Let $f(x)=a x^2+b x+c$, where $a, b, c$ are integers, Suppose $f(1)=0,40 < f(6) < 50,60 < f(7) < 70$ and $1000 t < f(50) < 1000(t+1)$ for some integer $t$. Then, the value of $t$ is

Consider the quadratic equation $n x^2+7 \sqrt{n x+n}=0$ where $n$ is a positive integer. Which of the following statements are necessarily correct?

$I$. For any $n$, the roots are distinct.

$II$. There are infinitely many values of $n$ for which both roots are real.

$III$. The product of the roots is necessarily an integer.

If $a, b, c, d$ are four distinct numbers chosen from the set $\{1,2,3, \ldots, 9\}$, then the minimum value of $\frac{a}{b}+\frac{c}{d}$ is

Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :

If $a, b, c \in R$ and $1$ is a root of equation $ax^2 + bx + c = 0$, then the curve y $= 4ax^2 + 3bx+ 2c, a \ne 0$ intersect $x-$ axis at