Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

The following defines a relation on N:

x y is square of an integer x, y ∈ N

Determine which of the above relations are reflexive, symmetric and transitive.

The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.

Let A = {1, 2, 3, ... 9} and R be the relation in A × A defined by (a, b) R(c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.

Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is ______.

The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.

If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.

Let us define a relation R in R as aRb if a ≥ b. Then R is ______.

Let A = {1, 2, 3} and consider the relation R = {1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1,3)}. Then R is ______.

Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.

Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a² – b²| < 8. Then R is given by ______.

Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.

Every relation which is symmetric and transitive is also reflexive.

An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.

The relation R on the set A = {1, 2, 3} defined as R = {{1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.

Find the real solutions of the equation
`tan^-1 sqrt(x(x + 1)) + sin^-1 sqrt(x^2 + x + 1) = pi/2`

Find the simplified form of `cos^-1 (3/5 cosx + 4/5 sin x)`, where x ∈ `[(-3pi)/4, pi/4]`

Without expanding, show that Δ = `|("cosec"^2theta, cot^2theta, 1),(cot^2theta, "cosec"^2theta, -1),(42, 40, 2)|` = 0

Show that Δ = `|(x, "p", "q"),("p", x, "q"),("q", "q", x)| = (x - "p")(x^2 + "p"x - 2"q"^2)` 

If Δ = `|(0, "b" - "a", "c" - "a"),("a" - "b", 0, "c" - "b"),("a" - "c", "b" - "c", 0)|`, then show that ∆ is equal to zero.