The equation $e^{\sin x}-e^{-\sin x}-4=0$ has

Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of real roots of the equation $5+\left|2^x-1\right|=2^x\left(2^x-2\right)$ is : $\mathrm{ }$

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Let $n\ge 3$ be an integer. For a permutation $\sigma =\left(a_1, a_2, \dots ., a_n\right)$ of $\left(1, 2, \dots ., n\right)$ we let $f_{\sigma }\left(x\right)=a_n{x}^{n-1}+a_{n-1}{x}^{n-2}+\dots +a_{2}x+a_1$ . Let $S_{\sigma }$ be the sum of the roots of $f_{\sigma }\left(x\right)=0$ and let $S$ denotes the sum over all permutations $\sigma$ of $\left(1, 2, \dots \dots , n\right)$ of the number $S_{\sigma }.$ Then-

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

If $x$ is a solution of the equation $\sqrt{2x+1}- \sqrt{2x-1}=1, \left(x\ge \frac{1}{2}\right)$ , then $\sqrt{4x^2-1}$  is equal to :

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

If $2+3i$ is one of the roots of the equation $2x^3-9x^2+kx-13=0, k\in R,$ then the real root of this equation (where $i^2=-1$) :

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Let $a,b,c,d$ be real numbers between $-5$ and $5$ such that $\left|a\right|=\sqrt{4-\sqrt{5-a}}, \left|b\right|=\sqrt{4+\sqrt{5-b}}, \left|c\right|=\sqrt{4-\sqrt{5+c}}, \left|d\right|=\sqrt{4+\sqrt{5+d}}$ , Then the product $abcd$ is

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of real roots of the equation ${\left(x+\frac{1}{x}\right)}^3+x+\frac{1}{x}=0$ is

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Let $S=\left\{x\in R :x\ge 0 \& 2\left|\sqrt{x}-3\right|+\sqrt{x} \left(\sqrt{x}-6\right)+6=0\right\}$ . Then $S$:

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Which of the following is not a root of $f\left(x\right)=x^3-11x^2+36x-36?$

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The real number $k$ for which the equation, $2x^3+3x+k=0$ has two distinct real roots in $[0, 1]$ belongs to

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

For a real number $x$, let $\left[x\right]$ denote the largest integer less than or equal to $x$, and let $\left\{x\right\}=x-\left[x\right]$. The number of solutions $x$ to be equation $\left[x\right]\left\{x\right\}=5$ with $0\le x\le 2015$ is

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The set of all positive real values of $k,$ for which the equation $x^3-9x^2+24x-k=0$ has three distinct real roots is the interval

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Suppose $a_2, a_3, a_4, a_5, a_6, a_7$ are integers such that $\frac{5}{7}=\frac{a_2}{2!}+\frac{a_3}{3!}+\frac{a_4}{4!}+\frac{a_5}{5!}+\frac{a_6}{6!}+\frac{a_7}{7!}$, where $0\le a_j<j$for $j=2, 3, 4, 5, 6 7$. The sum $a_2+a_3+a_4+a_5+a_6+a_7$ is

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

In the real number system, the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

Let $S$ be the set of all real roots of the equation $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$, then

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

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

MEDIUM

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

If $\alpha , \beta , \gamma$ are the roots of the equation $x^3+a{x}^2-bx+c=0$, then $\sum {\beta }^2\left(\gamma +\alpha \right)=$

EASY

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

If $\alpha , \beta , \gamma$ are the roots of $x^3-2x^2+3x-4=0$, then find $\sum \alpha \beta (\alpha +\beta )$

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of integers $n$ for which $3x^3-25x+n=0$ has three real roots is-

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The number of integers $a$ in the interval $\left[1,2014\right]$ for which the system of equations $x+y=a,\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is -

HARD

Mathematics>Algebra>Theory of Equation>Algebraic Equations of Higher Degree

The polynomial equation $x^3-3a{x}^2+(27a^2+9)x+2016=0$ has