Area of Triangle with Coordinates of Vertices Given

Q. Tangent at a point $P_1$ (other than origin) on the curve $y=x^3$ meets the curve again at $P_2.$ The tangent at $P_2$ meets the curve at $P_3$ and so on. Then which of the following is/are correct?

1. Abscissa of $P_1,P_2,\cdots ,P_3$ are in G.P.
   
2. Ordinate of $P_1,P_2,\cdots ,P_3$ are in G.P.
3. Ratio of area of $\Delta P_1{P}_2{P}_3$ and $\Delta P_2{P}_3{P}_4$ is $\frac{1}{8}$
4. Ratio of area of $\Delta P_1{P}_2{P}_3$ and $\Delta P_2{P}_3{P}_4$ is $\frac{1}{16}$

Q.

What is the area of the triangle formed by the vertices $(0,0),(2,1)\text{and}(5,3).$

1. $1$

2. $\frac{1}{2}$

3. $2$

4. $4$

Q. Let two points be $A(1,-1)$ and $B(0,2)$. If a point $P(x^{\prime },y^{\prime })$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line, $3x+y-4\lambda =0$, then the value of $\lambda$ is:

1. $4$
2. $1$
3. $-3$
4. $3$

Q. The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value(s) of k is/are:

1. 3
2. -3
3. 9
4. -9

Q.

Three distinct points $P(3u^2,2u^3);Q(3v^2,2v^3)andR(3w^2,2w^3)$ are collinear and equation $a{x}^3+b{x}^2+cx+d=0$ has roots u, v and w, then which of the following is true

1. b = 0

2. c = 0

3. d = 0

4. ab = 0