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

Write the order and the degree of the following differential equation: `"x"^3 ((d^2"y")/(d"x"^2))^2 + "x" ((d"y")/(d"x"))^4 = 0`

Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.

Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Find the value of λ, so that the lines `(1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5)` are at right angles. Also, find whether the lines are intersecting or not.

Show that the relation R defined by (a, b)R(c,d) ⇒ a + d = b + c   on the A x A  , where A =  {1, 2,3,...,10}  is an equivalence relation. Hence write the equivalence class [(3, 4)]; a, b, c,d ∈ A.

Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing. 

Find the order and the degree of the differential equation `x^2 (d^2y)/(dx^2) = { 1 + (dy/dx)^2}^4`

Choose the correct option from the given alternatives :

Let f(x) = x³ – 6x² + 9x + 18, then f(x) is strictly decreasing in ______.

Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?

In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R

Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.

For real numbers x and y, define xRy if and only if x – y + `sqrt(2)` is an irrational number. Then the relation R is ______.

Consider the set A = {1, 2, 3} and R be the smallest equivalence relation on A, then R = ______

Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets

Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.

Let A = {a, b, c} and the relation R be defined on A as follows:
R = {(a, a), (b, c), (a, b)}.
Then, write minimum number of ordered pairs to be added in R to make R reflexive and transitive

Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric

If A = {1, 2, 3, 4 }, define relations on A which have properties of being: 
symmetric but neither reflexive nor transitive