Practice Questions

Q71.If the line y = mx + 7√3 is normal to the hyperbola x224 −y218 = 1 (1) √5 (2) 3 2 √5 (3) √15 (4) 2 2 √5

Q72. lim sin2𝑥 equals 𝑥→0 √2 - √1 + cos𝑥 (1) 4√2 (2) 2√2 (3) √2 (4) 4

Q72.Let 𝑓: 𝑅→𝑅 be a differentiable function satisfying 𝑓'3 + 𝑓'2 = 0 . Then lim is equal to 𝑥→0 1 + 𝑓2 - 𝑥- 𝑓2 (1) 1 (2) e (3) 𝑒2 (4) e-1

Q72.For any two statement p and q, the negative of the expression p ∨(~p ∧q) is (1) ~p ∨~q (2) p ∧q (3) ~p ∧~q (4) p ↔q

Q72.Let P(4, −4) and Q(9, 6) be two points on the parabola, y2 = 4x and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of ΔPXQ is maximum. Then this maximum area (in sq. units) is : (1) 625 (2) 75 4 2 (3) 125 (4) 125 4 2

Q72.If x3−k3 , then k is lim lim x−1 = x2−k2 x→1 x→k (1) 3 (2) 4 2 3 (3) 3 (4) 8 8 3

Q72.If 5𝑥+ 9 = 0 is the directrix of the hyperbola 16𝑥2 - 9𝑦2 = 144, then its corresponding focus is: (1) -5, 0 (2) 5, 0 5 5 (3) - 3, 0 (4) 3, 0

Q72.If the tangent to the parabola y2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse, x2 + 2y2 = 1 then α is equal to: (1) √2 −1 (2) 2√2 + 1 (3) √2 + 1 (4) 2√2 −1

Q72.The equation of a tangent to the hyperbola, 4x2 −5y2 = 20, parallel to the line x −y = 2, is (1) x −y + 7 = 0 (2) x −y −3 = 0 (3) x −y + 1 = 0 (4) x −y + 9 = 0 (1−|x|+sin|1−x|)sin([1−x] π2 )

Q72.An ellipse, with foci at (0,2) and (0, −2) and minor axis of length 4 , passes through which of the following points? (1) (1, 2√2) (2) (2, √2) (3) (√2, 2) (4) (2, 2√2)

Q72.If the truth value of the statement 𝑝→~𝑞∨𝑟 is false 𝐹, then the truth values of the statements 𝑝, 𝑞, 𝑟 are respectively JEE Main 2019 (12 Apr Shift 1) JEE Main Previous Year Paper (1) 𝑇, 𝐹, 𝑇 (2) 𝑇, 𝐹, 𝐹 (3) 𝑇, 𝑇, 𝐹 (4) 𝐹, 𝑇, 𝑇

Q72.Let 0 < 𝜃< 𝜋 . If the eccentricity of the hyperbola 𝑥2 𝑦2 1 is greater than 2, then the length of its 2 cos2⁡𝜃- sin2⁡𝜃= latus rectum lies in the interval: (1) 3, ∞ (2) 1, 3 2 3 (3) 2, 3 (4) 2, 2 Q73. √1 + √1 + 𝑦4 - √2 The value of lim 𝑦→0 𝑦4 JEE Main 2019 (09 Jan Shift 1) JEE Main Previous Year Paper 1 1 (1) exists and equals (2) exists and equals 2√2 4√2 1 (3) does not exist (4) exists and equals 2√2√2 + 1

Q72.If the mean and standard deviation of 5 observations x1, x2, x3, x4, x5 are 10 and 3, respectively, then the variance of 6 observations x1, x2, … , x5 and −50 is equal to (1) 582.5 (2) 507.5 (3) 509.5 (4) 586.5

Q73.A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x −axis. Then the eccentricity of the hyperbola is: (1) √3 (2) 32 (3) 2 (4) 2 √3

Q73.Given b+c 11 = c+a12 = a+b13 for a ΔABC with usual notation. If cosα A = cosβ B = cosγ C , then the ordered triad (α, β, γ) has a value (1) (7,19,25) (2) (3,4,5) (3) (5,12,13) (4) (19,7,25)

Q73.Which one of the following Boolean expression is a tautology? (1) (p ∨q) ∧(~p ∨~q) (2) (p ∧q) ∨(p ∧~q) (3) (p ∨q) ∧(p ∨~q) (4) (p ∨q) ∨(~p ∨~q)

Q73.If the standard deviation of the numbers −1, 0, 1, k is √5 where k > 0, then k is equal to JEE Main 2019 (09 Apr Shift 1) JEE Main Previous Year Paper (1) √6 (2) 4√53 (3) 2√103 (4) 2√6 then the inverse of is: … . =

Q73.The equation of a common tangent to the curves, y2 = 16x and xy = −4, is: (1) x −2y + 16 = 0 (2) x −y + 4 = 0 (3) 2x −y + 2 = 0 (4) x + y + 4 = 0 JEE Main 2019 (12 Apr Shift 2) JEE Main Previous Year Paper

Q73.If f(x) = [x] −[ x4 ], x ∈R, where [x] denotes the greatest integer function, then: (1) x→4+f(x)lim exists but x→4−f(x)lim does not exist (2) f is continuous at x = 4 (3) x→4−f(x)lim exists but x→4+f(x)lim does not exist (4) Both x→4−f(x)lim and x→4+f(x)lim exist but are not equal

Q73.The expression ~(~p →q) is logically equivalent to (1) p ∧~q (2) ~p ∧~q (3) p ∧q (4) ~p ∧q

Q73.Which one of the following statements is not a tautology? (1) 𝑝∨𝑞→𝑝∨( ~𝑞) (2) 𝑝∧𝑞→( ~𝑝∨𝑞) (3) 𝑝→𝑝∨𝑞 (4) 𝑝∧𝑞→𝑝 JEE Main 2019 (08 Apr Shift 2) JEE Main Previous Year Paper

Q73.If the data 𝑥1, 𝑥2, … 𝑥10 is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2000, then the standard deviation of this data is: (1) 2√2 (2) 4 (3) 2 (4) √2 Q74. 5 2𝛼 1 If 𝐵= 0 2 1 is the inverse of a 3 × 3 matix 𝐴, then the sum of all values of 𝛼 for which 𝑑𝑒𝑡𝐴+ 1 = 0, 𝛼 3 -1 is: (1) 2 (2) 1 (3) 0 (4) -1

Q73.With the usual notation in ΔABC , if ∠A + ∠B = 120°, a = √3 + 1 units and b = √3 −1 units, then the ratio ∠A : ∠B is (1) 7 : 1 (2) 9 : 7 (3) 3 : 1 (4) 5 : 3 Q74. 2 b 1 is: Let A = ⎡ b b2 + 1 b ⎤ , where b > 0 . Then the minimum value of det(A)b 1 b 2 ⎣ ⎦ (1) 2√3 (2) −2√3 (3) √3 (4) −√3

Q73.For each t ∈R, let [t] be the greatest integer less than or equal to t. Then, lim x→1+ |1−x|[1−x] (1) equals 0 (2) equals −1 (3) does not exist (4) equal 1

Q73.If the vertices of a hyperbola be at (−2, 0) and (2, 0) and one of its foci be at (−3, 0), then which one of the following points does not lie on this hyperbola ? (1) (6, 5√2) (2) (−6, 2√10) (3) (2√6, 5) (4) (4, √15)