If $p,\;q,\;r$ are in $A.P.$ and are positive, the roots of the quadratic equation $p{x^2} + qx + r = 0$ are all real for
$\left| {\,\frac{r}{p} - 7\;} \right|\; \ge 4\sqrt 3 $
$\left| {\;\frac{p}{r} - 7\;} \right|\; < 4\sqrt 3 $
All $p$ and $r$
No $p$ and $r$
If $\log 2,\;\log ({2^n} - 1)$ and $\log ({2^n} + 3)$ are in $A.P.$, then $n =$
The sum of $1 + 3 + 5 + 7 + .........$ upto $n$ terms is
If $a _{1}, a _{2}, a _{3} \ldots$ and $b _{1}, b _{2}, b _{3} \ldots$ are $A.P.$ and $a_{1}=2, a_{10}=3, a_{1} b_{1}=1=a_{10} b_{10}$ then $a_{4} b_{4}$ is equal to
Find the sum of all two digit numbers which when divided by $4,$ yields $1$ as remainder.
Let $\frac{1}{{{x_1}}},\frac{1}{{{x_2}}},\frac{1}{{{x_3}}},.....,$ $({x_i} \ne \,0\,for\,\,i\, = 1,2,....,n)$ be in $A.P.$ such that $x_1 = 4$ and $x_{21} = 20.$ If $n$ is the least positive integer for which $x_n > 50,$ then $\sum\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}}} \right)} $ is equal to.