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Quadratic Equation and Inequalities

299 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
1989 JEE Advanced MCQ
IIT-JEE 1989
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 JEE Advanced Numerical
IIT-JEE 1988
Solve $\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$
1987 JEE Advanced MCQ
IIT-JEE 1987
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 JEE Advanced Numerical
IIT-JEE 1987
Find the set of all $x$ for which ${{2x} \over {\left( {2{x^2} + 5x + 2} \right)}}\, > \,{1 \over {\left( {x + 1} \right)}}$
1986 JEE Advanced MCQ
IIT-JEE 1986
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 JEE Advanced MSQ
IIT-JEE 1986
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 JEE Advanced Numerical
IIT-JEE 1986
For $a \le 0,$ determine all real roots of the equation $${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$$
1986 JEE Advanced Numerical
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 numerical value of a + b is..........................
1986 JEE Advanced Numerical
IIT-JEE 1986
The solution of equation ${\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$ is ........................
1985 JEE Advanced MCQ
IIT-JEE 1985
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 JEE Advanced Numerical
IIT-JEE 1985
Solve for $x$ ; ${\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10$
1985 JEE Advanced MCQ
IIT-JEE 1985
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 JEE Advanced MCQ
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 integers among them is odd.
A.
TRUE
B.
FALSE
1984 JEE Advanced MCQ
IIT-JEE 1984
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 JEE Advanced MCQ
IIT-JEE 1984
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 JEE Advanced MSQ
IIT-JEE 1984
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 JEE Advanced Numerical
IIT-JEE 1984
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 JEE Advanced MCQ
IIT-JEE 1984
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 JEE Advanced Numerical
IIT-JEE 1983
Find all real values of $x$ which satisfy ${x^2} - 3x + 2 > 0$ and ${x^2} - 2x - 4 \le 0$
1983 JEE Advanced Numerical
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 that $${\left( {a{c^n}} \right)^{{1 \over {n + 1}}}} + {\left( {{a^n}c} \right)^{{1 \over {n + 1}}}} + b = 0$$
1983 JEE Advanced MCQ
IIT-JEE 1983
The equation $2{x^2} + 3x + 1 = 0$ has an irrational root.
A.
TRUE
B.
FALSE
1982 JEE Advanced MCQ
IIT-JEE 1982
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 JEE Advanced MCQ
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 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 JEE Advanced MCQ
IIT-JEE 1982
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 JEE Advanced MCQ
IIT-JEE 1982
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 JEE Advanced Numerical
IIT-JEE 1982
$mn$ 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 JEE Advanced Numerical
IIT-JEE 1982
Show that the equation ${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$ has no real solution.
1982 JEE Advanced Numerical
IIT-JEE 1982
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 JEE Advanced Numerical
IIT-JEE 1982
The coeffcient of ${x^{99}}$ in the polynomial (x -1) (x - 2)...(x - 100) is ..............
1981 JEE Advanced MCQ
IIT-JEE 1981
For every integer n > 1, the inequality ${(n!)^{1/n}} < {{n + 1} \over 2}$ holds.
A.
TRUE
B.
FALSE
1980 JEE Advanced MCQ
IIT-JEE 1980
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 JEE Advanced MCQ
IIT-JEE 1980
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 JEE Advanced MCQ
IIT-JEE 1980
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 JEE Advanced Numerical
IIT-JEE 1980
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 JEE Advanced Numerical
IIT-JEE 1980
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 JEE Advanced Numerical
IIT-JEE 1980
Given ${n^4} < {10^n}$ for a fixed positive integer $n \ge 2,$ prove that ${\left( {n + 1} \right)^4} < {10^{n + 1}}.$
1980 JEE Advanced Numerical
IIT-JEE 1980
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 JEE Advanced MCQ
IIT-JEE 1979
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 JEE Advanced MCQ
IIT-JEE 1979
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 JEE Advanced MCQ
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.
A.
non negative
B.
zero
C.
non positive
D.
none of these
1979 JEE Advanced MCQ
IIT-JEE 1979
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 JEE Advanced Numerical
IIT-JEE 1979
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 JEE Advanced Numerical
IIT-JEE 1978
Sketch the solution set of the following system of inequalities: $${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.$$
1978 JEE Advanced Numerical
IIT-JEE 1978
Solve for $x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}$
1978 JEE Advanced Numerical
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 number.
1978 JEE Advanced Numerical
IIT-JEE 1978
Solve the following equation for $x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0$
1978 JEE Advanced Numerical
IIT-JEE 1978
If $\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n + 1}}} \right)} \over {\left( {1 - x} \right)\left( {1 - {x^2}} \right).........\left( {1 - {x^n}} \right)}}$

where $m$ and $n$ are positive integers $\left( {n \le m} \right),$ show that
$\left( {m,n + 1} \right) = \left( {m - 1,\,n + 1} \right) + {x^{m - n - 1}}\left( {m - 1,n} \right).$

1978 JEE Advanced Numerical
IIT-JEE 1978
Find all integers $x$ for which $\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).$
1978 JEE Advanced Numerical
IIT-JEE 1978
Solve for $x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.$