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

Verify that the function y = e^−3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + \frac{dy}{dx} - 6y = 0.\]

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = e^x + 1            y'' − y' = 0

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

`y=sqrt(a^2-x^2)`              `x+y(dy/dx)=0`

Form the differential equation representing the family of curves y = a sin (x + b), where a, b are arbitrary constant.

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

Form the differential equation of the family of circles having centre on y-axis and radius 3 unit.

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