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If all integers satisfying the inequality $\frac{(x-1{)}^2(x-2{)}^3(x-4{)}^5(x-5{)}^5}{(x-5{)}^2}\ge 0$ are arranged in increasing order then the quadratic equation with the first and fifth integers in the list as roots is -

If $\alpha ,\beta$ are the roots of the equation $x^2-2x+4=0$, then the equation whose roots are ${\alpha }^3,{\beta }^3$ is

The range of $f\left(x\right)=-x^2+7x+60$ in $x\in [-3,2]$ is

The linear factor(s) of the equation $x^2+4xy+4y^2+3x+6y-4=0$ is/are

If a, b, c are positive rational numbers such that a > b > c and the quadratic equation
$(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root in the interval (-1, 0), then

If $a,b,c$ are non-zero real numbers and $a{x}^2+bx+c=0,$ $b{x}^2+cx+a=0$ have one root in common, then $\frac{a^3+b^3+c^3}{abc}=$

For non-zero distinct real numbers $a_1$ and $a_2$, let $f\left(x\right)=a_1{x}^2+b_{1}x+c_1,$$g\left(x\right)=a_2{x}^2+b_{2}x+c_2$ and $p\left(x\right)=f\left(x\right)-g\left(x\right)$. If $p\left(x\right)=0$ only at $x=-1$ and $p(-2)=2$, then the value of $p\left(2\right)$ is

The values of $m$ for which $y=\frac{m{x}^2+3x-4}{-4x^2+3x+m}$ has range $R$ is

The number of integral value(s) of $a$ for which ${\log }_e(x^2+5x)={\log }_e(x+a+3)$ has exactly one solution is

The sum of the roots of equation ${\log }_2(3^{2+x}-6^x)=3+x{\log }_2\left(\frac{3}{2}\right)$ is

The sum of roots of the polynomial equation $(x-1)(x-2)(x-3)=2(x-2)(x-3)$ is

The number of values of $x$ satisfying the equation $(x^2+7x+11{)}^{(x^2-4x-21)}=1$ is

If the roots of the equation $\frac{x^2+4x}{8x+6}=\frac{k-1}{k+1}$ are equal in magnitude and opposite in sign, then the value of $k$ is

The quadratic equations $x^2-6x+a=0,$ $x^2-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$ . Then common root is :

The value(s) of 'p' for which the parabola represented by quadratic function $y=(p-2)x^2+8x+(p+4)$ will remain below X-axis for all real values of x is .

Let $-\frac{\pi }{6}<\theta <-\frac{\pi }{12}$. Suppose ${\alpha }_1$ and ${\beta }_1$ are the roots of the equation $x^2-2x\sec \theta +1=0$ and ${\alpha }_2$ and ${\beta }_2$ are the roots of the equation $x^2+2x\tan \theta -1=0$. If ${\alpha }_1>{\beta }_1$ and ${\alpha }_2>{\beta }_2$, then ${\alpha }_1+{\beta }_2$ equals:

If $\alpha$, $\beta$ are the roots of $x^2+px+q=0$, then the value of ${\alpha }^3+{\beta }^3$ is

If $x^2$ + 2(a - 1)x + a + 5 =0 has real roots belonging to the interval (1, 3) then a$ϵ$

If $5,5r,5r^2$ are the side lengths of a triangle, then the possible value(s) of $r$ is/are

If both the roots of $a{x}^2+bx+c=0$ are real, positive and distinct, then
(where $\Delta =b^2-4ax$)

Solve the following quadratics $4x^2+1=0$

If $a^2(a+p)=b^2(b+p)=c^2(c+p),$ where $a,b,c,p\in R,$ then value of $ab+bc+ca$ is

The largest natural number ${}^{\prime }{a}^{\prime }$ for which the maximum value of $f\left(x\right)=a-1+2x-x^2$ is always smaller than the minimum value of $g\left(x\right)=x^2-2ax+10-2a$ is

A quadratic equation whose difference of roots is $3$ and the sum of the squares of the roots is $29,$ is given by

Set of values of $a$ for which both the roots of the quadratic polynomial $f\left(x\right)=a{x}^2+(a-3)x+1$ lie on one side of the $y-$axis is

If the equation $x^4-(k-1)x^2+(2-k)=0$ has three distinct real roots, then the possible value(s) of $k$ is/are

Which of the following will have their graph plotted in first and second quadrant only?

If atleast one of the root of the equation $x^2-(a-3)x+a=0$ is greater than $2$, then $a$ lies in the interval

If $2x^2+5x+2b=0$ and $2x^3+7x^2+5x+1=0$ have atleast one common root for three values of $b$, then the sum of all three values of $b$ is

The solutions of quadratic equation $3x^2-5x+2=0$ are

Let $\alpha ,\beta$ are the roots of the equation $2x^2-3x-7=0$, then the quadratic equation whose roots are $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ is

If the roots of the equation $4x^2+ax+3=0$ are in ratio of $1:2$, then value(s) of $a$ is/are

If $x^2+bx-a=0$ and $x^2-ax+b=0$ have only one common root, then

If $a:b=1:5,$ then the roots of the equation $a{x}^2-bx+4a=0$ is/are

The number of integral roots of the equation $x^4+\sqrt{x^4+20}=22$ is

Which of the following represents the graph of $f\left(x\right)=a{x}^2+bx+c$ where $a<b<0<c$ ?

The number of integral values of $a$ such that the difference between the roots of the equation $x^2+ax-a=0$ is less than $1,$ is

If the polynomial equation $(x^2+x+1{)}^2-(m-3\left)\right(x^2+x+1)+m=0,m\in R$ has two distinct real roots, then $m$ lies in the interval

Solve the following quadratics $21x^2+9x+1=0$

Let $f\left(x\right)=Ax+B,A,B\in R$ and $y=f\left(x\right)$ passes through the points $(A,2A-B^2)$ and $(2B+3,(A+B{)}^2-1)$. If $B_1,B_2\cdots B_n,n\in N,$ are different possible value(s) of $B$, then the value of $\sum _{r=1}^n{B}_r$ is