Practice Questions

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

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

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.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.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

Q68.If P and Q are two statements, then which of the following compound statement is a tautology? (1) ((P ⇒Q) ∧~Q) ⇒Q (2) ((P ⇒Q) ∧~Q) ⇒~P (3) ((P ⇒Q) ∧~Q) ⇒P (4) ((P ⇒Q) ∧~Q) ⇒(P ∧Q)

Q68.If the curves, x2 intersect each other at an angle of 90°, then which of the a + b = 1 and x2c + y2d = 1 following relations is TRUE? (1) a −c = b + d (2) a −b = c −d (3) a + b = c + d (4) ab = a+bc+d 1 1 n 1+ 2 +……+ n

Q68.On the ellipse x2 8 + 4 = 1, let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S′ be the foci of the ellipse and e be its eccentricity. If A is the JEE Main 2021 (26 Aug Shift 1) JEE Main Previous Year Paper area of the triangle SPS′ , then the value of (5 −e2) ⋅A is (1) 12 (2) 6 (3) 14 (4) 24

Q68.Which of the following is equivalent to the Boolean expression p ∧~q ? (1) ~p →~q (2) ~ ( q →p ) (3) ~ ( 𝑝→𝑞) (4) ~ ( 𝑝→~𝑞)

Q68.If α, β are the distinct roots of x2 + bx + c = 0, then lim e2(x2+bx+c)−1−2(x2+bx+c) is equal to x→β (x−β)2 (1) 2(b2 + 4c) (2) b2 −4c (3) 2(b2 −4c) (4) b2 + 4c

Q68.Let R = {(P, Q)|P and Q are at the same distance from the origin } be a relation, then the equivalence class of (1, −1) is the set JEE Main 2021 (26 Feb Shift 1) JEE Main Previous Year Paper (1) S = {(x, y) x2 + y2 = 1} (2) S = {(x, y) x2 + y2 = 2} (3) S = {(x, y) x2 + y2 = √2} (4) S = {(x, y) x2 + y2 = 4}

Q68.Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is: (1) 25 (2) 30 (3) 20√3 (4) 25√3

Q68.Let the equation of the pair of lines, y = px and y = qx, can be written as (y −px)(y −qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 −4xy −5y2 = 0 is: (1) x2 −3xy + y2 = 0 (2) x2 + 4xy −y2 = 0 (3) x2 + 3xy −y2 = 0 (4) x2 −3xy −y2 = 0