A The roots of the given equations are \(\sqrt{3}+\sqrt{2} i \text { and } \sqrt{3}-\sqrt{2}\) \(x=\sqrt{3}+\sqrt{2} i\) \(x-\sqrt{3}=\sqrt{2} i\) On squaring both sides \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=2 \mathrm{i}^2\) \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=-2\) \(\mathrm{x}^2+5=2 \sqrt{3} \mathrm{x}\) Again on squaring both sides \(x^4+25+10 x^2=12 x^2\) \(x^4-2 x^2+25=0\) Also, \(x=\sqrt{3}-\sqrt{2}\) On squaring both sides \(x^2=(\sqrt{3}-\sqrt{2})^2\) \(x^2=3+2-2 \sqrt{6}\) \(x^2-5=-2 \sqrt{6}\) Again on squaring both sides \(\left(x^2-5\right)^2=(-2 \sqrt{6})^2\) \(x^4+25-10 x^2=24\) \(x^4-10 x^2+1=0\) \(\therefore\) The required equation is \(\left(\mathrm{x}^4-2 \mathrm{x}^2+25\right)\left(\mathrm{x}^4-10 \mathrm{x}^2+1\right)=0\)
TS EAMCET-19.07.2022
Complex Numbers and Quadratic Equation
118474
\(p\) is non-zero real number. If the equation whose roots are the squares of the roots of the equation \(\mathbf{x}^3-p x^2+p x-1=0\) is identical with the given equation, then \(p=\)
1 \(\frac{1}{2}\)
2 2
3 3
4 -1
Explanation:
C Let \(\alpha, \beta, \gamma\) are roots of equation \(\mathrm{x}^3-\mathrm{px}^2+\mathrm{px}-1=0\) \(\alpha+\beta+\gamma=\mathrm{p}\) \(\alpha \beta+\beta \gamma+\alpha \gamma=\mathrm{p}\) \(\alpha \beta \gamma=1\) Also, given, \(\alpha^2+\beta^2+\gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2+\beta^2 \gamma^2+\alpha^2 \gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2 \gamma^2=1\) \(\therefore(\alpha+\beta+\gamma)^2=\alpha^2+\beta^2+\gamma^2+2(\alpha \beta+\beta \gamma+\gamma \alpha)\) \(p^2=p+2 p\) \(p^2=3 p\) \(p=3\)
TS EAMCET-10.09.2020
Complex Numbers and Quadratic Equation
118475
The equation whose roots are squares of the roots of \(x^4-2 x^3+6 x-21=0\) is
1 \(x^4-4 x^3-18 x^2-36 x+441=0\)
2 \(x^4+18 x^3-4 x^2+36 x+441=0\)
3 \(x^4-2 x^3+4 x^2+6 x+441=0\)
4 \(x^4+3 x^3-5 x^2+6 x+441=0\)
Explanation:
Exp: (A): Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^4-2 \mathrm{x}^3+6 \mathrm{x}-21=0\) Required equation where roots are squares of the roots of \(f(x)\) is. \(\mathrm{f}(\sqrt{\mathrm{x}})=0\) \((\sqrt{\mathrm{x}})^4-2(\sqrt{\mathrm{x}})^3+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-2 \mathrm{x} \sqrt{\mathrm{x}}+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \mathrm{x} \sqrt{\mathrm{x}}-6 \sqrt{\mathrm{x}}\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \left(x^2-21\right)^2=4 x(x-3)^2\) \(\Rightarrow x^4-42 x^2+441=4 x\left(x^2-6 x+9\right)\) \(\Rightarrow x^4-42 x^2+441=4 x^3-24 x^2+36 x\) \(\Rightarrow x^4-4 x^3-18 x^2-36 x+441=0\)
TS EAMCET-04.08.2021
Complex Numbers and Quadratic Equation
118476
If \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) are the roots of the quadratic equation \(x^2+7 x+3=0\), then the equation having roots \(\alpha\) and \(\beta\) is
1 \(3 x^2-x-3=0\)
2 \(11 x^2+13 x+3=0\)
3 \(13 x^2+11 x+13=0\)
4 \(11 x^2+3 x+13=0\)
Explanation:
B Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^2+7 \mathrm{x}+3=0\) Roots are \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) Equation having roots \(\alpha\) and \(\beta\) \(\text { Is, } \quad f\left(\frac{x}{x+1}\right)=0\) \(\left(\frac{x}{x+1}\right)^2+7\left(\frac{x}{x+1}\right)+3=0\) \(x^2+7 x(x+1)+3(x+1)^2=0\) \(x^2+7 x^2+7 x+3\left(x^2+1+2 x\right)=0\) \(x^2+7 x^2+7 x+3 x^2+3+6 x=0\) \(11 x^2+13 x+3=0\)
A The roots of the given equations are \(\sqrt{3}+\sqrt{2} i \text { and } \sqrt{3}-\sqrt{2}\) \(x=\sqrt{3}+\sqrt{2} i\) \(x-\sqrt{3}=\sqrt{2} i\) On squaring both sides \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=2 \mathrm{i}^2\) \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=-2\) \(\mathrm{x}^2+5=2 \sqrt{3} \mathrm{x}\) Again on squaring both sides \(x^4+25+10 x^2=12 x^2\) \(x^4-2 x^2+25=0\) Also, \(x=\sqrt{3}-\sqrt{2}\) On squaring both sides \(x^2=(\sqrt{3}-\sqrt{2})^2\) \(x^2=3+2-2 \sqrt{6}\) \(x^2-5=-2 \sqrt{6}\) Again on squaring both sides \(\left(x^2-5\right)^2=(-2 \sqrt{6})^2\) \(x^4+25-10 x^2=24\) \(x^4-10 x^2+1=0\) \(\therefore\) The required equation is \(\left(\mathrm{x}^4-2 \mathrm{x}^2+25\right)\left(\mathrm{x}^4-10 \mathrm{x}^2+1\right)=0\)
TS EAMCET-19.07.2022
Complex Numbers and Quadratic Equation
118474
\(p\) is non-zero real number. If the equation whose roots are the squares of the roots of the equation \(\mathbf{x}^3-p x^2+p x-1=0\) is identical with the given equation, then \(p=\)
1 \(\frac{1}{2}\)
2 2
3 3
4 -1
Explanation:
C Let \(\alpha, \beta, \gamma\) are roots of equation \(\mathrm{x}^3-\mathrm{px}^2+\mathrm{px}-1=0\) \(\alpha+\beta+\gamma=\mathrm{p}\) \(\alpha \beta+\beta \gamma+\alpha \gamma=\mathrm{p}\) \(\alpha \beta \gamma=1\) Also, given, \(\alpha^2+\beta^2+\gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2+\beta^2 \gamma^2+\alpha^2 \gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2 \gamma^2=1\) \(\therefore(\alpha+\beta+\gamma)^2=\alpha^2+\beta^2+\gamma^2+2(\alpha \beta+\beta \gamma+\gamma \alpha)\) \(p^2=p+2 p\) \(p^2=3 p\) \(p=3\)
TS EAMCET-10.09.2020
Complex Numbers and Quadratic Equation
118475
The equation whose roots are squares of the roots of \(x^4-2 x^3+6 x-21=0\) is
1 \(x^4-4 x^3-18 x^2-36 x+441=0\)
2 \(x^4+18 x^3-4 x^2+36 x+441=0\)
3 \(x^4-2 x^3+4 x^2+6 x+441=0\)
4 \(x^4+3 x^3-5 x^2+6 x+441=0\)
Explanation:
Exp: (A): Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^4-2 \mathrm{x}^3+6 \mathrm{x}-21=0\) Required equation where roots are squares of the roots of \(f(x)\) is. \(\mathrm{f}(\sqrt{\mathrm{x}})=0\) \((\sqrt{\mathrm{x}})^4-2(\sqrt{\mathrm{x}})^3+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-2 \mathrm{x} \sqrt{\mathrm{x}}+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \mathrm{x} \sqrt{\mathrm{x}}-6 \sqrt{\mathrm{x}}\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \left(x^2-21\right)^2=4 x(x-3)^2\) \(\Rightarrow x^4-42 x^2+441=4 x\left(x^2-6 x+9\right)\) \(\Rightarrow x^4-42 x^2+441=4 x^3-24 x^2+36 x\) \(\Rightarrow x^4-4 x^3-18 x^2-36 x+441=0\)
TS EAMCET-04.08.2021
Complex Numbers and Quadratic Equation
118476
If \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) are the roots of the quadratic equation \(x^2+7 x+3=0\), then the equation having roots \(\alpha\) and \(\beta\) is
1 \(3 x^2-x-3=0\)
2 \(11 x^2+13 x+3=0\)
3 \(13 x^2+11 x+13=0\)
4 \(11 x^2+3 x+13=0\)
Explanation:
B Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^2+7 \mathrm{x}+3=0\) Roots are \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) Equation having roots \(\alpha\) and \(\beta\) \(\text { Is, } \quad f\left(\frac{x}{x+1}\right)=0\) \(\left(\frac{x}{x+1}\right)^2+7\left(\frac{x}{x+1}\right)+3=0\) \(x^2+7 x(x+1)+3(x+1)^2=0\) \(x^2+7 x^2+7 x+3\left(x^2+1+2 x\right)=0\) \(x^2+7 x^2+7 x+3 x^2+3+6 x=0\) \(11 x^2+13 x+3=0\)
A The roots of the given equations are \(\sqrt{3}+\sqrt{2} i \text { and } \sqrt{3}-\sqrt{2}\) \(x=\sqrt{3}+\sqrt{2} i\) \(x-\sqrt{3}=\sqrt{2} i\) On squaring both sides \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=2 \mathrm{i}^2\) \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=-2\) \(\mathrm{x}^2+5=2 \sqrt{3} \mathrm{x}\) Again on squaring both sides \(x^4+25+10 x^2=12 x^2\) \(x^4-2 x^2+25=0\) Also, \(x=\sqrt{3}-\sqrt{2}\) On squaring both sides \(x^2=(\sqrt{3}-\sqrt{2})^2\) \(x^2=3+2-2 \sqrt{6}\) \(x^2-5=-2 \sqrt{6}\) Again on squaring both sides \(\left(x^2-5\right)^2=(-2 \sqrt{6})^2\) \(x^4+25-10 x^2=24\) \(x^4-10 x^2+1=0\) \(\therefore\) The required equation is \(\left(\mathrm{x}^4-2 \mathrm{x}^2+25\right)\left(\mathrm{x}^4-10 \mathrm{x}^2+1\right)=0\)
TS EAMCET-19.07.2022
Complex Numbers and Quadratic Equation
118474
\(p\) is non-zero real number. If the equation whose roots are the squares of the roots of the equation \(\mathbf{x}^3-p x^2+p x-1=0\) is identical with the given equation, then \(p=\)
1 \(\frac{1}{2}\)
2 2
3 3
4 -1
Explanation:
C Let \(\alpha, \beta, \gamma\) are roots of equation \(\mathrm{x}^3-\mathrm{px}^2+\mathrm{px}-1=0\) \(\alpha+\beta+\gamma=\mathrm{p}\) \(\alpha \beta+\beta \gamma+\alpha \gamma=\mathrm{p}\) \(\alpha \beta \gamma=1\) Also, given, \(\alpha^2+\beta^2+\gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2+\beta^2 \gamma^2+\alpha^2 \gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2 \gamma^2=1\) \(\therefore(\alpha+\beta+\gamma)^2=\alpha^2+\beta^2+\gamma^2+2(\alpha \beta+\beta \gamma+\gamma \alpha)\) \(p^2=p+2 p\) \(p^2=3 p\) \(p=3\)
TS EAMCET-10.09.2020
Complex Numbers and Quadratic Equation
118475
The equation whose roots are squares of the roots of \(x^4-2 x^3+6 x-21=0\) is
1 \(x^4-4 x^3-18 x^2-36 x+441=0\)
2 \(x^4+18 x^3-4 x^2+36 x+441=0\)
3 \(x^4-2 x^3+4 x^2+6 x+441=0\)
4 \(x^4+3 x^3-5 x^2+6 x+441=0\)
Explanation:
Exp: (A): Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^4-2 \mathrm{x}^3+6 \mathrm{x}-21=0\) Required equation where roots are squares of the roots of \(f(x)\) is. \(\mathrm{f}(\sqrt{\mathrm{x}})=0\) \((\sqrt{\mathrm{x}})^4-2(\sqrt{\mathrm{x}})^3+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-2 \mathrm{x} \sqrt{\mathrm{x}}+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \mathrm{x} \sqrt{\mathrm{x}}-6 \sqrt{\mathrm{x}}\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \left(x^2-21\right)^2=4 x(x-3)^2\) \(\Rightarrow x^4-42 x^2+441=4 x\left(x^2-6 x+9\right)\) \(\Rightarrow x^4-42 x^2+441=4 x^3-24 x^2+36 x\) \(\Rightarrow x^4-4 x^3-18 x^2-36 x+441=0\)
TS EAMCET-04.08.2021
Complex Numbers and Quadratic Equation
118476
If \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) are the roots of the quadratic equation \(x^2+7 x+3=0\), then the equation having roots \(\alpha\) and \(\beta\) is
1 \(3 x^2-x-3=0\)
2 \(11 x^2+13 x+3=0\)
3 \(13 x^2+11 x+13=0\)
4 \(11 x^2+3 x+13=0\)
Explanation:
B Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^2+7 \mathrm{x}+3=0\) Roots are \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) Equation having roots \(\alpha\) and \(\beta\) \(\text { Is, } \quad f\left(\frac{x}{x+1}\right)=0\) \(\left(\frac{x}{x+1}\right)^2+7\left(\frac{x}{x+1}\right)+3=0\) \(x^2+7 x(x+1)+3(x+1)^2=0\) \(x^2+7 x^2+7 x+3\left(x^2+1+2 x\right)=0\) \(x^2+7 x^2+7 x+3 x^2+3+6 x=0\) \(11 x^2+13 x+3=0\)
A The roots of the given equations are \(\sqrt{3}+\sqrt{2} i \text { and } \sqrt{3}-\sqrt{2}\) \(x=\sqrt{3}+\sqrt{2} i\) \(x-\sqrt{3}=\sqrt{2} i\) On squaring both sides \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=2 \mathrm{i}^2\) \(\mathrm{x}^2+3-2 \sqrt{3} \mathrm{x}=-2\) \(\mathrm{x}^2+5=2 \sqrt{3} \mathrm{x}\) Again on squaring both sides \(x^4+25+10 x^2=12 x^2\) \(x^4-2 x^2+25=0\) Also, \(x=\sqrt{3}-\sqrt{2}\) On squaring both sides \(x^2=(\sqrt{3}-\sqrt{2})^2\) \(x^2=3+2-2 \sqrt{6}\) \(x^2-5=-2 \sqrt{6}\) Again on squaring both sides \(\left(x^2-5\right)^2=(-2 \sqrt{6})^2\) \(x^4+25-10 x^2=24\) \(x^4-10 x^2+1=0\) \(\therefore\) The required equation is \(\left(\mathrm{x}^4-2 \mathrm{x}^2+25\right)\left(\mathrm{x}^4-10 \mathrm{x}^2+1\right)=0\)
TS EAMCET-19.07.2022
Complex Numbers and Quadratic Equation
118474
\(p\) is non-zero real number. If the equation whose roots are the squares of the roots of the equation \(\mathbf{x}^3-p x^2+p x-1=0\) is identical with the given equation, then \(p=\)
1 \(\frac{1}{2}\)
2 2
3 3
4 -1
Explanation:
C Let \(\alpha, \beta, \gamma\) are roots of equation \(\mathrm{x}^3-\mathrm{px}^2+\mathrm{px}-1=0\) \(\alpha+\beta+\gamma=\mathrm{p}\) \(\alpha \beta+\beta \gamma+\alpha \gamma=\mathrm{p}\) \(\alpha \beta \gamma=1\) Also, given, \(\alpha^2+\beta^2+\gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2+\beta^2 \gamma^2+\alpha^2 \gamma^2=\mathrm{p}\) \(\alpha^2 \beta^2 \gamma^2=1\) \(\therefore(\alpha+\beta+\gamma)^2=\alpha^2+\beta^2+\gamma^2+2(\alpha \beta+\beta \gamma+\gamma \alpha)\) \(p^2=p+2 p\) \(p^2=3 p\) \(p=3\)
TS EAMCET-10.09.2020
Complex Numbers and Quadratic Equation
118475
The equation whose roots are squares of the roots of \(x^4-2 x^3+6 x-21=0\) is
1 \(x^4-4 x^3-18 x^2-36 x+441=0\)
2 \(x^4+18 x^3-4 x^2+36 x+441=0\)
3 \(x^4-2 x^3+4 x^2+6 x+441=0\)
4 \(x^4+3 x^3-5 x^2+6 x+441=0\)
Explanation:
Exp: (A): Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^4-2 \mathrm{x}^3+6 \mathrm{x}-21=0\) Required equation where roots are squares of the roots of \(f(x)\) is. \(\mathrm{f}(\sqrt{\mathrm{x}})=0\) \((\sqrt{\mathrm{x}})^4-2(\sqrt{\mathrm{x}})^3+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-2 \mathrm{x} \sqrt{\mathrm{x}}+6 \sqrt{\mathrm{x}}-21=0\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \mathrm{x} \sqrt{\mathrm{x}}-6 \sqrt{\mathrm{x}}\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \quad \mathrm{x}^2-21=2 \sqrt{\mathrm{x}}(\mathrm{x}-3)\) \(\Rightarrow \left(x^2-21\right)^2=4 x(x-3)^2\) \(\Rightarrow x^4-42 x^2+441=4 x\left(x^2-6 x+9\right)\) \(\Rightarrow x^4-42 x^2+441=4 x^3-24 x^2+36 x\) \(\Rightarrow x^4-4 x^3-18 x^2-36 x+441=0\)
TS EAMCET-04.08.2021
Complex Numbers and Quadratic Equation
118476
If \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) are the roots of the quadratic equation \(x^2+7 x+3=0\), then the equation having roots \(\alpha\) and \(\beta\) is
1 \(3 x^2-x-3=0\)
2 \(11 x^2+13 x+3=0\)
3 \(13 x^2+11 x+13=0\)
4 \(11 x^2+3 x+13=0\)
Explanation:
B Let, \(\mathrm{f}(\mathrm{x})=\mathrm{x}^2+7 \mathrm{x}+3=0\) Roots are \(\frac{\alpha}{\alpha+1}\) and \(\frac{\beta}{\beta+1}\) Equation having roots \(\alpha\) and \(\beta\) \(\text { Is, } \quad f\left(\frac{x}{x+1}\right)=0\) \(\left(\frac{x}{x+1}\right)^2+7\left(\frac{x}{x+1}\right)+3=0\) \(x^2+7 x(x+1)+3(x+1)^2=0\) \(x^2+7 x^2+7 x+3\left(x^2+1+2 x\right)=0\) \(x^2+7 x^2+7 x+3 x^2+3+6 x=0\) \(11 x^2+13 x+3=0\)
