{"id":20116,"date":"2024-04-15T05:48:36","date_gmt":"2024-04-15T05:48:36","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20116"},"modified":"2024-04-15T05:48:36","modified_gmt":"2024-04-15T05:48:36","slug":"for-the-set-of-equations-x1-2x2-x3-4x4-2-3x1-6x2-3x3-12x4-6-the-following-statement-is-true-a-only-the-trivial-solution-x1-x2-x3-x4-0-exists-b-there-are-no-solution-c-a-uni","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/for-the-set-of-equations-x1-2x2-x3-4x4-2-3x1-6x2-3x3-12x4-6-the-following-statement-is-true-a-only-the-trivial-solution-x1-x2-x3-x4-0-exists-b-there-are-no-solution-c-a-uni\/","title":{"rendered":"For the set of equations x1 + 2&#215;2 + x3 + 4&#215;4 = 2 3&#215;1 + 6&#215;2 + 3&#215;3 + 12&#215;4 = 6 the following statement is true: A. Only the trivial solution x1 = x2 = x3 = x4 = 0 exists B. There are no solution C. A unique non-trivial solution exists D. Multiple non-trivial solutions exist"},"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_6a9716ff0de48\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    Only the trivial solution x1 = x2 = x3 = x4 = 0 exists                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    There are no solution                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    A unique non-trivial solution exists                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    Multiple non-trivial solutions exist                <\/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: A. Only the trivial solution x1 = x2 = x3 = x4 = 0 exists.<\/p>\n<p>To solve this system of equations, we can use Gaussian elimination. First, we can eliminate $x_1$ by adding $-3$ times the first equation to the second equation:<\/p>\n<p>$$-3x_1 &#8211; 6x_2 &#8211; 3x_3 &#8211; 12x_4 = -6$$<\/p>\n<p>This gives us the following system of equations:<\/p>\n<p>$$x_1 + 2x_2 + x_3 + 4x_4 = 2$$<br \/>\n$$0x_2 + 3x_2 + 0x_3 + 9x_4 = 0$$<\/p>\n<p>We can then eliminate $x_2$ by adding $-\\frac{3}{3}x_2$ to the second equation:<\/p>\n<p>$$0x_2 + 0x_2 + 0x_3 + 6x_4 = 0$$<\/p>\n<p>This gives us the following system of equations:<\/p>\n<p>$$x_1 + 2x_2 + x_3 + <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> 4x_4 = 2$$<br \/>\n$$0x_2 + 0x_2 + 0x_3 + 0x_4 = 0$$<\/p>\n<p>Since the second equation is <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> now a trivial equation, we know that $x_2 = x_3 = x_4 = 0$. Substituting this into the first equation, we get $x_1 = 2$. Therefore, the only solution to the system of equations is $x_1 = x_2 = x_3 = x_4 = 0$. This is the trivial solution.<\/p>\n<p>The other options are incorrect because:<\/p>\n<ul>\n<li>Option B is incorrect because there is a solution.<\/li>\n<li>Option C is incorrect because there is only one solution.<\/li>\n<li>Option D is incorrect because there is not more than one solution.<\/li>\n<\/ul>\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>For the set of equations x1 + 2x2 + x3 + 4x4 = 2 3x1 + 6x2 + 3x3 + 12x4 = 6 the following statement is true: A. Only the trivial solution x1 = x2 = x3 = x4 = 0 exists B. There are no solution C. A unique non-trivial solution exists D. Multiple non-trivial solutions exist<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/for-the-set-of-equations-x1-2x2-x3-4x4-2-3x1-6x2-3x3-12x4-6-the-following-statement-is-true-a-only-the-trivial-solution-x1-x2-x3-x4-0-exists-b-there-are-no-solution-c-a-uni\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"For the set of equations x1 + 2x2 + x3 + 4x4 = 2 3x1 + 6x2 + 3x3 + 12x4 = 6 the following statement is true: A. Only the trivial solution x1 = x2 = x3 = x4 = 0 exists B. There are no solution C. A unique non-trivial solution exists D. 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Only the trivial solution x1 = x2 = x3 = x4 = 0 exists B. There are no solution C. A unique non-trivial solution exists D. 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