Let $f\left(x\right)=x^2+4x+1$, then -

Important Questions on Quadratic Expressions

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

For all $n\in \mathrm{\mathbb{N}}, n\in \mathrm{\mathbb{N}},$, if $1^2+2^2+3^2+\dots +n^2>x$, then $x=$

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The maximum value of $\mathrm{z}=7\mathrm{x}+5\mathrm{y}$ subject to $2\mathrm{x}+\mathrm{y}\le 100, 4\mathrm{x}+3\mathrm{y}\le 240, \mathrm{x}\ge 0, \mathrm{y}\ge 0$ is

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The smallest prime number satisfying the inequality $\frac{2\mathrm{n}-3}{3}\ge \frac{\mathrm{n}-1}{6}+1$ is

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The number of non-negative integer solutions of the equations $6x+4y+z=200$ and $x+y+z=100$ is

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The least positive integer $n$ for which $\sqrt{n+1}-\sqrt{n-1}<0.2$ is

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

For $0\le p\le 1$ and for any positive $a, b ;$ let $I\left(p\right)=(a+b{)}^p, J(p)=a^p+b^p,$ then

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solution set of the rational inequality $\frac{\mathrm{x}+9}{\mathrm{x}-6}\le 0$ is

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

If $\left|x+5\right|\ge 10$, then

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The least positive integer $n$ for which $\sqrt[3]{n+1}-\sqrt[3]{n}<\frac{1}{12}$ is-

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

If $\frac{6x^2-5x-3}{x^2-2x+6}\le 4,$ then the least and the highest values of $4x^2$ are

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let $N$ be the set of positive integers. For all $n\in N,$ let $f_n={\left(n+1\right)}^{1/3}-n^{1/3}$ and $A=\left\{n\in N:f_{n+1}<\frac{1}{3{\left(n+1\right)}^{2/3}}<f_n\right\}$
Then,

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

If $5, 5r, 5r^2$ are the lengths of the sides of a triangle, then $r$ can not be equal to:

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let $x=\left[\frac{a+2b}{a+b}\right]$ and $y=\frac{a}{b}$, where $a$ and $b$ are positive integers. If $y^2>2$, then

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

$\left\{x\in \mathrm{ℝ}:\frac{\sqrt{6+x-x^2}}{2x+5}\ge \frac{\sqrt{6+x-x^2}}{x+4}\right\}=$

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solution of the inequality $\left|x^2-4x\right|<5$ is

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

If $\beta$ satisfies the equation $x^2-x-6>0$, then a value exists for

EASY

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The solution of $\frac{6x}{4x-1}<\frac{1}{2}$is

MEDIUM

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The number of integers satisfying the inequality $\left|n^2-100\right|<50$ is

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

Let $a,b,c,d,e$ be real numbers such that $a+b<c+d, b+c<d+e, c+d<e+a, d+e<a+b.$ Then

HARD

Mathematics>Algebra>Quadratic Expressions>Algebraic Inequalities

The number of integral values of $m$ for which the quadratic expression $\left(1+2m\right) x^2-2\left(1+3m\right)x+4\left(1+m\right), x\in R$ is always positive, is