If y ( x )= begin array ccc sin x & cos x & sin x +cos x +1\\ 27 & 28 & 27\\ 1 & 1 & 1end array , x in mathbb R , then frac d 2 y dx 2 + y is equal to:

Mathematics · Methods of Differentiation

JEE Main 2025 — 3 April, Morning Shift — Question 30

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

Option A:
-1
Correct
Option B:
28
Option C:
27
Option D:
1

Answer: A

Step-by-step solution

$\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}-\mathrm{C}_{1}$

$y(x)=\left|\begin{array}{ccc}\sin x & \cos x & 1+\cos x\\ 27 & 28 & 0\\ 1 & 1 & 0\end{array}\right|$

$y(x)=-(1+\cos x)$

$\frac{d y}{d x}=\sin x$

$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\cos \mathrm{x}$

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

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 3 April, Morning Shift
Subject: Mathematics
Chapter: Methods of Differentiation
Topic: Differentiation of Determinants