Probability

Step 1: Let the position vectors of the vertices , , and be . Given midpoints:
Step 2: Add equations (2) and (3): Using equation (1), , hence:
Step 3: From equation (3): From equation (2):
Step 4: Coordinates of and are known. Slope of :
Step 5: Equation of line : Simplifying:

Step 1: Let the coin be tossed times. The probability of getting no head (i.e., all tails) is:
Step 2: Therefore, the probability of getting at least one head is:
Step 3: Given that this probability is more than :
Step 4: Rearranging: Taking logarithms:
Step 5: Using logarithmic values:
Step 6: Since must be a whole number, the minimum value satisfying the condition is: Final Answer (up to two decimal places):

Step 1: Write the word LONDON: The total number of successive letter pairs is: The pair ON appears twice.
Step 2: Write the word CLIFTON: The total number of successive letter pairs is: The pair ON appears once.
Step 3: Assuming the envelope is equally likely to come from either place:
Step 4: Using Bayes’ theorem:
Step 5: Simplify: Final Answer (up to two decimal places):

Step 1: Digits 1,3,7,9 are units digits coprime to 10. These arise only when the product contains neither factor 2 nor 5 in excess powers.
Step 2: Among digits 0–9, favourable residues modulo 10 are 4 out of 10 possibilities → probability per digit .
Step 3: For independent random integers, multiply probabilities pattern using Euler totient:
Step 4: Count of favourable sequences of length n: Boolean reduction supplied in option differentiation leads: Hence → (C).

Step 1: List all possible outcomes. Assume that the probability of having a boy or a girl is equal and independent for each child. Possible combinations for two children are: Step 2: Count total and favourable outcomes.

Total possible outcomes
Favourable outcome (both are boys)
Step 3: Calculate the probability. Step 4: Final conclusion. The probability that both children are boys is: