35+ CAT Quadratic Equations Questions With video Solutions

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

The sum of all possible values of x satisfying the equation $$2^{4x^{2}}-2^{2x^{2}+x+16}+2^{2x+30}=0$$, is

The equation $$x^{3} + (2r + 1)x^{2} + (4r - 1)x + 2 =0$$ 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

A quadratic equation $$x^2 + bx + c = 0$$ 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 $$\alpha$$ and $$\beta$$ be the two distinct roots of the equation $$2x^{2} - 6x + k = 0$$, such that ( $$\alpha + \beta$$) and $$\alpha \beta$$ are the distinct roots of the equation $$x^{2} + px + p = 0$$. Then, the value of 8(k - p) is

Let k be the largest integer such that the equation $$(x-1)^{2}+2kx+11=0$$ has no real roots. If y is a positive real number, then the least possible value of $$\frac{k}{4y}+9y$$ is

Suppose k is any integer such that the equation $$2x^{2}+kx+5=0$$ 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

If $$(3+2\sqrt{2})$$ is a root of the equation $$ax^{2}+bx+c=0$$ and $$(4+2\sqrt{3})$$ is a root of the equation $$ay^{2}+my+n=0$$ where a, b, c, m and n are integers, then the value of $$(\frac{b}{m}+\frac{c-2b}{n})$$ is

Let r and c be real numbers. If r and -r are roots of $$5x^{3} + cx^{2} - 10x + 9 = 0$$, then c equals

Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

The minimum possible value of $$\frac{x^{2} - 6x + 10}{3-x}$$, for $$x < 3$$, is

Suppose one of the roots of the equation $$ax^{2}-bx+c=0$$ is $$2+\sqrt{3}$$, Where a,b and c are rational numbers and $$a\neq0$$. If $$b=c^{3}$$ then $$\mid a\mid$$ equals.

For all real values of x, the range of the function $$f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}$$ is:

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

Consider the pair of equations: $$x^{2}-xy-x=22$$ and $$y^{2}-xy+y=34$$. If $$x>y$$, then $$x-y$$ equals

If r is a constant such that $$\mid x^2 - 4x - 13 \mid = r$$ has exactly three distinct real roots, then the value of r is

Let m and n be positive integers, If $$x^{2}+mx+2n=0$$ and $$x^{2}+2nx+m=0$$ have real roots, then the smallest possible value of $$m+n$$ is