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JEE Advanced

Quadratic Equation and Inequalities

Mathematics

Previous Years Questions

Numerical

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The product of all positive real values of $x$ satisfying the equation $$ x^{\left(16\left(\log _{5} x\right)^{3}-68 \l...
JEE Advanced 2022 Paper 2 Online
For x $$\in$$ R, the number of real roots of the equation $$3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0$$ is ______...
JEE Advanced 2021 Paper 1 Online
Let a, b, c three non-zero real numbers such that the equation $$\sqrt 3 a\cos x + 2b\sin x = c,x \in \left[ { - {\pi \...
JEE Advanced 2018 Paper 1 Offline
The value of $$6 + {\log _{3/2}}\left( {{1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt 2 }}\sqrt {4 - {1 \over {3\sqrt...
IIT-JEE 2012 Paper 1 Offline
The number of distinct real roots of $${x^4} - 4{x^3} + 12{x^2} + x - 1 = 0$$
IIT-JEE 2011 Paper 2 Offline
The minimum value of the sum of real numbers $${a^{ - 5}},\,{a^{ - 4}},\,3{a^{ - 3}},\,1,\,{a^8}$$ and $${a^{10}}$$ wher...
IIT-JEE 2011 Paper 1 Offline
Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation: $$...
IIT-JEE 2009
The smallest value of $$k$$, for which both the roots of the equation $$${x^2} - 8kx + 16\left( {{k^2} - k + 1} \right)...
IIT-JEE 2009

MCQ (Single Correct Answer)

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Suppose a, b denote the distinct real roots of the quadratic polynomial x2 + 20x $$-$$ 2020 and suppose c, d denote the ...
JEE Advanced 2020 Paper 1 Offline
If a4 = 28, then p + 2q =
JEE Advanced 2017 Paper 2 Offline
a12 = ?
JEE Advanced 2017 Paper 2 Offline
Let $$ - {\pi \over 6} < \theta < - {\pi \over {12}}.$$ Suppose $${\alpha _1}$$ and $${B_1}$$ are the roots o...
JEE Advanced 2016 Paper 1 Offline
The quadratic equation $$p(x)$$ $$ = 0$$ with real coefficients has purely imaginary roots. Then the equation $$p(p(x))...
JEE Advanced 2014 Paper 2 Offline
Let $$\alpha$$(a) and $$\beta$$(a) be the roots of the equation $$(\root 3 \of {1 + a} - 1){x^2} + (\sqrt {1 + a} - 1)...
IIT-JEE 2012 Paper 2 Offline
A value of $$b$$ for which the equations $$$\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr } $$$...
IIT-JEE 2011 Paper 2 Offline
Let $$\left( {{x_0},{y_0}} \right)$$ be the solution of the following equations $$\matrix{ {{{\left( {2x} \right)}^...
IIT-JEE 2011 Paper 1 Offline
Let $$\alpha $$ and $$\beta $$ be the roots of $${x^2} - 6x - 2 = 0,$$ with $$\alpha > \beta .$$ If $${a_n} = {\al...
IIT-JEE 2011 Paper 1 Offline
Let $$p$$ and $$q$$ be real numbers such that $$p \ne 0,\,{p^3} \ne q$$ and $${p^3} \ne - q.$$ If $${p^3} \ne - q.$$ ...
IIT-JEE 2010 Paper 1 Offline
Let $$a,\,b,c$$, $$p,q$$ be real numbers. Suppose $$\alpha ,\,\beta $$ are the roots of the equation $${x^2} + 2px + q =...
IIT-JEE 2008
Let $$\alpha ,\,\beta $$ be the roots of the equation $${x^2} - px + r = 0$$ and $${\alpha \over 2},\,2\beta $$ be the ...
IIT-JEE 2007
Let $$a,\,b,\,c$$ be the sides of triangle where $$a \ne b \ne c$$ and $$\lambda \in R$$. If the roots of the equation...
IIT-JEE 2006
For all $$'x',{x^2} + 2ax + 10 - 3a > 0,$$ then the interval in which '$$a$$' lies is
IIT-JEE 2004 Screening
If one root is square of the other root of the equation $${x^2} + px + q = 0$$, then the realation between $$p$$ and $$q...
IIT-JEE 2004 Screening
If $$\,\alpha \in \left( {0,{\pi \over 2}} \right)\,\,then\,\,\sqrt {{x^2} + x} + {{{{\tan }^2}\alpha } \over {\sqrt ...
IIT-JEE 2003 Screening
The set of all real numbers x for which $${x^2} - \left| {x + 2} \right| + x > 0$$, is
IIT-JEE 2002 Screening
If $${a_1},{a_2}.......,{a_n}$$ are positive real numbers whose product is a fixed number c, then the minimum value of $...
IIT-JEE 2002 Screening
For the equation $$3{x^2} + px + 3 = 0$$. p > 0, if one of the root is square of the other, then p is equal to
IIT-JEE 2000 Screening
If $$\alpha \,and\,\beta $$ $$(\alpha \, < \,\beta )$$ are the roots of the equation $${x^2} + bx + c = 0\,$$, where ...
IIT-JEE 2000 Screening
If b > a, then the equation (x - a) (x - b) - 1 = 0 has
IIT-JEE 2000 Screening
If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation
IIT-JEE 2000 Screening
If the roots of the equation $${x^2} - 2ax + {a^2} + a - 3 = 0$$ are real and less than 3, then
IIT-JEE 1999
Number of divisor of the form 4$$n$$$$ + 2\left( {n \ge 0} \right)$$ of the integer 240 is
IIT-JEE 1998
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 ro...
IIT-JEE 1994
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
IIT-JEE 1994
The number of points of intersection of two curves y = 2 sin x and y $$ = 5{x^2} + 2x + 3$$ is
IIT-JEE 1994
Let $$\alpha \,,\,\beta $$ be the roots of the equation (x - a) (x - b) = c, $$c \ne 0$$. Then the roots of the equation...
IIT-JEE 1992
The product of $$n$$ positive numbers is unity. Then their sum is
IIT-JEE 1991
The number of solutions of the equation sin$${(e)^x} = {5^x} + {5^{ - x}}$$ is
IIT-JEE 1990
If $$a,\,b,\,c,\,d$$ and p are distinct real numbers such that $$$\left( {{a^2} + {b^2} + {c^2}} \right){p^2} - 2\left(...
IIT-JEE 1987
If $$a,\,b$$ and $$c$$ are distinct positive numbers, then the expression $$\left( {b + c - a} \right)\left( {c + a - b...
IIT-JEE 1986
If $${\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)$$, then x lies in the interval-
IIT-JEE 1985
If $$\,{a^2} + {b^2} + {c^2} = 1$$, then ab + bc + ca lies in the interval
IIT-JEE 1984
The equation $$x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}$$ has
IIT-JEE 1984
The largest interval for which $${x^{12}} - {x^9} + {x^4} - x + 1 > 0$$ is
IIT-JEE 1982
If p, q, r are any real numbers, then
IIT-JEE 1982
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...
IIT-JEE 1982
The number of real solutions of the equation $${\left| x \right|^2} - 3\left| x \right| + 2 = 0$$ is
IIT-JEE 1982
If $$\,({x^2} + px + 1)\,$$ is a factor of $$(a{x^3} + bx + c)$$, then
IIT-JEE 1980
The least value of the expression $$2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$$ for x > 1, is
IIT-JEE 1980
Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always
IIT-JEE 1980
If $$\ell $$, m, n are real, $$\ell \ne m$$, then the roots by the equation : $$(\ell - m)\,{x^2} - 5\,(\ell + m)\,x...
IIT-JEE 1979
Let a > 0, b > 0 and c > 0. Then the roots of the equation $$a{x^2} + bx + c = 0$$
IIT-JEE 1979
If x, y and z are real and different and $$\,u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy$$, then u is always.
IIT-JEE 1979
The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have
IIT-JEE 1979

MCQ (More than One Correct Answer)

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Let $$\alpha $$ and $$\beta $$ be the roots of$${x^2} - x - 1 = 0$$, with $$\alpha $$ > $$\beta $$. For all positive ...
JEE Advanced 2019 Paper 1 Offline
Let $$S$$ be the set of all non-zero real numbers $$\alpha $$ such that the quadratic equation $$\alpha {x^2} - x + \alp...
JEE Advanced 2015 Paper 2 Offline
If $${3^x}\, = \,{4^{x - 1}},$$ then $$x\, = $$
JEE Advanced 2013 Paper 2 Offline
Let a, b, c be real numbers, $$a \ne 0$$. If $$\alpha \,$$ is a root of $${a^2}{x^2} + bx + c = 0$$. $$\beta \,$$ is the...
IIT-JEE 1989
If $$\alpha $$ and $$\beta $$ are the roots of $${x^2}$$+ px + q = 0 and $${\alpha ^4},{\beta ^4}$$ are the roots of $$\...
IIT-JEE 1989
The equation $${x^{3/4{{\left( {{{\log }_2}\,\,x} \right)}^2} + {{\log }_2}\,\,x - 5/4}} = \sqrt 2 $$ has
IIT-JEE 1989
If $$S$$ is the set of all real $$x$$ such that $${{2x - 1} \over {2{x^3} + 3{x^2} + x}}$$ is positive, then $$S$$ conta...
IIT-JEE 1986
For real $$x$$, the function $$\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}$$ will assume all real v...
IIT-JEE 1984

Subjective

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Let $$a$$ and $$b$$ be the roots of the equation $${x^2} - 10cx - 11d = 0$$ and those $${x^2} - 10ax - 11b = 0$$ are $$c...
IIT-JEE 2006
If $$a,\,b,c$$ are positive real numbers. Then prove that $$${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left...
IIT-JEE 2004
If $${x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0$$ where $$a,\,b\, \in \,R$$ then find the values o...
IIT-JEE 2003
Let $$a,\,b,\,c$$ be real numbers with $$a \ne 0$$ and let $$\alpha ,\,\beta $$ be the roots of the equation $$a{x^2} + ...
IIT-JEE 2001
If $$\alpha ,\,\beta $$ are the roots of $$a{x^2} + bx + c = 0$$, $$\,\left( {a \ne 0} \right)$$ and $$\alpha + \delta...
IIT-JEE 2000
Let $$S$$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $$S$$. If $$a,\,b,\,...
IIT-JEE 1997
Let $$a,\,b,\,c$$ be real. If $$a{x^2} + bx + c = 0$$ has two real roots $$\alpha $$ and $$\beta ,$$ where $$\alpha &lt...
IIT-JEE 1995
Solve $$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$
IIT-JEE 1988
Find the set of all $$x$$ for which $${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} ...
IIT-JEE 1987
For $$a \le 0,$$ determine all real roots of the equation $$${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$$
IIT-JEE 1986
Solve for $$x$$ ; $${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$$...
IIT-JEE 1985
Find all real values of $$x$$ which satisfy $${x^2} - 3x + 2 > 0$$ and $${x^2} - 2x - 4 \le 0$$
IIT-JEE 1983
If one root of the quadratic equation $$a{x^2} + bx + c = 0$$ is equal to the $$n$$-th power of the other, then show tha...
IIT-JEE 1983
Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.
IIT-JEE 1982
$$mn$$ squares of equal size are arranged to from a rectangle of dimension $$m$$ by $$n$$, where $$m$$ and $$n$$ are nat...
IIT-JEE 1982
Find the solution set of the system $$$\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y ...
IIT-JEE 1980
For what values of $$m,$$ does the system of equations $$$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr } $...
IIT-JEE 1980
Let $$y = \sqrt {{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)}}} $$ Find all the real ...
IIT-JEE 1980
Given $${n^4} < {10^n}$$ for a fixed positive integer $$n \ge 2,$$ prove that $${\left( {n + 1} \right)^4} < {10^{...
IIT-JEE 1980
If $$\alpha ,\,\beta $$ are the roots of $${x^2} + px + q = 0$$ and $$\gamma ,\,\delta $$ are the roots of $${x^2} + rx ...
IIT-JEE 1979
Sketch the solution set of the following system of inequalities: $$${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,...
IIT-JEE 1978
Find all integers $$x$$ for which $$\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right)...
IIT-JEE 1978
Show that the square of $$\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}$$ is a rational num...
IIT-JEE 1978
Solve the following equation for $$x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0$$
IIT-JEE 1978
Solve for $$x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.$$
IIT-JEE 1978
Solve for $$x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}$$
IIT-JEE 1978
If $$\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n...
IIT-JEE 1978

Fill in the Blanks

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The sum of all the real roots of the equation $${\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0$$ is ..........
IIT-JEE 1997
Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_...
IIT-JEE 1996
If $$\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}$$ and $$(x + y)\,{x \over y} = - {1 \over 2}$$, then...
IIT-JEE 1990
The solution of equation $${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is ......................
IIT-JEE 1986
If the quadratic equations $${x^2} + ax + b = 0$$ and $${x^2} + bx + a = 0$$ $$(a \ne b)$$ have a common root, then the...
IIT-JEE 1986
If the product of the roots of the equation $$\,{x^2} - 3\,k\,x + 2\,{e^{2lnk}} - 1 = 0\,\,\,\,is\,7$$, then the roots a...
IIT-JEE 1984
If $$2 + i\sqrt 3 $$ is root of the equation $${x^2} + px + q = 0$$, where p and q are real, then (p, q) = (..........,....
IIT-JEE 1982
The coeffcient of $${x^{99}}$$ in the polynomial (x -1) (x - 2)...(x - 100) is ..............
IIT-JEE 1982

True or False

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If x and y are positive real numbers and m, n are any positive integers, then $${{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(...
IIT-JEE 1989
If $$P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c$$, where $$ac \ne \,0$$, then P(x) Q(x) = 0 has at least...
IIT-JEE 1985
If $${n_1}$$, $${n_2}$$,.......$${n_p}$$ are p positive integers, whose sum is an even number, then the number of odd in...
IIT-JEE 1985
If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct...
IIT-JEE 1984
The equation $$2{x^2} + 3x + 1 = 0$$ has an irrational root.
IIT-JEE 1983
For every integer n > 1, the inequality $${(n!)^{1/n}} < {{n + 1} \over 2}$$ holds.
IIT-JEE 1981

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