Quadratic Equations

The set of all values of $x$ for which the inequalities $x^2-7 x+10 \geq 0$ and $2 x+3-x^2>0$ hold simultaneously is

Let the equations $a x^2-7 x+c=0$ and $a x^2+5 x-c=0$ have a common root and $a c \neq 0$. If 3 is a root of $a x^2-7 x+c=0$ other than the common root, then the common root of the given equations is

The sum of the roots of the equation $e^{4 t}-10 e^{3 t}+29 e^{2 t}-20 e^t+4=0$ is

Two numbers $b$ and $c$ are chosen at random in succession without replacement from the set $\{1,2,3, \ldots \ldots, 9\}$. Then, the probability that $x^2+b x+c>0, \forall x \in R$ i

If one root of the equation $4 x^2-2 x+k-4=0$ is the reciprocal of the other, then the value of $k$ is

If $(x-2)$ is a common factor of the expressions $x^2+a x+b$ and $x^2+c x+d$, then $\frac{b-d}{c-a}=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+2 x+2=0$, then $\alpha^{15}+\beta^{15}=$

If the equation whose roots are $P$ times the roots of the equation $x^4-2 a x^3+4 b x^2+8 a x+16=0$ is a reciprocal equation, then $|P|=$

The quadratic equations $x^2-6 x+a=0$ and $x^2-c x+6=0$ have one root in common. If the other roots of the first and second equations are integers and are in the ratio $4: 3$, then their common root is

The set of all values of $x$ which satisfy both the inequations $x^2-1 \leq 0$ and $x^2-x-2 \geq 0$ simultaneously is

The set of all real values of the expression $\frac{x^2-x+2}{x^2+x-2} \forall x \in R-\{-2,1\}$ is

$\left(x^4+1\right)=\frac{1}{a}(x+1)^4$ is a reciprocal equation

If $\alpha$ and $\beta$ are the roots of a quadratic equation $x^2+b x+c=0$ such that $\alpha^2+\beta^2=5$ and $\alpha^3+\beta^3=9$, then $b+c=$

If $\alpha, \beta$ and $2 \beta$ are the real roots of the equation $x^3-9 x^2+k=0$ and $k \in R-\{0\}$, then $14 \beta=$

The sum of all distinct roots of the equation $x^5-3 x^4+5 x^3-5 x^2+3 x-1=0$ is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-9 x^2+23 x-15=0$, then $\alpha^3+\beta^3+\gamma^3=$

If $6 x-x^2+12$ attains its extreme value $\beta$ at $x=\alpha$, then $\beta=$

If $\alpha$ and $\beta$ are the irrational roots of the equation $3 p^2 x^3+p x^2+q x+3=0$ when $p=1$ and $q=-7$, then $|\alpha-\beta|=$

The roots of a cubic equation $f(x)=0$ are diminished by $\frac{-3}{2}$ so, as to remove the term containing $x^2$ and the transformed equation is $8 x^3-54 x-78=0$. Then, the equation $f(x)=0$ is

Statement I The set of solutions of $|x|^2-4|x|+3<0$ is the interval $(-3,3)$
Statement II If $x<3$ or $x>5$, then $x^2-8 x+15>0$
Which of the above statements is (are) true?

If the sum of two roots of the equation $x^3-7 p x^2+5 q x-6 r=0$ is zero, then

Let $a$ be a common root of the equations $x^3-2 x-25 \lambda=0,3 x^3-8 x-\frac{175}{3} \lambda=0$ and $\lambda>0$. Then, $\lambda=$

The number of non-real roots of the equation $x^{10}-3 x^8+5 x^6-5 x^4+3 x^2-1=0$ is

If the roots of $x^5-a x^4+b x^3-c x^2+d x-1=0$ are all positive such that their arithmetic mean and geometric mean are equal, then $a+b+c+d=$

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-2 x-4=0$, then $\alpha^3+\beta^3+\gamma^3=$

Let $x$ be a real number. Malch the following:


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When $b=17$, it is found that the roots of the equation $x^2+b x+c=0$ are -2 and -15 . If $\alpha$ and $\beta$ are the roots of the same equation when $b=13$, then $|\alpha-\beta|=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2 \sqrt{3} x+4=0$, then $\alpha^6+\beta^6=$

If $m$ and $M$ are respectively, the smallest and greatest rational roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $M-m=$

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-3 x^2+2 x-4=0$, then $\Sigma \alpha^2 \beta^2=$

If $f(x)=\frac{2 x-3}{(x-2)(x-3)}$ is a real valued function, then the value that $f(x)$ does not take is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+4 x^2-9 x-36=0$ such that $\alpha+\beta=0$, then $\alpha^2+2 \beta^2+3 \gamma^2=$

Let $f(x)=A x^2+B x, g(x)=L x^2+M x+N$. Given that $f(2)-g(2)=1, f(3)-g(3)=4, f(4)-g(4)=9$. Then, a root of $f(x)-g(x)=0$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $4 x^3+12 x^2-7 x+165=0$ and $\alpha+5, \beta+5, \gamma+5$ are the roots of the equation $a x^3+b x^2+c x+d=0$ then the product of the roots of the second equation is

Let $p(x)$ be a quadratic polynomial with real coefficients. If $p(x)=0$ has only purely imaginary roots, then the zeroes of the polynomial $p(p(x))$ are

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-5 x^2-2 x+24=0$, then $\frac{\beta \gamma}{\alpha}+\frac{\gamma \alpha}{\beta}+\frac{\alpha \beta}{\gamma}=$

If the extreme value of $3 x-2 x^2+1$ is $k$, then the set of all real values of $x$ for which $k x^2+2 x+1>0$ is

If $\frac{5}{2}$ is the sum of two roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$ then the sum of all non-real roots of the equation is

If $1-\sqrt{2}$ and $2+i$ are the roots of the equation $x^4+b x^3+c x^2+d x+e=0$ where $b, c, d, e$ are rational numbers, then the roots of the equation $b x^2+c x+d=0$ are

If $\tan 15^{\circ}$ and $\tan 30^{\circ}$ are the roots of equation $x^2+p x+q=0$, then $p q=$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+x^2+x+r=0$ and $\alpha^3+\beta^3+\gamma^3=5$, then $r=$

II. Let $f(x)=\frac{6 x^2-18 x+21}{6 x^2-18 x+17}$. If $m$ is the maximum value of $f(x)$ and $f(x)>n \forall x \in R$. Then, $14 m-7 n=$

If the quadratic equations $x^2-7 x+3 c=0$ and $x^2+x-5 c=0$ have a common root, then for non-zero real value of $c$ the sign of the expression $x^2-3 x+c$ is

Let the transformed equation of $2 x^4-8 x^3+3 x^2-1=0$ so that the term containing the cubic power of $x$ is absent be $2 x^4+b x^2+c x+d=0$. Then, $b=$

If $2+\sqrt{3}$ is a root of the equation $f(x)=x^4+2 x^3-16 x^2-22 x+7=0$, then which one of the following is not a root of $f(x)=0$ ?

The number of integral values of $x$ satisfying $9 x-2<(x+2)^2<12 x-3$ is

If $\alpha$ and $\beta$ are two complex roots of the equation $6 x^6-25 x^5+31 x^4-31 x^2+25 x-6=0$, then $\alpha+\beta=$

When $\mathbf{R}$ is the set of all real numbers,
\(\left\{x \in \mathbf{R}: \frac{\sqrt{12-x-x^2}}{x+10} \leq \frac{\sqrt{12-x-x^2}}{2 x+9}\right\}=\)

If $x$ is real, then the maximum and minimum values of $\frac{x^2+14 x+9}{x^2+2 x+3}$ are respectively

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+3 x^2-x-3=0$, then $\left(1+\alpha^2\right)\left(1+\beta^2\right)\left(1+\gamma^2\right)=$