Arts (English Medium) Class 11 - CBSE Question Bank Solutions for Mathematics

Three vertices of a parallelogram taken in order are (−1, −6), (2, −5) and (7, 2). The fourth vertex is

The following example is the null set example or not?

Set of even prime numbers

The following example is the null set example or not?

{x : x is a natural numbers, x < 5 and x > 7}

The following example is the null set example or not?

{y : y is a point common to any two parallel lines}

Let T = `{x | (x + 5)/(x - 7) - 5 = (4x - 40)/(13 - x)}`. Is T an empty set? Justify your answer.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

1 + 3 + 5 + ... + (2n – 1) = n² 

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

`sum_(t = 1)^(n - 1) t(t + 1) = (n(n - 1)(n + 1))/3`, for all natural numbers n ≥ 2.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

`(1 - 1/2^2).(1 - 1/3^2)...(1 - 1/n^2) = (n + 1)/(2n)`, for all natural numbers, n ≥ 2. 

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2²ⁿ – 1 is divisible by 3.

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2n + 1 < 2ⁿ, for all natual numbers n ≥ 3.

Define the sequence a₁, a₂, a₃ ... as follows:
a₁ = 2, a_n = 5 a_n–1, for all natural numbers n ≥ 2.

Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula a_n = 2.5^n–1 for all natural numbers.

The distributive law from algebra says that for all real numbers c, a₁ and a₂, we have c(a₁ + a₂) = ca₁ + ca₂.

Use this law and mathematical induction to prove that, for all natural numbers, n ≥ 2, if c, a₁, a₂, ..., a_n are any real numbers, then c(a₁ + a₂ + ... + a_n) = ca₁ + ca₂ + ... + ca_n.

Prove by induction that for all natural number n sinα + sin(α + β) + sin(α + 2β)+ ... + sin(α + (n – 1)β) = `(sin (alpha + (n - 1)/2 beta)sin((nbeta)/2))/(sin(beta/2))`

Prove by the Principle of Mathematical Induction that 1 × 1! + 2 × 2! + 3 × 3! + ... + n × n! = (n + 1)! – 1 for all natural numbers n.

Show by the Principle of Mathematical Induction that the sum S_n of the n term of the series 1² + 2 × 2² + 3² + 2 × 4² + 5² + 2 × 6² ... is given by

S_n = `{{:((n(n + 1)^2)/2",",  "if n is even"),((n^2(n + 1))/2",",  "if n is odd"):}`

Let P(n): “2ⁿ < (1 × 2 × 3 × ... × n)”. Then the smallest positive integer for which P(n) is true is ______.

A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also that P(5) is true. On the basis of this he could conclude that P(n) is true ______.

State whether the following proof (by mathematical induction) is true or false for the statement.

P(n): 1² + 2² + ... + n² = `(n(n + 1) (2n + 1))/6`

Proof By the Principle of Mathematical induction, P(n) is true for n = 1,

1² = 1 = `(1(1 + 1)(2*1 + 1))/6`. Again for some k ≥ 1, k² = `(k(k + 1)(2k + 1))/6`. Now we prove that

(k + 1)² = `((k + 1)((k + 1) + 1)(2(k + 1) + 1))/6`

Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer