Practice Questions

Q10.Let the ellipse E1 : x2a2 + y2b2 A2 √3 product of their lengths of latus rectums be 32 , and the distance between the foci of E1 be 4. If E1 and E2 √3 meet at A, B, C and D , then the area of the quadrilateral ABCD equals : (1) 12√6 (2) 6√6 5 (3) 18√6 (4) 24√6 5 5

Q10.Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC , divides the arc −−−length of arc AB 1 → → → AC such that length of arc BC = 5 , and OC = αOA + βOB , then α + √2(√3 −1)β is equal to (1) 2√3 (2) 2 −√3 (3) 5√3 (4) 2 + √3 f ∘g is

Q11.If limx→∞(( 1−e ) ( e − 1+x )) = α, then the value of 1+loge α equals : (1) e−1 (2) e2 (3) e−2 (4) e

Q11. Let f : R →R be a twice differentiable function such that f(2) = 1. If F(x) = xf(x) for all x ∈R, ∫20 x F′(x)dx = 6 and ∫20 x2 F′′(x)dx = 40, then F′(2) + ∫20 F(x)dx is equal to : (1) 11 (2) 13 (3) 15 (4) 9 507S2025 is :

Q11.Let the range of the function f(x) = 6 + 16 cos x ⋅cos ( π3 −x) ⋅cos ( π3 + x) ⋅sin 3x ⋅cos 6x, x ∈R be [α, β] . Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is : (1) 11 (2) 8 (3) 10 (4) 9 sin y > 0 and x(1) = π2 . Then

Q11.Let the position vectors of three vertices of a triangle be 4→p + →q −3→r, −5→p + →q + 2→r and 2→p−→q+ 2→r. If the →p+→q+→r position vectors of the orthocenter and the circumcenter of the triangle are and α→p + β→q + γ→r 4 respectively, then α + 2β + 5γ is equal to : (1) 3 (2) 4 (3) 1 (4) 6 → →

Q12.Let f : R →R be a twice differentiable function such that f(x + y) = f(x)f(y) for all x, y ∈R. If f ′(0) = 4a and f satisfies f ′′(x) −3af ′(x) −f(x) = 0, a > 0, then the area of the region R = {(x, y) ∣0 ≤y ≤f(ax), 0 ≤x ≤2} is: (1) e2 −1 (2) e2 + 1 (3) e4 + 1 (4) e4 −1

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

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.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.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

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.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.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 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

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.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.Let (2, 3) be the largest open interval in which the function f(x) = 2 loge(x −2) −x2 + ax + 1 is strictly increasing and (b, c) be the largest open interval, in which the function g(x) = (x −1)3(x + 2 −a)2 is strictly decreasing. Then 100(a + b −c) is equal to : (1) 420 (2) 360 (3) 160 (4) 280

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

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

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

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