# Recent questions and answers in 2001

### $\frac{2}{2!} + \frac{2+4}{3!} + \frac{2+4+6}{4!}$ + ...$is equal to: ### If$\frac{x-4}{x^2-5x-2k} = \frac{2}{x-2} - \frac{1}{x+k'}$then$k$is equal to : ### If$(1+x)^n = C_0 + C_1x + C_2x^2 + ....+ C_nx^n$, then$C_0+2C_1 + 3C_2 + ....+(n+1)C_n$is equal to : ### The co-efficient of$x^4$in the expansion of$\frac{(1-3x)^2}{(1-2x)}$is equal to : ###$1+\frac{1}{4} + \frac{1.3}{4.8} + \frac{1.3.5}{4.8.12} + ...$is equal to : ### Using the digits$0, 2, 4, 6, 8$not more than once in any number, the number of 5 digited numbers that can be formed, is : ### The number of ways in which 5 boys and 4 girls sit around a circular tables. So that no two girls sit together is : ### If$y = A \cos nx + B \sin nx$, then$y_2 = n^2y$is equal to : ### If$2^3 + 4^3 + 6^3 + ....+(2n)^3 = hn^2 (n+1)^2$, then$h$is equal to : ### If$x =\log_{0.1} 0.001, \; y = \log_9 81 $, then$\sqrt{x -2\sqrt{y}}$is equal to : ###$\frac{\sqrt{8 +\sqrt{28} } + \sqrt{8 - \sqrt{28}}}{\sqrt{8 +\sqrt{28} } - \sqrt{8 - \sqrt{28}}} $is equal to : ### Two functions$f : R \to R, \; g: R \to R$are defined as follows <br>$ f(x) = \begin{cases} 0 , & \quad \text{x } \text{ is rational}\\ 1, & \quad \text{x} \text{ is irrational} \end{cases}$<br>$ g(x) = \begin{cases} -1 , & \quad \text{x } \text{ is rational}\\ 0, & \quad \text{x} \text{ is irrational} \end{cases}$<br> Then$(fog) (\pi) + (gof)(e)$is equal to : ### Let$f : R \to R$is defined by <br>$ f(x) = \begin{cases} x+2, & x \leq -1 \\ x^2 , & -1 < x <1 \\ 2-x, & x \geq 1 \end{cases}$<br> Then the value of$f(-1.75) + f(0.5) + f(1.5)\$ is :

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