Mathematics Mock Test - 10

The ratio in which the line segment joining the points $A(4,8,10)$ and $B(6,10,-8)$ is divided by the $yz$ -plane is:

Given matrix $A=\left(\begin{array}{ccc}1& -2& 1\\ 1& 1& -1\\ 3& 6& -5\end{array}\right)$ and $\text{adj }A=\left(\begin{array}{ccc}1& -4& 1\\ 2& 8& 2\\ 3& -12& 3\end{array}\right)$, the system of equations:$x-2y+z=5$$x+y-z=6$$3x+6y-5z=7$ has:

A time-series data is represented by the line $y=a+bx$. If the middle time period is taken as origin ($\sum x=0$), and it is found that $\sum xy=6$ and $\sum x^2=5$, what is the value of '$b$ '?

The maximum value of $f(x)=\sin (x+\frac{\pi }{6})+\cos (x+\frac{\pi }{6})$ in the interval $[0,\frac{\pi }{2}]$ occurs at $x=$:

For $x\ge 1$, the value of $f(x)=2{\tan }^{-1}x+{\sin }^{-1}\frac{2x}{1+x^2}$ is: