For some a, b , let f(x)= begin array ccc a+sin x/x & 1 & b \a & 1+sin x/x & b\\ a & 1 & b+sin x/x end array , x neq 0, lim x rightarrow 0 f(x)=λ+μ a+v b . Then (λ+μ+v) 2 is equal to:

Mathematics · Determinants

JEE Main 2025 — 24 January, Evening Shift — Question 13

For some $a, b$ , let

f(x)=\left|\begin{array}{ccc} a+\frac{\sin x}{x} & 1 & b \\a & 1+\frac{\sin x}{x} & b\\ a & 1 & b+\frac{\sin x}{x} \end{array}\right|, \quad x \neq 0,

$\lim _{x \rightarrow 0} f(x)=\lambda+\mu a+v b$ . Then $(\lambda+\mu+v)^{2}$ is equal to:

$\lim _{x \rightarrow 0} f(x)=\left|\begin{array}{ccc}a+1 & 1 & b \\a & 1+1 & b \\a & 1 & b+1\end{array}\right|$

$=(a+1)(2(b+1)-b)-1(a(b+1)-ab)+b(-a)$ $=(a+1)(b+2)-a-a b$

$=b+a+2=\lambda+\mu a+v b$

$\lambda=2, \mu=1, v=1 \Rightarrow(\lambda+\mu+v)^{2}=16$

Answer key and solution verified before publishing.

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.