If y(x)= | sin x cos x sin x+cos x+1 27 28 27 1 1 1 |, x ∈ R , then (d^2 y)/(d x^2)+y is equal to

Correct answer: (A) -1. C_3 arrow C_3-C_1 y(x)= | sin x cos x 1+cos x 27 28 0 1 1 0 | y(x)=-(1+cos x) (d y)/(d x)=sin x (d^2 y)/(d x^2)=cos x (d^2 y)/(d x^2)+y=-1

Q32 / 75 · JEE Main 2025 (3 Apr Shift 1) · Mathematics

If $\quad y(x)=\left|\begin{array}{ccc}\sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1\end{array}\right|, x \in \mathbb{R}$, then $\frac{d^2 y}{d x^2}+y$ is equal to

Correct answer
A
$-1$
B
$28$
C
$27$
D
$1$

Solution

$\begin{aligned} & C_3 \rightarrow C_3-C_1 \\ & y(x)=\left|\begin{array}{ccc}\sin x & \cos x & 1+\cos x \\ 27 & 28 & 0 \\ 1 & 1 & 0\end{array}\right| \\ & y(x)=-(1+\cos x) \\ & \frac{d y}{d x}=\sin x \\ & \frac{d^2 y}{d x^2}=\cos x \\ & \frac{d^2 y}{d x^2}+y=-1\end{aligned}$

Hence, the correct answer is option: 'A'.

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