Practice Questions

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

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 :

Q67. lim sin2(π cos4 x) is equal to : x4 x→0 (1) 2π2 (2) π2 (3) 4π2 (4) 4π

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.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 hyperbola H : x2 −2y2 = 4 . Let the tangent at a point P(4, √6) meet the rectum at R(x1, y1), x1 > 0 . If F is a focus of H which is nearer to the point P , then the area of ΔQFR (in sq. units) is equal to (1) 4√6 (2) √6 −1 (3) 7 −2 (4) 4√6 −1 √6

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

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

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. 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.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.Which of the following Boolean expressions is not a tautology? (1) (p ⇒q) ∨(~q ⇒p) (2) (q ⇒p) ∨(~q ⇒p) (3) (p ⇒~q) ∨(~q ⇒p) (4) (~p ⇒q) ∨(~q ⇒p)

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.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.For the system of linear equations: x −2y = 1, x −y + kz = −2, ky + 4z = 6, k ∈R Consider the following statements: (A) The system has unique solution if k ≠2, k ≠−2. (B) The system has unique solution if k = −2. (C) The system has unique solution if k = 2. JEE Main 2021 (24 Feb Shift 2) JEE Main Previous Year Paper (D) The system has no-solution if k = 2. (E) The system has infinite number of solutions if k ≠−2. Which of the following statements are correct? (1) (A) and (E) only (2) (B) and (E) only (3) (A) and (D) only (4) (C) and (D) only

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. 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.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.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