Quadratic Equations

Let $\displaystyle\lim_{x\to2}\frac{\left(\tan(x-2)\right)\left(rx^2+(p-2)x-2p\right)}{(x-2)^2}=5$ for some $r,p\in\mathbf R$. If the set of all possible values of q, such that the roots of the equation $rx^2-px+q=0$ lie in $(0,2)$, be the interval $(\alpha,\beta]$, then $4(\alpha+\beta)$ equals :

Let $\dfrac{x^2}{f\left(a^2+7a+3\right)}+\dfrac{y^2}{f(3a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on $\mathbf R$. If the set of all possible values of a is $\mathbf R-[\alpha,\beta]$, then $\alpha^2+\beta^2$ is equal to :

The sum of squares of all the real solutions of the equation $\log_{(x+1)}\left(2x^2+5x+3\right)=4-\log_{(2x+3)}\left(x^2+2x+1\right)$ is equal to ________.

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$. Define the functions $f : \mathbb{R} \to \mathbb{R}$, $g : \mathbb{R} \to \mathbb{R}$, and $h : \mathbb{R} \to \mathbb{R}$ by

$$\begin{aligned} f(x) &= a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\\ g(x) &= b_1 + 3x + b_2x^2 + b_3x^3 + x^4,\\ h(x) &= f(x+1) - g(x+2). \end{aligned}$$

If $f(x) \neq g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is

Let $\mathbb{R}^2$ denote $\mathbb{R}\times\mathbb{R}$. Let

$$S = \left\{(a,b,c) : a,b,c \in \mathbb{R} \text{ and } ax^2 + 2bxy + cy^2 > 0 \text{ for all } (x,y) \in \mathbb{R}^2 - \{(0,0)\}\right\}.$$

Then which of the following statements is (are) TRUE?

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$, where $i = \sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x) = 0$, then $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to _____.

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 + x - 1 = 0$. Consider the set $T = \{1, \alpha, \beta\}$. For a $3\times 3$ matrix $M = (a_{ij})_{3\times 3}$, define $R_i = a_{i1} + a_{i2} + a_{i3}$ and $C_j = a_{1j} + a_{2j} + a_{3j}$ for $i = 1, 2, 3$ and $j = 1, 2, 3$.

Match each entry in List-I to the correct entry in List-II.

The correct option is

Let the function $f : \mathbb{R} \to \mathbb{R}$ be defined by

$$f(x) = \frac{\sin x}{e^{\pi x}}\cdot\frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)} + \frac{2}{e^{\pi x}}\cdot\frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)}.$$

Then the number of solutions of $f(x) = 0$ in $\mathbb{R}$ is _____.

For any $y \in \mathbb{R}$, let $\cot^{-1}(y) \in (0,\pi)$ and $\tan^{-1}(y) \in \left(-\dfrac\pi2, \dfrac\pi2\right)$. Then the sum of all the solutions of the equation $\tan^{-1}\left(\dfrac{6y}{9-y^2}\right) + \cot^{-1}\left(\dfrac{9-y^2}{6y}\right) = \dfrac{2\pi}{3}$ for $0 < |y| < 3$, is equal to

Consider the equation
$$\int_1^e \frac{(\log_e x)^{1/2}}{x\left(a - (\log_e x)^{3/2}\right)^2}\,dx = 1, \qquad a \in (-\infty, 0)\cup(1,\infty).$$
Which of the following statements is/are TRUE?

Let $|M|$ denote the determinant of a square matrix $M$. Let $g : \left[0, \dfrac\pi2\right] \to \mathbb{R}$ be the function defined by
$$g(\theta) = \sqrt{f(\theta) - 1} + \sqrt{f\left(\frac\pi2 - \theta\right) - 1}$$
where
$$f(\theta) = \frac12\begin{vmatrix}1 & \sin\theta & 1\\ -\sin\theta & 1 & \sin\theta\\ -1 & -\sin\theta & 1\end{vmatrix} + \begin{vmatrix}\sin\pi & \cos\left(\theta + \frac\pi4\right) & \tan\left(\theta - \frac\pi4\right)\\ \sin\left(\theta - \frac\pi4\right) & -\cos\frac\pi2 & \log_e\left(\frac4\pi\right)\\ \cot\left(\theta + \frac\pi4\right) & \log_e\left(\frac\pi4\right) & \tan\pi\end{vmatrix}.$$
Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$, and $p(2) = 2 - \sqrt2$. Then, which of the following is/are TRUE?

The product of all positive real values of $x$ satisfying the equation
$$x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16}$$
is _____.

Let $\bar z$ denote the complex conjugate of a complex number $z$. If $z$ is a non-zero complex number for which both real and imaginary parts of
$$(\bar z)^2 + \frac{1}{z^2}$$
are integers, then which of the following is/are possible value(s) of $|z|$?

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3x^2+\lambda x-1=0$ is 15, then $6(\alpha^3+\beta^3)^2$ is equal to :

The sum of all the real roots of the equation $\left(e^{2x}-4\right)\left(6e^{2x}-5e^x+1\right)=0$ is

The number of distinct real roots of the equation $x^7-7x-2=0$ is

Let $\lambda^*$ be the largest value of $\lambda$ for which the function $f_\lambda(x)=4\lambda x^3-36\lambda x^2+36x+48$ is increasing for all $x\in\mathbb R$. Then $f_{\lambda^*}(1)+f_{\lambda^*}(-1)$ is equal to :

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be two functions defined by $f(x)=\log_e\left(x^2+1\right)-e^{-x}+1$ and $g(x)=\dfrac{1-2e^{2x}}{e^x}$. Then, for which of the following range of $\alpha$, the inequality
$$f\left(g\left(\frac{(\alpha-1)^2}3\right)\right)>f\left(g\left(\alpha-\frac53\right)\right)$$
holds ?

Let the abscissae of the two points P and Q be the roots of $2x^2-rx+p=0$ and the ordinates of P and Q be the roots of $x^2-sx-q=0$. If the equation of the circle described on PQ as diameter is $2\left(x^2+y^2\right)-11x-14y-22=0$, then $2r+s-2q+p$ is equal to ______.

For a natural number n, let $\alpha_n=19^n-12^n$. Then, the value of $\dfrac{31\alpha_9-\alpha_{10}}{57\alpha_8}$ is ______.