Practice Questions

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

Q67.Consider the following two propositions : 𝑃1: ~𝑝→~𝑞 𝑃2: 𝑝∧~𝑞∧~𝑝∨𝑞 If the proposition 𝑝→~𝑝∨𝑞 is evaluated as FALSE, then (1) 𝑃1 is TRUE and 𝑃2 is FALSE (2) 𝑃1 is FALSE and 𝑃2 is TRUE (3) Both 𝑃1 and 𝑃2 are FALSE (4) Both 𝑃1 and 𝑃2 are TRUE

Q67.A circle touches both the 𝑦-axis and the line 𝑥+ 𝑦= 0. Then the locus of its center (1) 𝑦= √2𝑥 (2) 𝑥= √2𝑦.. (3) 𝑦2 - 𝑥2 = 2𝑥𝑦 (4) 𝑥2 −𝑦2 = 2𝑥𝑦

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 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.Let f(x) = ax2 + bx + c be such that f(1) = 3, f(−2) = λ and f(3) = 4. If f(0) + f(1) + f(−2) + f(3) = 14 , then λ is equal to JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper (1) −4 (2) 132 (3) 23 (4) 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 operations * , ⊙∈∧, ∨. If 𝑝* 𝑞⊙𝑝⊙~𝑞 is a tautology, then the ordered pair * , ⊙ is (1) ∨, ∧ (2) ∨, ∨ (3) ∧, ∧ (4) ∧, ∨ JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q68.The statement 𝑝⇒𝑞∨𝑝⇒𝑟 is NOT equivalent to: (1) 𝑝∧~𝑟⇒𝑞 (2) ~𝑞⇒~𝑟∨𝑝 (3) 𝑝⇒𝑞∨𝑟 (4) 𝑝∧~𝑞⇒𝑟

Q68.The line 𝑦= 𝑥+ 1 meets the ellipse 𝑥2 + 𝑦2 = 1 at two points 𝑃 and 𝑄. If 𝑟 is the radius of the circle with 𝑃𝑄 4 2 as diameter then 3𝑟2 is equal to (1) 20 (2) 12 (3) 11 (4) 8 Q69. 12 12 lim tan2𝑥2sin2𝑥+ 3sin𝑥+ 4 - sin2𝑥+ 6sin𝑥+ 2 is equal to 𝑥→𝜋 2 1 1 (1) (2) - 12 18 (3) - 1 (4) 1 12 6

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 system of linear equations x + 2y + z = 2, αx + 3y −z = α, −αx + y + 2z = −α be inconsistent. Then α is equal to (1) 2 5 (2) −52 (3) 2 7 (4) −72

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.A tower 𝑃𝑄 stands on a horizontal ground with base 𝑄 on the ground. The point 𝑅 divides the tower in two parts such that 𝑄𝑅= 15m. If from a point 𝐴 on the ground the angle of elevation of 𝑅 is 60° and the part 𝑃𝑅 of the tower subtends an angle of 15° at 𝐴, then the height of the tower is (1) 52√3 + 3m (2) 5√3 + 3m (3) 10√3 + 1m (4) 102√3 + 1m

Q68.Which of the following statement is a tautology? (1) ((~q) ∧p) ∧q (2) ((~q) ∧p) ∧(p ∧(~p)) (3) ((~q) ∧p) ∨(p ∨(~p)) (4) (p ∧q) ∧(~(p ∧q))

Q68.Let the system of linear equations x + y + az = 2 3x + y + z = 4 x + 2z = 1 have a unique solution ( x∗, y∗, z∗). If ( (a, x∗), (y∗, α) and ( x∗, −y∗) are collinear points, then the sum of absolute values of all possible values of α is: (1) 4 (2) 3 (3) 2 (4) 1

Q68. (p ∧r) ⇔(p ∧(~q)) is equivalent to (~p) when r is (1) p (2) ~p (3) q (4) ~q

Q68.The number of choices for Δ ∈{∧, ∨, ⇒, ⇔} , such that (pΔq) ⇒((pΔ~q) ∨((~p)Δq)) is a tautology, is (1) 1 (2) 2 (3) 3 (4) 4 Q69. ⎡ 1 0 a ⎤ Let S ={ √n : 1 ⩽n ⩽50 and n is odd}. Let a ∈S and A = −1 1 0 . If Σ det (adj A) = 100λ, then λ ⎣−a 0 1 ⎦ a∈S is equal to (1) 218 (2) 221 (3) 663 (4) 1717

Q68.The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is . If the distance of the point B from the tower is 15 units, then cot α is cos−1( √133 ) equal to (1) 6 (2) 9 5 5 (3) 4 (4) 7 3 3