Area Under Curves PYQs Set 2 (PDF)

Set 2 of 7 · Questions 31-60 of 201

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Area Under Curves · Set 2 (Q31-60)
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31. $The area (in sq. units) bounded by the curve y = 2 x - x^{2} and the line y = - x is$ (1) $\frac{9}{2}$ (2) $\frac{11}{2}$ (3) $\frac{16}{3}$ (4) $\frac{22}{5}$ 32. $Let S (α) = ((x, y) : y^{2} \leq x, 0 \leq x \leq α) and A (α) is area of the region S (α) . If for aλ, 0 < λ < 4, A (λ) : A (4) = 2 : 5, then λ equals :$ (1) $2 (\frac{4}{25})^{\frac{1}{3}}$ (2) $2 (\frac{2}{5})^{\frac{1}{3}}$ (3) $4 (\frac{2}{5})^{\frac{1}{3}}$ (4) $4 (\frac{4}{25})^{\frac{1}{3}}$ 33. $The area bounded by the curves y - 1 = cos x, y = sin x and the X -axis between x = 0 and x = π is$ (1) $2 + \frac{π}{2}$ (2) $- \frac{π}{2}$ (3) $2 - \frac{π}{2}$ (4) $\frac{π}{2}$ 34. $The area of the region described by ((x, y) / x^{2} + y^{2} \leq 1) and (y^{2} \leq 1 - x) is$ (1) $\frac{π}{2} - \frac{2}{3}$ (2) $\frac{π}{2} + \frac{2}{3}$ (3) $\frac{π}{2} + \frac{4}{3}$ (4) $\frac{π}{2} - \frac{4}{3}$ 35. $The area of the region bounded by y = 16 - x^{2} and x axis is$ (1) $8 π sq. units$ (2) $20 π sq. units$ (3) $16 π sq. units$ (4) $256 π sq. units$ 36. $The area (in sq. units) of the region A = ((x, y) \in R \times R ∣ 0 \leq x \leq 3, 0 \leq y \leq 4, y \leq x^{2} + 3 x) is :$ (1) $\frac{53}{6}$ (2) $8$ (3) $\frac{59}{6}$ (4) $\frac{26}{3}$ 37. $The area of the region enclosed by the curves y = x^{2} - 4 x + 4 and y^{2} = 16 - 8 x is:$ (1) $\frac{8}{3}$ (2) $\frac{4}{3}$ (3) $8$ (4) $5$ 38. $The area of the region ((x, y) : x^{2} + 4 x + 2 \leq y \leq \| x + 2 \|) is equal to$ (1) $7$ (2) $5$ (3) $24/5$ (4) $20/3$ 39. $The area enclosed by the curves xy + 4 y = 16 and x + y = 6 is equal to :$ (1) $28 - 30 log_{e} 2$ (2) $30 - 28 log_{e} 2$ (3) $30 - 32 log_{e} 2$ (4) $32 - 30 log_{e} 2$ 40. $The area (in sq. units) of the region A = {(x, y) : \| x \| + \| y \| \leq 1, 2 y^{2} \geq \| x \|} is:$ (1) $\frac{1}{6}$ (2) $\frac{7}{6}$ (3) $\frac{5}{6}$ (4) $\frac{1}{3}$ 41. $The area of the region bounded by the curve y^{2} = 8 x and the line y = 2 x is$ (1) $\frac{4}{3} sq. units$ (2) $\frac{3}{4} sq. units$ (3) $\frac{8}{3} sq. units$ (4) $\frac{16}{3} sq. units$ 42. $Let C_{1} be the curve obtained by the solution ofdifferential equation 2 xy \frac{dy}{dx} = y^{2} - x^{2}, x > 0 & Let the curve C_{2} be the solution of \frac{2 xy}{x ^{2} - y ^{2}} = \frac{dy}{dx} . If both the curves pass through (1, 1), then the area enclosed by the curves C_{1} and C_{2} is equal to:$ (1) $π - 1$ (2) $\frac{π}{2} - 1$ (3) $π + 1$ (4) $\frac{π}{4} + 1$ 43. $If the area of the region ((x, y) : 1 - 2 x ⩽ y ⩽ 4 - x^{2}, x ⩾ 0, y ⩾ 0) is \frac{α}{β}, α, β \in N, gcd (α, β) = 1, then the value of (α + β) is:$ (1) $67$ (2) $91$ (3) $73$ (4) $85$ 44. $The area bounded by the curves y = 2 x^{2}, y = max . x- [x] + \| x \| and the lines x = 0, x = 2 (in sq. units), is$ (1) $2$ (2) $\frac{1}{2}$ (3) $\frac{1}{3}$ (4) $\frac{4}{3}$ 45. $The area of the region bounded by the parabola (y - 2)^{2} = (x - 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :$ (1) $9$ (2) $10$ (3) $4$ (4) $6$ 46. $The area (in sq. units) of the part of the circle x^{2} + y^{2} = 36, which is outside the parabola y^{2} = 9 x, is:$ (1) $24 π + 33$ (2) $12 π + 33$ (3) $12 π - 33$ (4) $24 π - 33$ 47. $The area of the region bounded by Y -axis, y = cos x and y = sin x; 0 \leq x \leq \frac{π}{2} is$ (1) $(2 - 1) sq. unit$ (2) $(2 + 1) sq. units$ (3) $2 sq. units$ (4) $(2 - 2) sq. unit$ 48. $The area of the region bounded by the lines y = 2 x + 1, y = 3 x + 1 and x = 4 is$ (1) $16 sq. units$ (2) $\frac{121}{3} sq. units$ (3) $\frac{121}{6} sq. units$ (4) $8 sq. units$ 49. $The area of the region, enclosed by the circle x^{2} + y^{2} = 2 which is not common to the region bounded by the parabola y^{2} = x and the straight line y = x, is:$ (1) $\frac{1}{12} (24 π - 1)$ (2) $\frac{1}{6} (12 π - 1)$ (3) $\frac{1}{12} (6 π - 1)$ (4) $\frac{1}{12} (12 π - 1)$ 50. $Let A = ((x, y) ϵ R : y \geq 0, 2 x \leq y \leq 4 - (x - 1)^{2}) and B {(x, y) ϵ R \times R : 0 \leq min {2 x 4 - (x - 1)^{2}}} Then the ratio of the area of A to the area of B is$ (1) $\frac{π - 1}{π + 1}$ (2) $\frac{π}{π - 1}$ (3) $\frac{π}{π + 1}$ (4) $\frac{π + 1}{π - 1}$ 51. $The area of the region {(x,y): x^{2} \leq y \leq 8 - x^{2}, y \leq 7} is$ (1) $27$ (2) $18$ (3) $20$ (4) $21$ 52. $The area of the region {(x, y) : \| x - y \| \leq y \leq 4 x} is$ (1) $\frac{512}{3}$ (2) $\frac{2048}{3}$ (3) $512$ (4) $\frac{1024}{3}$ 53. $The area of the region enclosed between pair of the lines xy = 0 and the lines xy + 5 x - 4 y - 20 = 0, is$ (1) $20 square units$ (2) $10 square units$ (3) $4/5 square unit$ (4) $6 square units$ 54. $Let the area of the region ((x, y) : 0 \leq x \leq 3, 0 \leq y \leq min (x^{2} + 2, 2 x + 2)) be A . Then 12 A is equal to________$ 55. $Let the area enclosed by the lines x + y = 2, y = 0, x = 0 and the curve f (x) = min (x^{2} + \frac{3}{4}, 1 + [x]) where [x] denotes the greatest integer \leq x, be A . Then the value of 12 A is$ 56. $The area of the region enclosed by the parabolas y = x^{2} - 5 x and y = 7 x - x^{2} is$ 57. $The area of the region bounded by the curve y = max \| x \|, x \| x-2 \|, the x -axis and the lines x = - 2 and x = 4 is equal to ______$ 58. $The area (in sq. units) of the part of circle x^{2} + y^{2} = 169 which is below the line 5 x - y = 13 is \frac{πα}{2 β} - \frac{65}{2} + \frac{α}{β} sin^{- 1} (\frac{12}{13}) where α, β are coprime numbers. Then α + β is equal to$ 59. $Let a be the area of the region ((x, y) : y \geq x^{2}, y \geq (1 - x)^{2}, y \leq 2 x (1 - x)) Then 540 A is equal to$ 60. $Let the curve y = y (x) be the solution of the differential equation, \frac{dy}{dx} = 2 (x + 1) . If the numerical value of area bounded by the curve y = y (x) and x - axis is \frac{4 8}{3}, then the value of y (1) is equal to_____$