For some a, b, let f(x)=a+xsinxaa11+xsinx1bbb+xsinx, x=0, limx→0f(x)=λ+μa+νb. Then (λ+μ+ν)2 is equal to:
Get the complete, step-by-step math solution for: "For some a, b, let f(x) = {ccc} a + (sin x)/(x) & 1 & b \\ a & 1 + (sin x)/(x) & b \\ a & 1 & b + (sin x)/(x) , x ≠ 0, _{x 0} f(x) = + a + b. Then ( +...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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 (R2→R2−R1) and the first row from the third row (R3→R3−R1) will introduce zeros, making the determinant easier to calculate. Let k=xsinx.
Step 2: Perform row operations
After applying the row operations R2→R2−R1 and R3→R3−R1, 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 k2 from the terms.
Step 5: Evaluate the limit as x→0
We know that limx→0xsinx=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 λ+μa+νb, we can identify the coefficients: λ=2, μ=1, and ν=1.
Step 7: Calculate (λ+μ+ν)2
Finally, substitute the values of λ, μ, and ν into the expression (λ+μ+ν)2 to find the final answer.