Linear and Quadratic Equations

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Modulus of a number cannot be negative. A simple idea that somehow escapes our mind every now and then.

Unique Solutions

    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?
    ii. ;
    iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3 are irrational numbers


  • Correct Answer
    Statement i.

Explanatory Answer

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Detailed Solution

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.


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. and

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.

Correct Answer: Only Statement 'i'.

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More Questions from Linear & Quad. Equations

  1. Integer Values; Modulus
  2. Integer Values; Modulus
  3. Integer Values; Modulus
  4. Quadratic Discriminant
  5. Quadratic Discriminant
  6. Quadratics - Counting
  7. Integer Solutions
  8. Linear Equations - Median
  9. Condition for Unique Soln.
Solving equations well is an integral part to cracking any competitive exam. Get cartloads of practice on these two topics.