The remainder obtained when the polynomial $1 + x + x^3 + x^9 + x^{27} + x^{81} + x^{243}$ is divided by $x - 1$ is

State which of the following sets are finite or infinite: $\{ x:x \in N \text{ and } 2x - 1 = 0 \}$

Match each of the set on the left in the roster form with the same set on the right described in set-builder form:

$(i)$ $\{1, 2, 3, 6\}$	$(a)$ $\{x : x \text{ is a prime number and a divisor of } 6\}$
$(ii)$ $\{2, 3\}$	$(b)$ $\{x : x \text{ is an odd natural number less than } 10\}$
$(iii)$ $\{M, A, T, H, E, I, C, S\}$	$(c)$ $\{x : x \text{ is a natural number and divisor of } 6\}$
$(iv)$ $\{1, 3, 5, 7, 9\}$	$(d)$ $\{x : x \text{ is a letter of the word } MATHEMATICS\}$

Are the following pair of sets equal? Give reasons.
$A = \{ x : x \text{ is a letter in the word } \text{FOLLOW} \}$
$B = \{ y : y \text{ is a letter in the word } \text{WOLF} \}$

If $A \cap B = B$,then

From the sets given below,select equal sets:
$A=\{2,4,8,12\}, B=\{1,2,3,4\}, C=\{4,8,12,14\}, D=\{3,1,4,2\}$
$E=\{-1,1\}, F=\{0, a\}, G=\{1,-1\}, H=\{0,1\}$