Area Under Curves PYQs Set 6 (PDF)

Set 6 of 7 · Questions 151-180 of 201

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Area Under Curves · Set 6 (Q151-180)
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151. $The area of the triangle formed by the positive X -axis, the tangent and normal to the curve x^{2} + y^{2} = 16 a^{2} at the point (22 a, 22 a) is$ (1) $a^{2}$ (2) $16 a^{2}$ (3) $4 a^{2}$ (4) $8 a^{2}$ 152. $The area bounded by the curve y = (x^{2}, x < 0 x, x \geq 0) and the line y = 4 is$ (1) $\frac{8}{3}$ (2) $\frac{32}{3}$ (3) $\frac{16}{3}$ (4) $\frac{40}{3}$ 153. $The area of the region bounded by the curve y = 2 x - x^{2} and the line y = x is square units.$ (1) $1/6$ (2) $1/2$ (3) $1/3$ (4) $7/6$ 154. $The area (in sq. units) of the region A = ((x, y) : \frac{y ^{2}}{2} \leq x \leq y + 4) is:$ (1) $\frac{53}{3}$ (2) $30$ (3) $16$ (4) $18$ 155. $The area (in sq. units) of the region described by ((x, y) : y^{2} \leq 2 x), and (y \geq 4 x - 1) is$ (1) $\frac{11}{32}$ (2) $\frac{8}{9}$ (3) $\frac{11}{12}$ (4) $\frac{9}{32}$ 156. $Let q be the maximum integral value of p in [0, 10] for which the roots of the equation x^{2} - p x + \frac{5 p}{4} = 0 are rational. Then the area of the region {(x, y) : 0 \leq y \leq (x - q)^{2}, 0 \leq x \leq q} is$ (1) $243$ (2) $25$ (3) $\frac{125}{3}$ (4) $164$ 157. $If the area enclosed between the curves y = k x^{2} and x = k y^{2}, (k > 0), is 1 square unit. Then k is:$ (1) $\frac{3}{2}$ (2) $\frac{1}{3}$ (3) $3$ (4) $\frac{2}{3}$ 158. $Consider a curve y = y(x) in the first quadrant as shown in the figure. Let the area A_{1} is twice the area A_{2} . Then the normal to the curve perpendicular to the line 2x - 12y = 15 does NOT pass through the point. Y$ (1) $(6,21)$ (2) $(8, 9)$ (3) $(10,-4)$ (4) $(12,-15)$ 159. $The area of the region bounded by y = 2 x - x^{2} and the x -axis is$ (1) $\frac{7}{3} sq. units$ (2) $\frac{2}{3} sq. units$ (3) $\frac{8}{3} sq. units$ (4) $\frac{4}{3} sq. units$ 160. $Consider the region R = ((x, y) : x \leq y \leq 9 - \frac{11}{3} x^{2}, x \geq 0) . The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R, is:$ (1) $\frac{730}{119}$ (2) $\frac{625}{111}$ (3) $\frac{821}{123}$ (4) $\frac{567}{121}$ 161. $The area (in sq. units) of the region A = {(x, y) : (x - 1) [x] \leq y \leq 2 x, 0 \leq x \leq 2} Where [t] denotes the greatest integer function, is:$ (1) $\frac{8}{3} 2 - 1$ (2) $\frac{4}{3} 2 + 1$ (3) $\frac{8}{3} 2 - \frac{1}{2}$ (4) $\frac{4}{3} 2 - \frac{1}{2}$ 162. $The area of the region bounded by the curve y = 2 x - x^{2} and x -axis is$ (1) $\frac{2}{3} sq.units$ (2) $\frac{4}{3} sq.units$ (3) $\frac{5}{3} sq.units$ (4) $\frac{8}{3} sq.units$ 163. $The area (in sq. units) in the first quadrant bounded by the parabola, y = x^{2} + 1, the tangent to it at the point (2, 5) and the coordinate axes is :$ (1) $\frac{8}{3}$ (2) $\frac{37}{24}$ (3) $\frac{187}{24}$ (4) $\frac{14}{3}$ 164. $The area of the region in the first quadrant inside the circle x^{2} + y^{2} = 8 and outside the parabola y^{2} = 2 x is equal to:$ (1) $\frac{π}{2} - \frac{1}{3}$ (2) $π - \frac{1}{3}$ (3) $\frac{π}{2} - \frac{2}{3}$ (4) $π - \frac{2}{3}$ 165. $Let f : R \to R be a twice differentiable function such that f (x + y) = f (x) f (y) for all x, y R . If f^{’} (0) = 4 a and f satisfies f^{’ ’} (x) - 3 a f^{’} (x) - f (x) = 0, a > 0, then the area of the region R = {(x, y) ∣ 0 \leq y \leq f (a x), 0 \leq x \leq 2} is:$ (1) $e^{2} - 1$ (2) $e^{2} + 1$ (3) $e^{4} + 1$ (4) $e^{4} - 1$ 166. $The area of the region A = ((x, y) : 4 x^{2} + y^{2} ⩽ 8) and (y^{2} ⩽ 4 x) is:$ (1) $π + 4$ (2) $π + \frac{2}{3}$ (3) $\frac{π}{2} + 2$ (4) $\frac{π}{2} + \frac{1}{3}$ 167. $The area of the region bounded by x^{2} = 4 y, y = 1, y = 4 and the y -axis lying in the first quadrant is square units.$ (1) $\frac{22}{3}$ (2) $\frac{28}{3}$ (3) $30$ (4) $\frac{21}{4}$ 168. $Let the area enclosed between the curves ∣ y ∣ = 1 - x^{2} and x^{2} + y^{2} = 1 be α . If 9 α = β π + γ; β, γ are integers, then the value of ∣ β - γ ∣ equals.$ (1) $27$ (2) $33$ (3) $15$ (4) $18$ 169. $Given: f (x) = ⎩ ⎨ ⎧ x, \frac{1}{2}, 1 - x, 0 \leq x < \frac{1}{2} x = \frac{1}{2} \frac{1}{2} < x \leq 1 and g (x) = (x - \frac{1}{2})^{2}, x \in R . Then the area (in sq.units) of the region bounded by the curves, y = f (x) and y = g (x) between the lines, 2 x = 1 and 2 x = 3 is:$ (1) $\frac{1}{2} - \frac{3}{4}$ (2) $\frac{1}{3} + \frac{3}{4}$ (3) $\frac{1}{2} + \frac{3}{4}$ (4) $\frac{3}{4} - \frac{1}{3}$ 170. $The area of the region {(x, y) : ∣ x - 1∣ \leq y \leq 5 - x^{2}} is equal to :$ (1) $\frac{5}{2} sin^{- 1} (\frac{3}{5}) - \frac{1}{2}$ (2) $\frac{5 π}{4} - \frac{3}{2}$ (3) $\frac{3 π}{4} + \frac{3}{2}$ (4) $\frac{5 π}{4} - \frac{1}{2}$ 171. $Area of the region {(x, y) \in R^{2} : y \geq ∣ x + 3∣, 5 y \leq x + 9 \leq 15} is equal to$ (1) $\frac{1}{6}$ (2) $\frac{4}{3}$ (3) $\frac{3}{2}$ (4) $\frac{5}{3}$ 172. $If A = \frac{1}{2} [1 - 3 31], then :$ (1) $A^{30} - A^{25} = 2 I$ (2) $A^{30} + A^{25} + A = I$ (3) $A^{30} + A^{25} - A = I$ (4) $A^{30} = A^{25}$ 173. $Area of the region bounded by the curve y = ex and lines x = 0 and y = e is$ (1) $e - 1$ (2) $\int_{1}^{e} ln (e + 1 - y) d y$ (3) $e -$ (4) $\int_{1}^{e} ln y d y$ 174. $The area bounded by the curves y = cos x and y = sinx between the ordinates x = 0 and x = \frac{3 π}{2} is$ (1) $42 + 2$ (2) $42 - 1$ (3) $42 + 1$ (4) $42 - 2$ 175. $If the area of the region S ={(x, y) : 2y - 2y, x y} is equal to , then the natural number n is equal to_______$ 176. $If the area of the region ((x, y) : 0 \leq y \leq min (2 x, 6 x - x^{2})) is A, then 12 A is equal to.$ 177. $Let α be the area of the larger region bounded by the curve y^{2} = 8 x and the lines y = x and x = 2, which lies in the first quadrant. Then the value of 3 α is equal to ________$ 178. $Let the area of the region ((x, y) : x - 2 y + 4 \geq 0, x + 2 y^{2} \geq 0, x + 4 y^{2} \leq 8, y \geq 0) be \frac{m}{n}, where m and n are coprime numbers. Then m + n is equal to_______$ 179. $Let for x \in R, f (x) = \frac{x + ∣ x ∣}{2} and g (x) = {x, x^{2}, x < 0 x \geq 0 . Then the area bounded by the curve y = (f \circ g) (x) and the lines y = 0, 2 y - x = 15 is equal to _________$ 180. $Three points O (0, 0), P (a, a^{2}), Q (- b, b^{2}), a > 0, b > 0, are on the parabola y = x^{2} . Let S_{1} be the area of the region bounded by the line PQ and the parabola, and S_{2} be the area of the triangle OPQ . If the minimum value of \frac{s _{1}}{s} is \frac{m}{n}, gcd (m, n) = 1, then m + n is equal to:$