A square `P_1 P_2 P_3 P_4` is drawn in the complex plane with `P_1`, at `(1, 0) and P_3` at `(3, 0)`. Let `P_n` denotes the point `(x_n,y_n)n = 1, 2, 3, 4`. Find the numerical value of the product of complex numbers `(x_1 +iy_1)(x_2 +i y_2)(x_3 + i y_3)(x_4 +i y_4)`.

If the lines p_1 x + q_1 y = 1, p_2 x + q_2 y=1 and p_3 x + q_3 y = 1 be concurrent, show that the points (p_1 , q_1), (p_2 , q_2 ) and (p_3 , q_3) are colliner.

IF 2y-p^2 x=3 and 2y-(4px+1)=0 are parallel find the value of p.

The random variable X can the values 0, 1, 2, 3, Give P(X = 0) = P(X = 1)= p and P(X = 2) = P(X = 3) such that sum p_i x_i^2=2sum p_i x_i then find the value of p

Let (x,y) be any point on the parabola y^(2)=4x. Let P be the point that divides the line segment from (0,0) and (x,y) n the ratio 1:3. Then the locus of P is :

Given four points P_1,P_2,P_3a n dP_4 on the coordinate plane with origin O which satisfy the condition ( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n) (i) If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve (ii) If P1,P2,P3 lie on a circle x^2+y^2=1, then prove that P4 also lies on this circle.

If the circel x ^(2) + y^(2) -(3p +4) x - (p -2)y + 10p=0 passes through the point (3,1) , then value of p is-

Let (x,y) be any point on the parabola y^2 = 4x . Let P be the point that divides the line segment from (0,0) and (x,y) n the ratio 1:3. Then the locus of P is :

Let (x,y) be any point on the parabola y^2 = 4x . Let P be the point that divides the line segment from (0,0) and (x,y) n the ratio 1:3. Then the locus of P is :

Let (x,y) be any point on the parabola y^2 = 4x . Let P be the point that divides the line segment from (0,0) and (x,y) n the ratio 1:3. Then the locus of P is :