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

Write the following as intervals:  {x: x ∈ R, –12 < x < –10}

Write the following as intervals: {x : x ∈ R, 0 ≤ x < 7}

Write the following as intervals: {x : x ∈ R, 3 ≤ x ≤ 4}

Write the given intervals in set-builder form:

(–3, 0)

Write the given intervals in set-builder form:

[6, 12]

Write the following interval in set-builder form:

(6, 12]

Write the following interval in set-builder form:

[–23, 5)

Decide, among the following sets, which sets are subsets of one and another:

A = {x : x ∈ R and x satisfy x² – 8x + 12 = 0},

B = {2, 4, 6}, C = {2, 4, 6, 8, …}, D = {6}.

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x ∈ A and A ∈ B, then x ∈ B

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and B ∈ C, then A ∈ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and B ⊂ C, then A ⊂ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊄ B and B ⊄ C, then A ⊄ C

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If x ∈ A and A ⊄ B, then x ∈ B

Determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

If A ⊂ B and x ∉ B, then x ∉ A

Let f, g: R → R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and `f/g`

Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b²}. Is the following true?

(a, a) ∈ R, for all a ∈ N

Justify your answer in case.

If \[x = \frac{2 \sin x}{1 + \cos x + \sin x}\], then prove that

If \[\sin x = \frac{a^2 - b^2}{a^2 + b^2}\], then the values of tan x, sec x and cosec x

If \[\tan x = \frac{b}{a}\] , then find the values of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\].