Area between curves

Let $f: [0, \infty) \to \mathbb{R}$ be a continuous function such that

$$f(x) = 1 - 2x + \int_0^x e^{x-t} f(t)\, dt$$

for all $x \in [0, \infty)$. Then, which of the following statement(s) is (are) TRUE?

A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0, 0)$, $Q(1, 1)$ and $R(2, 0)$. From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y = x^n$ $(n > 1)$. If the area of the region taken away by the farmer $F_2$ is exactly 30% of the area of $\triangle PQR$, then the value of $n$ is _____ .

If the line $x = \alpha$ divides the area of region $R = \{(x, y) \in \mathbb{R}^2 : x^3 \le y \le x, 0 \le x \le 1\}$ into two equal parts, then

The area (in sq. units) of the region $\{(x,y):x\ge0,\ x+y\le3,\ x^2\le4y\text{ and }y\le1+\sqrt x\}$ is :

The area (in sq. units) of the smaller portion enclosed between the curves, $x^2+y^2=4$ and $y^2=3x$, is :

Area of the region $\left\{(x,y)\in\mathbb R^2:y\ge\sqrt{|x+3|},\ 5y\le x+9\le15\right\}$ is equal to

The area (in sq. units) of the region $\{(x,y):y^2\ge2x\text{ and }x^2+y^2\le4x,\ x\ge0,\ y\ge0\}$ is :

The area (in sq. units) of the region described by $A=\{(x,y)\mid y\ge x^2-5x+4,\ x+y\ge1,\ y\le0\}$ is :

The area (in sq. units) of the region described by $\{(x,y):y^2\le2x\text{ and }y\ge4x-1\}$ is :

The area (in square units) of the region bounded by the curves $y+2x^2=0$ and $y+3x^2=1$, is equal to :

For a point $P$ in the plane, let $d_1(P)$ and $d_2(P)$ be the distances of the point $P$ from the lines $x-y=0$ and $x+y=0$ respectively. The area of the region $R$ consisting of all points $P$ lying in the first quadrant of the plane and satisfying $2\le d_1(P)+d_2(P)\le4$, is

The area of the region described by $A=\{(x,y):x^2+y^2\le1\text{ and }y^2\le1-x\}$ is :

Let $A=\left\{(x,y):y^2\le4x,\ y-2x\ge-4\right\}$. The area (in square units) of the region A is :

The area of the region above the $x$-axis bounded by the curve $y=\tan x$, $0\le x\le\dfrac\pi2$ and the tangent to the curve at $x=\dfrac\pi4$ is :

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0,\frac\pi2\right]$ is

The area of the region between the curves $y=\sqrt{\dfrac{1+\sin x}{\cos x}}$ and $y=\sqrt{\dfrac{1-\sin x}{\cos x}}$ bounded by the lines $x=0$ and $x=\frac\pi4$ is