Practice Questions

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

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. lim sin(cos−1 x)−x is equal to 1−tan(cos−1 x) x→1 √2 (1) 1 (2) −1 √2 √2 (3) √2 (4) −1

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 x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to _____ (1) 16 (2) 885 (3) 72 (4) −8 is equal to

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.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.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.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 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.Let P(a, b) be a point on the parabola y2 = 8x such that the tangent at P passes through the centre of the circle x2 + y2 −10x −14y + 65 = 0 . Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to (1) 0 (2) 25 (3) 40 (4) 65 + [2 −x], a ∈R, where [t] is the greatest integer

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.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

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.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.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.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 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.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.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.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.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.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 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.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