Quadratic Equation and Inequalities

Let \(\alpha \,,\,\beta\) be the roots of the equation (x - a) (x - b) = c, \(c \ne 0\). Then the roots of the equation \((x - \alpha \,)\,(x - \beta ) + c = 0\) are

The product of \(n\) positive numbers is unity. Then their sum is

The number of solutions of the equation sin\({(e)^x} = {5^x} + {5^{ - x}}\) is

If \(\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}\) and \((x + y)\,{x \over y} = - {1 \over 2}\), then x =..........and y =.........

If x and y are positive real numbers and m, n are any positive integers, then \({{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(1 + {y^{2m}})}} > {1 \over 4}\)

If \(\alpha\) and \(\beta\) are the roots of \({x^2}\)+ px + q = 0 and \({\alpha ^4},{\beta ^4}\) are the roots of \(\,{x^2} - rx + s = 0\), then the equation \({x^2} - 4qx + 2{q^2} - r = 0\) has always

Let a, b, c be real numbers, \(a \ne 0\). If \(\alpha \,\) is a root of \({a^2}{x^2} + bx + c = 0\). \(\beta \,\) is the root of \({a^2}{x^2} - bx - c = 0\) and \(0 < \alpha \, < \,\beta\), then the equation \({a^2}{x^2} + 2bx + 2c = 0\) h...

The equation \({x^{3/4{{\left( {{{\log }_2}\,\,x} \right)}^2} + {{\log }_2}\,\,x - 5/4}} = \sqrt 2\) has

Solve \(\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0\)

If \(a,\,b,\,c,\,d\) and p are distinct real numbers such that
\($\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left( {ab + bc + cd} \right)p + \left( {{b^2} + {c^2} + {d^2}} \right) \le 0\)$
then \(a,\,b,\,c,\,d\)

Find the set of all \(x\) for which \({{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}\)

The solution of equation \({\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0\) is ........................

If \(a,\,b\) and \(c\) are distinct positive numbers, then the expression
\(\left( {b + c - a} \right)\left( {c + a - b} \right)\left( {a + b - c} \right) - abc\) is

If \(S\) is the set of all real \(x\) such that \({{2x - 1} \over {2{x^3} + 3{x^2} + x}}\) is positive, then \(S\) contains

If the quadratic equations \({x^2} + ax + b = 0\) and \({x^2} + bx + a = 0\) \((a \ne b)\) have a common root, then the numerical value of a + b is..........................

For \(a \le 0,\) determine all real roots of the equation \(${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0\)$

If \(P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c\), where \(ac \ne \,0\), then P(x) Q(x) = 0 has at least two real roots.

If \({n_1}\), \({n_2}\),.......\({n_p}\) are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.

If \({\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)\), then x lies in the interval-

Solve for \(x\) ; \({\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10\)

If the product of the roots of the equation \(\,{x^2} - 3\,k\,x + 2\,{e^{2lnk}} - 1 = 0\,\,\,\,is\,7\), then the roots are real for k = .................................

The equation \(x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}\) has

If \(\,{a^2} + {b^2} + {c^2} = 1\), then ab + bc + ca lies in the interval

For real \(x\), the function \(\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}\) will assume all real values provided

If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct.

The equation \(2{x^2} + 3x + 1 = 0\) has an irrational root.

Find all real values of \(x\) which satisfy \({x^2} - 3x + 2 > 0\) and \({x^2} - 2x - 4 \le 0\)

If one root of the quadratic equation \(a{x^2} + bx + c = 0\) is equal to the \(n\)-th power of the other, then show that
\(${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0\)$

If p, q, r are any real numbers, then

If \(2 + i\sqrt 3\) is root of the equation \({x^2} + px + q = 0\), where p and q are real, then (p, q) = (..........,....................).

The coeffcient of \({x^{99}}\) in the polynomial (x -1) (x - 2)...(x - 100) is ..............

The largest interval for which \({x^{12}} - {x^9} + {x^4} - x + 1 > 0\) is

Two towns A and B are 60 km apart. A school is to be built to serve 150 students in town A and 50 students in town B. If the total distance to be travelled by all 200 students is to be as small as possible, then the school should be built a...

Show that the equation \({e^{\sin x}} - {e^{ - \sin x}} - 4 = 0\) has no real solution.

\(mn\) squares of equal size are arranged to from a rectangle of dimension \(m\) by \(n\), where \(m\) and \(n\) are natural numbers. Two squares will be called ' neighbours ' if they have exactly one common side. A natural number is writte...

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

For every integer n > 1, the inequality \({(n!)^{1/n}} < {{n + 1} \over 2}\) holds.

If \(\,({x^2} + px + 1)\,\) is a factor of \((a{x^3} + bx + c)\), then

The least value of the expression \(2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)\) for x > 1, is

Given \({n^4} < {10^n}\) for a fixed positive integer \(n \ge 2,\) prove that \({\left( {n + 1} \right)^4} < {10^{n + 1}}.\)

Let \(y = \sqrt {{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)}}}\)
Find all the real values of \(x,\) for which \(y\) takes real values.

For what values of \(m,\) does the system of equations
\($\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr }\)$
has solution satisfying the conditions \(x > 0,\,y > 0.\)

Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always

Find the solution set of the system
\($\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \cr }\)$

If \(\alpha ,\,\beta\) are the roots of \({x^2} + px + q = 0\) and \(\gamma ,\,\delta\) are the roots of \({x^2} + rx + s = 0,\) evaluate $$\left( {\alpha - \gamma } \right)\left( {\alpha - \delta } \right)\left( {\beta - \gamma } \rig...

Let a > 0, b > 0 and c > 0. Then the roots of the equation \(a{x^2} + bx + c = 0\)

If x, y and z are real and different and \(\,u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy\), then u is always.

If \(\ell\), m, n are real, \(\ell \ne m\), then the roots by the equation :
\((\ell - m)\,{x^2} - 5\,(\ell + m)\,x - 2\,(\ell - m) = 0\) are

The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have

Find all integers \(x\) for which \(\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).\)