Quadratic Equation and Inequalities

Let a, b, c be positive integers in arithmetic progression such that the equation\(ax^2 + bx + c = 0\)has only integer solutions.Then which of the following statements is (are) TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$.
Define the functions $f: \mathbb{R} \to \mathbb{R}$, $g: \mathbb{R} \to \mathbb{R}$, and $h: \mathbb{R} \to \mathbb{R}$ by
$f(x) = a...

Let $a=3 \sqrt{2}$ and $b=\frac{1}{5^{1 / 6} \sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that

$$ \begin{aligned} & 3 x+2 y=\log _a(18)^{\frac{5}{4}} \quad \text { and } \\ & 2 x-y=\log _b(\sqrt{1080}), \end{aligned} $$

then $4 x+5 y$ is...

The product of all positive real values of $x$ satisfying the equation

\(x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}\)

is __________.

For x \(\in\) R, the number of real roots of the equation \(3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0\) is ________.

Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x \(-\) 2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x2 \(-\) 20x + 2020. Then the value of ac(a \(-\) c) + ad(a \(-\) d) ...

Let \(\alpha\) and \(\beta\) be the roots of\({x^2} - x - 1 = 0\), with \(\alpha\) > \(\beta\). For all positive integers n, define\({a_n} = {{{\alpha ^n} - {\beta ^n}} \over {\alpha - \beta }},\,n \ge 1\)$${b_1} = 1\,and\,{b_n} = {a_{...

Let a, b, c three non-zero real numbers such that the equation \(\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]\), has two distinct real roots \(\alpha\) and \(\beta\) with $$\alpha + \beta = {\pi...

If a4 = 28, then p + 2q =

a12 = ?

Let \(- {\pi \over 6} < \theta < - {\pi \over {12}}.\) Suppose \({\alpha _1}\) and \({\beta_1}\) are the roots of the equation \({x^2} - 2x\sec \theta + 1 = 0\) and \({\alpha _2}\) and \({\beta _2}\) are the roots of the equation $${...

Let \(S\) be the set of all non-zero real numbers \(\alpha\) such that the quadratic equation \(\alpha {x^2} - x + \alpha = 0\) has two distinct real roots \({x_1}\) and \({x_2}\) satisfying the inequality $$\left| {{x_1} - {x_2}} \right|...

The quadratic equation \(p(x)\) \(= 0\) with real coefficients has purely imaginary roots. Then the equation \(p(p(x))=0\) has

If \({3^x}\, = \,{4^{x - 1}},\) then \(x\, =\)

Let \(\alpha\)(a) and \(\beta\)(a) be the roots of the equation \((\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)x + (\root 6 \of {1 + a} - 1) = 0\) where \(a > - 1\). Then \(\mathop {\lim }\limits_{a \to {0^ + }} \alpha (a)\) and ...

The value of \(6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}...} } } } \right)\) is __________.

A value of \(b\) for which the equations
\($\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr }\)$
have one root in common is

The number of distinct real roots of \({x^4} - 4{x^3} + 12{x^2} + x - 1 = 0\)

The minimum value of the sum of real numbers \({a^{ - 5}},\,{a^{ - 4}},\,3{a^{ - 3}},\,1,\,{a^8}\) and \({a^{10}}\) where \(a > 0\) is

Let \(\left( {{x_0},{y_0}} \right)\) be the solution of the following equations
\(\matrix{ {{{\left( {2x} \right)}^{\ell n2}}\, = {{\left( {3y} \right)}^{\ell n3}}} \cr {{3^{\ell nx}}\, = {2^{\ell ny}}} \cr }\)
Then \({x_0}\)...

Let \(\alpha\) and \(\beta\) be the roots of \({x^2} - 6x - 2 = 0,\) with \(\alpha > \beta .\) If \({a_n} = {\alpha ^n} - {\beta ^n}\) for \(\,n \ge 1\) then the value of \({{{a_{10}} - 2{a_8}} \over {2{a_9}}}\) is

Let \(p\) and \(q\) be real numbers such that \(p \ne 0,\,{p^3} \ne q\) and \({p^3} \ne - q.\) If \({p^3} \ne - q.\) and \(\,\beta\) are nonzero complex numbers satisfying \(\alpha \, + \beta = - p\,\) and $${\alpha ^3} + {\beta ^3} =...

The smallest value of \(k\), for which both the roots of the equation
\(${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right) = 0\)$
are real, distinct and have values at least 4, is

Let \(a,\,b,c\), \(p,q\) be real numbers. Suppose \(\alpha ,\,\beta\) are the roots of the equation \({x^2} + 2px + q = 0\) and \(\alpha ,{1 \over \beta }\) are the roots of the equation \(a{x^2} + 2bx + c = 0,\) where $${\beta ^2} \in \le...

Let \(\alpha,\beta\) be the roots of the equation \(x^2-px+r=0\) and \(\frac{\alpha}{2},2\beta\) be the roots of the equation \(x^2-qx+r=0\). Then the value of r is

Let \(\alpha ,\,\beta\) be the roots of the equation \({x^2} - px + r = 0\) and \({\alpha \over 2},\,2\beta\) be the roots of the equation \({x^2} - qx + r = 0\). Then the value of \(r\)

If roots of the equation $x^2-10 c x-11 d=0$ are $a, b$ and those of $x^2-10 a x-11 b=0$ are $c, d$, then the value of $a+b+c+d$ is $(a, b, c$ and $d$ are distinct numbers)

Let \(a, b, c\) be the sides of a triangle. No two of them are equal and \(\lambda \in R\). If the roots of the equation \(x^{2}+2(a+b+c) x+3 \lambda(a b+b c+c a)=0\) are real, then,

For all \('x',{x^2} + 2ax + 10 - 3a > 0,\) then the interval in which '\(a\)' lies is

If one root is square of the other root of the equation \({x^2} + px + q = 0\), then the realation between \(p\) and \(q\) is

If \(a,\,b,c\) are positive real numbers. Then prove that
\(${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left( {c + 1} \right)^7} > {7^7}\,{a^4}{b^4}{c^4}\)$

If \(\,\alpha \in \left( {0,{\pi \over 2}} \right)\,\,then\,\,\sqrt {{x^2} + x} + {{{{\tan }^2}\alpha } \over {\sqrt {{x^2} + x} }}\) is always greater than or equal to

If \({x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0\) where \(a,\,b\, \in \,R\) then find the values of a for which equation has unequal real roots for all values of \(b\).

The set of all real numbers x for which \({x^2} - \left| {x + 2} \right| + x > 0\), is

If \({a_1},{a_2}.......,{a_n}\) are positive real numbers whose product is a fixed number c, then the minimum value of \({a_1} + {a_2} + ..... + {a_{n - 1}} + 2{a_n}\) is

Let \(a,\,b,\,c\) be real numbers with \(a \ne 0\) and let \(\alpha ,\,\beta\) be the roots of the equation \(a{x^2} + bx + c = 0\). Express the roots of \({a^3}{x^2} + abcx + {c^3} = 0\) in terms of \(\alpha ,\,\beta \,\).

If b > a, then the equation (x - a) (x - b) - 1 = 0 has

If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation

For the equation \(3{x^2} + px + 3 = 0\). p > 0, if one of the root is square of the other, then p is equal to

If \(\alpha \,\text{and}\,\beta\) \((\alpha \, < \,\beta )\) are the roots of the equation \({x^2} + bx + c = 0\,\), where \(c < 0 < b\), then

If \(\alpha ,\,\beta\) are the roots of \(a{x^2} + bx + c = 0\), \(\,\left( {a \ne 0} \right)\) and \(\alpha + \delta ,\,\,\beta + \delta\) are the roots of \(A{x^2} + Bx + c = 0,\) \(\left( {A \ne 0\,} \right)\,\) for some contant $...

If the roots of the equation \({x^2} - 2ax + {a^2} + a - 3 = 0\) are real and less than 3, then

Number of divisor of the form 4\(n\)\(+ 2\left( {n \ge 0} \right)\) of the integer 240 is

The sum of all the real roots of the equation \({\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0\) is ............................

Let \(S\) be a square of unit area. Consider any quadrilateral which has one vertex on each side of \(S\). If \(a,\,b,\,c\) and \(d\) denote the lengths of the sides of the quadrilateral, prove that $$2 \le {a^2} + {b^2} + {c^2} + {d^2} \le...

Let n and k be positive such that \(n \ge {{k(k + 1)} \over 2}\) . The number of solutions \(\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k\), all integers, satisfying $${x_1} + {x_2} + \,..... + {x...

Let \(a,\,b,\,c\) be real. If \(a{x^2} + bx + c = 0\) has two real roots \(\alpha\) and \(\beta ,\) where \(\alpha < - 1\) and \(\beta > 1,\) then show that \(1 + {c \over a} + \left| {{b \over a}} \right| < 0.\)

If p, q, r are + ve and are on A.P., the roots of quadratic equation \(p{x^2} + qx + r = 0\) are all real for

The number of points of intersection of two curves y = 2 sin x and y \(= 5{x^2} + 2x + 3\) is

Let \(p,q \in \left\{ {1,2,3,4} \right\}\,\). The number of equations of the form \(p{x^2} + qx + 1 = 0\) having real roots is