PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:

R = {(x, y) : x is father of and y}

Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.

Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.

Test whether the following relation R₁ is  (i) reflexive (ii) symmetric and (iii) transitive :

R₁ on Q₀ defined by (a, b) ∈ R₁ ⇔ a = 1/b.

Test whether the following relation R₂ is (i) reflexive (ii) symmetric and (iii) transitive:

R₂ on Z defined by (a, b) ∈ R₂ ⇔ |a – b| ≤ 5

Test whether the following relation R₃ is (i) reflexive (ii) symmetric and (iii) transitive:

R₃ on R is defined by (a, b) ∈ R₃ `⇔` a² – 4ab + 3b² = 0.

Let A = {1, 2, 3}, and let R₁ = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R₂ = {(2, 2), (3, 1), (1, 3)}, R₃ = {(1, 3), (3, 3)}. Find whether or not each of the relations R₁, R₂, R₃ on A is (i) reflexive (ii) symmetric (iii) transitive.

The following relation is defined on the set of real numbers.
aRb if a – b > 0

Find whether relation is reflexive, symmetric or transitive.

The following relation is defined on the set of real numbers.

aRb if 1 + ab > 0

Find whether relation is reflexive, symmetric or transitive.

The following relation is defined on the set of real numbers.  aRb if |a| ≤ b

Find whether relation is reflexive, symmetric or transitive.

Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,`  find the value of x² + y² + z² 

Find the domain of definition of `f(x)=cos^-1(x^2-4)`

Find the domain of  `f(x) =2cos^-1 2x+sin^-1x.`

Find the domain of `f(x)=cos^-1x+cosx.`

​Find the principal values of the following:
`cos^-1(-sqrt3/2)`

​Find the principal values of the following:

`cos^-1(-1/sqrt2)`

​Find the principal values of the following:

`cos^-1(sin   (4pi)/3)`

​Find the principal values of the following:

`cos^-1(tan  (3pi)/4)`

`sin^-1(sin  pi/6)`

`sin^-1(sin  (7pi)/6)`