Practice Questions

Q67.Let r ∈(P, q, ~p, ~q) be such that the logical statement r ∨(~p) ⇒(p ∧q) ∨r is a tautology. Then r is equal to (1) p (2) q (3) ~p (4) ~q

Q67.Let AB and PQ be two vertical poles, 160m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let π and θ be the angles of elevation from C to P and A , respectively. If 8 the height of pole PQ is twice the height of pole AB, then tan2 θ is equal to (1) 3−2√2 (2) 3+√2 2 2 (3) 3−2√2 (4) 3−√2 4 4

Q67.Let λx −2y = μ be a tangent to the hyperbola a2x2 −y2 = b2 . Then ( λa ) 2 −( μb )2 (1) −2 (2) −4 (3) 2 (4) 4

Q67.Let Δ, ∇∈{∧, ∨} be such that p∇q →((pΔq)∇r) is a tautology. Then (p∇q) Δ r is logically equivalent to (1) (pΔr) ∨q (2) (pΔr) ∧q (3) (p ∧r)Δq (4) (p∇r) ∧q

Q67.Let A be a 2 × 2 matrix with det(A) = −1 and det((A + I)(Adj(A) + I)) = 4 . Then the sum of the diagonal elements of A can be: (1) −1 (2) 2 (3) 1 (4) −√2

Q67.If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y −29 = 0, is x2 + ay2 + bxy + cx + dy + k = 0, then a + b + c + d + k is equal to (1) 575 (2) −575 (3) 576 (4) −576

Q67.If vertex of parabola is (2, −1) and equation of its directrix is 4x −3y = 21, then the length of latus rectum is (1) 2 (2) 8 (3) 12 (4) 16

Q67.If the line 𝑥- 1 = 0, is a directrix of the hyperbola 𝑘𝑥2 - 𝑦2 = 6, then the hyperbola passes through the point (1) -2√5, 6 (2) -√5, 3 (3) √5, - 2 (4) 2√5, 3√6

Q67.Let 𝐴𝛼, - 2, 𝐵𝛼, 6 and 𝐶𝛼 - 2 be vertices of a ∆𝐴𝐵𝐶. If 5, 𝛼 is the circumcentre of ∆𝐴𝐵𝐶, then which of the 4, 4 following is NOT correct about ∆𝐴𝐵𝐶 (1) ares is 24 (2) perimeter is 25 (3) circumradius is 5 (4) inradius is 2

Q67.The statement (p ∧q) ⇒(p ∧r) is equivalent to (1) q ⇒(p ∧r) (2) p ⇒(p ∧r) (3) (p ∧r) ⇒(p ∧q) (4) (p ∧q) ⇒r JEE Main 2022 (29 Jul Shift 1) JEE Main Previous Year Paper

Q67.The boolean expression (~(p ∧q)) ∨q is equivalent to (1) q →(p ∧q) (2) p →q (3) p →(p →q) (4) p →(p ∨q)

Q67.Consider the following statements: A: Rishi is a judge. B: Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is (1) B →(A ∨C) (2) (~B) ∧(A ∧C) (3) B →((~A) ∨(~C)) (4) B →(A ∧C)

Q67.Let Δ ∈{∧, ∨, ⇒, ⇔} be such that (p ∧q)Δ((p ∨q) ⇒q) is a tautology. Then Δ is equal to (1) ∧ (2) ∨ (3) ⇒ (4) ⇔

Q67.If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16 , then |a| is equal to (1) 2√2 (2) 2√3 (3) 4√2 (4) 4 JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper

Q67.Which of the following statements is a tautology? (1) ~𝑝∨𝑞⇒𝑝 (2) 𝑝⇒~𝑝∨𝑞 (3) ~𝑝∨𝑞⇒𝑞 (4) 𝑞⇒~𝑝∨𝑞

Q67.If the ellipse x2 = 1 on the y-axis, a2 + b2 = 1 meets the line x7 + 2√6y = 1 on the x-axis and the line x7 − 2√6y then the eccentricity of the ellipse is (1) 5 (2) 2√6 7 7 (3) 3 (4) 2√5 7 7 y2

Q67.If the tangents drawn at the points 𝑃 and 𝑄 on the parabola 𝑦2 = 2𝑥- 3 intersect at the point 𝑅0, 1, then the orthocentre of the triangle 𝑃𝑄𝑅 is (1) 0, 1 (2) 2, - 1 (3) 6, 3 (4) 2, 1

Q67.Let f : R →R be a function defined as f(x) = a sin( π[x]2 ) less than or equal to t. If lim f(x) exists, then the value of ∫40 f(x)dx is equal to x→−1 (1) −1 (2) −2 (3) 1 (4) 2

Q68.The value of lim (x2−1) sin2(πx) is equal to: x→1 x4−2x3+2x−1 (1) π2 (2) π2 6 3 (3) π2 (4) π2 2

Q68.Let the mean and the variance of 5 observations x1, x2, x3, x4, x5 be 245 and 19425 respectively. If the mean and variance of the first 4 observation are 27 and a respectively, then (4a + x5) is equal to (1) 13 (2) 15 (3) 17 (4) 18

Q68.Let A be a matrix of order 3 × 3 and det(A) = 2 . Then det(det (A) adj (5 adj (A3)) is equal to _____. (1) 256 × 106 (2) 1024 × 106 (3) 512 × 106 (4) 256 × 1011

Q68.The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6. 8. If M is the mean deviation of the numbers about the mean, then 25M is equal to (1) 60 (2) 55 (3) 50 (4) 75

Q68.Let the mean of 50 observations is 15 and the standard deviation is 2 . However, one observation was wrongly recorded. The sum of the correct and incorrect observations is 70 . If the mean of the correct set of observations is 16 , then the variance of the correct set is equal to (1) 10 (2) 36 (3) 43 (4) 60

Q68.Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola x2 −y2 = 1. Let e′ and l′ respectively the eccentricity and length of the latus rectum of its conjugate a2 b2 hyperbola. If e2 = 1411 l and (e′)2 = 118 l′ , then the value of 77a + 44b is equal to (1) 100 (2) 110 (3) 120 (4) 130

Q68.If the system of linear equations. JEE Main 2022 (26 Jul Shift 1) JEE Main Previous Year Paper 8x + y + 4z = −2 x + y + z = 0 λx −3y = μ has infinitely many solutions, then the distance of the point (λ, μ, −12 ) from the plane 8x + y + 4z + 2 = 0 is: (1) 3√5 (2) 4 (3) 26 (4) 10 9 3