Find the angle between hour-hand and minute hand in a clock at “ten past eleven”.

To find the angle between the hour hand and the minute hand at "ten past eleven," follow these steps:

Step 1: Calculate the angle gained by each hour gained
The total angle of the clock is 360 degrees. Since there are 12 hours on the clock, each hour corresponds to an angle of:

Angle per hour = 360° / 12 = 30°.

Step 2: Calculate the angle of the minute hand
At "ten past eleven," the minute hand is at the 2nd mark, as each number on the clock represents a 5-minute increment.

The angle between each minute mark is 6° (since 360° / 60 minutes = 6° per minute). So, the minute hand at the 2nd mark (10 minutes past the hour) will be at:

Angle of minute hand = 10 minutes × 6° = 60°.

Step 3: Calculate the angle of the hour hand
At "ten past eleven," the hour hand is not exactly at 11 but has moved slightly past the 11 mark. Each hour the hour hand moves 30° (since 360° / 12 = 30° per hour).

At 11:00, the hour hand is at 330° (11 × 30°). Since it is 10 minutes past 11, the hour hand has moved slightly forward. The hour hand moves 0.5° per minute (since 30° per hour / 60 minutes = 0.5° per minute).

In 10 minutes, the hour hand will have moved:

Movement of hour hand = 10 minutes × 0.5° = 5°.

Thus, the hour hand will be at:

Angle of hour hand = 330° + 5° = 335°.

Step 4: Calculate the angle between the two hands
Now, subtract the angle of the minute hand from the angle of the hour hand:

Angle between the hands = |335° - 60°| = 275°.

Since the maximum angle between the hands of the clock can be 360°, the smallest angle will be:

Smallest angle = 360° - 275° = 85°.

Final Answer:
The angle between the hour hand and the minute hand at "ten past eleven" is 85°.