximum and the minimum values of 1 + sin2 x cos2 x 4 sin 4x f(x) = sin2 x 1 + cos2 x 4 sin 4x , x ∈R Then M 4 −m4 is equal to : sin2 x cos2 x 1 + 4 sin 4x (1) 1280 (2) 1295 (3) 1215 (4) 1040

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

Q14. The function f : (−∞, ∞) →(−∞, 1), defined by f(x) = 2x−2−x2x+2−x is : (1) Neither one-one nor onto (2) Onto but not one-one (3) Both one-one and onto (4) One-one but not onto

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)

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.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.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.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. 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. 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

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

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.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

Q17.The number of non-empty equivalence relations on the set {1, 2, 3} is : (1) 6 (2) 5 (3) 7 (4) 4

Q17.A board has 16 squares as shown in the figure: Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is : (1) 7/10 (2) 4/5 (3) 23/30 (4) 3/5