Practice Questions

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.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

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.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.If lim 𝑥2 - 𝑎𝑥+ 𝑏 = 5, then 𝑎+ 𝑏 is equal to: 𝑥→1 𝑥- 1 (1) 1 (2) 5 (3) – 4 (4) – 7

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.Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is : JEE Main 2019 (11 Jan Shift 1) JEE Main Previous Year Paper (1) x + y + 1 = 0 (2) x −2y + 4 = 0 (3) x + 2y + 4 = 0 (4) 4x + 2y + 1 = 0

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)

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

Q74.The mean and the variance of five observations are 4 and 5.20, respectively. If three of the observations are 3, 4 and 4; then the absolute value of the difference of the other two observations, is : (1) 3 (2) 5 (3) 7 (4) 1

Q74.If the Boolean expression 𝑝⊕𝑞∧~𝑝⊙𝑞 is equivalent to 𝑝∧𝑞, where ⊕, ⊙∈∧, ∨, then the ordered pair ⊕, ⊙ is (1) ∨, ∧ (2) ∧, ∧ (3) ∨, ∨ (4) ∧, ∨ Q75.5 students of a class have an average height 150 𝑐𝑚 and variance 18 𝑐𝑚2 . A new student, whose height is 156 𝑐𝑚, joined them. The variance in 𝑐𝑚2 of the height of these six students is: (1) 22 (2) 16 (3) 18 (4) 20

Q74.The mean and variance for seven observations are 8 and 16 respectively. If 5 of the observations are 2, 4, 10, 12, 14, then the product of the remaining two observations is (1) 48 (2) 45 (3) 49 (4) 40

Q74. lim x+2sinx is x→0 √x2+2 sin x+1 − √sin2x−x+1 (1) 3 (2) 1 (3) 2 (4) 6

Q74.If [10 11 ][10 21 ][10 31 ] [10 n −11 ] [10 781 ], [10 n1 ] (1) [10 –121 ] (2) [121 10 ] (3) [131 10 ] (4) [10 –131 ]

Q74. a −b −c 2a 2a If 2b b −c −a 2b = (a + b + c)(x + a + b + c)2, x ≠0 and a + b + c ≠0, then x is 2c 2c c −a −b equal to (1) abc (2) −(a + b + c) (3) 2(a + b + c) (4) −2(a + b + c) = 8 and det (AB−1) = 8, then det

Q74.For each x ∈R, let [x] be the greatest integer less than or equal to x . Then lim x([x]+|x|) sin[x] is equal to x→0− |x| (1) 1 (2) 0 (3) −sin 1 (4) sin 1

Q74.If for some x ∈R, the frequency distribution of the marks obtained by 20 students in a test is: Marks 2 3 5 7 Frequency distribution (x + 1)2 (2x −5) x2 −3x x JEE Main 2019 (10 Apr Shift 1) JEE Main Previous Year Paper Then the mean of the marks is : (1) 3.0 (2) 2.5 (3) 3.2 (4) 2.8

Q74. lim cot3x−tanxπ is x→π4 cos(x+ 4 ) (1) 4√2 (2) 8√2 (3) 4 (4) 8

Q74.A student scores the following marks in five tests: 45,54,41,57,43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is: 10 100 (1) (2) 3 3 (3) 10 (4) 100 √3 √3

Q75.The Boolean expression ((p ∧q) ∨(p ∨~q)) ∧(~p ∧~q) is equivalent to (1) p ∧(~q) (2) (~p) ∧(~q) (3) p ∨(~q) (4) p ∧q

Q75.The logical statement [~(~p ∨q) ∨(p ∧r)] ∧(~q ∧r) is equivalent to (1) (~p ∧~q) ∧r (2) (p ∧r) ∧~q (3) (p ∧~q) ∨r (4) ~p ∨r