Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Verify using Venn diagram.

Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.

If a \(\in\) {-1, 2, 3, 4, 5} and b \(\in\) {0,3, 6}. Write the set of all ordered pairs (a, b) such that a + b = 5.

Find the sum and difference of the identity function and the modulus function?

Find the range of the function.
f = {1, x), (1, y), (2, x), (2, y), (3, z)}

Solve: \(\sqrt{x+5}+\sqrt{x+21}=\sqrt{6x+40}\)

If \({{{log}_{e}^{x}}\over{b-c}}={{{log}_{e}^{y}}\over{c-a}}={{{log}_{e}^{z}}\over{a-b}},\) show that xyz = 1

Factorize: x⁴-14x²y²-51y⁴

Solve : \(2\left(x+{{1}\over{x}} \right)^2-7\left(x+{{1}\over{x}} \right)+5=0.\)

Resolve into partial fractions \(\frac { 9 }{ (x-1)(x+2)^{ 2 } } \)

Resolve into partial fractions \(\frac { { x }^{ 3 }-1 }{ { x }^{ 2 }+x+1 } \)

In \(\Delta ABC\), if a = 18, b = 24, c = 30 and \(\angle\)c = 90^o, find \(\sin { A, } \sin { B } \) and \(\sin { C } \)

Two trees A and B are on the same side of a river. From a point C in the river the distance of trees A and B are 250 m and 300 m respectively. If the angle C is 45^o, find the distance between the trees.

Prove that (i) cos 20° cos 40° cos 80° \(=\frac{1}{8}\)

A + B + C =\(\pi\), prove that sin 2A - sin 2B + sin 2C = 4 cos A sin B cos C

In a triangle ABC, A = 35° 17' ; C = 45° 13' ; b = 42.1 Solve the triangle

Solve \({ tan }^{ -1 }\left( \frac { 2x }{ 1-{ x }^{ 2 } } \right) +{ cot }^{ -1 }\left( \frac { 1-{ x }^{ 2 } }{ 2x } \right) =\frac { \pi }{ 3 } ,wherex>0\)

Using principle of mathematical induction, prove that x²ⁿ-y²ⁿ is divisible by x+y for all n∈N.

Prove that \(\left( 1+\frac { 1 }{ 1 } \right) \left( 1+\frac { 1 }{ 2 } \right) \left( 1+\frac { 1 }{ 3 } \right) ...\left( 1+\frac { 1 }{ n } \right) =\left( n+1 \right) \) for all \(n\in N\) by the principle of mathematical induction.

A committee of 6 is to be choosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. In how many different ways can this be done it two particular women refuse to serve on the same committee?

Determine n if  ²ⁿC₃ : ⁿC₃ = 11 : 1

Prove by induction the inequality (1 + x)ⁿ\(\ge\) 1 + nx, whenever x is positive and n is a positive integer.

If (p+1) ^th  term of an A.P is twice the (q+1)^th terms prove that the (3p+1)^th term is twice the  (p+q+1)^th term

If S _n denotes that Sum of n terms of a G. P., prove that (s₁₀-s₂₀ )² = s₁₀ (s₃₀ - s₂₀)

If the first two terms of a H. P are \(\frac { 2 }{ 5 } \)  and\(\frac { 12 }{ 13 } \)  respectively, find the largest term of the H.P.