If `(x_i , 1/x_i), i = 1, 2, 3, 4` are four distinct points on a circle, then (A) `x_1 x_2 = x_3 x_4` (B) `x_1 x_2 x_3 x_4 = 1` (C) `x_1 + x_2 + x_3 + x_4 = 1` (D) `1/x_1 + 1/x_2 + 1/x_3 + 1/x_4 = 1`

If the normal at four points P_(i)(x_(i), (y_(i)) l, I = 1, 2, 3, 4 on the rectangular hyperbola xy = c^(2) meet at the point Q(h, k), prove that x_(1) + x_(2) + x_(3) + x_(4) = h, y_(1) + y_(2) + y_(3) + y_(4) = k x_(1)x_(2)x_(3)x_(4) =y_(1)y_(2)y_(3)y_(4) =-c^(4)

The equation x ^(4) -2x ^(3)-3x^2 + 4x -1=0 has four distinct real roots x _(1), x _(2), x _(3), x_(4) such that x _(1) lt x _(2) lt x _(3)lt x _(4) and product of two roots is unity, then : x _(1)x _(2) +x_(1)x_(3) + x_(2) x _(4) +x_(3) x _(4)=

A triangle has vertices A_i(x_i , y_i)fori=1,2,3 If the orthocentre of triangle is (0,0), then prove that |x_2-x_3 y_2-y_3 y_1(y_2-y_3)+x_1(x_2-x_3) x_3-x_1y_2-y_3y_2(y_3-y_1)+x_1(x_3-x_1) x_1-x_2y_2-y_3y_3(y_1-y_2)+x_1(x_1-x_2)|=0

The total number of distinct x in R for which |[x, x^2, 1+x^3] , [2x,4x^2,1+8x^3] , [3x, 9x^2,1+27x^3]|=10 is (A) 0 (B) 1 (C) 2 (D) 3

If ((1 +i)/(1 -i))^(x) =1 , then (A) x=2n+1 (B) x=4n (C) x=2n (D) x=4n+1, n in N.

If the circle x^2+y^2=a^2 intersects the hyperbola x y=c^2 at four points P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3), and S(x_4, y_4), then x_1+x_2+x_3+x_4=0 y_1+y_2+y_3+y_4=0 x_1x_2x_3x_4=C^4 y_1y_2y_3y_4=C^4

If the circle x^2+y^2=a^2 intersects the hyperbola x y=c^2 at four points P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3), and S(x_4, y_4), then x_1+x_2+x_3+x_4=0 y_1+y_2+y_3+y_4=0 x_1x_2x_3x_4=C^4 y_1y_2y_3y_4=C^4

Find, in each case, the remainder when : (i) x^(4)- 3x^2 + 2x + 1 is divided by x - 1. (ii) x^2 + 3x^2 - 12x + 4 is divided by x - 2. (iii) x^4+ 1 is divided by x + 1.

If (x-2)/3=(2x-1)/3-1, then x= (a) 2 (b) 4 (c) 6 (d) 8

If x_1, x_2& x_3 are three distinct numbers which satisfy the relation (a-1)x^2+(a^2-4a+3)x+(a^2+a-2)=0 then a=-1 (b) a=-2 a=3 (d) None of these