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

Evaluate the following:

`cosec^-1(cosec  (6pi)/5)`

Defines a relation on N :

x + y = 10, x, y∈ N

Determine the above relation is reflexive, symmetric and transitive.

Evaluate the following:

`cosec^-1(cosec  (11pi)/6)`

Defines a relation on N:

xy is square of an integer, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Evaluate the following:

`cosec^-1(cosec  (13pi)/6)`

Defines a relation on N:

x + 4y = 10, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.

Evaluate the following:

`cosec^-1{cosec  (-(9pi)/4)}`

Evaluate the following:

`cot^-1(cot  pi/3)`

Evaluate the following:

`cot^-1(cot  (4pi)/3)`

Evaluate the following:

`cot^-1(cot  (9pi)/4)`

Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b ∈ Z} is an equivalence relation.

Evaluate the following:

`cot^-1(cot  (19pi)/6)`

Evaluate the following:

`cot^-1{cot (-(8pi)/3)}`

Evaluate the following:

`cot^-1{cot  ((21pi)/4)}`

Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b},  is an equivalence relation.

Prove that the relation R on Z defined by
(a, b) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.

Let n be a fixed positive integer. Define a relation R on Z as follows:
(a, b) ∈ R ⇔ a − b is divisible by n.
Show that R is an equivalence relation on Z.

Let Z be the set of integers. Show that the relation
 R = {(a, b) : a, b ∈ Z and a + b is even}
is an equivalence relation on Z.

Write the following in the simplest form:

`cot^-1  a/sqrt(x^2-a^2),|  x  | > a`

m is said to be related to n if m and n are integers and m − n is divisible by 13. Does this define an equivalence relation?