Practice Questions

Q69.If a tangent to the circle x2 + y2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is: (1) x2 + y2–16x2y2 = 0 (2) x2 + y2–4x2y2 = 0 (3) x2 + y2–2xy = 0 (4) x2 + y2–2x2y2 = 0

Q70.Let 𝑂0,0 and 𝐴0,1 be two fixed points. Then, the locus of a point 𝑃 such that the perimeter of 𝛥𝐴𝑂𝑃 is 4 is (1) 8𝑥2 + 9𝑦2 - 9𝑦= 18 (2) 9𝑥2 - 8𝑦2 + 8𝑦= 16 (3) 8𝑥2 - 9𝑦2 + 9𝑦= 18 (4) 9𝑥2 + 8𝑦2 - 8𝑦= 16

Q70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1

Q70.If the line 𝑎𝑥+ 𝑦= 𝑐, touches both the curves 𝑥2 + 𝑦2 = 1 and 𝑦2 = 4√2𝑥, then 𝑐 is equal to: 1 (1) (2) √2 2 (3) 1 (4) 2 √2

Q70.Let the equations of two sides of a triangle be 3x −2y + 6 = 0 and 4x + 5y −20 = 0. If the orthocenter of this triangle is at (1, 1) then the equation of it's third side is: (1) 122y + 26x + 1675 = 0 (2) 26x −122y −1675 = 0 (3) 26x + 61y + 1675 = 0 (4) 122y −26x −1675 = 0

Q70.If the normal to the ellipse 3𝑥2 + 4𝑦2 = 12 at a point 𝑃 on it is parallel to the line, 2𝑥+ 𝑦= 4 and the tangent to the ellipse at 𝑃 passes through 𝑄( 4,4 ) then 𝑃𝑄 is equal to: (1) √61 (2) 5√5 2 2 (3) √157 (4) √221 2 2

Q71.If the parabolas y2 = 4b(x −c) and y2 = 8ax have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c) (1) (1, 1, 3) (2) ( 12 , 2, 0) (3) ( 12 , 2, 3) (4) All of above

Q71.Let 𝑃 be the point of intersection of the common tangents to the parabola 𝑦2 = 12𝑥 and the hyperbola 8𝑥2 - 𝑦2 = 8. If 𝑆 and 𝑆' denote the foci of the hyperbola where 𝑆 lies on the positive 𝑥-axis then 𝑃 divides 𝑆𝑆' in a ratio: (1) 5: 4 (2) 2: 1 (3) 13: 11 (4) 14: 13

Q71.If a variable line 3x + 4y −λ = 0 is such that the two circles x2 + y2 −2x −2y + 1 = 0 and x2 + y2 −18x −2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval : (1) [13, 23] (2) (23, 31) (3) [12, 21] (4) (2, 17)

Q71.The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2) and the x -axis is (1) 8π(3 −2√2) (2) 8π(2 −√2) + (3) 4π(3 √2) (4) 4π(2 −√2)

Q72.Let A(4, −4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of ΔACB is maximum. Then, the area (in sq. units) of ΔACB , is: (1) 32 (2) 31 34 (3) 30 12 (4) 31 14

Q72.Let P(4, −4) and Q(9, 6) be two points on the parabola, y2 = 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of ΔPXQ is maximum. Then this maximum area (in sq. units) is : (1) 625 (2) 75 4 2 (3) 125 (4) 125 4 2

Q72.If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the coordinate axes lie on the curve : y2 (1) 1 + 1 = 1 (2) x2 4x2 2y2 4 + 2 = 1 y2 (3) 1 + 1 = 1 (4) x2 2x2 4y2 2 + 4 = 1

Q73.The equation of a common tangent to the curves, y2 = 16x and xy = −4, is: (1) x −2y + 16 = 0 (2) x −y + 4 = 0 (3) 2x −y + 2 = 0 (4) x + y + 4 = 0 JEE Main 2019 (12 Apr Shift 2) JEE Main Previous Year Paper

Q74.Let [x] denote the greatest integer less than or equal to X . Then : limx→0 tan(π sin2 x)+(|x|−sin(x[x]))2x2 (1) does not exist (2) equals π (3) equals π + 1 (4) equals 0

Q75.Let 𝐴= cos𝛼-sin𝛼 𝑎∈𝑅 such that 𝐴32 = 0 -1 . Then, a value of 𝛼 is: sin𝛼 cos𝛼, 1 0 (1) 0 (2) 𝜋 (3) 𝜋 (4) 𝜋 16 64 32 JEE Main 2019 (08 Apr Shift 1) JEE Main Previous Year Paper

Q75. y + 1 α β Let α and β be the roots of the equation x2 + x + 1 = 0. Then for y ≠0 in R, α y + β 1 is equal β 1 y + α to (1) y3 (2) y(y2–1) (3) y3–1 (4) y(y2–3)

Q75. ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at, A and B are cot−1(3√2) and cosec−1(2√2) respectively, then the height of the tower (in metres) is (1) 100 (2) 20 3√3 (3) 25 (4) 10√5

Q75.The number of values of θ ∈(0, π) for which the system of linear equations x + 3y + 7z = 0 −x + 4y + 7z = 0 (sin 3θ)x + (cos 2θ)y + 2z = 0 has a non-trivial solution, is: (1) Two (2) Three (3) Four (4) One

Q76.Let a1, a2, a3 … , a10 be in G. P. with ai > 0 for i = 1, 2, … , 10 and S be the set of pairs (r, k), r, k ∈N (the set of natural numbers) for which JEE Main 2019 (10 Jan Shift 2) JEE Main Previous Year Paper loge ar1 ak2 loge ar2ak3 loge ar3ak4 loge ar4 ak5 loge ar5ak6 loge ar6ak7 = 0 loge ar7ak8 loge ar8ak9 loge ar9ak10 Then the number of elements in S, is: (1) Infinitely many (2) 4 (3) 10 (4) 2

Q76.If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is: (1) 3: 4: 5 (2) 5: 6: 7 (3) 5: 9: 13 (4) 4: 5: 6 Q77. 1 1 1 Let the numbers 2, 𝑏, 𝑐 be in an A.P. and 𝐴= 2 𝑏 𝑐 . If det ( 𝐴) ∈[2,16], then 𝑐 lies in the interval: 4 𝑏2 𝑐2 (1) 2,3 (2) 4,6 3 (3) 3,2 + 2 4 (4) 2 + 2 34, 4

Q77.The value of cot(∑19n=1 cot−1(1 + ∑np=1 2p)) is: (1) 21 (2) 19 19 21 (3) 2223 (4) 2223

Q78.The number of functions f from {1, 2, 3, … , 20} onto {1, 2, 3, … , 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4 is: (1) 65 × (15)! (2) 5! × 6! (3) (15)! × 6! (4) 56 × 15

Q79.If [x] denotes the greatest integer ≤x, then the system of linear equations [sinθ]x + [−cosθ]y = 0, [cotθ]x + y = 0 (1) has a unique solution if θ ∈( π2 , 2π3 ) ∪(π, 7π6 ) (2) have infinitely many solution if θ ∈( π2 , 2π3 ) ∪(π, 7π6 ) (3) has a unique if θ ∈( π2 , 2π3 ) and have infinitely (4) have infinitely many solutions if θ ∈( π2 , 2π3 ) many solutions if θ ∈(π, 7π6 ) and has a unique solution if θ ∈(π, 7π6 )

Q80.If f(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set S = {x ∈R : f(x) = f(0)} contains exactly (1) Two irrational and two rational numbers (2) Four rational numbers (3) Two irrational and one rational number (4) Four irrational numbers