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

Q65.In a triangle PQR, the co-ordinates of the points P and Q are (−2, 4) and (4, −2) respectively. If the equation of the perpendicular bisector of PR is 2x −y + 2 = 0, then the centre of the circumcircle of the ΔPQR is: (1) (–1, 0) (2) (–2, –2) (3) (0, 2) (4) (1, 4) JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper

Q65.The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S(> R) respectively from the origin, is : (1) 2( S −R) (2) 2(S + R) (3) 4(S −R) (4) 4(S + R)

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 𝐴 be the set of all points 𝛼, 𝛽 such that the area of triangle formed by the points 5, 6, 3, 2 and 𝛼, 𝛽 is 12 square units. Then the least possible length of a line segment joining the origin to a point in 𝐴, is : 8 12 (1) (2) √5 √5 (3) 16 (4) 4 √5 √5

Q65.If 𝑛 is the number of solutions of the equation 2cos𝑥4sin + 𝑥sin - 𝑥- 1 = 1, 𝑥∈0, 𝜋 and 𝑆 is the sum of all 4 4 these solutions, then the ordered pair 𝑛, 𝑆 is : (1) 2, 8𝜋 (2) 3, 13π 9 9 2𝜋 5𝜋 (3) 2, (4) 3, 3 3 JEE Main 2021 (01 Sep Shift 2) JEE Main Previous Year Paper 1 3 1

Q65.The point P(−2√6, √3) lies on the hyperbola x2a2 −y2b2 normal at P to the hyperbola intersect its conjugate axis at the points Q and R respectively, then QR is equal to: (1) 4√3 (2) 6 (3) 3√6 (4) 6√3

Q65.Let P be a variable point on the parabola y = 4x2 + 1. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is: (1) (3x −y)2 + (x −3y) + 2 = 0 (2) 2(3x −y)2 + (x −3y) + 2 = 0 (3) (3x −y)2 + 2(x −3y) + 2 = 0 (4) 2(x −3y)2 + (3x −y) + 2 = 0

Q65.The value of -15𝐶1 + 2 · 15𝐶2 - 3 ·15 𝐶3 + . . . . . - 15 · 15𝐶15 + 14𝐶1 + 14𝐶3 + 14𝐶5 + . . . . + 14𝐶11 is equal to (1) 214 (2) 213 - 13 (3) 216 - 1 (4) 213 - 14

Q65.For the statements p and q, consider the following compound statements: (a) (~q ∧(p →q)) →~p (b) ((p ∨q) ∧~p) →q Then which of the following statements is correct? (1) (b) is a tautology but not (a). (2) (a) and (b) both are tautologies. (3) (a) and (b) both are not tautologies. (4) (a) is a tautology but not (b).

Q65.Let E1 : x2a2 + y2b2 and the foci of E2 are the end points of minor axis of E1. If E1 and E2 have same eccentricities, then its value is: (1) −1+√5 (2) −1+√8 2 2 (3) −1+√3 (4) −1+√6 2 2

Q65.The number of roots of the equation, (81)sin2 x + (81)cos2 x = 30 in the interval [0, π] is equal to : JEE Main 2021 (16 Mar Shift 1) JEE Main Previous Year Paper (1) 3 (2) 4 (3) 8 (4) 2

Q65.Two tangents are drawn from the point P(−1, 1) to the circle x2 + y2 −2x −6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to: + (1) 2 (2) (3√2 2) (3) 4 (4) 3(√2 −1)

Q65.The intersection of three lines x −y = 0, x + 2y = 3 and 2x + y = 6 is a/an (1) Isosceles triangle (2) Equilateral triangle (3) Right angled triangle (4) None of the above

Q65.If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q , then the angle subtended by the line segment PQ at the origin is (1) π 2 −tan−1( 31 ) (2) π2 + tan−1( 31 ) (3) π 2 + tan−1( 41 ) (4) π2 −tan−1( 41 ) y2

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

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.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.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.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.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.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.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