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UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 89-90

Consider the following for the next two (02) items that follow

Let f(x) =

\begin{cases} x+6,x\le 1 \\ px+q,1 < x<2 \\ 5x,x\ge 2 \end{cases}

and f(x) is continuous

Question : 89 of 120

Marks: +1, -0

What is the value of p ?

Validate

Solution:

Concept:

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

Explanation:

x = 1

, the left piece

x+6

and the middle piece

px+q

must be equal.

1 + 6 = p(1) + q

gives

p + q = 7

. (Equation I)

x = 2

, the middle piece

px+q

and the right piece

5x

must be equal.

2p + q = 5(2) = 10

. (Equation II)

Solve Eq. I and II: Subtract I from II:

(2p+q) - (p+q) = 10 - 7

gives

p = 3

Answer:

p = 3

, which corresponds to option B.

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