Practice Questions

Q68.Let 𝑎, 𝑏 and 𝑐 be the length of sides of a triangle 𝐴𝐵𝐶 such that 𝑎+ 𝑏 = 𝑏+ 𝑐 = 𝑐+ 𝑎 . If 𝑟 and 𝑅 are the radius of 7 8 9 𝑅 incircle and radius of circumcircle of the triangle 𝐴𝐵𝐶, respectively, then the value of is equal to 𝑟 (1) 2 (2) 3 5 (3) 5 (4) 1 2

Q68.Which of the following statement is a tautology? (1) ((~q) ∧p) ∧q (2) ((~q) ∧p) ∧(p ∧(~p)) (3) ((~q) ∧p) ∨(p ∨(~p)) (4) (p ∧q) ∧(~(p ∧q))

Q68.If the truth value of the statement (P ∧(~R)) →((~R) ∧Q) is F , then the truth value of which of the following is F ? (1) P ∨Q →~R (2) R ∨Q →~P (3) ~(P ∨Q) →~R (4) ~(R ∨Q) →~P

Q68.A tower 𝑃𝑄 stands on a horizontal ground with base 𝑄 on the ground. The point 𝑅 divides the tower in two parts such that 𝑄𝑅= 15m. If from a point 𝐴 on the ground the angle of elevation of 𝑅 is 60° and the part 𝑃𝑅 of the tower subtends an angle of 15° at 𝐴, then the height of the tower is (1) 52√3 + 3m (2) 5√3 + 3m (3) 10√3 + 1m (4) 102√3 + 1m

Q68.Let the system of linear equations x + 2y + z = 2, αx + 3y −z = α, −αx + y + 2z = −α be inconsistent. Then α is equal to (1) 2 5 (2) −52 (3) 2 7 (4) −72

Q68.Let f(x) = ax2 + bx + c be such that f(1) = 3, f(−2) = λ and f(3) = 4. If f(0) + f(1) + f(−2) + f(3) = 14 , then λ is equal to JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper (1) −4 (2) 132 (3) 23 (4) 4 2

Q68.Let the system of linear equations x + y + az = 2 3x + y + z = 4 x + 2z = 1 have a unique solution ( x∗, y∗, z∗). If ( (a, x∗), (y∗, α) and ( x∗, −y∗) are collinear points, then the sum of absolute values of all possible values of α is: (1) 4 (2) 3 (3) 2 (4) 1

Q68.Let the operations * , ⊙∈∧, ∨. If 𝑝* 𝑞⊙𝑝⊙~𝑞 is a tautology, then the ordered pair * , ⊙ is (1) ∨, ∧ (2) ∨, ∨ (3) ∧, ∧ (4) ∧, ∨ JEE Main 2022 (28 Jul Shift 1) JEE Main Previous Year Paper

Q68.The number of choices for Δ ∈{∧, ∨, ⇒, ⇔} , such that (pΔq) ⇒((pΔ~q) ∨((~p)Δq)) is a tautology, is (1) 1 (2) 2 (3) 3 (4) 4 Q69. ⎡ 1 0 a ⎤ Let S ={ √n : 1 ⩽n ⩽50 and n is odd}. Let a ∈S and A = −1 1 0 . If Σ det (adj A) = 100λ, then λ ⎣−a 0 1 ⎦ a∈S is equal to (1) 218 (2) 221 (3) 663 (4) 1717

Q68.The angle of elevation of the top of a tower from a point A due north of it is α and from a point B at a distance of 9 units due west of A is . If the distance of the point B from the tower is 15 units, then cot α is cos−1( √133 ) equal to (1) 6 (2) 9 5 5 (3) 4 (4) 7 3 3

Q68. (p ∧r) ⇔(p ∧(~q)) is equivalent to (~p) when r is (1) p (2) ~p (3) q (4) ~q

Q68.Let the foci of the ellipse x2 coincide. Then the length of the 16 + 7 = 1 and the hyperbola 144x2 −y2α = 251 latus rectum of the hyperbola is: (1) 32 (2) 18 9 5 (3) 27 (4) 27 4 10 8√2−(cos x+sin x)7

Q68.The statement 𝑝⇒𝑞∨𝑝⇒𝑟 is NOT equivalent to: (1) 𝑝∧~𝑟⇒𝑞 (2) ~𝑞⇒~𝑟∨𝑝 (3) 𝑝⇒𝑞∨𝑟 (4) 𝑝∧~𝑞⇒𝑟

Q68.Let 𝛽= lim 𝛼𝑥- 𝑒3𝑥- 1 for some 𝛼∈ℝ. Then the value of 𝛼+ 𝛽 is: 𝑥→0 𝛼𝑥𝑒3𝑥- 1 14 3 (1) (2) 5 2 (3) 5 (4) 7 2 2

Q68.The line 𝑦= 𝑥+ 1 meets the ellipse 𝑥2 + 𝑦2 = 1 at two points 𝑃 and 𝑄. If 𝑟 is the radius of the circle with 𝑃𝑄 4 2 as diameter then 3𝑟2 is equal to (1) 20 (2) 12 (3) 11 (4) 8 Q69. 12 12 lim tan2𝑥2sin2𝑥+ 3sin𝑥+ 4 - sin2𝑥+ 6sin𝑥+ 2 is equal to 𝑥→𝜋 2 1 1 (1) (2) - 12 18 (3) - 1 (4) 1 12 6

Q69.Let a set A = A1 ∪A2 ∪… ∪Ak , where Ai ∩Aj = ϕ for i ≠j; 1 ≤i, j ≤k. Define the relation R from A to A by R ={ (x, y) : y ∈Ai if and only if x ∈Ai, 1 ≤i ≤k}. Then, R is: (1) reflexive, symmetric but not transitive (2) reflexive, transitive but not symmetric (3) reflexive but not symmetric and transitive (4) an equivalence relation JEE Main 2022 (29 Jun Shift 1) JEE Main Previous Year Paper

Q69. is equal to lim x→π4 √2−√2 sin 2x (1) 14 (2) 7 (3) 14√2 (4) 7√2

Q69.For 𝛼∈𝑁, consider a relation 𝑅 on 𝑁 given by 𝑅= {𝑥, 𝑦: 3𝑥+ 𝛼𝑦 is a multiple of 7}. The relation 𝑅 is an equivalence relation if and only if (1) 𝛼= 14 (2) 𝛼 is a multiple of 4 (3) 4is the remainder when 𝛼 is divided by 10 (4) 4 is the remainder when 𝛼 is divided by 7 Q70. 0 1 0 Let the matrix 𝐴= 1 0 0 and the matrix 𝐵0 = 𝐴49 + 2𝐴98. If 𝐵𝑛= Adj𝐵𝑛- 1 for all 𝑛≥1, then det 𝐵4 is 0 0 1 equal to (1) 328 (2) 330 (3) 332 (4) 336

Q69.If the system of linear equations 2x + 3y −z = −2 x + y + z = 4 x −y + |λ|z = 4λ −4 where λ ∈R, has no solution, then (1) λ = 7 (2) λ = −7 (3) λ = 8 (4) λ2 = 1 Q70. ⎡ 2n, n = 2, 4, 6, 8, … . . Let a function f : N →N be defined by f(n) = n −1, n = 3, 7, 11, 15, … . . n+1 ⎣ 2 , n = 1, 5, 9, 13, … . . then, f is (1) One-one and onto (2) One-one but not onto (3) Onto but not one-one (4) Neither one-one nor onto JEE Main 2022 (28 Jun Shift 1) JEE Main Previous Year Paper Q71. ⎡[ex], x < 0 aex + [x −1], 0 ≤x < 1 Let f : R →R be defined as f(x) = b + [sin(πx)], 1 ≤x < 2 ⎣[e−x] −c, x ≥2 where a, b, c ∈R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true? (1) There exists a, b, c ∈R such that f is continuous (2) If f is discontinuous at exactly one point, then of R. a + b + c = 1 (3) If f is discontinuous at exactly one point, then (4) f is discontinuous at atleast two points, for any a + b + c ≠1 . values of a, b and c.

Q69.Let 𝐴= 0 -2 . If 𝑀 and 𝑁 are two matrices given by 𝑀= ∑𝑘=10 1 𝐴2𝑘 and 𝑁= ∑𝑘=10 1 𝐴2𝑘- 1 then 𝑀𝑁2 2 0 is (1) a non-identity symmetric matrix (2) a skew-symmetric matrix (3) neither symmetric nor skew-symmetric matrix (4) an identity matrix JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper Q70. 1 1 1 -1 0 1 Let 𝐴 be a 3 × 3 real matrix such that 𝐴 1 = 1 ; 𝐴 0 = 0 and 𝐴 0 = 1 . If 𝑋= 𝑥1 𝑥2 𝑥3𝑇 0 0 1 1 1 2 4 and 𝐼 is an identity matrix of order 3, then the system 𝐴- 2𝐼𝑋= 1 has 1 (1) no solution (2) infinitely many solutions (3) unique solution (4) exactly two solutions

Q69.If the system of equations αx + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β. Has infinitely many solutions, then the ordered pair (α, β) is equal to (1) (1, −3) (2) (−1, 3) (3) (1, 3) (4) (−1, −3)

Q69.Let x × y = x2 + y3 and (x × 1) × 1 = x × (1 × 1). Then a value of 2 sin−1( x4+x2−2x4+x2+2 ) is (1) π (2) π 4 3 (3) π (4) π 6 JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper Q70. , x ∈(−2, −1) ⎧ sin(x−[x])x−[x] Let f(x) = max(2x, 3[|x|]), |x| < 1 ⎨ ⎩1, otherwise where [t] denotes greatest integer ≤t. If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, the ordered pair (m, n) is: (1) (3, 3) (2) (2, 4) (3) (2, 3) (4) (3, 4)

Q69.Which of the following matrices can NOT be obtained from the matrix -1 2 by a single elementary row 1 -1 operation? (1) 0 1 (2) 1 -1 1 -1 -1 2 (3) -1 2 (4) -1 2 -2 7 -1 3

Q69.Negation of the Boolean statement (p ∨q) ⇒((~r) ∨p) is equivalent to: (1) p ∧(~q) ∧r (2) (~p) ∧(~q) ∧r (3) (~p) ∧q ∧r (4) p ∧q ∧(~r)

Q69.Let R1 = {(a, b) ∈N × N : |a −b| ≤13} and R2 = {(a, b) ∈N × N : |a −b| ≠13} Then on N : (1) Both R1 and R2 are equivalence relations (2) Neither R1 nor R2 is an equivalence relation (3) R1 is an equivalence relation but R2 is not (4) R2 is an equivalence relation but R1 is not