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Let $r,s,t$ and $u$ be the roots of the equation $x^4+A{x}^3+B{x}^2+Cx+D=0;$
$A,B,C,D\in R.$ If $rs=tu$, then $A^{2}D$ is equal to

Let $m_1,m_2,m_3$ be the slopes of all the three straight lines represented by an equation $y^3+(2a+5)x{y}^2-6x^{2}y-2a{x}^3=0.$ If $a,m_1,m_2,m_3$ are all integers, then which of the following holds good?

The range of $k$ for which both the roots of the quadratic equation $(k+1)x^2-3kx+4k=0$ are greater than $1$, is

Of $\alpha ,\beta$ are the roots of $a{x}^2+c=bx,$ then the equation $(a+cy{)}^2=b^{2}y$ in y has the roots

If $x^2$ + 5 = 2x - 4 cos (a + bx) where a, b $ϵ$ (0, 5) is satisfied for alteast one real x, then the minimum value of a + bx is

If all the roots of the equation $p{x}^4+q{x}^2+r=0,p\ne 0,q^2\ge 9pr$ are real, then which of the following option(s) is/are correct?

The number of distinct positive real roots of the equation $(x^2+6{)}^2-35x^2=2x(x^2+6)$ is

For $x^2-(a+3)|x|+4=0$ to have real solutions, the range of $a$ is

Consider the graph of the quadratic polynomial $y=a{x}^2+bx+c$ as shown below.

Which among the following is/are correct?

$x^2+x+\frac{1}{\sqrt{2}}=0$

Number of real solution(s) of the equation ${|x-3|}^{3x^2-10x+3}=1$ is

Let S be the set of all non - zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1-x_2|<1$. Which of the following interval(s) is/are a subset of S?

If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval :

The difference between a natural number and twice its reciprocal is $\frac{47}{7}$, then the number is

The value(s) of $m$ such that the roots of the quadratic equation $(m+1)x^2+(m+1)x-m+1=0$ are equal is

If $a,b,c$ are three distinct positive real numbers, then the number of real root(s) of $a{x}^2+2b|x|+c=0$ is

The value of $P$ for which the equation $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$ has exactly one root at infinity is

If the roots of the equation $x^2-2ax+a^2+a-3=0$ are real and less than 3, then

$2x^2+x+1=0$

Solve the following quadratics $27x^2-10x+1=0$

The value of
$x=1+\frac{1}{3+\frac{1}{2+\frac{1}{3+\frac{1}{2+.....\infty }}}}$ is

Solve the following quadratics $x^2-4x+7=0$

The number of integral values of $m$ for which the quadratic expression,
$(1+2m)x^2-2(1+3m)x+4(1+m),x\in R,$ is always positive, is

The roots of the equation $\frac{3}{x-5}+\frac{2x}{x-3}=5$ are (where $x\ne 3,5$)

$-x^2+x-2=0$

The quadratic equation whose roots are $-4$ and $6$ is given by

If the equations $2x^2+kx-5=0$ and $x^2-3x-4=0$ have a common root, then the value of k is

If a, b, c are distinct positive real numbers such that b(a + c) = 2ac then the roots of $a{x}^2+bx+c=0$ are

If $a{x}^2+2bx+3c=0,a\ne 0,c>0$ does not have any real roots, then which of the following is/are true?

Let ${\overrightarrow{p}}_1=6x\hat{i}+2m\hat{j}-\hat{k}$ and ${\overrightarrow{p}}_2=-m^{2}x\hat{i}+3x\hat{j}+2\hat{k}$, if the angle between them is obtuse, then $m$ belongs to

The root(s) of the equation $({\log }_{3}x{)}^2-{\log }_{3}x=6$ is/are

If the roots of $a{x}^2-bx-c=0$ are increased by same quantity then which of the following expression does not change?

Solve the following quadratics $21x^2-28x+10=0$

The set of values of $a$ for which $a{x}^2-(4-2a)x-8<0$ for exactly three integral values of $x$ is -

If the equations $4x^2-x-1=0$ and $3x^2+(\lambda +\mu )x+\lambda -\mu =0$ have a common root, then the rational values of $\lambda$ and $\mu$ are

If $\alpha$ and $\beta$ are the roots of the quadratic equation $(x-2)(x-3)+(x-3)(x+1)+(x+1)(x-2)=0$, then the value of $\frac{1}{(\alpha +1)(\beta +1)}+\frac{1}{(\alpha -2)(\beta -2)}+\frac{1}{(\alpha -3)(\beta -3)}$ is

The set of values of $k$ for which the equation $x^4+(k-1)x^3+x^2+(k-1)x+1=0$ has only $2$ real roots which are negative is

If $\alpha$ is the greater root of $x^2-5x+4=0$ and $\alpha +m=2$, then the value of $m$ is