JEE Advanced · 2009 · 12 Apr 2009 · Paper 2 · Relations And Functions XII

More JEE Advanced questions on Relations And Functions XII

1. Let $\mathrm{S}=\{1,2,3,\dots ,10\}$. Consider the set
   $\mathrm{X}=\{\mathrm{R}:\mathrm{R}$ is an equivalence relation on the set S such that R has exactly 42 elements $\}$.
   Then the number of elements in X is ____.
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2. Let $\mathbb{N}$ denote the set of all positive integers. Consider the sets
   

   $$\mathrm{A}=\{1,2,3,4,5\}\text{ and }\mathrm{B}=\{1,2,3,4,5,6,7\}.$$

   
   Let S be the set of all functions $f:\mathrm{A}\to \mathrm{B}$ such that $f(2)\ne 2$ and $f(4)\ne 4$. Consider the s …
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3. Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f:\mathbb{N}\to \mathbb{Z}$ and $g:\mathbb{Z}\to \mathbb{N}$ defined by …
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4. Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x,y\in \mathbb{R}$, and $g:\mathbb{R}\to (0,\infty )$ be a function such that $g(x+y)=g(x)g(y)$ for all $x,y\in \mathbb{R}$. If $f\left(\frac{-3}{5}\right)=12$ and …
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5. Let $\mathrm{S}=(0,1)\cup (1,2)\cup (3,4)$ and $\mathrm{T}=\{0,1,2,3\}$. Then which of the following statements is(are) true ?
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6. Let the function $f:[0,1]\to \mathbb{R}$ be defined by $f(x)=\frac{4^x}{4^x+2}$
   Then the value of $f\left(\frac{1}{40}\right)+f\left(\frac{2}{40}\right)+f\left(\frac{3}{40}\right)+\dots +f\left(\frac{39}{40}\right)-f\left(\frac{1}{2}\right)$ is ____
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