go to home page when i select default in select-box

You can try this

<script type="text/javascript">
    var dropdown = document.getElementById("cat");
    function onCatChange() {
        if ( dropdown.selectedIndex == 0 ) {
            location.href = "https://wordpress.stackexchange.com/questions/90797/<?php echo get_option("home');?>";
        }else if ( dropdown.selectedIndex > 0 ) {
            location.href = "https://wordpress.stackexchange.com/questions/90797/<?php echo get_option("home');?>/?cat="+dropdown.options[dropdown.selectedIndex].value;
        }
    }
    dropdown.onchange = onCatChange;
</script>