{"id":20015,"date":"2024-04-15T05:47:14","date_gmt":"2024-04-15T05:47:14","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20015"},"modified":"2024-04-15T05:47:14","modified_gmt":"2024-04-15T05:47:14","slug":"consider-the-following-system-of-linear-equations-3x-2ky-2-kx-6y-2-here-x-and-y-are-the-unknown-and-k-is-a-real-constant-the-value-of-k-for-which-there-are-infinite-number-of-solutions-is","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/consider-the-following-system-of-linear-equations-3x-2ky-2-kx-6y-2-here-x-and-y-are-the-unknown-and-k-is-a-real-constant-the-value-of-k-for-which-there-are-infinite-number-of-solutions-is\/","title":{"rendered":"Consider the following system of linear equations: 3x + 2ky = -2 kx + 6y = 2 Here, x and y are the unknown and k is a real constant. The value of k for which there are infinite number of solutions is A. 3 B. 1 C. -3 D. -6"},"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_6a9a017e307a5\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    3                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    1                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    -3                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    -6                <\/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 $\\boxed{-3}$.<\/p>\n<p>To find the value of $k$ for which there are infinite number of solutions, we can try to solve the system of equations for $x$ and $y$. However, we can quickly see that this is not possible, since the two equations are not independent. In other words, there is no value of $x$ and $y$ that will satisfy both equations.<\/p>\n<p>This means that the system of equations has no solution, or infinitely many solutions. Intuitively, this makes sense, since the two equations are essentially saying the same thing: $3x+2ky=-2$ and $kx+6y=2$. So, no matter what value we choose for $x$, we can always find a value for $y$ that will satisfy both equations.<\/p>\n<p>More formally, we can say that the system of equations has infinitely many solutions if the determinant of the coefficient matrix is equal to zero. In this case, the coefficient matrix is $$\\begin{bmatrix} 3 &amp; 2k \\\\ k &amp; 6 \\end{bmatrix}$$ and the determinant is $$3\\times 6 &#8211; 2k\\times k = 18 &#8211; 2k^2$$<\/p>\n<p>So, the system of equations has infinitely many solutions if $18 &#8211; 2k^2 = 0$. This means that $k^2 = 9$, or $k = \\pm 3$.<\/p>\n<p>However, only the value $k=-3$ makes sense in the context of the original system of <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     <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>        <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> equations. This is because the original system of equations is asking us to find values for $x$ and $y$ that satisfy both equations. If we choose $k=3$, then the first equation becomes $3x+6y=-2$, which has no solution. So, the only value of $k$ for which there are infinite number of solutions is $\\boxed{-3}$.<\/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 following system of linear equations: 3x + 2ky = -2 kx + 6y = 2 Here, x and y are the unknown and k is a real constant. 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