Practice Questions

Q65.If 20C1 + (22) 20C2 + (32) 20C3+. . . . . +(202) 20C20 = A(2β), then the ordered pair (A, β) is equal to JEE Main 2019 (12 Apr Shift 2) JEE Main Previous Year Paper (1) (380, 19) (2) (420, 18) (3) (420, 19) (4) (380, 18) − 3 ) 6 is equal to x2

Q65.If sin4α + 4cos4β + 2 = 4√2sinαcosβ, α, β ∈[0, π] , then cos(α + β) −cos(α −β) is equal to (1) −1 (2) −√2 (3) √2 (4) 0 JEE Main 2019 (12 Jan Shift 2) JEE Main Previous Year Paper

Q65.The sum of the following series 1 + 6 + 9(12+22+32)7 + 12(12+22+32+42)9 + 15(12+22+…+52)11 +. . . . is: (1) 7520 (2) 7510 (3) 7830 (4) 7820

Q65.The sum ∑ 𝑘 is equal to 𝑘= 1 2𝑘 11 21 (1) 1 – (2) 2 – 220 220 3 11 (3) 2 – (4) 2 – 217 219 6 Q66. 1 1 If the fourth term in the binomial expansion of √𝑥 1 + log10𝑥+ 𝑥 12 is equal to 200, and 𝑥> 1, then the value of 𝑥 is (1) 100 (2) 104 (3) 103 (4) 10

Q67.All the pairs (x, y), that satisfy the inequality 2√sin2x−2sinx+5 ⋅ 1 ≤1 also satisfy the equation: 4sin2y (1) 2 sin x = sin y (2) sin x = 2 sin y (3) |sin x| = |sin y| (4) 2|sin x| = 3 sin y

Q69.The locus of the centres of the circles, which touch the circle, 𝑥2 + 𝑦2 = 1 externally, also touch the 𝑦-axis and lie in the first quadrant, is: (1) 𝑦= √1 + 2𝑥, 𝑥≥0 (2) 𝑦= √1 + 4𝑥, 𝑥≥0 (3) 𝑥= √1 + 2𝑦, 𝑦≥0 (4) 𝑥= √1 + 4𝑦, 𝑦≥0

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

Q69.If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is : (1) (x2 + y2)(x + y) = R2xy (2) (x2 + y2)3 = 4R2x2y2 (3) (x2 + y2) 2 = 4R2x2y2 (4) (x2 + y2) 2 = 4Rx2y2

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

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 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

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.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.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.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. 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. 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.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