Quadratic Equation and Inequalities

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation x2
sin \(\theta\) – x(sin \(\theta\) cos \(\theta\) + 1) + cos \(\theta\) = 0 (0 < \(\theta\) < 45o), and \(\alpha\) < \(\beta\). Then $$\sum\limits_{n = 0}^\inft...

Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c \(\ne\) 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then t...

The value of \(\lambda\) such that sum of the squares of the roots of the quadratic equation, x2 + (3 – \(\lambda\))x + 2 = \(\lambda\) has the least value is -

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation,
x2 + x sin \(\theta\) - 2 sin \(\theta\) = 0, \(\theta \in \left( {0,{\pi \over 2}} \right)\), then
$${{{\alpha ^{12}} + {\beta ^{12}}} \over {\left( {{\alpha ^{ - ...

All the pairs (x, y) that satisfy the inequality
\({2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}}}} \le 1\)
also satisfy the equation

The number of real roots of the equation
5 + |2x
– 1| = 2x
(2x
– 2) is

Let p, q and r be real numbers (p \(\ne\) q, r \(\ne\) 0), such that the roots of the equation \({1 \over {x + p}} + {1 \over {x + q}} = {1 \over r}\) are equal in magnitude but opposite in sign, then the sum of squares of these roots...

If an angle A of a \(\Delta\)ABC satiesfies 5 cosA + 3 = 0, then the roots of the quadratic equation, 9x2 + 27x + 20 = 0 are :

If \(\lambda\) \(\in\) R is such that the sum of the cubes of the roots of the equation,
x2 + (2 \(-\) \(\lambda\)) x + (10 \(-\) \(\lambda\)) = 0 is minimum, then the magnitude of the difference of the roots of this equation is :

If tanA and tanB are the roots of the quadratic equation, 3x2 \(-\) 10x \(-\) 25 = 0, then the value of 3 sin2(A + B) \(-\) 10 sin(A + B).cos(A + B) \(-\) 25 cos2(A + B) is :

If f(x) is a quadratic expression such that f (1) + f (2) = 0, and \(-\) 1 is a root of f (x) = 0, then the other root of f(x) = 0 is :

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

The sum of all the real values of x satisfying the equation
2(x\(-\)1)(x2 + 5x \(-\) 50) = 1 is :

Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x\(-\) 1 and it leaves remainder 6 when divided by x + 1; then :

If for a positive integer n, the quadratic equation
\(x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right)\)\(+ .... + \left( {x + \overline {n - 1} } \right)\left( {x + n} \right)\)\(= 10n\)
has two consecutive integral...

If the equations x2 + bx−1 = 0 and x2 + x + b = 0 have a common root different from −1, then \(\left| b \right|\) is equal to :

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

The sum of all real values of \(x\) satisfying the equation \({\left( {{x^2} - 5x + 5} \right)^{{x^2} + 4x - 60}}\, = 1\) is :

Let \(\alpha\) and \(\beta\) be the roots of equation \({x^2} - 6x - 2 = 0\). 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 equal to :

If \(a \in R\) and the equation \(- 3{\left( {x - \left[ x \right]} \right)^2} + 2\left( {x - \left[ x \right]} \right) + {a^2} = 0\) (where [\(x\)] denotes the greater integer \(\le x\)) has no integral solution, then all possible values...

Let \(\alpha\) and \(\beta\) be the roots of equation \(p{x^2} + qx + r = 0,\) \(p \ne 0.\) If \(p,\,q,\,r\) in A.P. and \({1 \over \alpha } + {1 \over \beta } = 4,\) then the value of \(\left| {\alpha - \beta } \right|\) is :

If the equations \({x^2} + 2x + 3 = 0\) and \(a{x^2} + bx + c = 0,\) \(a,\,b,\,c\, \in \,R,\) have a common root, then \(a\,:b\,:c\,\) is

The equation \({e^{\sin x}} - {e^{ - \sin x}} - 4 = 0\) has:

If \(\alpha\) and \(\beta\) are the roots of the equation \({x^2} - x + 1 = 0,\) then \({\alpha ^{2009}} + {\beta ^{2009}} =\)

If the roots of the equation \(b{x^2} + cx + a = 0\) imaginary, then for all real values of \(x\), the expression \(3{b^2}{x^2} + 6bcx + 2{c^2}\) is :

The quadratic equations \({x^2} - 6x + a = 0\) and \({x^2} - cx + 6 = 0\) have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

STATEMENT - 1 : For every natural number \(n \ge 2,\)
\(${1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + ........ + {1 \over {\sqrt n }} > \sqrt n .\)$
STATEMENT - 2 : For every natural number \(n \ge 2,\),
$$$\sqrt {n\left( {n + 1} \right...

If the difference between the roots of the equation \({x^2} + ax + 1 = 0\) is less than \(\sqrt 5 ,\) then the set of possible values of \(a\) is

All the values of \(m\) for which both roots of the equation \({x^2} - 2mx + {m^2} - 1 = 0\) are greater than \(- 2\) but less then 4, lie in the interval

If \(x\) is real, the maximum value of \({{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}\) is

If the roots of the quadratic equation \({x^2} + px + q = 0\) are \(\tan {30^ \circ }\) and \(\tan {15^ \circ }\), respectively, then the value of \(2 + q - p\) is

In a triangle \(PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)\) and \(\tan \left( {{Q \over 2}} \right)\) are the roots of \(a{x^2} + bx + c = 0,\,\,a \ne 0\) then

If the roots of the equation \({x^2} - bx + c = 0\) be two consecutive integers, then \({b^2} - 4c\) equals

The value of \(a\) for which the sum of the squares of the roots of the equation \({x^2} - \left( {a - 2} \right)x - a - 1 = 0\)
assume the least value is :

The value of \(a\) for which the sum of the squares of the roots of the equation \({x^2} - \left( {a - 2} \right)x - a - 1 = 0\) assume the least value is

If both the roots of the quadratic equation \({x^2} - 2kx + {k^2} + k - 5 = 0\) are less than 5, then \(k\) lies in the interval

If the roots of the equation \({x^2} - bx + c = 0\) be two consecutive integers, then \({b^2} - 4c\) equals

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

If one root of the equation \({x^2} + px + 12 = 0\) is 4, while the equation \({x^2} + px + q = 0\) has equal roots,
then the value of \('q'\) is

If \(\left( {1 - p} \right)\) is a root of quadratic equation \({x^2} + px + \left( {1 - p} \right) = 0\) then its root are

If the sum of the roots of the quadratic equation \(a{x^2} + bx + c = 0\) is equal to the sum of the squares of their reciprocals, then \({a \over c},\,{b \over a}\) and \({c \over b}\) are in

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

The value of '\(a\)' for which one root of the quadratic equation
\($\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0\)$
is twice as large as the other is

If \(p\) and \(q\) are the roots of the equation \({x^2} + px + q = 0,\) then

Product of real roots of equation \({t^2}{x^2} + \left| x \right| + 9 = 0\)

If \(a,\,b,\,c\) are distinct \(+ ve\) real numbers and \({a^2} + {b^2} + {c^2} = 1\) then \(ab + bc + ca\) is

If \(\alpha \ne \beta\) but \({\alpha ^2} = 5\alpha - 3\) and \({\beta ^2} = 5\beta - 3\) then the equation having \(\alpha /\beta\) and \(\beta /\alpha \,\,\) as its roots is

Difference between the corresponding roots of \({x^2} + ax + b = 0\) and \({x^2} + bx + a = 0\) is same and \(a \ne b,\) then