Area Under Curves PYQs Set 1 (PDF)

Set 1 of 7 · Questions 1-30 of 201

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Area Under Curves · Set 1 (Q1-30)
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1. $Let A_{1} be the bounded area enclosed by the curves y = x^{2} + 2, x + y = 8 and y -axis that lies in the first quadrant. Let A_{2} be the bounded area enclosed by the curves y = x^{2} + 2, y^{2} = x, x = 2, and y -axis that lies in the first quadrant. Then A_{1} - A_{2} is equal to$ (1) $\frac{2}{3} (32 + 1)$ (2) $\frac{2}{3} (22 + 1)$ (3) $\frac{2}{3} (2 + 1)$ (4) $\frac{2}{3} (42 + 1)$ 2. $The curve y = a x^{2} + bx passes through the point (1, 2) and lies above the X -axis for 0 \leq x \leq 8 . If the area enclosed by this curve, the X -aixs and the line x = 6 is 108 square units, then 2 b - a =$ (1) $2$ (2) $0$ (3) $1$ (4) $-1$ 3. $If the area of the region bounded by the curves y = x^{2} and x = y^{2} is k, then the area of the region bounded by the curves \frac{x + 3 y}{2} = (\frac{3 x - y}{2})^{2} and \frac{3 x - y}{2} = (\frac{x + 3 y}{2})^{2}, is$ (1) $\frac{3}{2} k$ (2) $\frac{1}{2} k$ (3) $k$ (4) $(\frac{3 + 1}{2}) k$ 4. $The area (in sq. units) of the region bounded by the curves y = 4 \| cos x \| and y = - \| cos x \| from x = - \frac{π}{2} to \frac{π}{2} is$ (1) $6$ (2) $8$ (3) $12$ (4) $10$ 5. $The area of the region bounded by the lines y = mx, x = 1, x = 2, and x axis is 6 sq. units, then ' m' is$ (1) $3$ (2) $1$ (3) $2$ (4) $4$ 6. $The area enclosed by the closed curve C given by the differential equation \frac{dy}{dx} + \frac{x - 1}{y - 2} = 0, y (1) = 0 is 4 π . Let P and Q be the points of intersection of the curve C and the y -axis. If normals at P and Q on the curve C intersect x -axis at points R and S respectively, then the length of the line segment RS is$ (1) $23$ (2) $\frac{2 3}{3}$ (3) $2$ (4) $\frac{4 3}{3}$ 7. $The area bounded by the curve y = sin (\frac{x}{3}), x -axis and lines x = 0 and x = 3 π is$ (1) $9$ (2) $0$ (3) $6$ (4) $3$ 8. $The area of the region enclosed between the circles x^{2} + y^{2} = 4 and x^{2} + (y - 2)^{2} = 4 is:$ (1) $\frac{4}{3} (2 π - 33)$ (2) $\frac{2}{3} (2 π - 33)$ (3) $\frac{2}{3} (4 π - 33)$ (4) $\frac{4}{3} (2 π - 3)$ 9. $Area of the region bounded by the curve y = tan x, the x -axis and the line x = \frac{π}{3} is$ (1) $log \frac{1}{2}$ (2) $0$ (3) $log 2$ (4) $- log 2$ 10. $Let f (α) denote the area of the region in the first quadrant bounded by x = 0, x = 1, y^{2} = x and y = \| αx - 5 \| - \| 1 - αx \| + α x^{2} . Then (f (0) + f (1)) is equal to$ (1) $12$ (2) $14$ (3) $7$ (4) $9$ 11. $The area of the region, inside the ellipse x^{2} + 4 y^{2} = 4 and outside the region bounded by the curves y = \| x \| - 1 and y = 1 - \| x \|, is :$ (1) $2 π - 1$ (2) $3 (π - 1)$ (3) $2 π - 1 2$ (4) $2 (π - 1)$ 12. $Let f (x) be a positive function such that the area bounded by y = f (x), y = 0 from x = 0 to x = a > 0 is e^{- a} + 4 a^{2} + a - 1 . Then the differential equation, whose general solution is y = c_{1} f (x) + c_{2}, where c_{1} and c_{2} are arbitrary constants, is$ (1) $(8 e^{x} - 1) \frac{d ^{2} y}{d x ^{2}} + \frac{dy}{dx} = 0$ (2) $(8 e^{x} - 1) \frac{d ^{2} y}{d x ^{2}} - \frac{dy}{dx} = 0$ (3) $(8 e^{x} + 1) \frac{d ^{2} y}{d x ^{2}} - \frac{dy}{dx} = 0$ (4) $(8 e^{x} + 1) \frac{d ^{2} y}{d x ^{2}} + \frac{dy}{dx} = 0$ 13. $If the area of the ellipse is \frac{x ^{2}}{25} + \frac{y ^{2}}{λ ^{2}} = 1 is 20 π square units, then λ is$ (1) $\pm 4$ (2) $\pm 3$ (3) $\pm 2$ (4) $\pm 1$ 14. $Let f : [0, \infty) \to R be a differentiable function such that f (x) = 1 - 2 x + \int_{0}^{x} e^{x - t} f (t) dt for all x \in [0, \infty) . Then the area of the region bounded by y = f (x) and the coordinate axes is$ (1) $5$ (2) $2$ (3) $2$ (4) $\frac{1}{2}$ 15. $The area of the triangle formed by the tangent and the normal drawn to the curve y^{2} = 4 x at (1, 2) with Y -axis is (in square units)$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 16. $For a > 0, let the curves C_{1} : y^{2} = ax and C_{2} : x^{2} = ay intersect at origin O and a point P . Let the line x = b (0 < b < a) intersect the chord O P and the x -axis at points Q and R, respectively. If the line x = b bisects the area bounded by the curves, C_{1} and C_{2}, and the area of Δ OQR = \frac{1}{2}, then a satisfies the equation:$ (1) $a^{6} - 12 a^{3} - 4 = 0$ (2) $a^{6} + 12 a^{3} - 4 = 0$ (3) $a^{6} - 12 a^{3} + 4 = 0$ (4) $a^{6} + 12 a^{3} + 4 = 0$ 17. $The area included between the parabola y = \frac{x ^{2}}{4 a} and the curve y = \frac{8 a ^{3}}{( x ^{2} + 4 a ^{2} )} is$ (1) $a^{2} (2 π + \frac{2}{3})$ (2) $a^{2} (2 π - \frac{8}{3})$ (3) $a^{2} (π + \frac{4}{3})$ (4) $a^{2} (2 π - \frac{4}{3})$ 18. $The area of the region R = ((x, y) : xy \leq 8, 1 \leq y \leq x^{2}, x \geq 0) is$ (1) $\frac{2}{3} (24 log_{e} (2) - 7)$ (2) $\frac{1}{3} (40 log_{e} (2) + 27)$ (3) $\frac{2}{3} (20 log_{e} (2) + 9)$ (4) $\frac{1}{3} (49 log_{e} (2) - 15)$ 19. $The area enclosed between the parabola y = x^{2} - x + 2 and the line y = x + 2 in sq. units equals$ (1) $8/3$ (2) $1/3$ (3) $2/3$ (4) $4/3 .$ 20. $The parabola y^{2} = 4 x divides the area of the circle x^{2} + y^{2} = 5 in two parts. The area of the smaller part is equal to:$ (1) $\frac{1}{3} + 5 sin^{- 1} (\frac{2}{5})$ (2) $\frac{1}{3} + 5 sin^{- 1} (\frac{2}{5})$ (3) $\frac{2}{3} + 5 sin^{- 1} (\frac{2}{5})$ (4) $\frac{2}{3} + 5 sin^{- 1} (\frac{2}{5})$ 21. $The area (in sq. units) of the region bounded by the curve x^{2} = 4 y and the straight line x = 4 y - 2 is :$ (1) $\frac{5}{4}$ (2) $\frac{9}{8}$ (3) $\frac{7}{8}$ (4) $\frac{3}{4}$ 22. $Area of the region bounded by the curve y = cos x, x = 0 and x = π is$ (1) $2 sq, units$ (2) $4 sq. units$ (3) $3 sq. units$ (4) $1 sq. unit$ 23. $The area bounded by the line y = x, x -axis and ordinates x = - 1 and x = 2 is$ (1) $3/2 sq. units$ (2) $5/2 sq. units$ (3) $2 sq. units$ (4) $3 sq. units$ 24. $The area (in sq. units) of the region A = ((x, y) : x^{2} \leq y \leq x + 2) is:$ (1) $\frac{10}{3}$ (2) $\frac{9}{2}$ (3) $\frac{31}{6}$ (4) $\frac{13}{6}$ 25. $If the area of the region ((x, y) : 1 + x^{2} \leq y \leq min x+7, 11-3x) is A, then 3 A is equal to:$ (1) $50$ (2) $46$ (3) $49$ (4) $47$ 26. $The area of the region enclosed by the parabola y = 4 x - x^{2} and 3 y = (x - 4)^{2} is equal to$ (1) $\frac{32}{9}$ (2) $4$ (3) $6$ (4) $\frac{14}{3}$ 27. $Let the area of the region {(x, y) : ∣2 x - 1∣ \leq y \leq ∣ x^{2} - x ∣, 0 \leq x \leq 1} be A . Then (6 A + 11)^{2} is equal to ________$ 28. $Let y = y (x) be the solution of the differentialequation xdy - ydx = (x^{2} - y^{2}) dx, x \geq 1, with y (1) = 0 . If the area bounded by the line x = 1, x = e^{π}, y = 0 and y = y (x) is α e^{2 π} + β, then the value of 10 (α + β) is equal to_____$ 29. $Let A be the area bounded by the curve y = x (x - 3), the x -axis and the ordinates x = - 1 and x = 2. Then 12 A is equal to _______ .$ 30. $The sum of squares of all possible values of k, for which area of the region bounded by the parabolas 2 y^{2} = kx and ky^{2} = 2 (y - x) is maximum, is equal to :$