Question:
If \(\mathrm{f}(x)=2 x^3+\mathrm{m} x^2-13 x+\mathrm{n}\) and 2,3 are the roots of the equation \(\mathrm{f}(x)=0\) then the value of \(4 m+5 n\) is
If \(\mathrm{f}(x)=2 x^3+\mathrm{m} x^2-13 x+\mathrm{n}\) and 2,3 are the roots of the equation \(\mathrm{f}(x)=0\) then the value of \(4 m+5 n\) is
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Substitute x = 2 and x = 3 in f(x) = 0 to get two linear equations in m and n.
Updated On: Nov 13, 2026
- 30
- 100
- 130
- 150
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
Since 2 and 3 are roots, \(f(2)=0\) and \(f(3)=0\).
Step 2: Form equations
\(f(2)=16+4m-26+n=0\Rightarrow4m+n=10\).
\(f(3)=54+9m-39+n=0\Rightarrow9m+n=-15\).
Step 3: Solve
Subtracting: \(5m=-25\Rightarrow m=-5\), then \(n=30\).
Step 4: Compute
\(4m+5n=-20+150=130\).
Final Answer:
The value is 130. \[ \boxed{\text{(C) }130} \]
Since 2 and 3 are roots, \(f(2)=0\) and \(f(3)=0\).
Step 2: Form equations
\(f(2)=16+4m-26+n=0\Rightarrow4m+n=10\).
\(f(3)=54+9m-39+n=0\Rightarrow9m+n=-15\).
Step 3: Solve
Subtracting: \(5m=-25\Rightarrow m=-5\), then \(n=30\).
Step 4: Compute
\(4m+5n=-20+150=130\).
Final Answer:
The value is 130. \[ \boxed{\text{(C) }130} \]
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