NIMCET 2023 — Mathematics PYQ

1. Standard Formula Used

We use the standard algebraic limit formula:

$$\underset{x\to a}{\lim }\frac{x^n-a^n}{x-a}=n\cdot a^{n-1}$$

2. Evaluate the Left-Hand Side (LHS)

$$\text{LHS}=\underset{x\to 1}{\lim }\frac{x^4-1^4}{x-1}$$

Here, $n=4$ and $a=1$:

$$\text{LHS}=4\cdot (1{)}^{4-1}=4\cdot 1=4$$

3. Evaluate the Right-Hand Side (RHS)

$$\text{RHS}=\underset{x\to k}{\lim }\frac{x^3-k^3}{x^2-k^2}$$

Divide both the numerator and the denominator by $(x-k)$ to format it into the standard form:

$$\text{RHS}=\frac{{\lim }_{x\to k}\frac{x^3-k^3}{x-k}}{{\lim }_{x\to k}\frac{x^2-k^2}{x-k}}$$

Apply the standard formula to both parts:

• For the numerator: $n=3,a=k⟹3k^{3-1}=3k^2$

• For the denominator: $n=2,a=k⟹2k^{2-1}=2k^1$

Simplify the fraction:

$$\text{RHS}=\frac{3k^2}{2k}=\frac{3}{2}k$$

4. Equate LHS and RHS to Find $k$

$$\text{LHS}=\text{RHS}$$

$$4=\frac{3}{2}k$$

Solve for $k$:

$$k=4\cdot \frac{2}{3}$$

$$k=\frac{8}{3}$$