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

Answer the following:

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

1² + 4² + 7² + ... + (3n − 2)² = `"n"/2 (6"n"^2 - 3"n" - 1)`

Answer the following:

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

2 + 3.2 + 4.2² + ... + (n + 1)2^n–1 = n.2ⁿ 

Answer the following:

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

`1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "n"/(("n" + 2)("n" + 3)("n" + 4)) = ("n"("n" + 1))/(6("n" + 3)("n" + 4))`

Answer the following:

Given that t_n+1 = 5t_n − 8, t₁ = 3, prove by method of induction that t_n = 5ⁿ⁻¹ + 2

Answer the following:

Prove by method of induction

`[(3, -4),(1, -1)]^"n" = [(2"n" + 1, -4"n"),("n", -2"n" + 1)], ∀  "n" ∈ "N"`

Answer the following:

Prove by method of induction log_a xⁿ = n log_ax, x > 0, n ∈ N

Answer the following:

Prove by method of induction 15^2n–1 + 1 is divisible by 16, for all n ∈ N.

Answer the following:

Prove by method of induction 5²ⁿ − 2²ⁿ is divisible by 3, for all n ∈ N

Describe the following set in Roster form

A = {x/x is a letter of the word 'MOVEMENT'}

Describe the following set in Roster form

B = `{x//x  "is an integer", -3/2 < x < 9/2}`

Describe the following set in Roster form

C = {x/x = 2n + 1, n ∈ N}

Describe the following set in Set-Builder form

{0}

Describe the following set in Set-Builder form

{0, ±1, ±2, ±3}

Describe the following set in Set-Builder form

`{1/2, 2/5, 3/10, 4/17, 5/26, 6/37, 7/50}`

Describe the following set in Set-Builder form

{0, –1, 2, –3, 4, –5, 6, ...}

If A = {x/6x² + x – 15 = 0}, B = {x/2x² – 5x – 3 = 0}, C = {x/2x² – x – 3 = 0} then find (A ∪ B ∪ C).

If A = {x/6x² + x – 15 = 0}, B = {x/2x² – 5x – 3 = 0}, C = {x/2x² – x – 3 = 0} then find (A ∩ B ∩ C)

In a class of 200 students who appeared in certain examinations, 35 students failed in CET, 40 in NEET and 40 in JEE, 20 failed in CET and NEET, 17 in NEET and JEE, 15 in CET and JEE, and 5 failed in all three examinations. Find how many students, did not fail in any examination.

In a class of 200 students who appeared in certain examinations, 35 students failed in CET, 40 in NEET and 40 in JEE, 20 failed in CET and NEET, 17 in NEET and JEE, 15 in CET and JEE, and 5 failed in all three examinations. Find how many students, failed in NEET or JEE entrance

From amongst 2000 literate individuals of a town, 70% read Marathi newspapers, 50% read English newspapers and 32.5% read both Marathi and English newspapers. Find the number of individuals who read at least one of the newspapers