Quadratic expressions

Let $a,b,c\in\{1,2,3,4\}$. If the probability, that $ax^2+2\sqrt2\,bx+c>0$ for all $x\in\mathbb R$, is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.

Let the parabola $y=x^2+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of $p^2+q^2$ is :

The sum of all the integral values of $p$ such that the equation $3\sin^2x+12\cos x-3=p$, $x\in\mathbb R$, has at least one solution, is:

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 :

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 $\lambda^*$ 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 $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral $\displaystyle\int_0^1\left[-8x^2+6x-1\right]dx$ is equal to

The minimum value of the sum of the squares of the roots of $x^2+(3-a)x+1=2a$ is:

For $p,q\in\mathbf R$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf R$ and $q>0$. Let $a_1$, $a_2$, $a_3$ and $a_4$ be in an arithmetic progression with mean p and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ______.

Let $f(x)=ax^2+bx+c$ be such that $f(1)=3$, $f(-2)=\lambda$ and $f(3)=4$. If $f(0)+f(1)+f(-2)+f(3)=14$, then $\lambda$ is equal to :

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

If non-zero real numbers b and c are such that min $f(x)$ > max $g(x)$, where
$$f(x)=x^2+2bx+2c^2\ \text{and}\ g(x)=-x^2-2cx+b^2\ (x\in\mathbb R);$$
then $\left|\dfrac cb\right|$ lies in the interval :

The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\left\{x\mid x^2+20\le9x\right\}$ is