A) \[-3,\,\,-4\]
B) \[2,\,\,3\]
C) \[3,\,\,4\]
D) \[7,\,\,-7\]
Correct Answer: D
Solution :
We know that, the expression \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c\] ... (i) can be expressed as a product of two linear factors, if \[abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\] Now, we have the expression \[2{{x}^{2}}+kxy+3{{y}^{2}}-5y-2\] On comparing it with Eq. (i), we get \[a=2,\,\,h=\frac{k}{2},\,\,b=3,\,\,g=0,\,\,f=-\frac{5}{2},\,\,c=-2\] To express given expression as \[({{a}_{1}}x+{{b}_{1}}y+{{c}_{1}})({{a}_{2}}x+{{b}_{2}}y+{{c}_{2}})\], \[(2)(3)(-2)+0-2\left( -\frac{5}{2} \right)-0+2{{\left( \frac{k}{2} \right)}^{2}}=0\] \[\Rightarrow \] \[-12-\frac{25}{2}+\frac{{{k}^{2}}}{2}=0\] \[\Rightarrow \] \[{{k}^{2}}-49=0\] \[\Rightarrow \] \[k=\pm 7\]
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