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Quadratic Equation and Inequalities

106 Questions Start JEE Advanced Test
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET BITSAT
1992 Q51 JEE Advanced MCQ
14 Mar 2026
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
A.
a, c
B.
b, c
C.
a, b
D.
a + c, b + c
1991 Q52 JEE Advanced MCQ
14 Mar 2026
The product of $n$ positive numbers is unity. Then their sum is
A.
a positive integer
B.
divisible by $n$
C.
equal to $n + {1 \over n}$
D.
never less than $n$
1990 Q53 JEE Advanced MCQ
14 Mar 2026
The number of solutions of the equation sin${(e)^x} = {5^x} + {5^{ - x}}$ is
A.
0
B.
1
C.
2
D.
Infinitely many
1990 Q54 JEE Advanced Numerical
14 Mar 2026
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 =.........
1989 Q55 JEE Advanced MSQ
14 Mar 2026
The equation ${x^{3/4{{\left( {{{\log }_2}\,\,x} \right)}^2} + {{\log }_2}\,\,x - 5/4}} = \sqrt 2 $ has
A.
at least one real solution
B.
exactly three solutions
C.
exactly one irrational solution
D.
complex roots.
1989 Q56 JEE Advanced MCQ
14 Mar 2026
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
A.
two real roots
B.
two positive roots
C.
two negative roots
D.
one positive and one negative root.
1989 Q57 JEE Advanced MCQ
14 Mar 2026
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 ${a^2}{x^2} + 2bx + 2c = 0$ has a root $\gamma $ that always satisfies
A.
$\gamma = {{\alpha + \beta } \over 2}$
B.
$\gamma = \alpha + {\beta \over 2}$
C.
$\gamma = \alpha $
D.
$\alpha < \gamma < \beta $
1989 Q58 JEE Advanced MCQ
14 Mar 2026
If x and y are positive real numbers and m, n are any positive integers, then ${{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(1 + {y^{2m}})}} > {1 \over 4}$
A.
TRUE
B.
FALSE
1988 Q59 JEE Advanced Numerical
14 Mar 2026
Solve $\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$
1987 Q60 JEE Advanced MCQ
14 Mar 2026
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 $a,\,b,\,c,\,d$
A.
are in A. P.
B.
are in G P.
C.
are in H. P.
D.
satisfy $ab = cd$
1987 Q61 JEE Advanced Numerical
14 Mar 2026
Find the set of all $x$ for which ${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}$
1986 Q62 JEE Advanced MCQ
14 Mar 2026
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
A.
positive
B.
negative
C.
non-positive
D.
non-negative
1986 Q63 JEE Advanced MSQ
14 Mar 2026
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
A.
$\left( { - \infty ,\, - {\textstyle{3 \over 2}}} \right)$
B.
$\left( { - {3 \over 2},\, - {1 \over 4}} \right)$
C.
$\left( { - {1 \over 4},\,{1 \over 2}} \right)$
D.
$\left( {{1 \over 2},\,3} \right)\,\,\,\,$
1986 Q64 JEE Advanced Numerical
14 Mar 2026
For $a \le 0,$ determine all real roots of the equation $${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$
1986 Q65 JEE Advanced Numerical
14 Mar 2026
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..........................
1986 Q66 JEE Advanced Numerical
14 Mar 2026
The solution of equation ${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$ is ........................
1985 Q67 JEE Advanced MCQ
14 Mar 2026
If ${\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)$, then x lies in the interval-
A.
$(2,\infty )$
B.
(1, 2)
C.
(- 2, - 1)
D.
none of these
1985 Q68 JEE Advanced Numerical
14 Mar 2026
Solve for $x$ ; ${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$
1985 Q69 JEE Advanced MCQ
14 Mar 2026
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.
A.
TRUE
B.
FALSE
1985 Q70 JEE Advanced MCQ
14 Mar 2026
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.
A.
TRUE
B.
FALSE
1984 Q71 JEE Advanced MCQ
14 Mar 2026
The equation $x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}$ has
A.
no root
B.
one root
C.
two equal root
D.
infinitely many roots
1984 Q72 JEE Advanced MCQ
14 Mar 2026
If $\,{a^2} + {b^2} + {c^2} = 1$, then ab + bc + ca lies in the interval
A.
$[{1 \over 2},2]\,\,$
B.
$[ - 1,2]$
C.
$\,[ - {1 \over 2},1]\,$
D.
$\,[ - 1,{1 \over 2}]\,\,$
1984 Q73 JEE Advanced MSQ
14 Mar 2026
For real $x$, the function $\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}$ will assume all real values provided
A.
$a > b > c$
B.
$a < b < c$
C.
$a > c > b$
D.
$a < c < b$
1984 Q74 JEE Advanced Numerical
14 Mar 2026
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 = .................................
1984 Q75 JEE Advanced MCQ
14 Mar 2026
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.
A.
TRUE
B.
FALSE
1983 Q76 JEE Advanced Numerical
14 Mar 2026
Find all real values of $x$ which satisfy ${x^2} - 3x + 2 > 0$ and ${x^2} - 2x - 4 \le 0$
1983 Q77 JEE Advanced Numerical
14 Mar 2026
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 $${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$
1983 Q78 JEE Advanced MCQ
14 Mar 2026
The equation $2{x^2} + 3x + 1 = 0$ has an irrational root.
A.
TRUE
B.
FALSE
1982 Q79 JEE Advanced MCQ
14 Mar 2026
The number of real solutions of the equation ${\left| x \right|^2} - 3\left| x \right| + 2 = 0$ is
A.
4
B.
1
C.
3
D.
2
1982 Q80 JEE Advanced MCQ
14 Mar 2026
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 at
A.
town B
B.
45 km from town A
C.
town A
D.
45 km from town B
1982 Q81 JEE Advanced MCQ
14 Mar 2026
If p, q, r are any real numbers, then
A.
max (p, q) < max (p, q, r )
B.
min (p, q) = ${1 \over 2}\left( {p + q - \left| {p - q} \right|} \right)$
C.
max (p, q) < min (p, q, r)
D.
none of these
1982 Q82 JEE Advanced MCQ
14 Mar 2026
The largest interval for which ${x^{12}} - {x^9} + {x^4} - x + 1 > 0$ is
A.
$ - 4 < x \le 0$
B.
$\,0 < x < 1$
C.
$ - 100 < x < 100$
D.
$ - \infty < x < \infty $
1982 Q83 JEE Advanced Numerical
14 Mar 2026
$mn$ 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 written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares.Show that this is possible only if all the numbers used are equal.
1982 Q84 JEE Advanced Numerical
14 Mar 2026
Show that the equation ${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$ has no real solution.
1982 Q85 JEE Advanced Numerical
14 Mar 2026
If $2 + i\sqrt 3 $ is root of the equation ${x^2} + px + q = 0$, where p and q are real, then (p, q) = (..........,....................).
1982 Q86 JEE Advanced Numerical
14 Mar 2026
The coeffcient of ${x^{99}}$ in the polynomial (x -1) (x - 2)...(x - 100) is ..............
1981 Q87 JEE Advanced MCQ
14 Mar 2026
For every integer n > 1, the inequality ${(n!)^{1/n}} < {{n + 1} \over 2}$ holds.
A.
TRUE
B.
FALSE
1980 Q88 JEE Advanced MCQ
14 Mar 2026
Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always
A.
positive
B.
real
C.
negative
D.
none of these.
1980 Q89 JEE Advanced MCQ
14 Mar 2026
The least value of the expression $2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)$ for x > 1, is
A.
10
B.
2
C.
- 0.01
D.
none of these.
1980 Q90 JEE Advanced MCQ
14 Mar 2026
If $\,({x^2} + px + 1)\,$ is a factor of $(a{x^3} + bx + c)$, then
A.
${a^2} + {c^2} = - ab\,$
B.
${a^2} - {c^2} = - ab$
C.
${a^2} - {c^2} = ab$
D.
none of these
1980 Q91 JEE Advanced Numerical
14 Mar 2026
For what values of $m,$ does the system of equations $$\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr } $$

has solution satisfying the conditions $x > 0,\,y > 0.$

1980 Q92 JEE Advanced Numerical
14 Mar 2026
Find the solution set of the system $$\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \cr } $$
1980 Q93 JEE Advanced Numerical
14 Mar 2026
Given ${n^4} < {10^n}$ for a fixed positive integer $n \ge 2,$ prove that ${\left( {n + 1} \right)^4} < {10^{n + 1}}.$
1980 Q94 JEE Advanced Numerical
14 Mar 2026
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.

1979 Q95 JEE Advanced MCQ
14 Mar 2026
Let a > 0, b > 0 and c > 0. Then the roots of the equation $a{x^2} + bx + c = 0$
A.
are real and negative
B.
have negative real parts
C.
both (a) and (b)
D.
none of these
1979 Q96 JEE Advanced MCQ
14 Mar 2026
The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have
A.
only one solution
B.
only two solution
C.
Infinite number of solutions
D.
None of these.
1979 Q97 JEE Advanced MCQ
14 Mar 2026
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.
A.
non negative
B.
zero
C.
non positive
D.
none of these
1979 Q98 JEE Advanced MCQ
14 Mar 2026
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
A.
Real and equal
B.
Complex
C.
Real and unequal
D.
None of these.
1979 Q99 JEE Advanced Numerical
14 Mar 2026
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 $\left( {\alpha - \gamma } \right)\left( {\alpha - \delta } \right)\left( {\beta - \gamma } \right)$ $\left( {\beta - \delta } \right)$ in terms of $p,\,q,\,r$ and $s$.

deduce the condition that the equations have a common root.

1978 Q100 JEE Advanced Numerical
14 Mar 2026
Sketch the solution set of the following system of inequalities: $${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$