← Back to Mathematics

Quadratic Equation and Inequalities

299 Questions
All Exams JEE Mains JEE Advanced TS-EAMCET AP-EAPCET
2006 JEE Mains MCQ
AIEEE 2006
If the roots of the quadratic equation ${x^2} + px + q = 0$ are $\tan {30^ \circ }$ and $\tan {15^ \circ }$, respectively, then the value of $2 + q - p$ is
A.
2
B.
3
C.
0
D.
1
2006 JEE Mains MCQ
AIEEE 2006
All the values of $m$ for which both roots of the equation ${x^2} - 2mx + {m^2} - 1 = 0$ are greater than $ - 2$ but less then 4, lie in the interval
A.
$ - 2 < m < 0$
B.
$m > 3$
C.
$ - 1 < m < 3$
D.
$1 < m < 4$
2006 JEE Advanced MCQ
IIT-JEE 2006

Let $a, b, c$ be the sides of a triangle. No two of them are equal and $\lambda \in R$. If the roots of the equation $x^{2}+2(a+b+c) x+3 \lambda(a b+b c+c a)=0$ are real, then,

A.
$\lambda<\frac{4}{3}$
B.
$\lambda>\frac{5}{3}$
C.
$\lambda \in\left(\frac{1}{3}, \frac{5}{3}\right)$
D.
$\lambda \in\left(\frac{4}{3}, \frac{5}{3}\right)$
2006 JEE Advanced Numerical
IIT-JEE 2006

If roots of the equation $x^2-10 c x-11 d=0$ are $a, b$ and those of $x^2-10 a x-11 b=0$ are $c, d$, then the value of $a+b+c+d$ is $(a, b, c$ and $d$ are distinct numbers)

2005 JEE Mains MCQ
AIEEE 2005
In a triangle $PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$ and $ \tan \left( {{Q \over 2}} \right)$ are the roots of $a{x^2} + bx + c = 0,\,\,a \ne 0$ then
A.
$a = b + c$
B.
$c = a + b$
C.
$b = c$
D.
$b = a + c$
2005 JEE Mains MCQ
AIEEE 2005
The value of $a$ for which the sum of the squares of the roots of the equation
${x^2} - \left( {a - 2} \right)x - a - 1 = 0$ assume the least value is
A.
$1$
B.
$0$
C.
$3$
D.
$2$
2005 JEE Mains MCQ
AIEEE 2005
If the roots of the equation ${x^2} - bx + c = 0$ be two consecutive integers, then ${b^2} - 4c$ equals
A.
$-2$
B.
$3$
C.
$2$
D.
$1$
2005 JEE Mains MCQ
AIEEE 2005
If the roots of the equation ${x^2} - bx + c = 0$ be two consecutive integers, then ${b^2} - 4c$ equals
A.
$-2$
B.
$3$
C.
$2$
D.
$1$
2005 JEE Mains MCQ
AIEEE 2005
The value of $a$ for which the sum of the squares of the roots of the equation ${x^2} - \left( {a - 2} \right)x - a - 1 = 0$ assume the least value is :
A.
$1$
B.
$0$
C.
$3$
D.
$2$
2005 JEE Mains MCQ
AIEEE 2005
If both the roots of the quadratic equation ${x^2} - 2kx + {k^2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval
A.
$\left( {5,6} \right]$
B.
$\left( {6,\,\infty } \right)$
C.
$\left( { - \infty ,\,4} \right)$
D.
$\left[ {4,\,5} \right]$
2004 JEE Mains MCQ
AIEEE 2004
If one root of the equation ${x^2} + px + 12 = 0$ is 4, while the equation ${x^2} + px + q = 0$ has equal roots,
then the value of $'q'$ is
A.
4
B.
12
C.
3
D.
${{49} \over 4}$
2004 JEE Mains MCQ
AIEEE 2004
If $\left( {1 - p} \right)$ is a root of quadratic equation ${x^2} + px + \left( {1 - p} \right) = 0$ then its root are
A.
$ - 1,2$
B.
$ - 1,1$
C.
$ 0,-1$
D.
$0,1$
2004 JEE Mains MCQ
AIEEE 2004
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
A.
${x^2} - 18x - 16 = 0$
B.
${x^2} - 18x + 16 = 0$
C.
${x^2} + 18x - 16 = 0$
D.
${x^2} + 18x + 16 = 0$
2004 JEE Advanced MCQ
IIT-JEE 2004 Screening
For all $'x',{x^2} + 2ax + 10 - 3a > 0,$ then the interval in which '$a$' lies is
A.
$a < - 5$
B.
$ - 5 < a < 2$
C.
$a > 5$
D.
$2 < a < 5$
2004 JEE Advanced MCQ
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 $q$ is
A.
${p^3} - q\left( {3p - 1} \right) + {q^2} = 0$
B.
${p^3} - q\left( {3p + 1} \right) + {q^2} = 0$
C.
${p^3} + q\left( {3p - 1} \right) + {q^2} = 0$
D.
${p^3} + q\left( {3p + 1} \right) + {q^2} = 0$
2004 JEE Advanced Numerical
IIT-JEE 2004
If $a,\,b,c$ are positive real numbers. Then prove that $${\left( {a + 1} \right)^7}{\left( {b + 1} \right)^7}{\left( {c + 1} \right)^7} > {7^7}\,{a^4}{b^4}{c^4}$$
2003 JEE Mains MCQ
AIEEE 2003
The value of '$a$' for which one root of the quadratic equation $$\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0$$
is twice as large as the other is
A.
$ - {1 \over 3}$
B.
$ {2 \over 3}$
C.
$ - {2 \over 3}$
D.
$ {1 \over 3}$
2003 JEE Mains MCQ
AIEEE 2003
The number of real solutions of the equation ${x^2} - 3\left| x \right| + 2 = 0$ is
A.
3
B.
2
C.
4
D.
1
2003 JEE Mains MCQ
AIEEE 2003
If the sum of the roots of the quadratic equation $a{x^2} + bx + c = 0$ is equal to the sum of the squares of their reciprocals, then ${a \over c},\,{b \over a}$ and ${c \over b}$ are in
A.
Arithmetic - Geometric Progression
B.
Arithmetic Progression
C.
Geometric Progression
D.
Harmonic Progression
2003 JEE Advanced MCQ
IIT-JEE 2003 Screening
If $\,\alpha \in \left( {0,{\pi \over 2}} \right)\,\,then\,\,\sqrt {{x^2} + x} + {{{{\tan }^2}\alpha } \over {\sqrt {{x^2} + x} }}$ is always greater than or equal to
A.
$2\,\tan \alpha \,$
B.
1
C.
2
D.
${\sec ^2}\,\alpha $
2003 JEE Advanced Numerical
IIT-JEE 2003
If ${x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0$ where $a,\,b\, \in \,R$ then find the values of a for which equation has unequal real roots for all values of $b$.
2002 JEE Mains MCQ
AIEEE 2002
Product of real roots of equation ${t^2}{x^2} + \left| x \right| + 9 = 0$
A.
is always positive
B.
is always negative
C.
does not exist
D.
none of these
2002 JEE Mains MCQ
AIEEE 2002
If $a,\,b,\,c$ are distinct $ + ve$ real numbers and ${a^2} + {b^2} + {c^2} = 1$ then $ab + bc + ca$ is
A.
less than 1
B.
equal to 1
C.
greater than 1
D.
any real no.
2002 JEE Mains MCQ
AIEEE 2002
If $\alpha \ne \beta $ but ${\alpha ^2} = 5\alpha - 3$ and ${\beta ^2} = 5\beta - 3$ then the equation having $\alpha /\beta $ and $\beta /\alpha \,\,$ as its roots is
A.
$3{x^2} - 19x + 3 = 0$
B.
$3{x^2} + 19x - 3 = 0$
C.
$3{x^2} - 19x - 3 = 0$
D.
${x^2} - 5x + 3 = 0$
2002 JEE Mains MCQ
AIEEE 2002
If $p$ and $q$ are the roots of the equation ${x^2} + px + q = 0,$ then
A.
$p = 1,\,\,q = - 2$
B.
$p = 0,\,\,q = 1$
C.
$p = - 2,\,\,q = 0$
D.
$p = - 2,\,\,q = 1$
2002 JEE Mains MCQ
AIEEE 2002
Difference between the corresponding roots of ${x^2} + ax + b = 0$ and ${x^2} + bx + a = 0$ is same and $a \ne b,$ then
A.
$a + b + 4 = 0$
B.
$a + b - 4 = 0$
C.
$a - b - 4 = 0$
D.
$a - b + 4 = 0$
2002 JEE Advanced MCQ
IIT-JEE 2002 Screening
The set of all real numbers x for which ${x^2} - \left| {x + 2} \right| + x > 0$, is
A.
$( - \infty ,\, - 2) \cup (2,\infty )$
B.
$( - \infty ,\, - \sqrt 2 ) \cup (\sqrt 2 ,\infty )$
C.
$( - \infty ,\, - 1) \cup (1,\infty )$
D.
$(\sqrt 2 ,\infty )$
2002 JEE Advanced MCQ
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 ${a_1} + {a_2} + ..... + {a_{n - 1}} + 2{a_n}$ is
A.
$n{(2c)^{1/n}}$
B.
$(n + 1){c^{1/n}}$
C.
$2n{c^{1/n}}$
D.
$(n + 1)\,{(2c)^{1/n}}$
2001 JEE Advanced Numerical
IIT-JEE 2001
Let $a,\,b,\,c$ be real numbers with $a \ne 0$ and let $\alpha ,\,\beta $ be the roots of the equation $a{x^2} + bx + c = 0$. Express the roots of ${a^3}{x^2} + abcx + {c^3} = 0$ in terms of $\alpha ,\,\beta \,$.
2000 JEE Advanced MCQ
IIT-JEE 2000 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
A.
1/3
B.
1
C.
3
D.
2/3
2000 JEE Advanced MCQ
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
A.
$0 \le M \le 1$
B.
$1 \le M \le 2$
C.
$2 \le M \le 3$
D.
$3 \le M \le 4$
2000 JEE Advanced MCQ
IIT-JEE 2000 Screening
If $\alpha \,\text{and}\,\beta $ $(\alpha \, < \,\beta )$ are the roots of the equation ${x^2} + bx + c = 0\,$, where $c < 0 < b$, then
A.
$0 < \alpha \, < \,\beta \,$
B.
$\alpha \, < \,0 < \beta \,<\left| \alpha \right|$
C.
$\alpha \, < \beta \, < 0\,$
D.
$\alpha \, < \,0 < \left| \alpha \right| < \beta $
2000 JEE Advanced MCQ
IIT-JEE 2000 Screening
If b > a, then the equation (x - a) (x - b) - 1 = 0 has
A.
both roots in (a, b)
B.
both roots in (- $\infty $, a)
C.
both roots in (b, + $\infty $)
D.
one root in (- $\infty $, a) and the other in (b, + $\infty $)
2000 JEE Advanced Numerical
IIT-JEE 2000
If $\alpha ,\,\beta $ are the roots of $a{x^2} + bx + c = 0$, $\,\left( {a \ne 0} \right)$ and $\alpha + \delta ,\,\,\beta + \delta $ are the roots of $A{x^2} + Bx + c = 0,$ $\left( {A \ne 0\,} \right)\,$ for some contant $\delta $, then prove that ${{{b^2} - 4ac} \over {{a^2}}} = {{{B^2} - 4Ac} \over {{A^2}}}$.
1999 JEE Advanced MCQ
IIT-JEE 1999
If the roots of the equation ${x^2} - 2ax + {a^2} + a - 3 = 0$ are real and less than 3, then
A.
$a < 2$
B.
$2 \le a \le 3$
C.
$3 < a \le 4$
D.
$a > 4$
1998 JEE Advanced MCQ
IIT-JEE 1998
Number of divisor of the form 4$n$$ + 2\left( {n \ge 0} \right)$ of the integer 240 is
A.
4
B.
8
C.
10
D.
3
1997 JEE Advanced Numerical
IIT-JEE 1997
Let $S$ be a square of unit area. Consider any quadrilateral which has one vertex on each side of $S$. If $a,\,b,\,c$ and $d$ denote the lengths of the sides of the quadrilateral, prove that $2 \le {a^2} + {b^2} + {c^2} + {d^2} \le 4.$
1997 JEE Advanced Numerical
IIT-JEE 1997
The sum of all the real roots of the equation ${\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0$ is ............................
1996 JEE Advanced Numerical
IIT-JEE 1996
Let n and k be positive such that $n \ge {{k(k + 1)} \over 2}$ . The number of solutions $\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k$, all integers, satisfying ${x_1} + {x_2} + \,..... + {x_k} = n,\,$ is......................................
1995 JEE Advanced Numerical
IIT-JEE 1995
Let $a,\,b,\,c$ be real. If $a{x^2} + bx + c = 0$ has two real roots $\alpha $ and $\beta ,$ where $\alpha < - 1$ and $\beta > 1,$ then show that $1 + {c \over a} + \left| {{b \over a}} \right| < 0.$
1994 JEE Advanced MCQ
IIT-JEE 1994
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 roots is
A.
15
B.
9
C.
7
D.
8
1994 JEE Advanced MCQ
IIT-JEE 1994
The number of points of intersection of two curves y = 2 sin x and y $ = 5{x^2} + 2x + 3$ is
A.
0
B.
1
C.
2
D.
$\infty $
1994 JEE Advanced MCQ
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
A.
$\left| {{r \over p} - 7} \right| \ge 4\sqrt 3 $
B.
$\left| {{p \over r} - 7} \right| \ge 4\sqrt 3 $
C.
all p and r
D.
no p and r
1992 JEE Advanced MCQ
IIT-JEE 1992
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 JEE Advanced MCQ
IIT-JEE 1991
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 JEE Advanced MCQ
IIT-JEE 1990
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 JEE Advanced Numerical
IIT-JEE 1990
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 JEE Advanced MSQ
IIT-JEE 1989
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 JEE Advanced MCQ
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 $\,{x^2} - rx + s = 0$, 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 JEE Advanced MCQ
IIT-JEE 1989
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 $