CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

CAT Questions | Algebra | Linear Equations - Unique Solutions
CAT Questions

The question is about checking for consistency of the given equations. Knowing a few equations, we can find whether the equations have an unique solution or infinite solutions or no solution. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 9: a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.

Which of the following statements if true would imply that the above system of equations does not have a unique solution?
i. $$frac{a_{1}}{a_{2}}\\$ = $\frac{b_{1}}{b_{2}}\\$ = $\frac{c_{1}}{c_{2}}\\$ ≠ $\frac{d_{1}}{d_{2}}\\$ ii. $\frac{ a_{1} }{ a_{2} }\\$= $\frac{ a_{2} }{ a_{3} }\\$ ; $\frac{ b_{1} }{ b_{2} }\\$= $\frac{ b_{2} }{ b_{3} }\\$ iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3 are irrational numbers 1. Statement i 2. Statement ii 3. Statement iii 4. None 40 Hours of Sample classes. Signup to check now! 2IIM : Best Online CAT Coaching. Video Explanation Best CAT Coaching in Chennai CAT Coaching in Chennai - CAT 2020Online Batches Available Now! Explanatory Answer Method of solving this CAT Question from CAT Algebra - Linear Equations: How we would find out whether the equations are independent? If we have three independent equations, we will have a unique solution. In other words, we will not have unique solutions if: The equations are inconsistent or Two equations can be combined to give the third Now, let us move to the statements. i.$\frac{ a_{1} }{ a_{2} }\\$ = $\frac{ b_{1} }{ a_{2} }\\$ = $\frac{ c_{1} }{ c_{2} }\\$ ≠ $\frac{ d_{1} }{ d_{2} }\\$ This tells us that the first two equations cannot hold good at the same time. Think about this: x + y + z = 3; 2x + 2y + 2z = 5. Either the first or the second can hold good. Both cannot hold good at the same time. So, this will definitely not have any solution. ii.$\frac{ a_{1} }{ a_{2} }\\$= $\frac{ a_{2} }{ a_{3} }\\$ and $\frac{ b_{1} }{ b_{2} }\\$ = $\frac{ b_{2} }{ b_{3} }\\$ a1, a2, a3 are in GP, b1, b2, b3. This does not prevent the system from having a unique solution. For instance, if we have x + 9y + 5z = 11 2x + 3y – 6z = 17 4x + y – 3z = 15 This could very well have a unique solution. iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3 are irrational numbers This gives us practically nothing. This system of equations can definitely have a unique solution. So, only Statement i tells us that a unique solution is impossible. The question is "Which of the following statements if true would imply that the above system of equations does not have a unique solution?" Hence the answer is "Statement i: $\frac{a_{1}}{a_{2}}\\$ = $\frac{b_{1}}{b_{2}}\\$ = $\frac{c_{1}}{c_{2}}\\$ ≠ $\frac{d_{1}}{d_{2}}\\$" Statement i is the correct answer. Signup and Sample 9 full classes for Free Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! Already have an Account? CAT Coaching in ChennaiCAT 2020Enroll at 35,000/- Online Classroom Batches Starting Now! Best CAT Coaching in ChennaiRegister Online, get Rs 7000/- off Attend a Demo Class Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 How to reach 2IIM? Phone:$91) 44 4505 8484
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