Algebra Questions

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is:

If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to:

The number of values of $k$, for which the system of equations$(k+1)x+8y=4k$$kx+(k+3)y=3k-1$ has no solution, is:

Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is:

A multiple choice examination has $5$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $4$ or more correct answers just by guessing is:

If $z$ is a complex number of unit modulus and argument $\theta$, then $\text{arg}(\frac{1+z}{1+\bar{z}})$ equals:

If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$, $a,b,c\in \mathbb{R}$, have a common root, then $a:b:c$ is:

The term independent of $x$ in expansion of ${\left[\frac{x^{2/3}+x^{1/3}+1}{x^{2/3}-1}-\frac{x^{1/2}+1}{x^{1/2}-x}\right]}^{10}$ is: