Quadratic Equation and Inequalities

Let \(\alpha, \beta, \gamma\) be the three roots of the equation \(x^{3}+b x+c=0\). If \(\beta \gamma=1=-\alpha\), then \(b^{3}+2 c^{3}-3 \alpha^{3}-6 \beta^{3}-8 \gamma^{3}\) is equal to :

Let m and \(\mathrm{n}\) be the numbers of real roots of the quadratic equations \(x^{2}-12 x+[x]+31=0\) and \(x^{2}-5|x+2|-4=0\) respectively, where \([x]\) denotes the greatest integer \(\leq x\). Then $$\mathrm{m}^{2}+\mathrm{mn}+\mathr...

The sum of all the roots of the equation \(\left|x^{2}-8 x+15\right|-2 x+7=0\) is :

Let \(A = \{ x \in R:[x + 3] + [x + 4] \le 3\} ,\)
\(B = \left\{ {x \in R:{3^x}{{\left( {\sum\limits_{r = 1}^\infty {{3 \over {{{10}^r}}}} } \right)}^{x - 3}} < {3^{ - 3x}}} \right\},\) where [t] denotes greatest integer function. Then,

The number of real roots of the equation \(\sqrt{x^{2}-4 x+3}+\sqrt{x^{2}-9}=\sqrt{4 x^{2}-14 x+6}\), is :

The equation $\mathrm{e}^{4 x}+8 \mathrm{e}^{3 x}+13 \mathrm{e}^{2 x}-8 \mathrm{e}^{x}+1=0, x \in \mathbb{R}$ has :

If the value of real number $a>0$ for which $x^2-5 a x+1=0$ and $x^2-a x-5=0$
have a common real root is $\frac{3}{\sqrt{2 \beta}}$ then $\beta$ is equal to ___________.

Let \(\lambda \ne 0\) be a real number. Let \(\alpha,\beta\) be the roots of the equation \(14{x^2} - 31x + 3\lambda = 0\) and \(\alpha,\gamma\) be the roots of the equation \(35{x^2} - 53x + 4\lambda = 0\). Then $${{3\alpha } \over \bet...

Let \(\alpha_1,\alpha_2,....,\alpha_7\) be the roots of the equation \({x^7} + 3{x^5} - 13{x^3} - 15x = 0\) and \(|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|\). Then \(\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6\)...

Let \(\alpha \in\mathbb{R}\) and let \(\alpha,\beta\) be the roots of the equation \({x^2} + {60^{{1 \over 4}}}x + a = 0\). If \({\alpha ^4} + {\beta ^4} = - 30\), then the product of all possible values of \(a\) is ____________.

Let \(\lambda \in \mathbb{R}\) and let the equation E be \(|x{|^2} - 2|x| + |\lambda - 3| = 0\). Then the largest element in the set S = {\(x+\lambda:x\) is an integer solution of E} is ______

The equation \({x^2} - 4x + [x] + 3 = x[x]\), where \([x]\) denotes the greatest integer function, has :

The number of real solutions of the equation \(3\left( {{x^2} + {1 \over {{x^2}}}} \right) - 2\left( {x + {1 \over x}} \right) + 5 = 0\), is

Let \(S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}\). Then \(n(S)\) is equal to

The number of integral values of k, for which one root of the equation \(2x^2-8x+k=0\) lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is :

The set of all \(a \in \mathbb{R}\) for which the equation \(x|x-1|+|x+2|+a=0\) has exactly one real root, is :

Let \([\alpha]\) denote the greatest integer \(\leq \alpha\). Then \([\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots+[\sqrt{120}]\) is equal to __________

Let \(\alpha, \beta\) be the roots of the equation \(x^{2}-\sqrt{2} x+2=0\). Then \(\alpha^{14}+\beta^{14}\) is equal to

Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^{2}+\sqrt{6} x+3=0\). Then \(\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}\) is equal to :

If \(a\) and \(b\) are the roots of the equation \(x^{2}-7 x-1=0\), then the value of \(\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}\) is equal to _____________.

The number of points, where the curve \(f(x)=\mathrm{e}^{8 x}-\mathrm{e}^{6 x}-3 \mathrm{e}^{4 x}-\mathrm{e}^{2 x}+1, x \in \mathbb{R}\) cuts \(x\)-axis, is equal to _________.

Let S be the set of all integral values of \(\alpha\) for which the sum of squares of two real roots of the quadratic equation \(3{x^2} + (\alpha - 6)x + (\alpha + 3) = 0\) is minimum. Then S :

Let \({S_1} = \left\{ {x \in R - \{ 1,2\} :{{(x + 2)({x^2} + 3x + 5)} \over { - 2 + 3x - {x^2}}} \ge 0} \right\}\) and \({S_2} = \left\{ {x \in R:{3^{2x}} - {3^{x + 1}} - {3^{x + 2}} + 27 \le 0} \right\}\). Then, \({S_1} \cup {S_2}\) is equ...

Let \(\alpha\) be a root of the equation 1 + x2 + x4 = 0. Then, the value of \(\alpha\)1011 + \(\alpha\)2022 \(-\) \(\alpha\)3033 is equal to :

If \(\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots+\frac{1}{(180-a)(200-a)}=\frac{1}{256}\), then the maximum value of \(\mathrm{a}\) is :

Let \(\alpha, \beta(\alpha>\beta)\) be the roots of the quadratic equation \(x^{2}-x-4=0 .\) If \(P_{n}=\alpha^{n}-\beta^{n}\), \(n \in \mathrm{N}\), then \(\frac{P_{15} P_{16}-P_{14} P_{16}-P_{15}^{2}+P_{14} P_{15}}{P_{13} P_{14}}\) is equ...

The number of real solutions of the equation \({e^{4x}} + 4{e^{3x}} - 58{e^{2x}} + 4{e^x} + 1 = 0\) is ___________.

Let f(x) be a quadratic polynomial such that f(\(-\)2) + f(3) = 0. If one of the roots of f(x) = 0 is \(-\)1, then the sum of the roots of f(x) = 0 is equal to :

The sum of all real values of \(x\) for which \(\frac{3 x^{2}-9 x+17}{x^{2}+3 x+10}=\frac{5 x^{2}-7 x+19}{3 x^{2}+5 x+12}\) is equal to __________.

Let \(\alpha\), \(\beta\) be the roots of the equation \(x^{2}-\sqrt{2} x+\sqrt{6}=0\) and \(\frac{1}{\alpha^{2}}+1, \frac{1}{\beta^{2}}+1\) be the roots of the equation \(x^{2}+a x+b=0\). Then the roots of the equation $$x^{2}-(a+b-2) x+(a...

\(\text { Let } S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\} \text { and }\)\(T=\left\{x \in \mathbb{Z}: x^{2}-7|x|+9 \leq 0\right\} \text {. }\)
Then the number of elements in \(\mathrm{S} \cap \mathrm{T}\) is :

If the sum of all the roots of the equation \({e^{2x}} - 11{e^x} - 45{e^{ - x}} + {{81} \over 2} = 0\) is \({\log _e}p\), then p is equal to ____________.

The number of distinct real roots of x4 \(-\) 4x + 1 = 0 is :

Let \(\alpha\), \(\beta\) be the roots of the equation \({x^2} - 4\lambda x + 5 = 0\) and \(\alpha\), \(\gamma\) be the roots of the equation \({x^2} - \left( {3\sqrt 2 + 2\sqrt 3 } \right)x + 7 + 3\lambda \sqrt 3 = 0\), \(\lambda\) > 0. ...

If \(\alpha, \beta\) are the roots of the equation
\(x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0\),
then the e...

The sum of the cubes of all the roots of the equation \({x^4} - 3{x^3} - 2{x^2} + 3x + 1 = 0\) is _________.

Let p and q be two real numbers such that p + q = 3 and p4 + q4 = 369. Then \({\left( {{1 \over p} + {1 \over q}} \right)^{ - 2}}\) is equal to _________.

The number of distinct real roots of the equation \(x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0\) is ______________.

If for some \(\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathbf{R}\), not all have same sign, one of the roots of the equation $$\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right) x^{2}-2 \mathrm{q}(\mathrm{p}+\mathrm{r}) x+\mathrm{q}^{2}+\mathrm{r}^{...

The minimum value of the sum of the squares of the roots of \(x^{2}+(3-a) x+1=2 a\) is:

Let a, b \(\in\) R be such that the equation \(a{x^2} - 2bx + 15 = 0\) has a repeated root \(\alpha\). If \(\alpha\) and \(\beta\) are the roots of the equation \({x^2} - 2bx + 21 = 0\), then \({\alpha ^2} + {\beta ^2}\) is equal to :

Let \(A = \{ x \in R:|x + 1| < 2\}\) and \(B = \{ x \in R:|x - 1| \ge 2\}\). Then which one of the following statements is NOT true?

If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x^{4}+x^{3}+x^{2}+x+1=0\), then \(\alpha^{2021}+\beta^{2021}+\gamma^{2021}+\delta^{2021}\) is equal to :

If the sum of the squares of the reciprocals of the roots \(\alpha\) and \(\beta\) of the equation 3x2 + \(\lambda\)x \(-\) 1 = 0 is 15, then 6(\(\alpha\)3 + \(\beta\)3)2 is equal to :

The number of distinct real roots of the equation x7 \(-\) 7x \(-\) 2 = 0 is

The sum of all the real roots of the equation \(({e^{2x}} - 4)(6{e^{2x}} - 5{e^x} + 1) = 0\) is

cosec18\(^\circ\) is a root of the equation :

The sum of the roots of the equation \(x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0\), is :

Let \(\alpha\), \(\beta\) be two roots of the equation x2 + (20)1/4x + (5)1/2 = 0. Then \(\alpha\)8 + \(\beta\)8 is equal to