For some $a$, $b$, let $f(x) = \left| \begin{array}{ccc} a + \frac{\sin x}{x} & 1 & b \\ a & 1 + \frac{\sin x}{x} & b \\ a & 1 & b + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to:

Step 1: Simplify the determinant using row operations

To simplify the determinant, we can perform row operations. Subtracting the first row from the second row ($R_2 \to R_2 - R_1$) and the first row from the third row ($R_3 \to R_3 - R_1$) will introduce zeros, making the determinant easier to calculate. Let $k = \frac{\sin x}{x}$.

Step 2: Perform row operations

After applying the row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$, the determinant becomes simpler. Now we can expand the determinant along the first row or first column.

Step 3: Expand the determinant

Expanding the determinant along the first row, we multiply each element by its corresponding cofactor. This gives us a polynomial in terms of $a$, $b$, and $k$.

Step 4: Simplify the expression for $f(x)$

Combine the terms to simplify the expression for $f(x)$. We can factor out $k^2$ from the terms.

Step 5: Evaluate the limit as $x \to 0$

We know that $\lim_{x \to 0} \frac{\sin x}{x} = 1$. Substitute this limit into the simplified expression for $f(x)$.

Step 6: Calculate the limit and identify coefficients

Comparing this result with the given form $\lambda + \mu a + \nu b$, we can identify the coefficients: $\lambda = 2$, $\mu = 1$, and $\nu = 1$.

Step 7: Calculate $(\lambda + \mu + \nu)^2$

Finally, substitute the values of $\lambda$, $\mu$, and $\nu$ into the expression $(\lambda + \mu + \nu)^2$ to find the final answer.

More from this chapter

• Evaluate the determinant 1 & 0 & 4 3 & 5 & -1 0 & 1 & 2 (i) with the help of second row. (ii) with the help of third column.
• If A = [[1, 2], [3, 4]] find det(A) and adj(A).
• The value of 1 & 2 & 3 4 & 5 & 6 7 & 8 & 9 is:
• Using Cramer's rule, show that the following system of linear equations is consistent and hence solve it: x + y + z = 1, 2x + 3y + 2z = 2, x + y + 2z = 4.
• Let a R and A be a matrix of order 3 3 such that det(A)=-4 and A+I= 1 & a & 1 2 & 1 & 0 a & 1 & 2, where I is the 3 3 identity. If det ((a+1)adj((a-1)A))=2^m3^n
• Let A be a 3 × 3 matrix such that |adj (adj(adj A))| = 81. If S = n Z: |adj(adj A)|^(((n-1)²)/(2)) = |A|³n² - 5n - 4, then Σ_n S | A^(n² + n) | is equal to:
• Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k, 0) is a point such that area of triangle ABD is 3 sq units.
• If the system of linear equations 3x + y + β z = 3, 2x + α y - z = -3, x + 2y + z = 4 has infinitely many solutions, then the value of 22β - 9α is:

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