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

Select the correct answer from the given alternatives.

A college has 7 courses in the morning and 3 in the evening. The possible number of choices with the student if he wants to study one course in the morning and one in the evening is -

How many words can be formed by writing letters in the word CROWN in different order?

Answer the following:

A hall has 12 lamps and every lamp can be switched on independently. Find the number of ways of illuminating the hall.

Prove by method of induction, for all n ∈ N:

2 + 4 + 6 + ..... + 2n = n (n+1)

Prove by method of induction, for all n ∈ N:

3 + 7 + 11 + ..... + to n terms = n(2n+1)

Prove by method of induction, for all n ∈ N:

1² + 2² + 3² + .... + n² = `("n"("n" + 1)(2"n" + 1))/6`

Prove by method of induction, for all n ∈ N:

1² + 3² + 5² + .... + (2n − 1)² = `"n"/3 (2"n" − 1)(2"n" + 1)`

Prove by method of induction, for all n ∈ N:

1³ + 3³ + 5³ + .... to n terms = n²(2n² − 1)

Prove by method of induction, for all n ∈ N:

1.2 + 2.3 + 3.4 + ..... + n(n + 1) = `"n"/3 ("n" + 1)("n" + 2)`

Prove by method of induction, for all n ∈ N:

1.3 + 3.5 + 5.7 + ..... to n terms = `"n"/3(4"n"^2 + 6"n" - 1)`

Prove by method of induction, for all n ∈ N:

`1/(1.3) + 1/(3.5) + 1/(5.7) + ... + 1/((2"n" - 1)(2"n" + 1)) = "n"/(2"n" + 1)`

Prove by method of induction, for all n ∈ N:

`1/(3.5) + 1/(5.7) + 1/(7.9) + ...` to n terms = `"n"/(3(2"n" + 3))`

Prove by method of induction, for all n ∈ N:

(2³ⁿ − 1) is divisible by 7

Prove by method of induction, for all n ∈ N:

(2⁴ⁿ−1) is divisible by 15

Prove by method of induction, for all n ∈ N:

3ⁿ − 2n − 1 is divisible by 4

Prove by method of induction, for all n ∈ N:

5 + 5² + 5³ + .... + 5ⁿ = `5/4(5^"n" - 1)`

Prove by method of induction, for all n ∈ N:

(cos θ + i sin θ)ⁿ = cos (nθ) + i sin (nθ)

Prove by method of induction, for all n ∈ N:

Given that t_n+1 = 5t_n + 4, t₁ = 4, prove that t_n = 5ⁿ − 1

Prove by method of induction, for all n ∈ N:

`[(1, 2),(0, 1)]^"n" = [(1, 2"n"),(0, 1)]` ∀ n ∈ N

Answer the following:

Prove, by method of induction, for all n ∈ N

8 + 17 + 26 + … + (9n – 1) = `"n"/2(9"n" + 7)`