Mathematics Mock Test - 7

If ${\int }_{-3}^{2}f(x)dx=\frac{7}{3}$ and ${\int }_{-3}^{9}f(x)dx=-\frac{5}{6}$, then what is the value of ${\int }_2^{9}f(x)dx$?

Determine the vector equation for the line, given the Cartesian equation is $\frac{x+5}{3}=\frac{y-7}{2}=\frac{z+3}{2}$?

Determine the vector equation of the plane passing through the intersection of the planes $\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})=6$ and $\vec{r}\cdot (2\hat{i}+3\hat{j}+4\hat{k})=-5$, and through the point $(1,1,1)$?

Determine the direction cosines of the unit vector perpendicular to the plane $\vec{r}\cdot (2\hat{i}-6\hat{j}-3\hat{k})+1=0$ passing through the origin?

If the plane $2x-y+z=0$ is parallel to the line $\frac{2x-1}{2}=\frac{2-y}{2}=\frac{z+1}{a}$, then the value of $a$ is: