If one root of the equation $${x^2} + (1 - 3i)x - 2(1 + i) = 0$$ is $$-$$1 + i, then the other root is

The equation $${x^2} - 3|x| + 2 = 0$$ has

If $$\alpha$$, $$\beta$$ be the roots of $${x^2} - a(x - 1) + b = 0$$, then the value of $${1 \over {{\alpha ^2} - a\alpha }} + {1 \over {{\beta ^2} - a\beta }} + {2 \over {a + b}}$$ is

The sum of all real roots of the equation $$|x - 2{|^2} + |x - 2| - 2 = 0$$ is

The quadratic equation whose roots are three times the roots of $$3a{x^2} + 3bx + c = 0$$ is

If a, b, c are real, then both the roots of the equation $$(x - b)(x - c) + (x - c)(x - a) + (x - a)(x - b) = 0$$ are always

If $$\alpha$$, $$\beta$$ be the roots of the quadratic equation x² + x + 1 = 0 then the equation whose roots are $$\alpha$$¹⁹, $$\beta$$⁷ is

The roots of the quadratic equation $${x^2} - 2\sqrt 3 x - 22 = 0$$ are

The quadratic equation $${x^2} + 15|x| + 14 = 0$$ has

If sin $$\theta$$ and cos $$\theta$$ are the roots of equation ax² $$-$$ bx + c = 0, then a, b and c satisfy the relation

Let a, b, c be three real numbers such that a + 2b + 4c = 0. Then the equation ax² + bx + c = 0

If the ratio of the roots of the equation $$p{x^2} + qx + r = 0$$ is $$a:b$$, then $${{ab} \over {{{(a + b)}^2}}}$$ =

If $$\alpha$$ and $$\beta$$ are the roots of the equation x² + x + 1 = 0, then the equation whose roots ar $$\alpha$$¹⁹ and $$\beta$$⁷ is

If the sum of the squares of the roots of the equation $x^2-(a-2) x-(a+1)=0$ is least for an appropriate value of the variable parameter $a$, then that value of ' $a$ ' will be

If $$\mathrm{P}(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}$$ and $$\mathrm{Q}(x)=-\mathrm{a} x^2+\mathrm{d} x+\mathrm{c}$$ where $$\mathrm{ac} \neq 0$$, then $$\mathrm{P}(x) \cdot \mathrm{Q}(x)=0$$ has (a, b, c, d are real)

Let $$\mathrm{N}$$ be the number of quadratic equations with coefficients from $$\{0,1,2, \ldots, 9\}$$ such that 0 is a solution of each equation. Then the value of $$\mathrm{N}$$ is

If $$a, b, c$$ are distinct odd natural numbers, then the number of rational roots of the equation $$a x^2+b x+c=0$$

If one root of $${x^2} + px - {q^2} = 0,p$$ and $$q$$ are real, be less than 2 and other be greater than 2, then

If a, b are odd integers, then the roots of the equation $$2a{x^2} + (2a + b)x + b = 0$$, $$a \ne 0$$ are

The value of a for which the sum of the squares of the roots of the equation $${x^2} - (a - 2)x - a - 1 = 0$$ assumes the least value is

Find the values of a for which the expression $${x^2} - (3a - 1)x + 2{a^2} + 2a - 11$$ is always positive.

Determine the sum of imaginary roots of the equation $$(2{x^2} + x - 1)(4{x^2} + 2x - 3) = 6$$

If the quadratic equation $$a x^2+b x+c=0(a>0)$$ has two roots $$\alpha$$ and $$\beta$$ such that $$\alpha<-2$$ and $$\beta>2$$, then