Answer validation and no skipping questions (for Quiz)

Answer validation and no skipping questions (for Quiz)

I'm building a quiz and there are two functions I can't seem to work into the mostly complete code. I followed several tutorials and forums to put this together, and cant seem to work in what I need. If someone could point me in right direction?

What it does now:
1. Allows questions to be skipped
2. Checks answers after the last question, and displays the question and correct answer at the end.

What I'd like it to do:
1. Cannot proceed to next question without a selection (cannot leave radio empty)
2. Check if correct answer before moving to next question, if incorrect then display questions[j].w
3. Click next again to move to next question

HTML: 
  2. <div id="wrapper">
  3.       <div id="main">
  4.         <h1>Quiz</h1>
  5.         <p>Test your knowledge.</p>
  6.         <button type="button" class="begin" value="begin">begin</button>
  7.       </div>
  8.       
  9.       <div id="questions"></div>
  10. </div>
JS:

  2. $(document).ready(function() {
  3.   $(".begin").click(function(){
  4.     $("#main").hide();
  5.     return quiz();
  6. });

  8. function startOver() {
  9.  location.reload(true);
  10. }

  12. function quiz() {
  13.   var score = [];
  14.   var questions = [{
  15.     q: "1 + 1",
  16.     s: ["1", 
  17.         "2", 
  18.         "3"],
  19.     a: "2",
  20.     w: "Incorrect",
  21.     correct: 0

  23. }, {
  24.     q: "2 + 1",
  25.     s: ["1", 
  26.         "2", 
  27.         "3"],
  28.     a: "3",
  29.     w: "Incorrect",
  30.     correct: 0

  32. }, {
  33.     q: "3 + 1",
  34.     s: ["1", 
  35.         "2", 
  36.         "3"],
  37.     a: "4",
  38.     w: "Incorrect",
  39.     correct: 0
  40. }];
  41.     
  42.     var counter = questions.length;
  43.     
  44. function renderQuestion(questions) {
  45.     for (var i = 0; i < questions.length; i++) {
  46.         $("#questions").append(
  47.           '<form id="'+i+'">' +
  48.           '<p class="page">Question ' + (i + 1) + ' of ' + questions.length + '</p>' +
  49.           '<hr class="row">' +
  50.           '<h3 class="question">' + questions[i].q + '</h3>' + 
  51.           radioButtons(questions[i].s, i) + 
  52.           '<hr class="row">' +
  53.           '<span class="nav">' +
  54.           '<button type="button" class="reset" value="reset">start over</button>' +
  55.           '<button type="submit" class="next">next</button>' +
  56.           '</span></form>');
  57.       }
  58.     for (var k = questions.length - 1; k > 0; k--) {
  59.       $('#' + k).hide();
  60.       }
  61.       
  62.     $('.reset').click(startOver);
  63.     }
  64.     
  65. function radioButtons(ary, qNum) {
  66.     var answers = [];
  67.     for (i = 0; i < ary.length; i++) {
  68.         answers.push(
  69.           '<div class="choices">' +
  70.           '<input type="radio" name="'+qNum+'" value="' + ary[i] + '" id="'+qNum+'">' +
  71.           '<label for="'+qNum+'">' +ary[i]+ '</label>' +
  72.           '</div>');
  73.       }
  74.     return answers.join("");
  75.     }
  76.     
  77. function sumCorrect(questions) {
  78.     return score.reduce(function (previousValue, currentValue, index, array) {
  79.     return previousValue + currentValue;
  80.       });
  81.     }

  83. function checkAnswer(answer, qNum, questions) {
  84.     if (answer == questions[qNum].a) {
  85.       questions[qNum].correct = 1;
  86.       score.push(questions[qNum].correct);
  87.       
  88.     } else {
  89.       score.push(questions[qNum].correct);
  90.     }
  91. }

  93.     
  94.     renderQuestion(questions);
  95.     
  96.     $(".next").click(function (event) {
  97.       event.preventDefault();
  98.     var qNum = $(this).closest("form").attr("id");
  99.     
  100.     var userInput = $('input[name='+qNum+']:radio:checked').val();
  101.     if (counter > 1) {
  102.       checkAnswer(userInput, qNum, questions);
  103.       $("#" + qNum).hide();
  104.       $("#" + qNum).next().show();
  105.       counter--;
  106.     } else if (counter == 1) {
  107.       checkAnswer(userInput, qNum, questions);
  108.       $("#questions").find("form").remove();
  109.       $("#questions").append(
  110.       '<p class="page result"></p>' +
  111.       '<hr class="row">' +
  112.       '<div class="nav nextLast">' +
  113.       '<button type="button" class="reset" value="reset">start over</button>' +
  114.       '</div>');
  115.       $(".result").text('You answered ' + sumCorrect(questions) + ' questions correctly out of 3.');
  116.         for (j = 0; j < score.length; j++) {
  117.           if (score[j] === 0) {
  118.             console.log(questions[j].q, questions[j].a);
  119.       $("#questions").append('<div>' + 
  120.       '<h3 class="question">' + questions[j].q + '</h3>' +
  121.       '<p>' + questions[j].a + '</p>' +
  122.       '</div>');
  123.             }
  124.       $('.reset').click(startOver);
  125.         }
  126.       } else {
  127.         return false;
  128.       }
  129.     });
  130.   }
  131. });