NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

If $[x]$ represents the greatest integer not exceeding $x$ , then $\int_{0}^{9} [x] dx$ is:

Choose the correct answer:

Explanation

Solution

Concept:

Definite Integral Property: If $\int f(x) dx = g(x) + C$ , then $\int_{a}^{b} f(x) dx = [g(x)]_{a}^{b} = g(b) - g(a)$ .
Additive Property: $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$ .
Sum of first $n$ natural numbers: $\sum n = \frac{n(n+1)}{2}$ .

Calculation:

The greatest integer function $[x]$ is a step function that takes constant integer values over specific intervals:

$[x] = 0$ for $x \in [0, 1)$
$[x] = 1$ for $x \in [1, 2)$
$[x] = 2$ for $x \in [2, 3)$
And so on, up to $[x] = 8$ for $x \in [8, 9)$ .

Therefore, we can split the integral into unit intervals:

\int_{0}^{9} [x] dx = \int_{0}^{1} 0 dx + \int_{1}^{2} 1 dx + \dots + \int_{8}^{9} 8 dx

Evaluating each part:

\Rightarrow 0[x]_{0}^{1} + 1[x]_{1}^{2} + \dots + 8[x]_{8}^{9}

\Rightarrow 0[1 - 0] + 1[2 - 1] + \dots + 8[9 - 8]

\Rightarrow 0 + 1 + 2 + \dots + 8

Using the sum formula $\frac{n(n+1)}{2}$ where $n = 8$ :

\Rightarrow \frac{8(8+1)}{2}

\Rightarrow \frac{8 \times 9}{2} = 36

Correct Option: 2. (36)

Choose the correct answer:

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