Practice Questions

Q65.Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x−axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to: (1) 25 (2) 35 2 2 (3) 15 (4) 25 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.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.If the points of intersection of the ellipse x216 + y2b2 y2 = 3x2 , then b is equal to : (1) 12 (2) 5 (3) 6 (4) 10

Q66.Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to 3 + (1) {(4, 0), (0, 6)} (2) {(2 + 2√2, 3 −√5), (2 −2√2, √5)} + 2√2, 3 + −2√2, 3 (3) {(2 √5), (2 −√5)} (4) {(−1, 5), (5, 1)}

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.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.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 angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 20 seconds at the speed of 432 km / hour, the angle of elevation changes to 30°. If the jet plane is flying at a constant height, then its height is: (1) 1200√3 m (2) 2400√3 m (3) 1800√3 m (4) 3600√3 m

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.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.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.Let a tangent be drawn to the ellipse x2 cos θ, sin θ ∈(0, π2 ). Then the value of θ 27 + y2 = 1 at (3√3 θ) where such that the sum of intercepts on axes made by this tangent is minimum is equal to : (1) π (2) π 8 4 (3) π (4) π 6 3 x-axis at Q and latus

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.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 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.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.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.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.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

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.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.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.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