Area Under Curves PYQs Set 4 (PDF)

Set 4 of 7 · Questions 91-120 of 201

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Area Under Curves · Set 4 (Q91-120)
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91. $Let the area of the region ((x, y) : 2 y \leq x^{2} + 3, y + \| x \| \leq 3, y ⩾ \| x - 1 \|) be A. Then 6 A is equal to:$ (1) $16$ (2) $12$ (3) $14$ (4) $18$ 92. $Area of the region (in sq. units) bounded by the curves y = x, x = y and the lines x = 1, x = 4, is$ (1) $\frac{8}{3}$ (2) $\frac{49}{3}$ (3) $\frac{16}{3}$ (4) $\frac{14}{3}$ 93. $Area of the region bounded by two parabolas y = x^{2} and x = y^{2} is$ (1) $\frac{1}{4}$ (2) $\frac{1}{3}$ (3) $4$ (4) $3$ 94. $The area bounded between the parabola y^{2} = 4 x and the line y = 2 x - 4 is equal to$ (1) $9 sq. units$ (2) $15 sq. units$ (3) $\frac{17}{3} sq. units$ (4) $\frac{19}{3} sq. units$ 95. $The area (in sq. units) enclosed between the curves y = x^{2} and y = \| x \| is$ (1) $\frac{2}{3}$ (2) $\frac{1}{6} .$ (3) $\frac{1}{3}$ (4) $1$ 96. $The area bounded by the curves y = sin x, y = cos x and x = 0 is$ (1) $(2 - 1) sq. units$ (2) $1 sq. unit$ (3) $2 sq. units$ (4) $(1 + 2) sq. units$ 97. $The area of the region described by A = {(x, y) : x2 + y2 \leq 1 and y2 \leq 1 - x} is :$ (1) $\frac{π}{2} - \frac{2}{3}$ (2) $\frac{π}{2} + \frac{2}{3}$ (3) $\frac{π}{2} + \frac{4}{3}$ (4) $\frac{π}{2} - \frac{4}{3}$ 98. $The area of the region bounded by the line y = 2 x + 1, x -axis and the ordinates x = - 1 and x = 1 is$ (1) $2$ (2) $5/2$ (3) $5$ (4) $9/4$ 99. $Let g :_{(} 0, \infty_{)} \to R be a differentiable function such that \int (\frac{x ( c o s x - s i n x )}{e ^{x} + 1} + \frac{g ( x ) ( e ^{x} + 1 - x e ^{x} )}{( e ^{x} + 1 )}) d x = \frac{xg ( x )}{e ^{x} + 1} + c, for all x > 0, where c is an arbitrary constant. Then$ (1) $g is decreasing in$ (2) $g’ is increasing in$ (3) $g + g' is increasing in$ (4) $g - g' is increasing in$ 100. $The area of the region enclosed by the curves y = e^{x}, y = ∣ e^{x} - 1∣ and y -axis is:$ (1) $1 - log_{e} 2$ (2) $log_{e} 2$ (3) $1 + log_{e} 2$ (4) $2 log_{e} 2 - 1$ 101. $The area (in sq. units) bounded by the parabola y = x^{2} - 1, the tangent at the point (2, 3) to it and the y -axis is:$ (1) $\frac{8}{3}$ (2) $\frac{32}{3}$ (3) $\frac{56}{3}$ (4) $\frac{14}{3}$ 102. $Consider a region R = ((x, y) \in R^{2} : x^{2} \leq y \leq 2 x) if a line y = a divides the area of region R into two equal parts, then which of the following is true?$ (1) $3 α^{2} - 8 α + 8 = 0$ (2) $α^{3} - 6 α^{3/2} - 16 = 0$ (3) $3 α^{2} - 8 α^{3/2} + 8 = 0$ (4) $α^{3} - 6 α^{2} + 16 = 0$ 103. $The area bounded by y - 1 = - \| x \| and y + 1 = \| x \| is$ (1) $\frac{1}{2}$ (2) $1$ (3) $2$ (4) $0$ 104. $The area bounded by the curve 4 y^{2} = x^{2} (4 - x) (x - 2) is equal to:$ (1) $\frac{π}{8}$ (2) $\frac{3 π}{8}$ (3) $\frac{3 π}{2}$ (4) $\frac{π}{16}$ 105. $Let the area of the region enclosed by the curves y = 3 x, 2 y = 27 - 3 x and y = 3 x - x x be A . Then 10 A is equal to$ (1) $172$ (2) $162$ (3) $154$ (4) $184$ 106. $If the area bounded by the curves y = a x^{2} and x = a y^{2}, (a > 0) is 3 sq units, then the value of a is$ (1) $\frac{2}{3}$ (2) $\frac{1}{3}$ (3) $1$ (4) $4$ 107. $The area of the region bounded by the curves y = 9 x^{2} and y = 5 x^{2} + 4 (in sq units) is$ (1) $64$ (2) $\frac{64}{3}$ (3) $\frac{32}{3}$ (4) $\frac{16}{3}$ 108. $The region represented by \| x - y \| \leq 2 and \| x + y \| \leq 2 is bounded by a :$ (1) $square of side length 22 units$ (2) $rhombus of side length 2 units$ (3) $square of area 16 sq. units$ (4) $rhombus of area 82 sq. units$ 109. $The area (in sq. units) of the region bounded by the curves y = 2^{x} and y = \| x + 1 \|, in the first quadrant is :$ (1) $log_{e} 2 + \frac{3}{2}$ (2) $\frac{3}{2}$ (3) $\frac{1}{2}$ (4) $\frac{3}{2} - \frac{1}{log _{e} 2}$ 110. $The area (in sq. units) of the region (x, y) \in R^{2}; x^{2} \leq y \leq 3 - 2 x} is :$ (1) $\frac{16}{3}$ (2) $\frac{32}{3}$ (3) $\frac{29}{3}$ (4) $343$ 111. $The area (in square units) of the region bounded by the curve y = \| sin 2 x \| and the X -axis in [0, 2 π] is$ (1) $0$ (2) $3$ (3) $4$ (4) $1$ 112. $The area of the triangle formed by the tangent to the curve xy = a^{2} at (x_{1}, y_{1}) on it and the axes is$ (1) $a^{2} sq. units$ (2) $\frac{3 a ^{2}}{2} sq. units$ (3) $2 a^{2} sq. units$ (4) $4 a^{2} sq. units$ 113. $The area bounded by the curve x = 4 - y^{2} and the Y -axis is$ (1) $16 sq. units$ (2) $32 sq. units$ (3) $\frac{32}{3} sq. units$ (4) $\frac{16}{3} sq. units$ 114. $If the area bounded by the curve 2 y^{2} = 3x, lines x + y = 3, y = 0 and outside the circle (x - 3)^{2} + y^{2} = 2 is A, then 4(π + 4A) is equal to__________.$ 115. $If the area of the region bounded by the curves y^{2} - 2 y = - x, x + y = 0 is A, then 8A is equal to$ 116. $Let the area of the region enclosed by the curve y = min sin x, cos x and the x axis between x = - π to x = π be A . Then A^{2} is equal to_____$ 117. $The area of the region enclosed by the parabola (y - 2)^{2} = x - 1, the line x - 2 y + 4 = 0 and the positive coordinate axes is______$ 118. $Let the area of the region bounded by the curve y = max sin x, cos x, lines x = 0, x = \frac{3 π}{2}, and the x -axis be A. Then, A + A^{2} is equal to____.$ 119. $Let f : [- 3, 1] \to R be given as f (x) = {min ((x + 6), x^{2}), max (x, x^{2}), - 3 \leq x \leq 0 0 \leq x \leq 1 If the area bounded by y = f (x) and x -axis is A, then the value of 6 A is equal to____$ 120. $If the area of the region ((x, y) : (4 - x^{2}) \leq y \leq x^{2}, y \leq 4, x \geq 0) is (\frac{80 2}{α} - β), α, β \in N, then α + β is equal to ________ .$