The equation of base $BC$ of an equilateral triangle is $3x + 4y = 1$ and vertex is $(-3,2),$ then the area of triangle is-

Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line $2x + y = 5$ . Then the area of the triangle is :

Two lines are drawn through $(3, 4)$, each of which makes angle of $45^\circ$ with the line $x - y = 2$, then area of the triangle formed by these lines is

There are two candles of same length and same size. Both of them burn at uniform rate. The first one burns in $5\,hr$ and the second one burns in $3\,h$. Both the candles are lit together. After how many minutes the length of the first candle is $3$ times that of the other?

A straight the through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points $P$ and $Q.$ If $O$ is the origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is:

Let $O=(0,0)$ : let $A$ and $B$ be points respectively on $X$-axis and $Y$-axis such that $\angle O B A=60^{\circ}$. Let $D$ be a point in the first quadrant such that $A D$ is an equilateral triangle. Then, the slope of $D B$ is