If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $f(x) = x^2 - x - 2$, find a polynomial whose zeroes are $2\alpha + 1$ and $2\beta + 1$.
Step-by-Step Solution
Key Concept: $\alpha + \beta = 1, \alpha \beta = -2$. New sum $= 2(\alpha + \beta) + 2 = 4$. New product $= (2\alpha+1)(2\beta+1) = 4\alpha\beta + 2(\alpha+\beta) + 1 = -8 + 2 + 1 = -5$.
Original $\alpha + \beta = 1, \alpha \beta = -2$. [0.5 Mark]
New Sum $= (2\alpha + 1) + (2\beta + 1) = 2(\alpha + \beta) + 2 = 2(1) + 2 = 4$. [0.5 Mark]
New Product $= 4\alpha\beta + 2(\alpha + \beta) + 1 = 4(-2) + 2(1) + 1 = -5$. [0.5 Mark]
Polynomial is $x^2 - 4x - 5$. [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Finding new sum $= 4$: 0.5 Mark
Finding new product $= -5$: 0.5 Mark
Writing polynomial $x^2 - 4x - 5$: 1.0 Mark
Correct Answer: