Practice Questions

Q66.A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of a circle C1 . Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to (1) 1 : 4 (2) 1 : 5 (3) 2 : 5 (4) 1 : 3

Q66.A horizontal park is in the shape of a triangle OAB with AB = 16 . A vertical lamp post OP is erected at the point O such that ∠PAO = ∠PBO = 15° and ∠PCO = 45° , where C is the midpoint of AB. Then (OP)2 is equal to (1) √3 32 (√3 −1) (2) √332 (2 −√3) (3) 16 (4) 16 √3 (√3 −1) √3 (2 −√3)

Q66.Let 𝑓𝑥 be a polynomial function such that 𝑓𝑥+ 𝑓'𝑥+ 𝑓''𝑥= 𝑥5 + 64. Then, the value of lim 𝑓𝑥 is equal to 𝑥→1 𝑥- 1 (1) -15 (2) 15 (3) -60 (4) 60

Q66.If 𝑛→∞√𝑛2lim - 𝑛- 1 + 𝑛𝛼+ 𝛽= 0 then 8𝛼+ 𝛽 is equal to JEE Main 2022 (25 Jul Shift 1) JEE Main Previous Year Paper (1) 4 (2) -8 (3) -4 (4) 8

Q66.The statement (~(p ⇔~q)) ∧q is: (1) a tautology (2) a contradiction (3) equivalent to (p ⇒q) ∧q (4) equivalent to (p ⇒q) ∧p

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.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 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.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 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 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.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.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.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

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 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.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 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.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 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 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 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 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