Practice Questions

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

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.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 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.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)

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.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.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.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.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.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.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.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

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.The negation of the Boolean expression ~𝑠∨~𝑟∧𝑠 is equivalent to (1) 𝑟 (2) 𝑠∧𝑟 (3) 𝑠∨𝑟 (4) ~𝑠∧~𝑟

Q74.If p ⇒(q ∨r) is False , then the truth values of p, q, r are respectively, (where T is True and F is False ) (1) T, F, F (2) F, T, T (3) F, F, F (4) T, T, F

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.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.Let [x] denote the greatest integer less than or equal to X . Then : limx→0 tan(π sin2 x)+(|x|−sin(x[x]))2x2 (1) does not exist (2) equals π (3) equals π + 1 (4) equals 0

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