Let $x, y,$ and $z$ be real numbers satisfying $4(x^{2}+y^{2}+z^{2})=a$, $4(x-y-z)=3+a$ The a equals
lf the equations $x^{2}+mx+9=0, x^{2}+nx+17=0$ and $x^{2}+(m+n)x+35=0$ have a common negative root, then the value of $(2m+3n)$ is
The roots $\alpha, \beta$ of the equation $3x^2 + \lambda x - 1 = 0$, satisfy $\cfrac{1}{\alpha^2} + \cfrac{1}{\beta^2} = 15$.The value of $(\alpha^3 + \beta^3)^2$, is
If x and y are real numbers such that $4x^2 + 4y^2 - 4xy - 6y + 3 = 0$, then the value of $(4x + 5y)$ is
Let α and β be the two distinct roots of the equation $2 x^2-6 x+k=0$, such that (α+β) and αβ are the distinct roots of the equation $x^2+p x+p=0$. Then, the value of 8(k−p) is
The equation $x^3+(2 r+1) x^2+(4 r-1) x+2=0$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is
The sum of all possible values of x satisfying the equation $2^{4 x^2}-2^{2 x^2+x+16}+2^{2 x+30}=0$, is
Let k be the largest integer such that the equation $(x-1)^2+2 k x+11=0$ has no real roots. If y is a positive real number, then the least possible value of $\frac{k}{4 y}+9 y$ is
A quadratic equation $x^2+b x+c=0$ has two real roots. If the difference between the reciprocals of the roots is $\frac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $\frac{5}{9}$, then the largest possible value of (b + c) is
Let a,b,c be non-zero real numbers such that $b^2 \lt 4 a c$ and $f(x)=a x^2+b x+c$. If the set S consists of al integers m such that f(m) < 0, then the set S must necessarily be
Let r and c be real numbers. If r and −r are roots of $5 x^3+c x^2-10 x+9=0$, then c equals
Suppose k is any integer such that the equation $2 x^2+k x+5=0$ has no real roots and the equation $x^2+(k-5) x+1=0$ has two distinct real roots for x . Then, the number of possible values of k is
The minimum possible value of $\frac{x^2-6 x+10}{3-x}$, for x<3, is
If $(3+2 \sqrt{2})$ is a root of the equation $a x^2+b x+c=0$, and $(4+2 \sqrt{3})$ is a root of the equation $a y^2+m y+n=0$, where a, b, c, m and n are integers, then the value of $\left(\frac{b}{m}+\frac{c-2 b}{n}\right)$ is
Suppose one of the roots of the equation a $x^2$ - b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = $c^3$ then |a| equals
The number of non-negative integer values of $k$ for which the quadratic equation $x^2 - 5x + k = 0$ has only integer roots, is .