Practice Questions

Q64.A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the 1 coordinate axes is 4. Three stones 𝐴, 𝐵 and 𝐶 are placed at the points 1, 1, 2, 2 and 4, 4 respectively. Then which of these stones is / are on the path of the man? (1) 𝐶 only (2) All the three (3) 𝐵 only (4) 𝐴 only

Q64.If the fourth term in the expansion of (x + xlog2 x) 7 is 4480, then the value of x where x ∈N is equal to: (1) 2 (2) 4 (3) 3 (4) 1

Q65.If nP r = nP r+1 and nCr = nCr−1, then the value of r is equal to: (1) 1 (2) 4 (3) 2 (4) 3

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.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)

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.If x→∞(√x2 (1) (1, −12 ) (2) (−1, 21 ) (3) (−1, −12 ) (4) (1, 21 )

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.The sum of solutions of the equation 1+sin x = |tan 2x|, x ∈(−π2 , π2 ) −{−π4 , π4 } is: (1) 10 π (2) −7π30 (3) −π15 (4) −11π30

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.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 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.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.Two tangents are drawn from a point P to the circle x2 + y2 −2x −4y + 4 = 0, such that the angle between these tangents is tan−1( 125 ), where tan−1( 125 ) ∈(0, π). If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of ΔPAB and ΔCAB is : (1) 11 : 4 (2) 9 : 4 (3) 3 : 1 (4) 2 : 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 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.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.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.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.All possible values of θ ∈[0, 2π] for which sin 2θ + tan 2θ > 0 lie in : (1) (0, π2 ) ∪(π, 3π2 ) (2) (0, π2 ) ∪( π2 , 3π4 ) ∪(π, 7π6 ) (3) (0, π4 ) ∪( π2 , 3π4 ) ∪(π, 5π4 ) ∪( 3π2 , 7π4 ) (4) (0, π4 ) ∪( π2 , 3π4 ) ∪( 3π2 , 11π6 )

Q65.Let A(1, 4) and B(1, −5) be two points. Let P be a point on the circle ((x −1))2 + (y −1)2 = 1 , such that (PA)2 + (PB)2 have maximum value, then the points, P, A and B lie on (1) a hyperbola (2) a straight line (3) an ellipse (4) a parabola xf(a)−af(x) equals:

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 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 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.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