Quadratic Equations - Algebra - Mathematics Previous Year Questions

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

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 $1-\sqrt{2}$ 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

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

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, $b=$

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 $\alpha, \beta, \gamma$ 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

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

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

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 $|P|=$

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 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

If $x^2+3 x-2 k=0$ and $x^2-2 x-7 k=0$ have a non-zero common root, then the positive root of the equation $k x^2+(k+2) x-(k+1)=0$ is

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

If the sum of two particular roots of the equation $x^4-4 x^3-7 x^2+22 x+24=0$ is equal to the sum of the remaining two roots, then the sum of the cubes of all the roots of this equation is