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Q77.Let y(x) be the solution of the differential equation 2x2 dy + (ey βˆ’2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to: (1) loge(2e) (2) loge 2 (3) 2 (4) 0

202126 Aug Shift 2Differential Equations
MathsHard

Q78.Let β†’a,β†’b and β†’cbe three vectors such that β†’a =β†’b Γ— (β†’ β†’ Γ—β†’c). β†’ Ο€ 2 respectively and the angle between b and β†’cis ΞΈ(0 < ΞΈ < 2 ), then the value of 1 + tan ΞΈ is equal to : (1) √3 + 1 (2) 2 (3) 1 (4) √3+1 √3 JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper

202127 Jul Shift 2Vectors
MathsMedium

Q78.The vector equation of the plane passing through the intersection of the planes β†’rβ‹…(Λ†i +Λ†j + Λ†k) = βˆ’2, and the point (1, 0, 2) is: β†’rβ‹…(Λ†i βˆ’2Λ†j) = 73 (1) β†’rβ‹…(Λ†i + 7Λ†j + 3Λ†k) = 7 (2) β†’rβ‹…(Λ†i βˆ’7Λ†j + 3Λ†k) = 7 = 37 (4) β†’rβ‹…(3Λ†i + 7Λ†j + 3Λ†k) (3) β†’rβ‹…(Λ†i + 7Λ†j + 3Λ†k)

202124 Feb Shift 23D Geometry
MathsMedium

Q78.The equation of the line through the point (0, 1, 2) and perpendicular to the line xβˆ’12 = y+13 = zβˆ’1βˆ’2 is : yβˆ’1 (1) x 3 = βˆ’4 = zβˆ’23 (2) x3 = yβˆ’14 = zβˆ’23 (3) βˆ’3x = yβˆ’14 = zβˆ’23 (4) x3 = yβˆ’14 = zβˆ’2βˆ’3

202125 Feb Shift 13D Geometry
MathsMedium

Q78.A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, βˆ’1, 1) βˆ’β†’ , then the projection of OP on this plane is of length: (1) √25 (2) √27 (3) √23 (4) √211

202125 Feb Shift 23D Geometry
MathsHard

Q78.Let β†’π‘Ž, →𝑏, →𝑐 be three vectors mutually perpendicular to each other and have same magnitude. If a vector β†’π‘Ÿ satisfies β†’π‘ŽΓ— {β†’π‘Ÿ- →𝑏× β†’π‘Ž} + →𝑏× {β†’π‘Ÿ- →𝑐× →𝑏} + →𝑐× {β†’π‘Ÿ- β†’π‘ŽΓ— →𝑐} = β†’0, then β†’π‘Ÿ is equal to: (1) 1 (β†’π‘Ž+ →𝑏+ →𝑐) (2) 1 (2β†’π‘Ž+ →𝑏- →𝑐) 3 3 (3) 1 (β†’π‘Ž+ →𝑏+ →𝑐) (4) 1 ( β†’π‘Ž+ →𝑏+ 2 →𝑐) 2 2

202131 Aug Shift 2Vectors
MathsMedium

Q78.The distance of the point 1, 1, 9 from the point of intersection of the line = = and the plane 1 2 2 π‘₯+ 𝑦+ 𝑧= 17 is: (1) 19√2 (2) 2√19 (3) √38 (4) 38

202124 Feb Shift 13D Geometry
MathsMedium

Q78.The lines x = ay βˆ’1 = z βˆ’2 and x = 3y βˆ’2 = bz βˆ’2, (ab β‰ 0) are coplanar, if: (1) b = 1, a ∈R βˆ’{0} (2) a = 1, b ∈R βˆ’{0} (3) a = 2, b = 2 (4) a = 2, b = 3

202120 Jul Shift 23D Geometry
MathsMedium

Q78.Let y = y(x) be the solution of the differential equation ex√1 βˆ’y2 dx + ( xy )dy = 0, y(1) = βˆ’1 Then the value of (y(3))2 is equal to: (1) 1 βˆ’4e3 (2) 1 βˆ’4e6 (3) 1 + 4e3 (4) 1 + 4e6 β†’

202120 Jul Shift 1Differential Equations
MathsHard

Q78.Let β†’a = Λ†i + Λ†j + 2Λ†k and b = βˆ’Λ†i + 2Λ†j + 3Λ†k. Then the vector product Γ— Γ— is equal to : (β†’a+β†’b) ((β†’a ((β†’aβˆ’β†’b) Γ—β†’b)) Γ—β†’b) + + (1) 5(34Λ†i βˆ’5Λ†j 3Λ†k) (2) 7(34Λ†i βˆ’5Λ†j 3Λ†k) + + (3) 7(30Λ†i βˆ’5Λ†j 7Λ†k) (4) 5(30Λ†i βˆ’5Λ†j 7Λ†k)

202127 Jul Shift 1Differential Equations
MathsMedium

Q78.A hall has a square floor of dimension 10 m Γ— 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is cosβˆ’1 15 , then the height of the hall (in meters) is: (1) 5√2 (2) 5√3 (3) 5√10 (4) 5

202126 Aug Shift 2Vectors
MathsMedium

Q78.Let the vectors 2 + π‘Ž+ 𝑏 ^𝑖+ π‘Ž+ 2𝑏+ 𝑐 ^𝑗- 𝑏+ 𝑐 ^π‘˜, 1 + 𝑏 ^𝑖+ 2𝑏 ^𝑗- 𝑏 ^π‘˜ and 2 + 𝑏 ^𝑖+ 2𝑏 ^𝑗+ 1 - 𝑏 ^π‘˜, βˆ€π‘Ž, 𝑏, π‘βˆˆπ‘… be co-planar. Then which of the following is true? (1) 2𝑏= π‘Ž+ 𝑐 (2) 3𝑐= π‘Ž+ 𝑏 (3) π‘Ž= 𝑏+ 2𝑐 (4) 2π‘Ž= 𝑏+ 𝑐

202125 Jul Shift 1Vectors
MathsMedium

Q78.Let β†’a = 2Λ†i βˆ’3Λ†j + 4Λ†k and b = 7Λ†i + Λ†j βˆ’6Λ†k If β†’rΓ—β†’a =β†’rΓ— b,β†’rβ‹…(Λ†i Λ†k) equal to: (1) 12 (2) 8 (3) 13 (4) 10

202117 Mar Shift 1Vectors
MathsMedium

Q78.Let L be the line of intersection of planes β†’rβ‹…(Λ†i βˆ’Λ†j + 2Λ†k) = 2 and β†’rβ‹…(2Λ†i + Λ†j βˆ’Λ†k) foot of perpendicular on L from the point (1, 2, 0), then the value of 35(Ξ± + Ξ² + Ξ³) is equal to: (1) 101 (2) 119 (3) 143 (4) 134

202122 Jul Shift 13D Geometry
MathsHard

Q78.Let the position vectors of two points P and Q be 3Λ†i βˆ’Λ†j + 2Λ†k and Λ†i + 2Λ†j βˆ’4Λ†k, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, βˆ’1, 2) and (βˆ’2, 1, βˆ’2), respectively. Let βˆ’βˆ’βˆ’β†’ β†’ β†’ lines PR and QS intersect at T . If the vector TA is perpendicular to both PR and QS and the length of vector βˆ’β†’ TA is √5 units, then the modulus of a position vector of A is : (1) √482 (2) √171 (3) √5 (4) √227 P divides the line

202116 Mar Shift 1Vectors
MathsHard

Q78.In a triangle ABC , if BC→ = 8, CA→ = 7, AB→ = 10 , then the projection of the vector AB→ on AC→ is equal to : (1) 25 (2) 85 4 14 (3) 127 (4) 115 20 16 → → →

202118 Mar Shift 2Vectors
MathsEasy

Q78.Let a, b and c be distinct positive numbers. If the vectors aΛ†i + aΛ†j + cΛ†k,Λ†i + Λ†k and cΛ†i + cΛ†j + bΛ†k are co-planar, then c is equal to: 2 (1) (2) a+b 1 2 1 + a b (3) a 1 + 1b (4) √ab

202125 Jul Shift 2Vectors
MathsMedium

Q78.The equation of the plane passing through the line of intersection of the planes β†’rβ‹…(Λ†i + Λ†j + Λ†k) + 4 = 0 and parallel to the x-axis, is β†’rβ‹…(2Λ†i + 3Λ†j βˆ’Λ†k) + + 6 = 0 (1) β†’rβ‹…(Λ†i 3Λ†k) + 6 = 0 (2) β†’rβ‹…(Λ†i βˆ’3Λ†k) + 6 = 0 (3) β†’rβ‹…(Λ†j βˆ’3Λ†k) βˆ’6 = 0 (4) β†’rβ‹…(Λ†j βˆ’3Λ†k)

202127 Aug Shift 23D Geometry
MathsMedium

Q78.Let O be the origin. Let OPβ†’ = xΛ†i + yΛ†j βˆ’Λ†k and OQβ†’ = βˆ’Λ†i + 2Λ†j + 3xΛ†k, x, y ∈R, x > 0, be such that βˆ’βˆ’βˆ’βˆ’βˆ’β†’ β†’ β†’ β†’ β†’ PQ = √20 and the vector OP is perpendicular to OQ. If OR = 3Λ†i + zΛ†j βˆ’7Λ†k, z ∈R, is coplanar with OP βˆ’β†’ and OQ, then the value of x2 + y2 + z2 is equal to (1) 7 (2) 9 (3) 2 (4) 1

202117 Mar Shift 2Vectors
MathsMedium

Q78.The distance of the point (1, βˆ’2, 3) from the plane x βˆ’y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, βˆ’6 , is (1) 2 (2) 5 (3) 3 (4) 1 units from the origin, which contains the line of intersection of the

202127 Aug Shift 13D Geometry
MathsMedium

Q78.The integral ∫ (2xβˆ’1) cos √(2xβˆ’1)2+5 dx is equal to (where c is a constant of integration) √4x2βˆ’4x+6 (1) 2 1 sin √(2x βˆ’1)2 + 5 + c (2) 21 cos √(2x + 1)2 + 5 + c (3) 1 2 cos √(2x βˆ’1)2 + 5 + c (4) 12 sin √(2x + 1)2 + 5 + c

202118 Mar Shift 1Indefinite Integration
MathsMedium

Q78.The distance of line 3𝑦- 2𝑧- 1 = 0 = 3π‘₯- 𝑧+ 4 from the point ( 2, - 1, 6 ) is : (1) 2√5 (2) 2√6 (3) √26 (4) 4√2

202101 Sep Shift 23D Geometry
MathsMedium

Q78.If (x, y, z) be an arbitrary point lying on a plane P which passes through the point (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of expression 3 + xβˆ’11 + yβˆ’19 + zβˆ’12 βˆ’ 14(xβˆ’11)(yβˆ’19)(zβˆ’12)x+y+z is (yβˆ’19)2(zβˆ’12)2 (xβˆ’11)2(zβˆ’12)2 (xβˆ’11)2(yβˆ’19)2 (1) 0 (2) 3 (3) 39 (4) βˆ’45

202116 Mar Shift 23D Geometry
MathsMedium

Q78.Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4 . If point P(Ξ±, Ξ², Ξ³) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21(Ξ± + Ξ² + Ξ³) equals: (1) 102 (2) 142 (3) 68 (4) 136

202126 Feb Shift 23D Geometry
MathsMedium

Q78.Let β†’a = Λ†i + Λ†j + Λ†k andβ†’b = Λ†j βˆ’Λ†k. If β†’cis a vector such that β†’aΓ—β†’c=β†’b and β†’aβ‹…β†’c= 3, then β†’aβ‹…(β†’ Γ—β†’c) to: (1) 6 (2) βˆ’2 (3) 2 (4) βˆ’6

202126 Aug Shift 1Vectors
MathsMedium

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