PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

Which of the following are examples of empty set? 

Set of all even natural numbers divisible by 5

Which of the following are examples of empty set? 

Set of all even prime numbers

Which of the following are examples of empty set? 

 {x : x² −2 = 0 and x is rational}

Which of the following are examples of empty set? 

{x : x is a natural number, x < 8 and simultaneously x > 12};

Which of the following are examples of empty set? 

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

If P (n) is the statement "n(n + 1) is even", then what is P(3)?

If P (n) is the statement "n3 + n is divisible by 3", prove that P (3) is true but P (4) is not true.

If P (n) is the statement "n² − n + 41 is prime", prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.

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.

1 + 2 + 3 + ... + n =  \[\frac{n(n + 1)}{2}\] i.e. the sum of the first n natural numbers is \[\frac{n(n + 1)}{2}\] .

1² + 2² + 3² + ... + n² =\[\frac{n(n + 1)(2n + 1)}{6}\] .

1 + 3 + 3² + ... + 3ⁿ⁻¹ = \[\frac{3^n - 1}{2}\]

\[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + . . . + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\]

\[\frac{1}{2 . 5} + \frac{1}{5 . 8} + \frac{1}{8 . 11} + . . . + \frac{1}{(3n - 1)(3n + 2)} = \frac{n}{6n + 4}\]