Practice Questions

Q65.Let an ellipse 𝐸: 𝑥2 + 𝑦2 = 1, 𝑎2 > 𝑏2, passes through 3 1 and has eccentricity 1 If a circle, centered at √ 2, √3. 𝑎2 𝑏2 2 focus 𝐹( 𝛼, 0 ) , 𝛼> 0, of 𝐸 and radius √3, intersects 𝐸 at two points 𝑃 and 𝑄, then 𝑃𝑄2 is equal to : (1) 8 (2) 4 3 3 16 (3) (4) 3 3

Q65.The point P(a, b) undergoes the following three transformations successively: (a) reflection about the line y = x. (b) translation through 2 units along the positive direction of x− axis. (c) rotation through angle π4 about the origin in the anti-clockwise direction. , 2a + b is equal to: 7 ), then the value of If the co-ordinates of the final position of the point P are (−1√2 √2 (1) 13 (2) 9 (3) 5 (4) 7

Q65.Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is : (1) x −y = 1 (2) 2x + y = 5 (3) x + 3y = 5 (4) x + 2y = 4 = 1 and the circle x2 + y2 = 4 b, b > 4 lie on the curve

Q65.Let S1 : x2 + y2 = 9 and S2 : (x −2)2 + y2 = 1 . JEE Main 2021 (18 Mar Shift 2) JEE Main Previous Year Paper Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points : (1) (0, ±√3) (2) ( 12 , ± √52 ) (3) (2, ± 32 ) (4) (1, ±2)

Q66.The image of the point (3, 5) in the line x −y + 1 = 0, lies on : (1) (x −2)2 + (y −4)2 = 4 (2) (x −4)2 + (y −4)2 = 8 (3) (x −4)2 + (y + 2)2 = 16 (4) (x −2)2 + (y −2)2 = 12

Q66.The Boolean expression (p ∧~q) ⇒(q ∨~p) is equivalent to: (1) q ⇒p (2) p ⇒q (3) ~q ⇒p (4) p ⇒~q

Q66.Let (1 + x + 2x2) 20 = a0 + a1x + a2x2 + … + a40x40, then a1 + a3 + a5 + … + a37 is equal to (1) 220(220 −21) (2) 219(220 −21) (3) 219(220 + 21) (4) 220(220 + 21) Q67. 1 + sin2 x sin2 x sin2 x The solutions of the equation cos2 x 1 + cos2 x cos2 x = 0, (0 < x < π), are 4 sin 2x 4 sin 2x 1 + 4 sin 2x (1) 12 π , π6 (2) π6 , 5π6 (3) 5π 12 , 7π12 (4) 7π12 , 11π12

Q66.In the circle given below, let OA = 1 unit, OB = 13 unit and PQ ⊥OB. Then, the area of the triangle PQB (in square units) is : (1) 24√3 (2) 26√3 (3) 24√2 (4) 26√2 √3 sin( π6 +h)−cos( π6 +h) is :

Q66.The locus of mid-points of the line segments joining -3, - 5 and the points on the ellipse 𝑥2 + 𝑦2 = 1 is : 4 9 (1) 36𝑥2 + 16𝑦2 + 90𝑥+ 56𝑦+ 145 = 0 (2) 36𝑥2 + 16𝑦2 + 108𝑥+ 80𝑦+ 145 = 0 (3) 9𝑥2 + 4𝑦2 + 18𝑥+ 8𝑦+ 145 = 0 (4) 36𝑥2 + 16𝑦2 + 72𝑥+ 32𝑦+ 145 = 0

Q66.Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0 . If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point: (1) (1, 2) (2) (2, 2) (3) (2, 1) (4) (1, 3)

Q66.Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y−axis at C. The locus of the mid-point P of MC is (1) 3x2 + 2y −6 = 0 (2) 2x2 −3y + 9 = 0 (3) 3x2 −2y −6 = 0 (4) 2x2 + 3y −9 = 0

Q66.Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x -axis and y-axis at point P and Q , respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r2 is equal to (1) 529 (2) 125 64 72 (3) 625 (4) 585 72 66

Q66.The locus of the mid points of the chords of the hyperbola x2 −y2 = 4, which touch the parabola y2 = 8x, is : (1) y2(x −2) = x3 (2) x3(x −2) = y2 (3) x2(x −2) = y3 (4) y3(x −2) = x2 lim n=1 n(n+1)x2+2(2n+1)x+4x ) is equal to :

Q66.If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0), a ≠0, then a must be greater than : (1) 1 2 (2) −12 (3) −1 (4) 1

Q66.The value of cot 24π is: (1) √2 + √3 + 2 −√6 (2) √2 + √3 + 2 + √6 (3) √2 −√3 −2 + √6 (4) 3√2 −√3 −√6 JEE Main 2021 (25 Jul Shift 2) JEE Main Previous Year Paper

Q66.A hyperbola passes through the foci of the ellipse x2 = 1 and its transverse and conjugate axes coincide 25 + 16 with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is: (1) x2 = 9 9 −y216 = 1 (2) x2 −y2 (3) x2 9 −y225 = 1 (4) x29 −y24 = 1

Q66.The line 12x cos θ + 5y sin θ = 60 is tangent to which of the following curves ? (1) x2 + y2 = 30 (2) 144x2 + 25y2 = 3600 (3) x2 + y2 = 169 (4) 25x2 + 12y2 = 3600

Q66.The Boolean expression (p ∧q) ⇒((r ∧q) ∧p) is equivalent to: (1) (p ∧r) ⇒(p ∧q) (2) (q ∧r) ⇒(p ∧q) (3) (p ∧q) ⇒(r ∧q) (4) (p ∧q) ⇒(r ∨q)

Q66.Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 −y2 = 3. If L is also a tangent to the parabola y2 = αx, then α is equal to: (1) 12 (2) −12 (3) 24 (4) −24

Q66.Let f(x) be a differentiable function at x = a with f ′(a) = 2 and f(a) = 4. Then lim x−a x→a (1) a + 4 (2) 2a −4 (3) 4 −2a (4) 2a + 4

Q66.The locus of the centroid of the triangle formed by any point 𝑃 on the hyperbola 16𝑥2 - 9𝑦2 + 32𝑥+ 36𝑦- 164 = 0 and its foci is (1) 16𝑥2 - 9𝑦2 + 32𝑥+ 36𝑦- 36 = 0 (2) 9𝑥2 - 16𝑦2 + 36𝑥+ 32𝑦- 144 = 0 (3) 16𝑥2 - 9𝑦2 + 32𝑥+ 36𝑦- 144 = 0 (4) 9𝑥2 - 16𝑦2 + 36𝑥+ 32𝑦- 36 = 0

Q66.The line 2x −y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x −2y = 4. Then, the radius of the circle is: (1) 3√5 (2) 5√3 (3) 5√4 (4) 4√5

Q66.Let ABC be a triangle with A(−3, 1) and ∠ACB = θ, 0 < θ < π2 . If the equation of the median through B is 2x + y −3 = 0 and the equation of angle bisector of C is 7x −4y −1 = 0, then tan θ is equal to: (1) 3 (2) 4 4 3 (3) 2 (4) 12

Q66.Consider the following three statements: (A) If 3 + 3 = 7 then 4 + 3 = 8 (B) If 5 + 3 = 8 then earth is flat. (C) If both (A) and (B) are true then 5 + 6 = 17. Then, which of the following statements is correct? (1) (A) is false, but (B) and (C) are true (2) (A) and (C) are true while (B) is false (3) (A) is true while (B) and (C) are false (4) (A) and (B) are false while (C) is true

Q66.Consider the parabola with vertex 2, 4 and the directrix 𝑦= 2 . Let P be the point where the parabola meets the line 𝑥= - 12. If the normal to the parabola at P intersects the parabola again at the point Q . then ( PQ ) 2 is equal to : 25 75 (1) (2) 2 8 (3) 125 (4) 15 16 2