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

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

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. Also, obtain the equivalence class [(2, 5)].

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x − y = 0}.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric, or transitive.

Check whether the relation R in R defined by R = {(a, b) : a ≤ b³} is reflexive, symmetric or transitive.

Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.

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 and y work at the same place}

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 and y live in the same locality}

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 wife of y}

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.