Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Given an example of a relation. Which is Symmetric and transitive but not reflexive.

Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is the same as the distance of the point Q from the origin} is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with the origin as its centre.

Show that the relation R defined in the set A of all triangles as R = {(T₁, T₂) : T₁ is similar to T₂}, is an equivalence relation. Consider three right angle triangles T₁ with sides 3, 4, 5, T₂ with sides 5, 12, 13 and T₃ with sides 6, 8, and 10. Which triangles among T₁, T₂ and T₃ are related?

Show that the relation R defined in the set A of all polygons as R = {(P₁, P₂): P₁ and P₂have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Let L be the set of all lines in the 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.

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

Let R be the relation in the set N given by R = {(a, b) : a = b − 2, b > 6}. Choose the correct answer.

Given a non-empty set X, consider P(X), which is the set of all subsets of X. Define the relation R in P(X) as follows:

For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.

Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is

(A) 1 (B) 2 (C) 3 (D) 4

Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is

(A) 1

(B) 2

(C) 3

(D) 4

If A = `[(1,1,-2),(2,1,-3),(5,4,-9)]`, find |A|.

Find the value of x, if `|(2,4),(5,1)|=|(2x, 4), (6,x)|`.

Find the value of x, if `|(2,3),(4,5)|=|(x,3),(2x,5)|`.

Using the property of determinants and without expanding, prove that:

`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`

Let A be a square matrix of order 3 × 3, then | kA| is equal to

(A) k|A|

(B) k² | A |

(C) k³ | A |

(D) 3k | A |

Find the direction cosines of a line which makes equal angles with the coordinate axes.

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).

If l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂ − m₂n₁, n₁l₂ − n₂l₁, l₁m₂ ­− l₂m₁.