Practice Questions

Q64.Let a line L pass through the point of intersection of the lines bx + 10y −8 = 0 and 2x −3y = 0, b ∈R −{ 34 }. If the line L also passes through the point (1, 1) and touches the circle 17(x2 + y2) = 16, then x2 y2 the eccentricity of the ellipse 5 + b2 = 1 is (1) 2 (2) √5 √35 (3) 1 (4) √5 √25

Q64.Let C be a circle passing through the points A(2, −1) and B(3, 4). The line segment AB is not a diameter of C . If r is the radius of C and its centre lies on the circle (x −5)2 + (y −1)2 = 132 , then r2 is equal to (1) 32 (2) 652 (3) 61 (4) 30 2

Q64.The remainder when 72022 + 32022 is divided by 5 is (1) 0 (2) 2 (3) 3 (4) 4

Q64.The value of 2 sin 22π sin 3π22 sin 5π22 sin 7π22 sin 9π22 is equal to: (1) 1 (2) 5 16 16 (3) 7 (4) 3 16 16 JEE Main 2022 (25 Jul Shift 2) JEE Main Previous Year Paper

Q64.The distance between the two points A and A′ which lie on y = 2 such that both the line segments AB and A′B (where B is the point (2, 3)) subtend angle π4 at the origin, is equal to (1) 10 (2) 485 (3) 52 (4) 3 5

Q64.Let A(1, 1), B(−4, 3), C(−2, −5) be vertices of a triangle ABC, P be a point on side BC , and Δ1 and Δ2 be the areas of triangle APB and ABC . Respectively. If Δ1 : Δ2 = 4 : 7 , then the area enclosed by the lines AP, AC and the x -axis is (1) 1 (2) 3 4 4 (3) 1 (4) 1 2

Q64.If the tangents drawn at the point O(0, 0) and P(1 + √5, 2) on the circle x2 the point Q, then the area of the triangle OPQ is equal to (1) 3+√5 (2) 4+2√5 2 2 (3) 5+3√5 (4) 7+3√5 2 2

Q64.A point P moves so that the sum of squares of its distances from the points (1, 2) and (−2, 1) is 14 . Let f(x, y) = 0 be the locus of P , which intersects the x-axis at the points A, B and the y-axis at the point C, D. Then the area of the quadrilateral ACBD is equal to (1) 9 (2) 3√17 2 2 (3) 3√17 (4) 9 4

Q64.In an isosceles triangle ABC , the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4 . Let the point B lie on the line x + 3y = 7. If (α, β) is the centroid ΔABC , then 15(α + β) is equal to (1) 51 (2) 39 (3) 41 (4) 49 y2

Q64.Let n ≥5 be an integer. If 9n −8n −1 = 64α and 6n −5n −1 = 25 β, then α −β is equal to: (1) 1 + nC2(8 −5) + nC3(82 −52) + … + nCn(8n−1(2)−5n−2)1 + nC3(8 −5) + nC4(82 −52) + … + nCn(8n−2 −5n−2 (3) nC3(8 −5) + nC4(82 −52) + … + nCn(8n−2 −5n−2)(4) nC4(8 −5) + nC5(82 −52) + … + nCn(8n−3 −5n−3)

Q64.Let S = {θ ∈(0, π2 ) : ∑9m=1 sec(θ + (m −1) π6 ) sec(θ + mπ6 ) = −8√3 }. Then (1) S = { 12π } (2) S = { 2π3 } (3) ∑θ∈S θ = π2 (4) ∑θ∈S θ = 3π4

Q64.Let the abscissae of the two points 𝑃 and 𝑄 on a circle be the roots of 𝑥2 - 4𝑥- 6 = 0 and the ordinates of 𝑃 and 𝑄 be the roots of 𝑦2 + 2𝑦- 7 = 0. If 𝑃𝑄 is a diameter of the circle 𝑥2 + 𝑦2 + 2𝑎𝑥+ 2𝑏𝑦+ 𝑐= 0, then the value of 𝑎+ 𝑏- 𝑐 is (1) 12 (2) 13 (3) 14 (4) 16 JEE Main 2022 (26 Jul Shift 2) JEE Main Previous Year Paper

Q64.Let the hyperbola H : x2 −y2 = 1 pass through the point . A parabola is drawn whose focus is a2 b2 (2√2, −2√2) same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H . If the length of the latus rectum of the parabola is e times the length of the latus rectum of H , where e is the eccentricity of H , then which of the following points lies on the parabola? (1) (2√3, 3√2) (2) (3√3, −6√2) (3) (√3, −√6) (4) (3√6, 6√2)

Q64.The sum 1 + 2 · 3 + 3 · 32 + … … . . + 10 · 39 is equal to JEE Main 2022 (25 Jun Shift 2) JEE Main Previous Year Paper (1) 2 · 312 + 10 (2) 19 · 310 + 1 4 4 (3) 5 · 310 - 2 (4) 9 · 310 + 1 2

Q64.The locus of the mid-point of the line segment joining the point (4, 3) and the points on the ellipse x2 + 2y2 = 4 is an ellipse with eccentricity (1) √3 (2) 1 2 2√2 (3) 1 (4) 1 √2 2

Q64.A line, with the slope greater than one, passes through the point 𝐴4, 3 and intersects the line 𝑥- 𝑦- 2 = 0 at the point 𝐵. If the length of the line segment 𝐴𝐵 is √29 , then 𝐵 also lies on the line 3 (1) 2𝑥+ 𝑦= 9 (2) 3𝑥- 2𝑦= 7 (3) 𝑥+ 2𝑦= 6 (4) 2𝑥- 3𝑦= 3

Q64.Let the area of the triangle with vertices A(1, α), B(α, 0) and C(0, α) be 4 sq. units. If the points (α, −α), (−α, α) and (α2, β) are collinear, then β is equal to (1) 64 (2) −8 (3) −64 (4) 512

Q64.If 𝑦= 𝑚1𝑥+ 𝑐1 and 𝑦= 𝑚2𝑥+ 𝑐2, 𝑚1 ≠𝑚2 are two common tangents of circle 𝑥2 + 𝑦2 = 2 and parabola 𝑦2 = 𝑥, then the value of 8 𝑚1 𝑚2 is equal to (1) 3√2 - 4 (2) 6√2 - 4 (3) -5 + 6√2 (4) 3 + 4√2

Q64.The term independent of x in the expression of (1 −x2 + 11 5x2 1 ) 3x3)( 25 x3 (1) 7 (2) 33 40 200 (3) 39 (4) 11 200 50

Q64.If a1, a2, a3 … and b1, b2, b3 … . are A.P. and a1 = 2, a10 = 3, a1b1 = 1 = a10b10 then a4b4 is equal to (1) 28 (2) 28 27 24 (3) 23 (4) 22 26 23 Q65. α = sin 36° is a root of which of the following equation (1) 16x4 −20x2 + 5 = 0 (2) 16x4 + 20x2 + 5 = 0 (3) 10x4 −10x2 −5 = 0 (4) 16x4 −10x2 + 5 = 0

Q65.Let the focal chord of the parabola P : y2 = 4x along the line L : y = mx + c, m > 0 meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola H : x2 −y2 = 4. If O is the vertex of P and F is the focus of H on the positive x-axis, then the area of the quadrilateral OMFN is (1) 2√6 (2) 2√14 (3) 4√6 (4) 4√14 αex+βe−x+γ sin x 2

Q65.Let the normal at the point P on the parabola y2 = 6x pass through the point (5, −8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is (1) −9 (2) 9 4 4 (3) −5 (4) −3 2

Q65.Let the eccentricity of an ellipse x2 + = 1, a > b, be 14 . If this ellipse passes through the point a2 b2 5 , , then a2 + b2 is equal to (−4√2 3) (1) 29 (2) 31 (3) 32 (4) 34 a is equal to

Q65.The number of elements in the set 𝑥2 + 𝑥 𝑆= 𝑥∈ℝ: 2cos = 4𝑥+ 4-𝑥 is 6 (1) 1 (2) 3 (3) 0 (4) infinite JEE Main 2022 (29 Jul Shift 2) JEE Main Previous Year Paper

Q65.Let S = {θ ∈[−π, π] −{± π2 } : sin θ tan θ + tan θ = sin 2θ}. If T = ∑θ∈S cos 2θ, then T + n(S) is equal to (1) 7 + √3 (2) 5 (3) 8 + √3 (4) 9