{"id":20092,"date":"2024-04-15T05:48:18","date_gmt":"2024-04-15T05:48:18","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20092"},"modified":"2024-04-15T05:48:18","modified_gmt":"2024-04-15T05:48:18","slug":"consider-the-systems-each-consisting-of-m-linear-equations-in-n-variables-i-if-m-n-then-all-such-systems-have-a-solution-ii-if-m-n-then-none-of-these-systems-has-a-solution-iii-if-m-n","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/consider-the-systems-each-consisting-of-m-linear-equations-in-n-variables-i-if-m-n-then-all-such-systems-have-a-solution-ii-if-m-n-then-none-of-these-systems-has-a-solution-iii-if-m-n\/","title":{"rendered":"Consider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution. II. If m > n, then none of these systems has a solution. III. If m = n, then there exists a system which has a solution. Which one of the following is CORRECT? A. I, II and III are true B. Only II and III are true C. Only III is true D. None of them is true"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a991aee6a404\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    I, II and III are true                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    Only II and III are true                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    Only III is true                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    None of them is true                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is: <strong>C. Only III is true.<\/strong><\/p>\n<p>I. If m &lt; n, then all such systems have a solution.<\/p>\n<p>This is false. If m &lt; n, then the system of equations is said to be <strong>underdetermined<\/strong>. In this case, there are infinitely many solutions to the system.<\/p>\n<p>II. If m &gt; n, then none of these systems has a solution.<\/p>\n<p>This is true. If m &gt; n, then the system of equations is said to be <strong>overdetermined<\/strong>. In this case, <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n      <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div>       Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> the system of equations is <strong>inconsistent<\/strong> and has no solution.<\/p>\n<p>III. If m = n, then there exists a system which has a solution.<\/p>\n<p>This is true. If m = n, then the system of equations is said to be <strong>exactly determined<\/strong>. In this case, the system of equations is <strong>consistent<\/strong> and has a unique solution.<\/p>\n<p>Therefore, only III is true.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[489],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Consider the systems, each consisting of m linear equations in n variables. I. If m &lt; n, then all such systems have a solution. II. If m &gt; n, then none of these systems has a solution. III. 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