NIMCET 2025 — Mathematics PYQ

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Question

NIMCET 2025 — Mathematics PYQ

NIMCET | Mathematics | 2025

The maximum value of $\sin(x) + \sin(x+1)$ is $k \cos \frac{1}{2}$ . Then the value of $k$ is:

Choose the correct answer:

Explanation

Step 1: Apply the formula to the given expression

Let $f(x) = \sin(x) + \sin(x+1)$ . Here, $A = x+1$ and $B = x$ .

f(x) = 2 \sin \left( \frac{(x+1) + x}{2} \right) \cos \left( \frac{(x+1) - x}{2} \right)

f(x) = 2 \sin \left( \frac{2x+1}{2} \right) \cos \left( \frac{1}{2} \right)

f(x) = 2 \sin \left( x + \frac{1}{2} \right) \cos \left( \frac{1}{2} \right)

Step 2: Determine the maximum value

We know that the maximum value of the sine function, $\sin \theta$ , is $1$ .

Specifically, the maximum value of $\sin \left( x + \frac{1}{2} \right)$ is $1$ .

Therefore, the maximum value of $f(x)$ is:

f(x)_{max} = 2 \cdot (1) \cdot \cos \left( \frac{1}{2} \right)

f(x)_{max} = 2 \cos \frac{1}{2}

Step 3: Compare with the given form

The problem states the maximum value is $k \cos \frac{1}{2}$ .

Comparing $2 \cos \frac{1}{2}$ with $k \cos \frac{1}{2}$ :

k = 2

Final Answer:

The value of $k$ is 2. The correct option is (B).