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Q 71 :

Let the line $x = - 1$ divide the area of the region ${(x, y) : 1 + x^{2} \leq y \leq 3 - x}$ in the ratio $m : n$ , $\gcd (m, n) = 1$ . Then $m + n$ is equal to [2026]

(4)

Q 72 :

Let $P_{1} : y = 4 x^{2} a n d 𝑃_{2} : 𝑦 = 𝑥^{2} + 27$ be two parabolas. If the area of the bounded region enclosed between $P_{1} a n d P_{2}$ is six times the area of the bounded region enclosed between the line $y = α x, α > 0$ and $P_{1}$ , then $α$ is equal to : [2026]

(2)

Q 73 :

Let $f (α)$ denote the area of the region in the first quadrant bounded by $x = 0$ , $x = 1$ , $y^{2} = x$ and $y =$ $| α x - 5 |$ $-$ $| 1 - α x |$ $+ α x^{2} .$ Then $(f (0) + f (1))$ is equal to [2026]

(3)

Q 74 :

The area of the region enclosed between the circles $x^{2} + y^{2} = 4$ and $x^{2} + {(y - 2)}^{2} = 4$ is: [2026]

$\frac{2}{3} (4 π - 3 \sqrt{3})$
$\frac{4}{3} (2 π - \sqrt{3})$
$\frac{4}{3} (2 π - 3 \sqrt{3})$
$\frac{2}{3} (2 π - 3 \sqrt{3})$

(1)

Q 75 :

If the area of the region ${(x, y) : 1 - 2 x \leq y \leq 4 - x^{2}, x \geq 0, y \geq 0}$ is $\frac{α}{β}, α, β \in ℕ, \gcd (α, β) = 1$ , then the value of $(α + β)$ is: [2026]

(2)

Q 76 :

The area of the region $A = {(x, y) : 4 x^{2} + y^{2} \leq 8 and y^{2} \leq 4 x}$ is: [2026]

$\frac{π}{2} + 2$
$\frac{π}{2} + \frac{1}{3}$
$π + \frac{2}{3}$
$π + 4$

(3)

Q 77 :

The area of the region $R = {(x, y) : x y \leq 8, 1 \leq y \leq x^{2}, x \geq 0}$ is [2026]

$\frac{1}{3} (49 \log_{e} (2) - 15)$
$\frac{2}{3} (20 \log_{e} (2) + 9)$
$\frac{2}{3} (24 \log_{e} (2) - 7)$
$\frac{1}{3} (40 \log_{e} (2) + 27)$

(3)

Topic Question Set

Q 71 :

Let the line $x = - 1$ divide the area of the region ${(x, y) : 1 + x^{2} \leq y \leq 3 - x}$ in the ratio $m : n$ , $\gcd (m, n) = 1$ . Then $m + n$ is equal to [2026]

Q 72 :

Let $P_{1} : y = 4 x^{2} a n d 𝑃_{2} : 𝑦 = 𝑥^{2} + 27$ be two parabolas. If the area of the bounded region enclosed between $P_{1} a n d P_{2}$ is six times the area of the bounded region enclosed between the line $y = α x, α > 0$ and $P_{1}$ , then $α$ is equal to : [2026]

Q 73 :

Let $f (α)$ denote the area of the region in the first quadrant bounded by $x = 0$ , $x = 1$ , $y^{2} = x$ and $y =$ $| α x - 5 |$ $-$ $| 1 - α x |$ $+ α x^{2} .$ Then $(f (0) + f (1))$ is equal to [2026]

Q 74 :

The area of the region enclosed between the circles $x^{2} + y^{2} = 4$ and $x^{2} + {(y - 2)}^{2} = 4$ is: [2026]

Q 75 :

If the area of the region ${(x, y) : 1 - 2 x \leq y \leq 4 - x^{2}, x \geq 0, y \geq 0}$ is $\frac{α}{β}, α, β \in ℕ, \gcd (α, β) = 1$ , then the value of $(α + β)$ is: [2026]

Q 76 :

The area of the region $A = {(x, y) : 4 x^{2} + y^{2} \leq 8 and y^{2} \leq 4 x}$ is: [2026]

Q 77 :

The area of the region $R = {(x, y) : x y \leq 8, 1 \leq y \leq x^{2}, x \geq 0}$ is [2026]

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