NIMCET 2010 — Mathematics PYQ

A system of homogeneous equations $AX=0$ has a non-zero (non-trivial) solution if and only if the determinant of the coefficient matrix is equal to zero: $|A|=0$.

Step 2: Set up the determinant.

Writing the coefficients of $x,y,$ and $z$ from the three equations into a determinant:

Step 3: Apply Column Operations to simplify.

To make the determinant easier to calculate, we perform:

$C_2\to C_2-C_1$ and $C_3\to C_2$ (using the original $C_2$) or more simply $C_3\to C_3-C_2$ and $C_2\to C_2-C_1$.

Let's use:

1. $C_2\to C_2-C_1$

2. $C_3\to C_3-C_2$

The determinant becomes:

Step 4: Expand the determinant.

Expanding along the third row ($R_3$), which has two zeros:

Using the formula $x^3-y^3=(x-y)(x^2+xy+y^2)$:

• $(a+1{)}^3-a^3=1(a^2+2a+1+a^2+a+a^2)=3a^2+3a+1$

• $(a+2{)}^3-(a+1{)}^3=1((a+2{)}^2+(a+2)(a+1)+(a+1{)}^2)=3a^2+9a+7$

Now, solve the $2\times 2$ determinant:

Step 5: Conclusion.

The value of $a$ that satisfies the condition for a non-zero solution is $-1$.

Correct Option: (c) $-1$