The system of equations $3x + 2y + z = 6$,$3x + 4y + 3z = 14$ and $6x + 10y + 8z = a$ has an infinite number of solutions if $a$ is equal to

An insect crawls up a hemispherical surface very slowly (see figure). The coefficient of friction between the insect and the surface is $\mu = 1/3$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical,the maximum possible value of $\alpha$ is given by

If the slope of one of the pair of lines represented by $2x^2 + 3xy + Ky^2 = 0$ is $2$,then the angle between the pair of lines is

The pair$(s)$ of diamagnetic ions is$(are)$
$(A) \ La ^{3+}, Ce ^{4+}$ $(B) \ Yb ^{2+}, Lu ^{3+}$ $(C) \ La ^{2+}, Ce ^{3+}$ $(D) \ Yb ^{3+}, Lu ^{2+}$

The area (in square units) of the triangle formed by the points with polar coordinates $(1, 0)$,$(2, \frac{\pi}{3})$,and $(3, \frac{2\pi}{3})$ is

If $[x]$ denotes the greatest integer not exceeding $x$ and if the function $f$ defined by $f(x)= \begin{cases} \frac{a+2 \cos x}{x^2} & , x < 0 \\ b \tan \frac{\pi}{[x+4]} & , x \geq 0 \end{cases}$ is continuous at $x=0$,then the ordered pair $(a, b)$ is equal to