Factorize each of the following expressions: (1)`q r-p r+q s-p s` (2) `p^2q-p r^2-p q+r^2` (3)`1+x+x y+x^2y` (4) `a x+a y-b x-b y`

Factorize each of the following algebraic expressions: 4x^2+12 x y+9y^2 (2) 9a^2-24 a b+16 b^2 (3) p^2q^2-6p q r+9r^2 (4) 36 a^2+36 a+9

Match the columns and choose correct options. (1) a-q, b-r, c-s, d-p (2) a-s, b-p, c-q, d-r (3) a-q, b-p, c-s, d-r (4) a-s, b-p, c-r, d-q

If zeros of the polynomial f(x)=x^3-3p x^2+q x-r are in A.P., then (a) 2p^3=p q-r (b) 2p^3=p q+r (c) p^3=p q-r (d) None of these

Prove that the value of each the following determinants is zero: |[a-b,b-c,c-a],[ x-y, y-z, z-x],[ p-q, q-r ,r-p]|

Add: (i) p\ (p-q), q(q-r) and \ r\ (r-p) (ii) 2x\ (z-x-y)\ and \ 2y\ (z-y-x)

Tangent is drawn at any point (p ,q) on the parabola y^2=4a x .Tangents are drawn from any point on this tangant to the circle x^2+y^2=a^2 , such that the chords of contact pass through a fixed point (r , s) . Then p ,q ,r and s can hold the relation (A) r^2q=4p^2s (B) r q^2=4p s^2 (C) r q^2=-4p s^2 (D) r^2q=-4p^2s

If tangents P Qa n dP R are drawn from a variable point P to thehyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,(a > b), so that the fourth vertex S of parallelogram P Q S R lies on the circumcircle of triangle P Q R , then the locus of P is (a) x^2+y^2=b^2 (b) x^2+y^2=a^2 (c) x^2+y^2=a^2-b^2 (d) none of these

For p, q in R , the roots of (p^(2) + 2)x^(2) + 2x(p +q) - 2 = 0 are

Consider the following relations: R = {(x, y) | x, y are real numbers and x = wy for some rational number w}; S={(m/n , p/q)"m , n , p and q are integers such that n ,q"!="0 and q m = p n"} . Then (1) neither R nor S is an equivalence relation (2) S is an equivalence relation but R is not an equivalence relation (3) R and S both are equivalence relations (4) R is an equivalence relation but S is not an equivalence relation