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.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.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 A and B be any two 3 × 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true? (1) A4 −B4 is a symmetric matrix (2) AB −BA is a symmetric matrix (3) B5 −A5 is a skew-symmetric matrix (4) AB + BA is a skew-symmetric matrix

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 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.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)Δ((p ∨q) ⇒q) is a tautology. Then Δ is equal to (1) ∧ (2) ∨ (3) ⇒ (4) ⇔

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.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.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

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.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.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.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.Let 𝑎, 𝑏 and 𝑐 be the length of sides of a triangle 𝐴𝐵𝐶 such that 𝑎+ 𝑏 = 𝑏+ 𝑐 = 𝑐+ 𝑎 . If 𝑟 and 𝑅 are the radius of 7 8 9 𝑅 incircle and radius of circumcircle of the triangle 𝐴𝐵𝐶, respectively, then the value of is equal to 𝑟 (1) 2 (2) 3 5 (3) 5 (4) 1 2

Q68.Let the foci of the ellipse x2 coincide. Then the length of the 16 + 7 = 1 and the hyperbola 144x2 −y2α = 251 latus rectum of the hyperbola is: (1) 32 (2) 18 9 5 (3) 27 (4) 27 4 10 8√2−(cos x+sin x)7

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.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.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.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 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.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

Q68.Let 𝛽= lim 𝛼𝑥- 𝑒3𝑥- 1 for some 𝛼∈ℝ. Then the value of 𝛼+ 𝛽 is: 𝑥→0 𝛼𝑥𝑒3𝑥- 1 14 3 (1) (2) 5 2 (3) 5 (4) 7 2 2

Q69.Let R1 = {(a, b) ∈N × N : |a −b| ≤13} and R2 = {(a, b) ∈N × N : |a −b| ≠13} Then on N : (1) Both R1 and R2 are equivalence relations (2) Neither R1 nor R2 is an equivalence relation (3) R1 is an equivalence relation but R2 is not (4) R2 is an equivalence relation but R1 is not