Nature Of Roots Depending Upon Coefficients And Discriminant - Practice Questions & MCQ

If  has real roots  and  , where   then

 The number of real solutions of the equation , is

Let the function have a maxima for some value of and a minima for some value of . Then, the set of all values of is

Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14 \mathrm{x}^2-31 \mathrm{x}+3 \lambda=0$ and $\alpha, \gamma$ be the roots of the equation $35 \mathrm{x}^2-53 \mathrm{x}+4 \lambda=0$. Then $\frac{3 \alpha}{\beta}$ and $\frac{4 \alpha}{\gamma}$ are the roots of the equation

The number of real roots of the equation   , is

If for , then and are the roots of the equation:

Let $S$ be the set of positive integral values of a for which $\frac{a^2+2(a+1) x+9 a+4}{x^2-8 x+32}<0, \forall x \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is:

The sum of all the roots of the equation $\left|x^{2}-8 x+15\right|-2 x+7=0$ is:

If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then:

Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{px}^2+\mathrm{qx}-\mathrm{r}=0$, where $\mathrm{p} \neq 0$. If $\mathrm{p}, \mathrm{q}$ and $\mathrm{r}$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$, then the value of $(\alpha-\beta)^2$ is :