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Q77.If f(x) = {ax2 + b ; |x| < 1 respectively: (1) 1 2 , 12 (2) 12 , βˆ’32 (3) 2 5 , βˆ’32 (4) βˆ’12 , 32

202118 Mar Shift 1Limits & Continuity
MathsMedium

Q77.Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If dt, 0 ≀x ≀1 and f(0) = 0, then : lim x21 ∫x0 xβ†’0 ∫x0 √1 βˆ’(f β€²(t))2 dt = ∫x0 f(t) f(t)dt (1) does not exist (2) equals 0 (3) equals 1 (4) equals 21 2x+yβˆ’2x

202131 Aug Shift 1Indefinite Integration
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.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.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.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.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.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.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 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.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

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 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 (1, 5, 35), (7, 5, 5), (1, Ξ», 7) and (2Ξ», 1, 2) are coplanar, then the sum of all possible values of Ξ» is: (1) 445 (2) βˆ’445 (3) 395 (4) βˆ’395 JEE Main 2021 (26 Feb Shift 1) JEE Main Previous Year Paper

202126 Feb Shift 13D Geometry
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 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 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.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.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.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.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

Q79.Let β†’a and b be two non-zero vectors perpendicular to each other and β†’a = b , If β†’aΓ— b = β†’a , then the angle between the vectors and β†’a is equal to : + b + Γ— (β†’a β†’ β†’ (β†’a b)) JEE Main 2021 (18 Mar Shift 2) JEE Main Previous Year Paper (1) sinβˆ’1( √31 ) (2) cosβˆ’1( √31 ) (3) cosβˆ’1( √21 ) (4) sinβˆ’1( √61 )

202118 Mar Shift 2Vectors
MathsMedium

Q79.Let P be a plane lx + my + nz = 0 containing the line, 1βˆ’x1 = y+42 = z+23 . If plane segment AB joining points A(βˆ’3, βˆ’6, 1) and B(2, 4, βˆ’3) in ratio k : 1 then the value of k is equal to : (1) 1. 5 (2) 3 (3) 2 (4) 4

202116 Mar Shift 13D Geometry
MathsMedium

Q79.For real numbers Ξ± and Ξ² β‰ 0, if the point of intersection of the straight lines xβˆ’Ξ±1 = yβˆ’12 = zβˆ’13 and xβˆ’4 Ξ² = yβˆ’63 = zβˆ’73 lies on the plane x + 2y βˆ’z = 8, then Ξ± βˆ’Ξ² is equal to : (1) 5 (2) 9 (3) 3 (4) 7

202127 Jul Shift 23D Geometry
MathsMedium

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