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Nature of Roots - (Concept)
Let the quadratic equation is $a x^2+b x+c=0,(a, b, c \in R)$
$D$ (called the discriminant of the equation) $=b^2-4 a d$
The roots of this equation are given by
$x_1=\frac{-b+\sqrt{D}}{2 a}$ and $x_2=\frac{-b-\sqrt{D}}{2 a}$
i) if D < 0, then both roots are non-real (imaginary numbers), and the roots will be conjugate of each other, which means if p + iq is one of the roots then the other root will be p - iq
ii) If D > 0, then roots will be real and distinc
iii) D = 0, then roots will be real and equal, and they equal$\mathrm{x}_1=\mathrm{x}_2=\frac{-\mathrm{b}}{2 \mathrm{a}}$
Special cases of case ii (D > 0)
i) if a,b,c are rational numbers (Q) and
If D is a perfect square, then roots are rational
If D is not a perfect square then roots are irrational (in this case if$p+\sqrt{q}$ is one root of the quadratic equation then another root will be $p-\sqrt{q}$ )
ii) If a = 1 and b and c are integers and
If D is a perfect square, then roots are integers
If D is not a perfect square then roots are non-integer values
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
If, for a positive integer n, the quadratic equation,
has two consecutive integral solutions, then n is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the two roots of the equation,
are real and distinct, then the set of all values of ‘a’ is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Equation will have
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If has real and distinct roots then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If has real roots
and
, where
then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If has real and equal roots
. Then
will have
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If has real and distinct roots
then
will have roots
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are three consecutive terms of a non-constant G.P. such that the equations
and
have a common root, then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If three distinct numbers are in
and the equations
and
have a common root, then which one of the following statements is correct?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of the equation be imaginary, then for all real values of
. The expression
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If roots of are imaginary
then roots of equation
are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of integral values of m for which the equation has no real root is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are odd integers , then roots of equation
are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of all possible positive integral values of for which the roots of the quadratic equation,
are rational number is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is a root of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let If
is a root of the quadratic equation,
then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a, b \in \mathbf{R}, a \neq 0$ be such that the equation, $a x^2-2 b x+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^2-2 b x-10=0$. If $\beta$ is the other root of this equation, then $\alpha^2+\beta^2$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The least positive value of 'a' for which the equation, has real roots is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the equation, has conjugate complex roots and they satisfy
, then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of the real roots of the equation is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are such that 1 - 2i (here
) is root of
, then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The coefficients $\mathrm{a}, \mathrm{b}$ and c of the quadratic equation, $a x^2+b x+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the value of
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\mathrm{a}, \mathrm{b} \in \mathbb{R} {\text { be such that the equation } \mathrm{ax}^2-2 \mathrm{bx}+15=0 \text { has a repeated root } \mathrm{a} \text {. If } \mathrm{a} .}$ and $\beta$ are the roots of the equation $\mathrm{x}^2-2 \mathrm{bx}+21=0$, then $\mathrm{a}^2+\beta^2$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of ‘q’ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of the equation be imaginary, then for all real values of x, the expression
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f(x)=-x^2+a x+\left(a^2+1\right)$. Then number of values of ' $a^{\prime}$ for which $f(x)=0$ has real and equal roots is $(a \in R)$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
then the number of rational roots of equation
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
then the number of rational roots of equation
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of irrational roots of equation equals _________
_________ where
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then roots of equation
are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of real solutions of the equation , is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let .Then the maximum value of
for which the equation
has real roots, is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let the function $f(x)=2 x^3+(2 p-7) x^2+3(2 p-9) x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then, the set of all values of $p_{\text {is }}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and let
be the roots of the equation
If
, then the product of all possible values of a is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\lambda \neq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14 \mathrm{x}^2-31 \mathrm{x}+3 \lambda=0$ and $\alpha, \gamma$ be the roots of the equation $35 \mathrm{x}^2-53 \mathrm{x}+4 \lambda=0$. Then $\frac{3 \alpha}{\beta}$ and $\frac{4 \alpha}{\gamma}$ are the roots of the equation
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be the roots of the equation
and
. Then
is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of real roots of the equation , is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of integral values of $k$, for which one root of the equation $2 x^2-8 x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$, is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If for , then
and
are the roots of the equation:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of the equation $(2 b+3 c-5 a) x^2+(2 c+3 a-5 b) x+(2 a+3 b-5 c)$ $=0$ are real and equal, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta \in \mathbf{N}$ be roots of the equation $x^2-70 x+\lambda=0$, where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbf{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $S$ be the set of positive integral values of a for which $\frac{a^2+2(a+1) x+9 a+4}{x^2-8 x+32}<0, \forall x \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\right.$ $\left(\sin ^6 \theta+\cos ^6 \theta\right)=0$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $\mathrm{S}$, respectively, then $3\left((\alpha-2)^2+(\beta-1)^2\right)$ equals.....
| A. |
|
| B. |
|
| C. |
|
| D. |
|
'a' for which $x^2-a x+9=0$ can be written as square of a linear factor is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Both the roots of given equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are always:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$
\text { Find } \mathrm{C} \text { such that there is exactly one root of } x^2-2 x+c=0 \text { between }(-1,1) \text {. }
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the values of ' k ' so that $-2 x^2+4 x-k$ has completely below the $x$ - axis
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For all $x \epsilon R, a x^2+b x-c<0$, then $b^2+4 a c<k$ and $a<k$ for $\mathrm{k}=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For all $x \in R, 2 x^2-b x+4>0$, find b
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what value of $\mathrm{C}, x^2-C x+4=0$, has real & distinct roots?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what value of $\mathrm{K}, K x^2-4 x+3=0$, has real & equal roots?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what values of $\mathrm{a}, a x^2-2 x+a=0$, has complex roots with non-zero imaginary part?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what values of $\mathrm{a}, \mathrm{b}$ \& c , the given quadratic equation is an identity
$
(a+b+c) x^2-a x+3 c=2 b x-2 c+a+1
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what Values of C, roots of the equation: $2 x^2+C x-2 C=0$, have opposite signs?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For what values of K does the equation: $x^2-K x+4=0$ have both roots negative?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For $\alpha=16$ and $\beta=-\frac{33}{2}$, as roots of which quadratic equation?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $-2-3i$ is a root of a quadratic equation with a real coefficient what will be the product of roots
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $a x^2+b x+c=0$, has discriminant as perfect square of rational no., What should be the conditions $a, b \& c$ Such that the equation has rational roots?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $a x^2+b x+c=0$ and $a x^2+b x-c=0$ has $D_1$ and $D_2$ as discriminants respectively then: $(a, b, c \in R)$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$
\begin{aligned}
&\text { What is the value of discriminant of the quadratic equation : }\\
&3 x^2-5 x+2=0
\end{aligned}
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following has integer roots?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of the equation $b x^2+c x+a=0$ be imaginary, then for all real values of x , the expression $3 b^2 x^2+6 b c x+2 c^2$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The integer $m$ for which the inequality $x^2-2(4 m-1) x+15 m^2-2 m-7>0$ is valid for $x$ any, is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $a, b, c \in R$ then the roots of the equation $x^2+2 a x+a^2-b^2-c^2=0$ are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If both the roots of the equation $x^2-2 m x+m^2+m-5=0$ are greater than 5 , then $m$ lies in
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a, b$ and $c$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}=0$ has all real roots is $\frac{\mathrm{m}}{\mathrm{n}}$, $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to $\qquad$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then m+n is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The coefficients $\mathrm{a}, \mathrm{b}, \mathrm{c}$ in the quadratic equation $\mathrm{a x^2+b x+c=0}$ are from the set {1,2,3,4,5,6}. If the probability of this equation having one real root bigger than the other is P, then 216 p equals :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The coefficients a, b, c in the quadratic equation $\mathrm{ax^2+b x+c=0}$ are chosen from the set {1,2,3,4,5,6,7,8}. The probability of this equation having repeated roots is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of real solutions of the equation $\mathrm{x}|\mathrm{x}+5|+2|\mathrm{x}+7|-2=0$ is______.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Real value(s) of 'a' for which both roots of $a x^2+2 x-1=0$ are real is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The equation $2\left ( x-1 \right )\left ( x-2 \right )+\left ( x+1 \right )\left ( x-4 \right )=0$ has
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\alpha, \beta$ are the roots of the equation $x^2-4 x+3=0$. If $a_n=\alpha^n-\beta^n$, for $n \geq 1$, then the value of $\frac{a_8+3 a_6}{a_7}$ is, and $\alpha>\beta$.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$f(x)$ is a quadratic polynomial with positive integral coefficients such that for every $\mathrm{x}_1, \mathrm{x}_2 \in \mathrm{R}, \mathrm{x}_2>\mathrm{x}_1$ and $\int_{x_1}^{x_2} f(x) d x>0$. If $g(x)=f(x) f^{\prime \prime}(x)$ and $\mathrm{g}(0)=24$ number of such quadratics is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f(t)=t^3+a t^2+b t+c$, where a, b, c are integers, and $f(0)$ and $f(-1)$ are odd integers. Which of the following is/are correct
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is the value of $m$ so that both roots of the equation $x^2+m x+1$ are less than unity?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Define a function $f(x)=\frac{16 x^2-96 x+153}{x-3}$ for all real $x \neq 3$. The least positive value of $f(x)$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Define a function $f(x)=\frac{16 x^2-96 x+153}{x-3}$ for all real $x \neq 3$. The least positive value of $f(x)$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a, b, c, d$ be numbers in the set $\{1,2,3,4,5,6\}$ such that the curves $y=2 x^3+a x+b$ and $y=2 x^3+c x+d$ have no point in common. The maximum possible value of $(a-b)^2+b-d$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For $2 x^2-b x+c=0, b, c \epsilon Q$; we have $2+\sqrt{3}$ as one of its roots. Find the other root
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the value of a for which the equation $x^2-3 a x+2 a^2+a-3=0$ has real roots and both the roots are less than 1
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the value of k for which both roots of the equation $x^2-4 k x-4-k+9 k^2=0$ are positive
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Number of real solutions of $\small \sqrt{(2x-4)}-\sqrt{(2x-8)}=2$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Consider the cubic equation $x^3+a x^2+b x+c=0$, where $a, b, c$ are real numbers. Which of the following statements is correct?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The positive value of $k$ for which $4 x^2+4 k x+81=0$ and $x^2-6 x+k=0$ will have real roots is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the equation $a(b-c) x^2+b(c-a) x+c(a-b)=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and $\mathrm{b}=\frac{36}{5}$, then $\mathrm{a}^2+\mathrm{c}^2$ is equal to____________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
In $a x^2+b x+c=0, a, b, c \equiv Q ;$ and $2+\sqrt{3}$ is one of the roots, find another root.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the set of all $\mathrm{a} \in \mathrm{R}-\{1\}$, for which the roots of the equation $(1-a) x^{2}+2(a-3) x+9=0$ are positive is $(-\infty,-\alpha] \cup[\beta, \gamma)$, then $2 \alpha+\beta+\gamma$ is equal to________
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let the set of all values of $\mathrm{p} \in \mathbb{R}$, for which both the roots of the equation $x^2-(p+2) x+(2 p+9)=$ 0 are negative real numbers, be the interval $(\alpha, \beta]$. Then $\beta-2 \alpha$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the range of the function $f(x)=\frac{5-x}{x^2-3 x+2}$, $x \neq 1,2$, is $(-\infty, \alpha] \cup[\beta, \infty)$, then $\alpha^2+\beta^2$ is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the root of the equation $a x^2-2 b x+c=0$ are imaginary, the number of real roots of $4 e^x+(a+c)^2\left(x^3+x\right)=4 b^2 x$ is/are:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$, then $12\left(\alpha^2+\beta^2\right)$ is equal to_________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
