Question

At what time between $6\text{a.m.}$ and $7\text{a.m.}$ will the minute hand and hour hand of a clock make an angle ${59}^{\circ }$ ? $\left[3\text{Marks}\right]$

Answer 1

At $6:00\text{a.m.},$ the initial angle between minute and hour hand is ${180}^{\circ }$.
As we know ,
Angle made by hour hand in $1$ minute is ${\left(\frac{1}{2}\right)}^{\circ }$
Angle made by minute hand in $1$ minute is $6^{\circ }$. $\left[1\text{Mark}\right]$
Now let us assume that the angle between minute and hour hand is ${59}^{\circ }$ after $x$ minutes from $6:00\text{a.m.}.$
In these $x$ minutes , the hour hand will trace an angle of ${\left(\frac{x}{2}\right)}^{\circ }$ . And in the same time minute hand will move by ${\left(6x\right)}^{\circ }$ .
This condition can be represented as below - $\left[1\text{Mark}\right]$

From the above diagram, we can write that -
${59}^{\circ }={180}^{\circ }-{\left(6x\right)}^{\circ }+{\left(\frac{x}{2}\right)}^{\circ }$
$\Rightarrow {59}^{\circ }={180}^{\circ }-(6x{)}^{\circ }+(0.5x{)}^{\circ }$
$\Rightarrow x=22\text{min}$
Therefore, required time is $6:22\text{a.m.}$ $\left[1\text{Mark}\right]$