HSC Science (Electronics) ११ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. What is the probability that it lands head up?

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. If happens to be head, what is the probability that it is the two-headed coin?

There are three social media groups on a mobile: Group I, Group II and Group III. The probabilities that Group I, Group II and Group III sending the messages on sports are `2/5, 1/2`, and `2/3` respectively. The probability of opening the messages by Group I, Group II and Group III are `1/2, 1/4` and `1/4` respectively. Randomly one of the messages is opened and found a message on sports. What is the probability that the message was from Group III

(Activity):

Mr. X goes to office by Auto, Car, and train. The probabilities him travelling by these modes are `2/7, 3/7, 2/7` respectively. The chances of him being late to the office are `1/2, 1/4, 1/4` respectively by Auto, Car, and train. On one particular day, he was late to the office. Find the probability that he travelled by car.

Solution: Let A, C and T be the events that Mr. X goes to office by Auto, Car and Train respectively. Let L be event that he is late.

Given that P(A) = `square`, P(C) = `square`

P(T) = `square`

P(L/A) = `1/2`, P(L/C) = `square` P(L/T) = `1/4`

P(L) = P(A ∩ L) + P(C ∩ L) + P(T ∩ L)

`="P"("A")*"P"("L"//"A") + "P"("C")*"P"("L"//"C") + "P"("T")*"P"("L"//"T")`

`= square * square + square * square + square * square`

`= square + square + square`

`= square`

`"P"("C"//"L") = ("P"("L" ∩ "C"))/("P"("L"))`

= `("P"("C") * "P"("L"//"C"))/("P"("L"))`

`= (square * square)/square`

`= square`

Solve the following:

The chances of P, Q and R, getting selected as principal of a college are `2/5, 2/5, 1/5` respectively. Their chances of introducing IT in the college are `1/2, 1/3, 1/4` respectively. Find the probability that IT is introduced in the college after one of them is selected as a principal

Solve the following:

The chances of P, Q and R, getting selected as principal of a college are `2/5, 2/5, 1/5` respectively. Their chances of introducing IT in the college are `1/2, 1/3, 1/4` respectively. Find the probability that IT is introduced by Q

Solve the following:

The ratio of Boys to Girls in a college is 3:2 and 3 girls out of 500 and 2 boys out of 50 of that college are good singers. A good singer is chosen what is the probability that the chosen singer is a girl?

Solve the following:

Given three identical boxes, I, II, and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?

Solve the following:

In a factory which manufactures bulbs, machines A, B and C manufacture respectively 25%, 35% and 40% of the bulbs. Of their outputs, 5, 4 and 2 percent are respectively defective bulbs. A bulbs is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?

Find the value of ω¹⁸

Find the value of ω²¹

Find the value of ω^–30

Find the value of ω^–105

If ω is a complex cube root of unity, show that (2 − ω)(2 − ω²) = 7

If ω is a complex cube root of unity, show that (1 + ω − ω²)⁶ = 64

If ω is a complex cube root of unity, show that (1 + ω)³ − (1 + ω²)³ = 0

If ω is a complex cube root of unity, show that (2 + ω + ω²)³ − (1 − 3ω + ω²)³ = 65

If ω is a complex cube root of unity, show that (3 + 3ω + 5ω²)⁶ − (2 + 6ω + 2ω²)³ = 0

If ω is a complex cube root of unity, show that `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω²

If ω is a complex cube root of unity, show that (a + b) + (aω + bω²) + (aω² + bω) = 0