Quadratic Equations

For a polynomial $g(x)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by

$$S = \left\{(x^2-1)^2\left(a_0 + a_1x + a_2x^2 + a_3x^3\right) : a_0, a_1, a_2, a_3 \in \mathbb R\right\}.$$

For a polynomial $f$, let $f'$ and $f''$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'} + m_{f''}\right)$, where $f\in S$, is ______

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times2$ matrix such that the trace of $A$ is 3 and the trace of $A^3$ is $-18$, then the value of the determinant of $A$ is ______

Let $\alpha$ and $\beta$ be the roots of $x^2 - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define

$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},\quad n \ge 1,$$

$$b_1 = 1\quad\text{and}\quad b_n = a_{n-1} + a_{n+1},\quad n \ge 2.$$

Then which of the following options is/are correct?

The number of real solutions of the equation

$$\sin^{-1}\left(\sum_{i=1}^{\infty} x^{i+1} - x \sum_{i=1}^{\infty} \left(\frac{x}{2}\right)^i\right) = \frac{\pi}{2} - \cos^{-1}\left(\sum_{i=1}^{\infty} \left(-\frac{x}{2}\right)^i - \sum_{i=1}^{\infty} (-x)^i\right)$$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is _____ .

(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\cos^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

$a_{12} =$

If $a_4 = 28$, then $p + 2q =$

If, for a positive integer n, the quadratic equation,
$$x(x+1)+(x+1)(x+2)+\dots+(x+\overline{n-1})(x+n)=10n$$
has two consecutive integral solutions, then n is equal to :

Let p(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by $x-1$ and it leaves remainder 6 when divided by $x+1$; then :

The sum of all the real values of $x$ satisfying the equation $2^{(x-1)(x^2+5x-50)}=1$ is :

Let $-\dfrac{\pi}{6}<\theta<-\dfrac{\pi}{12}$. 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

The total number of distinct $x\in\mathbb R$ for which $\begin{vmatrix}x&x^2&1+x^3\\2x&4x^2&1+8x^3\\3x&9x^2&1+27x^3\end{vmatrix}=10$ is

The sum of all real values of $x$ satisfying the equation $\left(x^2-5x+5\right)^{x^2+4x-60}=1$ is :

If the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have a common root different from −1, then |b| is equal to :

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 intervals is(are) a subset(s) of $S$?

Let α and β be the roots of equation $x^2-6x-2=0$. If $a_n=\alpha^n-\beta^n$, for $n\ge1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is equal to :

If $2+3i$ is one of the roots of the equation $2x^3-9x^2+kx-13=0$, $k\in\mathbb R$, then the real root of this equation :

If $z$ is a non-real complex number, then the minimum value of $\dfrac{\text{Im }z^5}{(\text{Im }z)^5}$ is :

If the two roots of the equation, $(a-1)(x^4+x^2+1)+(a+1)(x^2+x+1)^2=0$ are real and distinct, then the set of all values of 'a' is :

Let $a\in\mathbb R$ and let $f:\mathbb R\to\mathbb R$ be given by
$$f(x)=x^5-5x+a.$$
Then

The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation
$$p(p(x))=0$$
has