Area Under Curves PYQs Set 3 (PDF)

Set 3 of 7 · Questions 61-90 of 201

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Area Under Curves · Set 3 (Q61-90)
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61. $The area of the region bounded by the curves x (1 + y^{2}) = 1 and y^{2} = 2 x is:$ (1) $2 (\frac{π}{2} - \frac{1}{3})$ (2) $\frac{π}{2} - \frac{1}{3}$ (3) $\frac{π}{4} - \frac{1}{3}$ (4) $\frac{1}{2} (\frac{π}{2} - \frac{1}{3})$ 62. $The area of the region ((x, y) : y^{2} \leq 4 x, x < 4, \frac{xy ( x - 1 ) ( x - 2 )}{( x - 3 ) ( x - 4 )} > 0, x \neq = 3) is$ (1) $\frac{16}{3}$ (2) $\frac{64}{3}$ (3) $\frac{8}{3}$ (4) $\frac{32}{3}$ 63. $The area of the region bounded by y^{2} = 8x and y^{2} = 16 (3 -x) is equal to :-$ (1) $\frac{32}{3}$ (2) $\frac{40}{3}$ (3) $16$ (4) $19$ 64. $Let ∆ be the area of the region {(x, y) ϵ R^{2} : x^{2} + y^{2} \leq 21, y^{2} \leq 4 x, x \geq 1} . then \frac{1}{2} (∆ - 21 sin^{- 1} \frac{2}{7}) is equal to$ (1) $23 - \frac{1}{3}$ (2) $3 - \frac{2}{3}$ (3) $23 - \frac{2}{3}$ (4) $3 - \frac{4}{3}$ 65. $The area bounded by the curves y =\|x - l\| + \|x - 2\| and y = 3 is equal to$ (1) $4$ (2) $6$ (3) $3$ (4) $5$ 66. $The area of the region, inside the circle (x - 23)^{2} + y^{2} = 12 and outside the parabola y^{2} = 23 x is:$ (1) $3 π + 8$ (2) $6 π - 16$ (3) $3 π - 8$ (4) $6 π - 8$ 67. $The area of the region A = {(x, y) : \| cos x - sin x \| \leq y \leq sin x, 0 \leq x \leq \frac{π}{2}}$ (1) $1 - \frac{3}{2} + \frac{4}{5}$ (2) $5 + 22 - 4.5$ (3) $\frac{3}{5} - \frac{3}{2} + 1$ (4) $5 - 22 + 1$ 68. $The area of the region enclosed by the curve f(x)= max{sin x, cos x}, - π \leq x \leq π and the x-axis is$ (1) $22 (2 + 1)$ (2) $4$ (3) $4 (2)$ (4) $2 (2 + 1)$ 69. $Area lying between the curves y^{2} = 2 x and y = x is$ (1) $\frac{2}{3} sq. units$ (2) $\frac{1}{3} sq. units$ (3) $\frac{1}{4} sq. units$ (4) $\frac{3}{4} sq. units$ 70. $If the area of the region ((x, y) : - 1 \leq x \leq 1, 0 \leq y \leq a + e^{\| x \|} - e^{- x}, a > 0) is \frac{e ^{2} + 8 e + 1}{e}, then the value of a is$ (1) $8$ (2) $7$ (3) $5$ (4) $6$ 71. $The area (in sq. units) bounded by x^{2} = y, y = x + 2 and the X -axis is$ (1) $\frac{2}{3}$ (2) $\frac{3}{5}$ (3) $\frac{5}{6}$ (4) $\frac{4}{5}$ 72. $The area of the region bounded by the line 2 y + x = 8, X -axis and the lines x = 2 and x = 4 is$ (1) $5 sq. units$ (2) $10 sq. units$ (3) $4 sq. units$ (4) $6 sq. units$ 73. $The area bounded by the curves x^{2} + y^{2} = 9 and y^{2} = 8 x is$ (1) $0 sq. unit$ (2) $(\frac{2 2}{3} + \frac{9 π}{2} - 9 sin^{- 1} \frac{1}{3}) sq. units$ (3) $16 π sq. units$ (4) $none of these$ 74. $The area (in sq. units) of the region {(x, y) \in R^{2} ∣ 4 x^{2} \leq y \leq 8 x + 12} is :$ (1) $\frac{127}{3}$ (2) $\frac{128}{3}$ (3) $\frac{124}{3}$ (4) $\frac{125}{3}$ 75. $If the area of the region bounded by the curves y = 4 - \frac{x ^{2}}{4} and y = \frac{x - 4}{2} is equal to α, then 6 α . Equals$ (1) $210$ (2) $250$ (3) $240$ (4) $220$ 76. $The area of the region bounded by the curves x = y^{2} - 2 and x = y is$ (1) $\frac{9}{4}$ (2) $9$ (3) $\frac{9}{2}$ (4) $\frac{9}{7}$ 77. $Let A_{1} be the area of the region bounded by the curves y = sin x, y = cos x and y - axis in the first quadrant. also, let A_{2} be the area of the region bounded by the curves y = sin x, y = cos x, x - axis and x = \frac{π}{2} in the first quadrant. Then,$ (1) $A_{1} = A_{2} and A_{1} + A_{2} = 2$ (2) $A_{1} : A_{2} = 1 : 2 and A_{1} + A_{2} = 1$ (3) $2 A_{1} = A_{2} and A_{1} + A_{2} = 1 + 2$ (4) $A_{1} : A_{2} = 1 : 2 and A_{1} + A_{2} = 1$ 78. $If the area of the circle x^{2} + y^{2} = 2 is divided into two parts by the parabola y = x^{2}, then the area (in sq units) of the larger part is$ (1) $\frac{3 π}{2} - \frac{1}{3}$ (2) $6 π - \frac{4}{3}$ (3) $\frac{4 π}{3} - \frac{2}{3}$ (4) $4 π - \frac{1}{4}$ 79. $The area (in sq. units) bounded by the curves x^{2} = 9 y, (x - 6)^{2} = 9 y and the X- axis is$ (1) $0$ (2) $1$ (3) $2$ (4) $4$ 80. $One of the points of intersection of the curves y = 1 + 3 x - 2 x^{2} and y = \frac{1}{x} is (\frac{1}{2}, 2) . Let the area of the region enclosed by these curves be \frac{1}{24} (l 5 + m) - n log_{e} (1 + 5), where l, m, n \in N . Then l + m + n is equal to$ (1) $29$ (2) $31$ (3) $30$ (4) $32$ 81. $The area bounded by the curves y2 = 4x and x2 = 4y is :$ (1) $\frac{32}{3}$ (2) $\frac{16}{3}$ (3) $\frac{8}{3}$ (4) $0$ 82. $The area bounded by the curves y = (x - 1)^{2}, y = (x + 1)^{2} and y = \frac{1}{4} is$ (1) $\frac{1}{3} sq. units.$ (2) $\frac{2}{3} sq. units.$ (3) $\frac{1}{4} sq. units.$ (4) $\frac{1}{5} sq. units.$ 83. $In the interval (0, \frac{π}{2}), area lying between the curves y = tan x and y = cot x and the X -axis is$ (1) $4 log 2 sq. units$ (2) $log 2 sq. units$ (3) $3 log 2 sq. units$ (4) $2 log 2 sq. units$ 84. $The area (in sq. units) of the region bounded by the parabola, y = x^{2} + 2 and the lines, y = x + 1, x = 0 and x = 3, is :$ (1) $\frac{15}{4}$ (2) $\frac{21}{2}$ (3) $\frac{17}{4}$ (4) $\frac{15}{2}$ 85. $Let and respectively be the points of local maximum and local minimum of the function f(x) = 2 x^{3} - 3 x^{2} - 12x. If is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to .$ 86. $A farmer F_{1} has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F_{2} takes away the region which lies between the side PQ and a curve of the form y = x^{n} (n > 1). If the area of the region taken away by the farmer F_{2} is exactly 30% of the area of ΔPQR, then the value of n is$ 87. $Let the area of the bounded region ((x, y) : 0 \leq 9 x \leq y^{2}, y \geq 3 x - 6) be A . Then 6 A is equal to ______.$ 88. $Let y = p(x) be the parabola passing through the points (-1, 0), (0, 1) and (1, 0). If the area of the region {(x, y): + 1, y p(x)} is A, then 12( - 4A) is equal to________.$ 89. $The graphs of sine and cosine functions, intersect each other at a number of points and between twoconsecutive points of intersection, the two graphs enclose the same area A. Then A^{4} is equal to$ 90. $if the area of the larger portion bounded between the curves x^{2} + y^{2} = 25 and y = \| x - 1 \| is \frac{1}{4} (bπ + c), b, c \in N, then b + c is equal to$