Practice Questions

Q13.Suppose that the number of terms in an A.P. is 2k, k ∈N . If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to : (1) 6 (2) 5 (3) 8 (4) 4 y+2

Q13.The area of the region, inside the circle (x −2√3)2 + y2 = 12 and outside the parabola y2 = 2√3x is : (1) 3π + 8 (2) 6π −16 (3) 3π −8 (4) 6π −8

Q13.Let L1 : x−11 = y−2−1 = z−12 and L2 : x+1−1 = y−22 = 1z be two lines. Let L3 be a line passing through the point (α, β, γ) and be perpendicular to both L1 and L2 . If L3 intersects L1 , then |5α −11β −8γ| equals : (1) 20 (2) 18 (3) 25 (4) 16

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 the foci of a hyperbola be (1, 14) and (1, −12). If it passes through the point (1, 6), then the length of its latus-rectum is : (1) 24 (2) 25 5 6 (3) 144 (4) 288 5 5 is equal to :

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 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.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. 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. 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

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.If A and B are the points of intersection of the circle x2 + y2 −8x = 0 and the hyperbola x29 −y24 = 1 point P moves on the line 2x −3y + 4 = 0, then the centroid of △PAB lies on the line : (1) x + 9y = 36 (2) 4x −9y = 12 (3) 6x −9y = 20 (4) 9x −9y = 32

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

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.Let the area of a △PQR with vertices P(5, 4), Q(−2, 4) and R(a, b) be 35 square units. If its orthocenter and centroid are O (2, 145 ) and C(c, d) respectively, then c + 2d is equal to (1) 8 (2) 7 3 3 (3) 2 (4) 3 ((loge x)2+1)−1 1 e is

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.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.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.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. 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. 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

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