Practice Questions

Q67.If the equation of the parabola, whose vertex is at (5, 4) and the directrix is 3x + y −29 = 0, is x2 + ay2 + bxy + cx + dy + k = 0, then a + b + c + d + k is equal to (1) 575 (2) −575 (3) 576 (4) −576

Q67.Let λx −2y = μ be a tangent to the hyperbola a2x2 −y2 = b2 . Then ( λa ) 2 −( μb )2 (1) −2 (2) −4 (3) 2 (4) 4

Q67.Consider the following two propositions : 𝑃1: ~𝑝→~𝑞 𝑃2: 𝑝∧~𝑞∧~𝑝∨𝑞 If the proposition 𝑝→~𝑝∨𝑞 is evaluated as FALSE, then (1) 𝑃1 is TRUE and 𝑃2 is FALSE (2) 𝑃1 is FALSE and 𝑃2 is TRUE (3) Both 𝑃1 and 𝑃2 are FALSE (4) Both 𝑃1 and 𝑃2 are TRUE

Q67.Let 𝐴𝛼, - 2, 𝐵𝛼, 6 and 𝐶𝛼 - 2 be vertices of a ∆𝐴𝐵𝐶. If 5, 𝛼 is the circumcentre of ∆𝐴𝐵𝐶, then which of the 4, 4 following is NOT correct about ∆𝐴𝐵𝐶 (1) ares is 24 (2) perimeter is 25 (3) circumradius is 5 (4) inradius is 2

Q67.If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is x + y = a, is 16 , then |a| is equal to (1) 2√2 (2) 2√3 (3) 4√2 (4) 4 JEE Main 2022 (27 Jul Shift 2) JEE Main Previous Year Paper

Q67.Consider the following statements: A: Rishi is a judge. B: Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is (1) B →(A ∨C) (2) (~B) ∧(A ∧C) (3) B →((~A) ∨(~C)) (4) B →(A ∧C)

Q67.Let A be a 2 × 2 matrix with det(A) = −1 and det((A + I)(Adj(A) + I)) = 4 . Then the sum of the diagonal elements of A can be: (1) −1 (2) 2 (3) 1 (4) −√2

Q67.If the ellipse x2 = 1 on the y-axis, a2 + b2 = 1 meets the line x7 + 2√6y = 1 on the x-axis and the line x7 − 2√6y then the eccentricity of the ellipse is (1) 5 (2) 2√6 7 7 (3) 3 (4) 2√5 7 7 y2

Q67.Let r ∈(P, q, ~p, ~q) be such that the logical statement r ∨(~p) ⇒(p ∧q) ∨r is a tautology. Then r is equal to (1) p (2) q (3) ~p (4) ~q

Q67.If the line 𝑥- 1 = 0, is a directrix of the hyperbola 𝑘𝑥2 - 𝑦2 = 6, then the hyperbola passes through the point (1) -2√5, 6 (2) -√5, 3 (3) √5, - 2 (4) 2√5, 3√6

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.The mean of the numbers a, b, 8, 5, 10 is 6 and their variance is 6. 8. If M is the mean deviation of the numbers about the mean, then 25M is equal to (1) 60 (2) 55 (3) 50 (4) 75

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

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.The value of lim (x2−1) sin2(πx) is equal to: x→1 x4−2x3+2x−1 (1) π2 (2) π2 6 3 (3) π2 (4) π2 2

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.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.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.Let the mean and the variance of 5 observations x1, x2, x3, x4, x5 be 245 and 19425 respectively. If the mean and variance of the first 4 observation are 27 and a respectively, then (4a + x5) is equal to (1) 13 (2) 15 (3) 17 (4) 18

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.Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola x2 −y2 = 1. Let e′ and l′ respectively the eccentricity and length of the latus rectum of its conjugate a2 b2 hyperbola. If e2 = 1411 l and (e′)2 = 118 l′ , then the value of 77a + 44b is equal to (1) 100 (2) 110 (3) 120 (4) 130

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 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