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1.Relation and Function
hard
The domain of the function $f(x){ = ^{16 - x}}{\kern 1pt} {C_{2x - 1}}{ + ^{20 - 3x}}{\kern 1pt} {P_{4x - 5}}$, where the symbols have their usual meanings, is the set
A
{$2, 3$}
B
{$2, 3, 4$}
C
{$1, 2, 3, 4$}
D
{$1, 2, 3, 4, 5$}
Solution
(a) For $f(x)$ to be defined
$(i)$ $16 – x > 0 \Rightarrow x < 16$
$(ii)$ $2x – 1 \ge 0 \Rightarrow x \ge \frac{1}{2}$
$(iii)$ $20 – 3x > 0 \Rightarrow x < \frac{{20}}{3}$
$(iv)$ $4x – 5 \ge 0 \Rightarrow x \ge \frac{5}{4}$
$(v)$ $16 – x \ge 2x – 1 \Rightarrow x \le \frac{{17}}{3}$
$(vi)$ $20 – 3x \ge 4x – 5 \Rightarrow x \le \frac{{25}}{7}$
$(vii)$ $16 – x$ is an integer, ? x should be an integer
$\therefore$ $\frac{5}{4} \le x \le \frac{{25}}{7}$ and $x \in I$ ==> $x = 2,\,3$
$\therefore$ Domain of $f = \{ 2,\,\,3\} $.
Standard 12
Mathematics
