Given:
- LM is a chord of a circle with centre C.
- CN is perpendicular to LM.
- MC produced intersects the circle at point P.
- PL = 10 cm.

To find:
The length of CN.

Solution:

Step 1: Identify the key points in the question:
- LM is a chord of the circle.
- CN is perpendicular to LM.
- MC produced intersects the circle at point P.
- PL = 10 cm.

Step 2: Identify the relevant concepts and formulas:
- In a circle, a line drawn from the centre of the circle to a point on the circle is called a radius.
- Perpendicular lines intersect at a right angle.
- In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This is known as the Pythagorean theorem.

Step 3: Apply the concepts and formulas to solve the problem:
- Since CN is perpendicular to LM, CN is the radius of the circle.
- Let's assume the length of CN is 'x' cm.
- Using the Pythagorean theorem, we can find the value of x.

- According to the Pythagorean theorem:
- (PL)^2 = (CN)^2 + (LP)^2

- Substituting the given values:
- (10)^2 = (x)^2 + (LP)^2

- Since LP is a chord of the circle, we can use the property that states that the product of the lengths of the segments of a chord intersecting outside the circle is equal. In other words:
- (LP)(MP) = (LM)(MP)

- Substituting the given values:
- (LP)(MP) = (LM)(MP)
- (LP)(MP) = (LM)(LM + LP)
- (LP)(MP) = (LM^2) + (LM)(LP)

- Since LM = 2(LP) (as LM is a diameter of the circle), we can substitute this value:
- (LP)(MP) = (4(LP)^2) + (2(LP))(LP)
- (LP)(MP) = 4(LP)^2 + 2(LP)^2
- (LP)(MP) = 6(LP)^2

- Substituting the given value of LP:
- (10)(MP) = 6(LP)^2
- 10(MP) = 6(10)^2
- 10(MP) = 600
- MP = 60 cm

- Now, substituting the value of LP and MP in the Pythagorean theorem equation:
- (10)^2 = (x)^2 + (60)^2
- 100 = x^2 + 3600
- x^2 = 100 - 3600
- x^2 = -3500

- Since the square of a length cannot be negative, there is no real solution for x. Therefore, there is no valid length for CN.

Step 4: Answer:
The length of CN cannot be determined based on the given information. Therefore, the answer is not applicable (N/A).