Practice Questions

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

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

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 f : R −{0} →R be a function such that f(x) −6f ( x1 ) = 3x35 −52 . If the limx→0 ( αx1 + f(x)) = β; α, β ∈R, then α + 2β is equal to (1) 5 (2) 3 (3) 4 (4) 6 n > 0, then I(9, 14) + I(10, 13) is

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.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

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)

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

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.The square of the distance of the point ( 157 , 327 , 7) from the line x+13 = y+35 = z+57 in the direction of the vector ^i + 4^j + 7^k is : (1) 54 (2) 44 (3) 41 (4) 66 y2

Q17.Let αθ and βθ be the distinct roots of 2x2 + (cos θ)x −1 = 0, θ ∈(0, 2π). If m and M are the minimum and the maximum values of α4θ + β4θ , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27

Q18.A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (α, β), then 3β −2α is equal to : (1) 10 (2) 15 (3) 12 (4) 14

Q18.Let y = y(x) be the solution of the differential equation cos x(loge(cos x))2dy + (sin x −3y sin x loge(cos x))dx = 0, x ∈(0, π2 ). If y ( π4 ) = loge−1 2 , then y ( π6 ) is equal to : (1) 1 (2) 2 loge(3)−loge(4) loge(3)−loge(4) (3) 1 (4) − 1 loge(4)−loge(3) loge(4)

Q18.If the midpoint of a chord of the ellipse x2 α 9 + 4 = 1 is (√2, 4/3), and the length of the chord is 2√α3 , then is : (1) 20 (2) 22 (3) 18 (4) 26 2025 (28 Jan Shift 2) JEE Main Previous Year Paper

Q18.Let α, β(α ≠β) be the values of m , for which the equations x + y + z = 1; x + 2y + 4z = m and x + 4y + 10z = m2 have infinitely many solutions. Then the value of ∑10n=1 (nα + nβ) is equal to : (1) 3080 (2) 560 (3) 3410 (4) 440

Q19.Let I(x) = ∫ 11 15 . If I(37) −I(24) = 4 1 − 1 b, c ∈N (x−11) 13 (x+15) 13 ( b 13 c 13 ), (1) 22 (2) 39 (3) 40 (4) 26

Q19.Consider the region R = {(x, y) : x ≤y ≤9 −113 x2, x ≥0}. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is: (1) 730 (2) 625 119 111 (3) 821 (4) 567 123 121

Q19.Let the curve z(1 + i) + ¯z(1 −i) = 4, z ∈C, divide the region |z −3| ≤1 into two parts of areas α and β . Then |α −β| equals : (1) 1 + π2 (2) 1 + π3 (3) 1 + π6 (4) 1 + π4

Q19.If in the expansion of (1 + x)p(1 −x)q , the coefficients of x and x2 are 1 and -2 , respectively, then p2 + q2 is equal to : (1) 18 (2) 13 (3) 8 (4) 20 a

Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

Q20.Let z1, z2 and z3 be three complex numbers on the circle |z| = 1 with arg (z1) = −π4 , arg (z2) = 0 and arg (z3) = π4 . If |z1¯z2 + z2¯z3 + z3¯z1|2 = α + β√2, α, β ∈Z, then the value of α2 + β2 is : (1) 24 (2) 29 (3) 41 (4) 31

Q20.Two equal sides of an isosceles triangle are along −x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is : (1) −2√10 (2) 12 (3) 6 (4) −6

Q20.Let E : x2 + y2 = 1, a > b and H : x2 − y2 = 1. Let the distance between the foci of E and the foci of H a2 b2 A2 B2 be 2√3. If a −A = 2, and the ratio of the eccentricities of E and H is 13 , then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7 = α × 229 , then α is equal to ______

Q20.Let the area of the region {(x, y) : 2y ≤x2 + 3, y + |x| ≤3, y ⩾|x −1|} be A. Then 6 A is equal to : (1) 16 (2) 12 (3) 14 (4) 18