Show that the expression `((ax-b)(dx-c))/((bx-a)(cx-d))` be capabie of all values when x is real, if `a^2 - b^2` and `c^(2)-d^(2)` have the same sign.

The expression y = ax^2+ bx + c has always the same sign as of a if (A) 4ac b^2 (C) 4ac= b2 (D) ac < b^2

If (a)/(b) = (c)/(d) , show that : (a + b) : (c + d) = sqrt(a^(2) + b^(2)) : sqrt(c^(2) + d^(2))

If (x-2) is a common factor of the expressions x^(2)+ax+b and x^(2)+cx+d , then (b-d)/(c-a) is equal to

If f(x)=x^3+bx^2+cx+d and 0< b^2 < c , then

If alpha and beta are the roots of the equation ax^2+bx+c=0 then ax^2+bx+c=a(x-alpha)(x-beta) .Also if a quadratic equation f(x)=0 has both roots between m and n then f(m) and f(n) must have same sign. It is given that all the quadratic equations are of form ax^2-bx+c=0 a,b,c epsi N have two distict real roots between 0 and 1 . The least value of c for which such a quadratic equation exists is (A) 1 (B) 2 (C) 3 (D) 4

If alpha and beta are the roots of the equation ax^2+bx+c=0 then ax^2+bx+c=a(x-alpha)(x-beta) .Also if a quadratic equation f(x)=0 has both roots between m and n then f(m) and f(n) must have same sign. It is given that all the quadratic equations are of form ax^2-bx+c=0 a,b,c epsi N have two distict real roots between 0 and 1 .The least value of b for which such a quadratic equation exists is (A) 3 (B) 4 (C) 5 (D) 6

If a ,b ,c ,d in R , then the equation (x^2+a x-3b)(x^2-c x+b)(x^2-dx+2b)=0 has a. 6 real roots b. at least 2 real roots c. 4 real roots d. none of these

int(ax^3+bx^2+cx+d)dx

If a, b, c are distinct positive real numbers such that the quadratic expression Q_(1)(x) = ax^(2) + bx + c , Q_(2)(x) = bx^(2) + cx + a, Q_(3)(x) = cx^(2) + ax + b are always non-negative, then possible integer in the range of the expression y = (a^(2)+ b^(2) + c^(2))/(ab + bc + ca) is

Evaluate: int (2ax+b)/(ax^2+bx+c)dx