Test Index

UPSC NDA II 2025 Mathematics Paper

Question Numbers: 83-84

For the following two (02) items:

Consider the function

f(x)=\begin{cases} 4(5^{x}), & x<0 \\ 8k+x, & x\ge 0 \end{cases}

Question : 83 of 120

Marks: +1, -0

If the function is continuous, then what is the value of k?

Solution:

Concept:

For a function to be continuous at a point, its left-hand limit must equal its right-hand limit and the function value at that point.

Explanation:

The piecewise function is

f(x)=4(5^{x})

for

x<0

and

f(x)=8k+x

for

x\ge 0

x=0

, the left-hand limit is

\lim_{x\to0^{-}}4(5^{x})=4(5^{0})=4

The right-hand limit is

\lim_{x\to0^{+}}(8k+x)=8k+0=8k

For continuity at

x=0

, these limits must be equal:

8k=4

Solving gives

k=\frac{1}{2}=0.5

Answer:

k=0.5

, which is option A.

Go to Question:

Prev Question

Next Question