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Q77.Let S be the set of all a ∈R for which the angle between the vectors u = a(loge b)Λ†i βˆ’6Λ†j + 3Λ†k and β†’v= (loge b)Λ†i + 2Λ†j + 2a(loge b)Λ†k, (b > 1) is acute. Then S is equal to (1) (βˆ’βˆž, βˆ’43 ) (2) Ξ¦ (3) (βˆ’43 , 0) (4) ( 127 , ∞) JEE Main 2022 (28 Jul Shift 2) JEE Main Previous Year Paper

202228 Jul Shift 2Vectors
MathsMedium

Q77.Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 tan x(cos x βˆ’y). if the curve passes Ο€ through the point ( Ο€4 , 0), then the value of ∫ 0 2 ydx is equal to (1) (2 βˆ’βˆš2) + √2Ο€ (2) 2 βˆ’ √2Ο€ (3) (2 + √2) + √2Ο€ (4) 2 + √2Ο€ β†’

202228 Jun Shift 2Differential Equations
MathsMedium

Q77.The area bounded by the curves 𝑦= π‘₯2 - 1 and 𝑦= 1 is (1) 2 + 1 (2) 4 - 1 3√2 3√2 8 (3) 2√2 - 1 (4) 3√2 - 1

202226 Jul Shift 2Definite Integration & Area
MathsMedium

Q77.Let β†’a and b be the vectors along the diagonal of a parallelogram having area 2√2. Let the angle between β†’a and β†’ β†’ β†’ β†’ β†’ β†’ Γ— βˆ’2b, then an angle between b and β†’cis b be acute. β†’a = 1 and β†’a. b = β†’aΓ— b . If β†’c= 2√2(β†’a b) (1) βˆ’Ο€ (2) 5Ο€ 4 6 (3) Ο€ (4) 3Ο€ 3 4 P . Then the

202227 Jun Shift 2Differential Equations
MathsMedium

Q77.Let β†’a = Λ†i + Λ†j βˆ’Λ†k and β†’c= 2Λ†i βˆ’3Λ†j + 2Λ†k. Then the number of vectors b such that b Γ—β†’c=β†’a and β†’ b ∈{1, 2, … , 10} is (1) 0 (2) 1 (3) 2 (4) 3

202227 Jun Shift 1Vectors
MathsHard

Q77.If two distinct point Q, R lie on the line of intersection of the planes βˆ’x + 2y βˆ’z = 0 and 3x βˆ’5y + 2z = 0 and PQ = PR = √18 where the point P is (1, βˆ’2, 3), then the area of the triangle PQR is equal to (1) 2 3 √38 (2) 43 √38 (3) 8 3 √38 (4) √1523

202228 Jun Shift 13D Geometry
MathsHard

Q77.If 2, 3, 9, 5, 2, 1, 1, πœ†, 8 and πœ†, 2, 3 are coplanar, then the product of all possible values of πœ† is (1) 21 (2) 59 2 8 57 95 (3) (4) 8 8

202229 Jul Shift 2Vectors
MathsMedium

Q77.Let β†’a = Λ†i βˆ’Λ†j + 2Λ†k and let b be a vector such that β†’aΓ— b = 2Λ†i βˆ’Λ†k and β†’aβ‹… b = 3 . Then the projection of b on the β†’ vector β†’aβˆ’ b is: (1) 2 (2) √21 2√37 (3) 2 (4) 2 3 3 √73

202225 Jul Shift 2Vectors
MathsHard

Q77.Let β†’a = Ξ±Λ†i + Λ†j βˆ’Λ†k and b = 2Λ†i + Λ†j βˆ’Ξ±Λ†k, Ξ± > 0 . If the projection of β†’aΓ— b on the vector βˆ’Λ†i + 2Λ†j βˆ’2Λ†k is 30 , then Ξ± is equal to (1) 15 (2) 8 2 (3) 13 (4) 7 2

202226 Jul Shift 1Vectors
MathsMedium

Q77.The area of the region enclosed between the parabolas 𝑦2 = 2π‘₯- 1 and 𝑦2 = 4π‘₯- 3 is. 1 1 (1) (2) 3 6 2 3 (3) (4) 3 4

202225 Jun Shift 2Definite Integration & Area
MathsMedium

Q77.Let β†’a = 3Λ†i + Λ†j andβ†’b = Λ†i + 2Λ†j + Λ†k. Let β†’cbe a vector satisfying β†’aΓ— (β†’ Γ—β†’c) parallel, then the value of Ξ» is (1) βˆ’5 (2) 5 (3) 1 (4) βˆ’1 ΞΈ is the angle between the vectors

202229 Jul Shift 1Vectors
MathsMedium

Q77.Let 𝐴𝐡𝐢 be a triangle such that 𝐡𝐢= β†’π‘Ž, 𝐢𝐴= 𝑏, 𝐴𝐡= →𝑐, β†’π‘Ž= 6√2, 𝑏= 2√3 and 𝑏· →𝑐= 12 Consider the statements : 𝑆1: β†’π‘ŽΓ— →𝑏+ →𝑐× →𝑏- →𝑐= 62√2 - 1 𝑆2: ∠𝐴𝐡𝐢= cos-1√ 23. Then (1) both 𝑆1 and 𝑆2are true (2) only 𝑆1 is true (3) only 𝑆2 is true (4) both 𝑆1 and 𝑆2 are false π‘₯- 3 𝑦+ 4 𝑧- 7

202225 Jul Shift 1Vectors
MathsMedium

Q77.Let β†’a = Ξ±Λ†i + Λ†j + Ξ²Λ†k and b = 3Λ†i βˆ’5Λ†j + 4Λ†k be two vectors, such that β†’aΓ— b = βˆ’Λ†i + 9Λ†i + 12Λ†k. Then the β†’ β†’ projection of b βˆ’2β†’a on b +β†’a is equal to (1) 2 (2) 395 (3) 9 (4) 465 β†’ β†’ β†’ 23 Γ— b Γ— 2Λ†j is equal to β‹…Λ†k = 2 , then

202227 Jul Shift 1Vectors
MathsMedium

Q77.If the length of the perpendicular drawn from the point P(a, 4, 2), a > 0 on the line x+12 = yβˆ’33 = zβˆ’1βˆ’1 is 2√6 units and Q(Ξ±1, Ξ±2, Ξ±3) is the image of the point P in this line, then a + βˆ‘3i=1 Ξ±i is equal to (1) 7 (2) 8 (3) 12 (4) 14

202227 Jul Shift 23D Geometry
MathsMedium

Q77.Let the vectors β†’π‘Ž= 1 + 𝑑 ^𝑖+ 1 - 𝑑 ^𝑗+ ^π‘˜, →𝑏= 1 - 𝑑 ^𝑖+ 1 + t ^𝑗+ 2 ^π‘˜ and →𝑐= 𝑑 ^𝑖- 𝑑 ^𝑗+ ^π‘˜, π‘‘βˆˆπ‘… be such that for 𝛼, 𝛽, π›Ύβˆˆπ‘…, 𝛼 β†’π‘Ž+ 𝛽 →𝑏+ 𝛾 →𝑐= β†’0 ⇒𝛼= 𝛽= 𝛾= 0. Then, the set of all values of 𝑑 is (1) a non-empty finite set (2) equal to 𝑁 (3) equal to 𝑅- 0 (4) equal to 𝑅

202228 Jul Shift 1Vectors
MathsMedium

Q77.Let a and b be two unit vectors such that |(a + b) + 2(a Γ— b)| = 2. If ΞΈ ∈(0, Ο€) is the angle between Λ†a and Λ†b , then among the statements: (S1) : 2 Λ†a Γ— Λ†b = Λ†a βˆ’Λ†b is 1 + (S2) : The projection of Λ†a on 2 (Λ†a Λ†b) (1) Only (S1) is true. (2) Only (S2) is true. (3) Both (S1) and (S2) are true. (4) Both (S1) and (S2) are false. JEE Main 2022 (24 Jun Shift 2) JEE Main Previous Year Paper

202224 Jun Shift 2Vectors
MathsMedium

Q77.If the solution curve 𝑦= 𝑦π‘₯ of the differential equation 𝑦2 dπ‘₯+ π‘₯2 - π‘₯𝑦+ 𝑦2d𝑦= 0, which passes through the point 1, 1 and intersects the line 𝑦= √3π‘₯ at the point 𝛼, √3𝛼, then value of logπ‘’βˆš3𝛼 is equal to πœ‹ πœ‹ (1) (2) 2 4 (3) πœ‹ (4) πœ‹ 6 12 JEE Main 2022 (25 Jun Shift 1) JEE Main Previous Year Paper

202225 Jun Shift 1Differential Equations
MathsHard

Q77.If dy + ex(x2 βˆ’2)y = (x2 βˆ’2x)(x2 βˆ’2)e2x and y(0) = 0 , then the value of y(2) is dx (1) βˆ’1 (2) 1 (3) 0 (4) e β†’

202226 Jun Shift 2Differential Equations
MathsMedium

Q77.If β†’aβ‹… b = 1, b β‹…β†’c= 2 and β†’cβ‹…β†’a = 3 , then the value of [β†’a ( Γ—β†’c) ( Γ—β†’a)] b (1) 0 (2) βˆ’6β†’aβ‹…(β†’ Γ—β†’c) β†’ βˆ’12b β‹…(β†’cΓ—β†’a) (3) 12β†’cβ‹…(β†’aΓ—β†’b) (4)

202226 Jun Shift 1Vectors
MathsMedium

Q77.If x = x(y) is the solution of the differential equation y dxdy = 2x + y3(y + 1)ey, x(1) = 0 ; then x(e) is equal to (1) ee(e3 βˆ’1) (2) e3(ee βˆ’1) (3) ee βˆ’1 (4) ee(e2 βˆ’1) Γ—

202224 Jun Shift 1Differential Equations
MathsHard

Q77.Let A, B, C be three points whose position vectors respectively are: β†’a = Λ†i + 4Λ†j + 3Λ†k β†’ b = 2Λ†i + Ξ±Λ†j + 4Λ†k, Ξ± ∈R β†’c= 3Λ†i βˆ’2Λ†j + 5Λ†k β†’ If Ξ± is the smallest positive integer for which β†’a, b, β†’care non-collinear, then the length of the median, β–³ABC , through A is: (1) √82 (2) √62 2 2 (3) √69 (4) √66 2 2 y+1

202229 Jun Shift 2Vectors
MathsMedium

Q77.The area enclosed by y2 = 8x and y = √2x that lies outside the triangle formed by y = √2x, x = 1, y = 2√2 , is equal to (1) 16√2 (2) 11√2 6 6 (3) 13√2 (4) 5√2 6 6

202229 Jun Shift 1Definite Integration & Area
MathsMedium

Q78.Let Λ†a and Λ†b be two unit vectors such that the angle between them is Ο€4 . If and + Γ— then the value of 164 cos2 ΞΈ is equal to (Λ†a Λ†b) (Λ†a + 2Λ†b + 2(Λ†a Λ†b)) (1) 90 + 27√2 (2) 45 + 18√2 (3) 90 + 3√2 (4) 54 + 90√2

202229 Jul Shift 1Vectors
MathsMedium

Q78.Let β†’a = Ξ±Λ†i + 2Λ†j βˆ’Λ†k and b = βˆ’2Λ†i + Ξ±Λ†j + Λ†k, where Ξ± ∈R. If the area of the parallelogram whose adjacent β†’ 2 β†’ β†’ 2 b is equal to β‹… sides are represented by the vectors β†’a and b is √15(Ξ±2 + 4), then the value of 2β†’a + (β†’a b) (1) 10 (2) 7 (3) 9 (4) 14 + = 2Λ†i βˆ’13Λ†j βˆ’4Λ†k, then

202228 Jun Shift 2Vectors
MathsMedium

Q78.If two straight lines whose direction cosines are given by the relations l + m βˆ’n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is (1) 6 (2) 4 (3) 3 (4) 2

202227 Jun Shift 13D Geometry
MathsMedium

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