Practice Questions

Q13.The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is (1) 36000 (2) 37000 (3) 34000 (4) 35000

Q13.The sum, of the squares of all the roots of the equation x2 + |2x −3| −4 = 0, is (1) 3(3 −√2) (2) 6(3 −√2) (3) 6(2 −√2) (4) 3(2 −√2)

Q13.Let f : R −{0} →(−∞, 1) be a polynomial of degree 2, satisfying f(x)f ( x1 ) = f(x) + f ( x1 ). If f(K) = −2K , then the sum of squares of all possible values of K is : (1) 7 (2) 6 (3) 1 (4) 9 and a

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.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.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.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. 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.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.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 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.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

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. 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 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. 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. 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.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)

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

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