Practice Questions

Q66.Let a triangle ABC be inscribed in the circle x2 −√2(x + y) + y2 = 0 such that ∠BAC = π2 . If the length of side AB is √2 , then the area of the △ABC is equal to: (1) 1 (2) (√6+√3) 2 (3) (√3+√3) (4) (√6+2√3) 2 4

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

Q66.Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of π2 at the point (3, 0). Let the x2 y2 line segment PQ be also a focal chord of the ellipse E : + = 1, a2 > b2 . If e is the eccentricity of the a2 b2 ellipse E , then the value of 1 is equal to e2 (1) 1 + √2 (2) 3 + 2√2 (3) 1 + 2√3 (4) 4 + 5√3

Q66.Let a be an integer such that lim 18−[1−x][x−3a] exists, where [t] is greatest integer ≤t . Then x→7 (1) −2 (2) 6 (3) −6 (4) −7

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.If lim = 3 , where α, β, γ ∈R, then which of the following is NOT correct? x sin2 x x→0 (1) α2 + β2 + γ 2 = 6 (2) αβ + βγ + γα + 1 = 0 (3) αβ2 + βγ 2 + γα2 + 3 = 0 (4) α2 −β2 + γ 2 = 4

Q66.Let 𝐶 be the centre of the circle 𝑥2 + 𝑦2 - 𝑥+ 2𝑦= and 𝑃 be a point on the circle. A line passes through the 4 point 𝐶, makes an angle of 𝜋 with the line 𝐶𝑃 and intersects the circle at the points 𝑄 and 𝑅. Then the area of 4 the triangle 𝑃𝑄𝑅 (in unit2) is (1) 2 (2) 2√2 𝜋 𝜋 (3) 8sin (4) 8cos 8 8

Q66.The tangents at the points A(1, 3) and B(1, −1) on the parabola y2 −2x −2y = 1 meet at the point P . Then the area (in unit2 ) of the triangle PAB is: (1) 4 (2) 6 (3) 7 (4) 8 y2

Q66.Let p, q, r be three logical statements. Consider the compound statements S1 : ((~p) ∨q) ∨((~p) ∨r) and S2 : p →(q ∨r) Then, which of the following is NOT true? (1) If S2 is True, then S1 is True (2) If S2 is False, then S1 is False (3) If S2 is False, then S1 is True (4) If S1 is False, then S2 is False

Q66.Let the maximum area of the triangle that can be inscribed in the ellipse x2 + 4 = 1, a > 2, having one of its a2 vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be 6√3. Then the eccentricity of the ellipse is: (1) √3 (2) 1 2 2 (3) 1 (4) √3 √2 4

Q66.Let a triangle be bounded by the lines L1 : 2x + 5y = 10 ; L2 : −4x + 3y = 12 and the line L3 , which passes through the point P(2, 3), intersect L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to (1) 110 (2) 132 13 13 (3) 142 (4) 151 13 13

Q66. lim cos(sin x)−cos x is equal to x→0 x4 (1) 1 (2) 1 3 6 (3) 1 (4) 1 4 12

Q66.The acute angle between the pair of tangents drawn to the ellipse 2𝑥2 + 3𝑦2 = 5 from the point 1, 3 is 16 24 (1) tan-1 (2) tan-1 7√5 7√5 32 + 8√5 (3) tan-1 (4) tan-13 7√5 35

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.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.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 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 P : y2 = 4ax, a > 0 be a parabola with focus S .Let the tangents to the parabola P make an angle of π4 with the line y = 3x + 5 touch the parabola P at A and B . Then the value of a for which A, B and S are collinear is: (1) 8 only (2) 2 only (3) 1 only (4) any a > 0 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.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 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 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.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 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