The number of real roots of the equation,
e^4x + e^3x – 4e^2x + e^x + 1 = 0 is :

Let S be the set of all real roots of the equation,
3^x(3^x – 1) + 2 = |3^x – 1| + |3^x – 2|. Then S :

Let $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$.
If $$a = \left( {1 + \alpha } \right)\sum\limits_{k = 0}^{100} {{\alpha ^{2k}}} $$ and
$$b = \sum\limits_{k = 0}^{100} {{\alpha ^{3k}}} $$, then a and b are the roots of the quadratic equation :

Let $$\alpha $$ and $$\beta $$ be the roots of the equation x² - x - 1 = 0.
If p_k = $${\left( \alpha \right)^k} + {\left( \beta \right)^k}$$ , k $$ \ge $$ 1, then which one of the following statements is not true?