If , then is equal to [2025]

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Solution

Q.
If $y (x)$ $=$ $| \begin{matrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{matrix} |$ $, x \in ℝ$ , then $\frac{d^{2} y}{d x^{2}} + y$ is equal to [2025]

1 –1

2 27

3 1

4 28

Ans.
(1)

We have,

$y (x)$ $=$ $| \begin{matrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{matrix} |$

$\Rightarrow$ y(x) = sin x (28 – 27) – cos x (27 – 27) + (sin x + cos x + 1)(27 – 28)

y(x) = – cos x – 1

On differentiate w.r.t. x, we get

$\frac{d y}{d x} = \sin x \Rightarrow \frac{d^{2} y}{d x^{2}} = \cos x$

$∴ \frac{d^{2} y}{d x^{2}} + y = \cos x - 1 - \cos x = - 1$ .

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