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

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`

If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?

If R and S are relations on a set A, then prove that R is reflexive and S is any relation ⇒ R ∪ S is reflexive ?

If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.

Write the following in the simplest form:

`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`

Write the following in the simplest form:

`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`

Let C be the set of all complex numbers and C₀ be the set of all no-zero complex numbers. Let a relation R on C₀ be defined as

`z_1 R  z_2  ⇔ (z_1 -z_2)/(z_1 + z_2)` is real for all z₁, z₂ ∈ C₀.

Show that R is an equivalence relation.