Quadratic Equation and Inequalities

The number of real roots of the equation e4x \(-\) e3x \(-\) 4e2x \(-\) ex + 1 = 0 is equal to ______________.

Let \(\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}\) and \(\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}\). If \(8{x^2} + bx + c = 0\) is a quadratic equation whose roots are...

The set of all values of K > \(-\)1, for which the equation \({(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0\) has real roots, is :

The sum of 162th power of the roots of the equation x3 \(-\) 2x2 + 2x \(-\) 1 = 0 is ________.

Let \(\alpha\) and \(\beta\) be two real numbers such that \(\alpha\) + \(\beta\) = 1 and \(\alpha\)\(\beta\) = \(-\)1. Let pn = (\(\alpha\))n + (\(\beta\))n, pn\(-\)1 = 11 and pn+1 = 29 for some integer n \(\ge\) 1. Then, the value of p$...

The sum of all integral values of k (k \(\ne\) 0) for which the equation \({2 \over {x - 1}} - {1 \over {x - 2}} = {2 \over k}\) in x has no real roots, is ____________.

Let \(\lambda\) \(\ne\) 0 be in R. If \(\alpha\) and \(\beta\) are the roots of the equation x2 \(-\) x + 2\(\lambda\) = 0, and \(\alpha\) and \(\gamma\) are the roots of equation 3x2 \(-\) 10x + 27\(\lambda\) = 0, then $${{\beta \gamma } \...

If \(\alpha\), \(\beta\) are roots of the equation \({x^2} + 5(\sqrt 2 )x + 10 = 0\), \(\alpha\) > \(\beta\) and \({P_n} = {\alpha ^n} - {\beta ^n}\) for each positive integer n, then the value of $$\left( {{{{P_{17}}{P_{20}} + 5\sqrt 2 {P_...

The number of real roots of the equation \({e^{6x}} - {e^{4x}} - 2{e^{3x}} - 12{e^{2x}} + {e^x} + 1 = 0\) is :

If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________.

The number of real solutions of the equation, x2 \(-\) |x| \(-\) 12 = 0 is :

If [x] be the greatest integer less than or equal to x, then \(\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]}\) is equal to :

The integer 'k', for which the inequality x2 \(-\) 2(3k \(-\) 1)x + 8k2 \(-\) 7 > 0 is valid for every x in R, is :

Let \(\alpha\) and \(\beta\) be the roots of x2 \(-\) 6x \(-\) 2 = 0. If an = \(\alpha\)n \(-\) \(\beta\)n for n \(\ge\) 1, then the value of \({{{a_{10}} - 2{a_8}} \over {3{a_9}}}\) is :

Let p and q be two positive numbers such that p + q = 2 and p4+q4 = 272. Then p and q are
roots of the equation :

The number of the real roots of the equation \({(x + 1)^2} + |x - 5| = {{27} \over 4}\) is ________.

Let [x] denote the greatest integer less than or equal to x. Then, the values of x\(\in\)R satisfying the equation \({[{e^x}]^2} + [{e^x} + 1] - 3 = 0\) lie in the interval :

If \(\alpha\) and \(\beta\) are the distinct roots of the equation \({x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0\), then the value of \({\alpha ^{96}}({\alpha ^{12}} - 1) + {\beta ^{96}}({\beta ^{12}} - 1)\) is equal to :

Let f(x) be a polynomial of degree 3 such that \(f(k) = - {2 \over k}\) for k = 2, 3, 4, 5. Then the value of 52 \(-\) 10f(10) is equal to :

The numbers of pairs (a, b) of real numbers, such that whenever \(\alpha\) is a root of the equation x2 + ax + b = 0, \(\alpha\)2 \(-\) 2 is also a root of this equation, is :

The value of \(3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}\) is equal to

The value of \(4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}\) is :

The number of real roots of the equation,
e4x + e3x – 4e2x + ex + 1 = 0 is :

Let a, b \(\in\) R, a \(\ne\) 0 be such that the equation,
ax2 – 2bx + 5 = 0 has a repeated root \(\alpha\), which
is also a root of the equation, x2 – 2bx – 10 = 0.
If \(\beta\) is the other root of this equation, then
\(\alpha\)2 +...

The least positive value of 'a' for which the
equation 2x2 + (a – 10)x + \({{33} \over 2}\)
= 2a has real
roots is

Let S be the set of all real roots of the equation,
3x(3x – 1) + 2 = |3x – 1| + |3x – 2|. Then S :

Let \(\alpha = {{ - 1 + i\sqrt 3 } \over 2}\). If \(a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}}\) and \(b = \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}}\), then a and b are the roots of the quadratic equation...

Let \(\alpha\) and \(\beta\) be two real roots of the equation (k + 1)tan2x - \(\sqrt 2\) . \(\lambda\)tanx = (1 - k), where k(\(\ne\) - 1)
and \(\lambda\) are real numbers. if tan2 (\(\alpha\) + \(\beta\)) = 50, then a value of $$...

Let \(\alpha\) and \(\beta\) be the roots of the equation x2
- x - 1 = 0. If pk
= \({\left( \alpha \right)^k} + {\left( \beta \right)^k}\)
, k \(\ge\) 1, then which one
of the following statements is not true?

If \(\alpha\) and \(\beta\) be two roots of the equation x2 – 64x + 256 = 0. Then the value of
\({\left( {{{{\alpha ^3}} \over {{\beta ^5}}}} \right)^{1/8}} + {\left( {{{{\beta ^3}} \over {{\alpha ^5}}}} \right)^{1/8}}\) is :

If \(\alpha\) and \(\beta\) are the roots of the equation
2x(2x + 1) = 1, then \(\beta\) is equal to :

The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is :

If \(\alpha\) and \(\beta\) are the roots of the equation,
7x2 – 3x – 2 = 0, then the value of
\({\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}\) is equal to :

Let [t] denote the greatest integer \(\le\) t. Then the equation in x,
[x]2 + 2[x+2] - 7 = 0 has :

Let \(\alpha\) and \(\beta\) be the roots of x2 - 3x + p=0 and \(\gamma\) and \(\delta\) be the roots of x2 - 6x + q = 0. If \(\alpha, \beta, \gamma, \delta\)
form a geometric progression.Then ratio (2q + p) : (2q - p) is:

Let \(\lambda \ne 0\) be in R. If \(\alpha\) and \(\beta\) are the roots of the equation, x2 - x + 2\(\lambda\) = 0 and \(\alpha\) and \(\gamma\) are the roots of the equation, \(3{x^2} - 10x + 27\lambda = 0\), then $${{\beta \gamma...

If \(\alpha\) and \(\beta\) are the roots of the equation
x2
+ px + 2 = 0 and \({1 \over \alpha }\) and \({1 \over \beta }\) are the roots of
the equation 2x2
+ 2qx + 1 = 0, then
$$\left( {\alpha - {1 \over \alpha }} \right)\left( {\be...

The set of all real values of \(\lambda\) for which the
quadratic equations, (\(\lambda\)2
+ 1)x2
– 4\(\lambda\)x + 2 = 0
always have exactly one root in the interval
(0, 1) is :

Let
\(\alpha\) and
\(\beta\) be the roots of the equation
5x2 + 6x – 2 = 0. If Sn
=
\(\alpha\)n +
\(\beta\)n, n = 1, 2, 3....,
then :

Let f(x) be a quadratic polynomial such that
f(–1) + f(2) = 0. If one of the roots of f(x) = 0
is 3, then its other root lies in :

The number of all possible positive integral values of \(\alpha\)  for which the roots of the quadratic equation, 6x2 \(-\) 11x + \(\alpha\) = 0 are rational numbers is :

If both the roots of the quadratic equation x2 \(-\) mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval :

Let p, q \(\in\) R. If 2 - \(\sqrt 3\) is a root of the quadratic
equation, x2 + px + q = 0, then :

If m is chosen in the quadratic equation
(m2 + 1)
x2 – 3x + (m2 + 1)2 = 0
such that the sum of its
roots is greatest, then the absolute difference of
the cubes of its roots is :-

The sum of the solutions of the equation
\(\left| {\sqrt x - 2} \right| + \sqrt x \left( {\sqrt x - 4} \right) + 2 = 0\)
(x > 0) is equal to:

The number of integral values of m for which the
equation
(1 + m2
)x2
– 2(1 + 3m)x + (1 + 8m) = 0
has no real root is :

If \(\lambda\) be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m – 4)x + 2 = 0, then the least value of m for which \(\lambda + {1 \over \lambda } = 1,\) is

The number of integral values of m for which the quadratic expression, (1 + 2m)x2 – 2(1 + 3m)x + 4(1 + m), x \(\in\) R, is always positive, is :

If \(\alpha\), \(\beta\) and \(\gamma\) are three consecutive terms of a non-constant G.P. such that the equations \(\alpha\)x
2
+ 2\(\beta\)x + \(\gamma\) = 0 and
x2
+ x – 1 = 0 have a common root, then \(\alpha\)(\(\beta\) + $$\...

If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is