NIMCET 2007 — Mathematics PYQ

Step 1: Understand the Absolute Value Property

The expression contains an absolute value sign $\left|\cdot \right|$, which means whatever value we get inside the bars, the final output will be its non-negative (positive) magnitude:

$$\left|-k\right|=k\text{and}\left|k\right|=k$$

Step 2: Substitute $x=-3$ into the Expression

Let's substitute $x=-3$ into the given algebraic fraction:

$$E=\left|\frac{3(-3{)}^3+1}{2(-3{)}^2+2}\right|$$

Step 3: Evaluate the Numerator and Denominator

1. Evaluate the Numerator:

   • Calculate the cube of $-3$:

     $$(-3{)}^3=-3\times -3\times -3=-27$$

   • Multiply by $3$ and add $1$:

     $$3(-27)+1=-81+1=-80$$

2. Evaluate the Denominator:

   • Calculate the square of $-3$:

     $$(-3{)}^2=-3\times -3=9$$

   • Multiply by $2$ and add $2$:

     $$2(9)+2=18+2=20$$

Step 4: Divide the Values and Apply Absolute Value

Substitute the calculated numerator and denominator back into the expression:

$$E=\left|\frac{-80}{20}\right|$$

Simplify the fraction inside the bars:

$$E=\left|-4\right|$$

Apply the absolute value property to get the positive numerical value:

$$E=4$$

Conclusion

The numerical value of the expression for $x=-3$ is $4$.

Correct Option: (a) $4$