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

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?

Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.

Write the following in the simplest form:

`tan^-1{x+sqrt(1+x^2)},x in R `

Show that the relation R on the set A = {x ∈ Z ; 0 ≤ x ≤ 12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1.

Write the following in the simplest form:

`tan^-1{sqrt(1+x^2)-x},x in R`

Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {L₁, L₂) : L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y= 2x + 4.

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)-1)/x},x !=0`

Show that the relation R, defined in the set A of all polygons as R = {(P₁, P₂) : P₁ and P₂ have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?

Write the following in the simplest form:

`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a² + b² = 1}
Prove that S is not an equivalence relation on R.

Write the following in the simplest form:

`tan^-1sqrt((a-x)/(a+x)),-a<x<a`

Let Z be the set of all integers and Z₀ be the set of all non-zero integers. Let a relation R on Z × Z₀be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z₀,
Prove that R is an equivalence relation on Z × Z₀.

Write the following in the simplest form:

`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`