Prove that the relation "less than or equal to" (<) on the set R of real numbers is antisymmetric.

On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is symmetric

Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

Find the range of the following functions given by  \( f(x) = \frac { 1 }{ 2-sin\ 3x } .\)

Consider the functions: 
i) \(f(x)=x^2,\)
ii) \(f(x)={1\over 2}x^2,\)
iii) \(f(x)=2x^2\)

If the letters of the word FUNNY are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, find the rank of the word FUNNY.

If p(n) is the statement "n² - n + 41 is prime". Prove that p(1), p(2), and p(3) are true. Prove also that p(41) is not true.

3²ⁿ - 1 is divisible by 8

Evaluate :\(\int(tan \ x+cot \ x)^2dx\)

Integrate the following functions with respect to x : \({sin 4x\over sin \ x}\)

Integrate the following with respect to x : \(\int{2x+4\over x^2+4x+6}dx\)

Integrate the following functions with respect to x : \(\sqrt{81+(2x+1)^2}\)