Practice Questions

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

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 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.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.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

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.Contrapositive of the statement "If two numbers are not equal, then their squares are not equal". is : (1) If the squares of two numbers are not equal, then (2) If the squares of two numbers are equal, then the the numbers are equal. numbers are not equal. (3) If the squares of two numbers are equal, then the (4) If the squares of two numbers are not equal, then numbers are equal. the numbers 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.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.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.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.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.The contrapositive of the statement “If you are born in India, then you are a citizen of India”, is (1) If you are not born in (2) If you are a citizen of (3) If you are born in (4) If you are not a citizen India, then you are not India, then you are India, then you are not of India, then you are a citizen of India. born in India. a citizen of India. not born in India.

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

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.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.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.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.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 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

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

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.Consider the statement: " P(n) : n2 −n + 41 is prime". Then which one of the following is true? (1) P(3) is false but P(5) is true (2) Both P(3) and P(5) are false (3) Both P(3) and P(5) are true (4) P(5) is false but P(3) is true