Practice Questions

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 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.The locus of the mid-point of the line segment joining the focus of the parabola 𝑦2 = 4𝑎𝑥 to a moving point of the parabola, is another parabola whose directrix is: (1) 𝑥= 𝑎 (2) 𝑥= 0 (3) 𝑥= - 𝑎 (4) 𝑥= 𝑎 2 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

Q67.If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and 203 , respectively, then the value of |a −b| is equal to: (1) 9 (2) 11 (3) 7 (4) 1

Q67.The value of lim cos h−sin h) } h→0{ √3h(√3 (1) 43 (2) √32 (3) 23 (4) 43

Q67.The contrapositive of the statement "If you will work, you will earn money" is: (1) If you will not earn money, you will not work (2) To earn money, you need to work (3) You will earn money, if you will not work (4) If you will earn money, you will work AAT = I2 , then the value of α4 + β4 is :

Q67.Let 𝜃 be the acute angle between the tangents to the ellipse 𝑥2 + 𝑦2 = 1 and the circle 𝑥2 + 𝑦2 = 3 at their 9 1 point of intersection in the first quadrant. Then tan𝜃 is equal to : (1) 5 (2) 4 2√3 √3 (3) 2 (4) 2 √3

Q67. x→2(∑9 (1) 5 (2) 7 24 36 (3) 1 (4) 9 5 44

Q67.The Boolean expression ( 𝑝⇒𝑞) ∧( 𝑞⇒~𝑝) is equivalent to : (1) ~𝑞 (2) 𝑞 (3) 𝑝 (4) ~𝑝

Q67.Let A = {2, 3, 4, 5, … . , 30} and ′≃′ be an equivalence relation on A × A, defined by (a, b) ≃(c, d), if and only if ad = bc. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair (4, 3) is equal to : (1) 5 (2) 6 (3) 8 (4) 7

Q67.Let L be a tangent line to the parabola y2 = 4x −20 at (6, 2). If L is also a tangent to the ellipse x2 y2 2 + b = 1, then the value of b is equal to : (1) 11 (2) 14 (3) 16 (4) 20 JEE Main 2021 (17 Mar Shift 2) JEE Main Previous Year Paper

Q67.If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (−30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is: (1) 5 (2) 7 (3) 3√5 (4) 5√3 y2

Q67.The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, x2 is : 9 −y216 = 1 (1) (x2 + y2)2 −16x2 + 9y2 = 0 (2) (x2 + y2)2 −9x2 + 144y2 = 0 2 2 (3) (x2 + y2) −9x2 −16y2 = 0 (4) (x2 + y2) −9x2 + 16y2 = 0

Q67.The statement among the following that is a tautology is: JEE Main 2021 (24 Feb Shift 1) JEE Main Previous Year Paper (1) 𝐴∨𝐴∧𝐵 (2) 𝐴∧𝐴∨𝐵 (3) 𝐵→𝐴∧𝐴→𝐵 (4) 𝐴∧𝐴→𝐵→𝐵

Q67.Two poles AB of length a metres and CD of length a + b(b ≠a) metres are erected at the same horizontal level with bases at B and D. If BD = x and tan ∠ACB = 12 , then: (1) x2 + 2(a + 2b)x −b(a + b) = 0 (2) x2 + 2(a + 2b)x + a(a + b) = 0 (3) x2 −2ax + b(a + b) = 0 (4) x2 −2ax + a(a + b) = 0 JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper

Q67.A tangent and a normal are drawn at the point P(2, −4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to (1) −12 (2) −20 (3) −16 (4) −18

Q67.If 𝛼= lim tan3𝑥- tan𝑥𝜋 and 𝛽= lim are the roots of the equation, 𝑎𝑥2 + 𝑏𝑥- 4 = 0, then the ordered 𝑥→𝜋/ 4 cos𝑥+ 4 𝑥→0cos𝑥cot𝑥 pair 𝑎, 𝑏 is : (1) -1, 3 (2) 1, - 3 (3) 1, 3 (4) -1, - 3

Q67.Consider a circle C which touches the y− axis at (0, 6) and cuts off an intercept 6√5 on the x− axis. Then the radius of the circle C is equal to : (1) √53 (2) 9 (3) 8 (4) √82 x lim x ) is equal to : 8√1−sin x−8√1+sin

Q67. sin x cos x cos x The number of distinct real roots of cos x sin x cos x = 0 in the interval −π4 ≤x ≤π4 is: cos x cos x sin x (1) 4 (2) 1 (3) 2 (4) 3

Q67.Choose the incorrect statement about the two circles whose equations are given below: x2 + y2 −10x −10y + 41 = 0 and x2 + y2 −16x −10y + 80 = 0 (1) Distance between two centres is the average of (2) Both circles' centres lie inside region of one radii of both the circles. another. (3) Both circles pass through the centre of each (4) Circles have two intersection points. other.

Q67.Let F1(A, B, C) = (A ∧~B) ∨[~C ∧(A ∨B)] ∨~A and F2(A, B) = (A ∨B) ∨(B →~A) be two logical expressions. Then : (1) F1 is a tautology but F2 is not a tautology (2) F1 is not a tautology but F2 is a tautology (3) Both F1 and F2 are not tautologies (4) F1 and F2 both are tautologies

Q67.A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1 . Which of the following points does NOT lie on it? (1) (0, 3) (2) (4, 5) (3) (5, 4) (4) (−6, 0) y2

Q67.Let A = {(x, y) ∈R × R ∣2x2 + 2y2 −2x −2y = 1} B = {(x, y) ∈R × R ∣4x2 + 4y2 −16y + 7 = 0} and C = {(x, y) ∈R × R ∣x2 + y2 −4x −2y + 5 ≤r2}. Then the minimum value of |r| such that A ∪B ⊆C is equal to (1) 3+√10 (2) 2+√10 2 2 (3) 3+2√5 (4) 1 + √5 2