HSC Commerce (English Medium) ११ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

The odds against a certain event are 5: 2 and odds in favour of another independent event are 6: 5. Find the chance that at least one of the events will happen.

The odds against a husband who is 55 years old living till he is 75 is 8: 5 and it is 4: 3 against his wife who is now 48, living till she is 68. Find the probability that the couple will be alive 20 years hence.

The odds against a husband who is 55 years old living till he is 75 is 8: 5 and it is 4: 3 against his wife who is now 48, living till she is 68. Find the probability that at least one of them will be alive 20 years hence.

The odds against student X solving a business statistics problem are 8: 6 and odds in favour of student Y solving the same problem are 14: 16 What is the chance that the problem will be solved, if they try independently?

The odds against student X solving a business statistics problem are 8: 6 and odds in favour of student Y solving the same problem are 14: 16 What is the probability that neither solves the problem?

If `omega` is a complex cube root of unity, show that `(2 - omega)(2 - omega^2)` = 7

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 `((a + bomega + comega^2))/("c" + aomega + bomega^2) = omega^2`.

If ω is a complex cube root of unity, find the value of `omega + 1/omega`

If ω is a complex cube root of unity, find the value of ω² + ω³ + ω⁴.

If ω is a complex cube root of unity, find the value of (1 + ω²)³

If ω is a complex cube root of unity, find the value of (1 - ω - ω²)³ + (1 - ω + ω²)³

If `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`

If α and β are the complex cube roots of unity, show that α² + β² + αβ = 0.

If x = a + b, y = αa + βb and z = aβ + bα, where α and β are the complex cube roots of unity, show that xyz = a³ + b³.

If ω is a complex cube root of unity, then prove the following: (ω² + ω - 1)³ = – 8

If ω is a complex cube root of unity, then prove the following:  (a + b) + (aω + bω²) + (aω² + bω) = 0.

Find A.M. of two positive numbers whose G.M. and H.M. are 4 and `16/5`. 

Find H.M. of two positive numbers whose A.M. and G.M. are `15/2` and 6.

Find G.M. of two positive numbers whose A.M. and H.M. are 75 and 48.