1Write down an A.P., sum of whose first 11 terms is zero.
2Find the probability of getting 53 Mondays in a leap year.
3What is the value of tan 45°+ cot 45°?
Answers
Question 1 :
Write down an A.P., sum of whose first 11 terms is zero.
AnswEr :
If we need Sum of 11 terms as Zero then, we must have to create Set of (5 Positive Term) and, (5 Negative Term). Middle Term must be 0 (Average) so that Sum can be Zero.
Let's Take 11 terms, in this Case.
⋆ Average = 0 /11 = 0
So Just Make Set of Numbers with 5 Negative Term, 0 in Middle and 5 Positive Term. Difference must be Equal.
• Let's take an Example :
- 5, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, 5
Here ; a = - 5 , l = 5 and, n = 11
⇒ S₁₁ = n /2(a + l)
⇒ S₁₁ = 11 /2(- 5 + 5)
⇒ S₁₁ = 5.5 × 0
⇒ S₁₁ = 0
∴ Therefore, Set Can be made like that.
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Question 2 :
Find the probability of getting 53 Mondays in a leap year.
AnswEr :
1 Normal Year = 365 Days
1 Leap Year = 366 Days
A year has 52 weeks. Hence there will be 52 Mondays for sure in Normal Yr or Leap Yr.
• Days in Normal Year :
⇝ 52 weeks = 52 x 7 = 364 days
• Extra Days in Leap Year :
⇝ 366 – 364 = 2 days
In a leap year there will be 52 Mondays and 2 days will be left.
⋆ These 2 days can be like this :
- Sunday, Monday
- Monday, Tuesday
- Tuesday, Wednesday
- Wednesday, Thursday
- Thursday, Friday
- Friday, Saturday
- Saturday, Sunday
Of these Total 7 outcomes, the favourable outcomes are 2.
• Probability of getting 53 Mondays :
⇒ P( E ) = Favorable Outcomes /Total Outcomes
⇒ P( E ) = 2 /7
∴ Therefore, probability of getting 53 Mondays in a leap year is 2 /7.
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Question 3 :
What is the value of tan 45°+ cot 45°?
AnswEr :
⇒ tan (45°) + cot (45°)
- value of both is 1
⇒ 1 + 1
⇒ 2
∴ Value of tan 45° + cot 45° is 2.