Nature and Number of Roots
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Complex Numbers and Quadratic Equation

118274 The equation \(6^x+8^x=10^x\) has

1 No real root.
2 Infinitely many rational roots.
3 Exactly one real root.
4 Two distinct real root.
WB JEE-2021
Complex Numbers and Quadratic Equation

118275 Roots of the equation \(x^3-3 x^2+3 x-9=0\) are...

1 \(3,1+2 \omega, 1+2 \omega^2\)
2 \(3,-1+2 \omega,-1-2 \omega^2\)
3 3,2 - \(\omega, 2\) - \(\omega^2\)
4 \(1,1+2 \omega, 1+2 \omega^2\)
AP EAMCET-17.09.2020
Complex Numbers and Quadratic Equation

118276 If \(\alpha, \beta\) are the roots of the equation
\(\mathbf{x}^2-\mathbf{2 x}+\mathbf{4}=\mathbf{0} \text {, then } \boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\ldots \ldots .\)
\(\cos \left(\frac{n \pi}{3}\right)\) for any \(n \in N\)

1 \(2^{\mathrm{n}}\)
2 \(2^{\mathrm{n}+1}\)
3 \(2^{\mathrm{n}-1}\)
4 \(2^{\mathrm{n}-2}\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118277 If \(\alpha, \beta\) are the roots of \(x^2+p x+q=0\), then the values of \(\alpha^3+\beta^3\) and \(\alpha^4+\alpha^2 \beta^2+\beta^4\) are respectively .... and .....

1 \(\left(3 p q-p^3\right)\) and \(\left(p^4-3 p^2 q+3 q^2\right)\)
2 \(-p\left(3 q-p^2\right)\) and \(\left(p^2-q\right)\left(p^2+3 q\right)\)
3 \((\mathrm{pq}-4)\) and \(\left(\mathrm{p}^4-\mathrm{q}^4\right)\)
4 \(\left(3 p q-p^3\right)\) and \(\left(p^2-q\right)\left(p^2-3 q\right)\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118278 Let \(\theta\) be an acute angle such that the equation \(x^3+4 x^2 \cos \theta+x \cot \theta=0\) has multiple roots. Then the value of \(\theta\) (in radians) is

1 \(\frac{\pi}{3}\)
2 \(\frac{\pi}{8}\)
3 \(\frac{\pi}{12}\) or \(\frac{5 \pi}{12}\)
4 \(\frac{\pi}{6}\) or \(\frac{5 \pi}{12}\)
AP EAMCET-21.04.2019
NEET DIGITAL LIBRARY
Complex Numbers and Quadratic Equation

118274 The equation \(6^x+8^x=10^x\) has

1 No real root.
2 Infinitely many rational roots.
3 Exactly one real root.
4 Two distinct real root.
WB JEE-2021
Complex Numbers and Quadratic Equation

118275 Roots of the equation \(x^3-3 x^2+3 x-9=0\) are...

1 \(3,1+2 \omega, 1+2 \omega^2\)
2 \(3,-1+2 \omega,-1-2 \omega^2\)
3 3,2 - \(\omega, 2\) - \(\omega^2\)
4 \(1,1+2 \omega, 1+2 \omega^2\)
AP EAMCET-17.09.2020
Complex Numbers and Quadratic Equation

118276 If \(\alpha, \beta\) are the roots of the equation
\(\mathbf{x}^2-\mathbf{2 x}+\mathbf{4}=\mathbf{0} \text {, then } \boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\ldots \ldots .\)
\(\cos \left(\frac{n \pi}{3}\right)\) for any \(n \in N\)

1 \(2^{\mathrm{n}}\)
2 \(2^{\mathrm{n}+1}\)
3 \(2^{\mathrm{n}-1}\)
4 \(2^{\mathrm{n}-2}\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118277 If \(\alpha, \beta\) are the roots of \(x^2+p x+q=0\), then the values of \(\alpha^3+\beta^3\) and \(\alpha^4+\alpha^2 \beta^2+\beta^4\) are respectively .... and .....

1 \(\left(3 p q-p^3\right)\) and \(\left(p^4-3 p^2 q+3 q^2\right)\)
2 \(-p\left(3 q-p^2\right)\) and \(\left(p^2-q\right)\left(p^2+3 q\right)\)
3 \((\mathrm{pq}-4)\) and \(\left(\mathrm{p}^4-\mathrm{q}^4\right)\)
4 \(\left(3 p q-p^3\right)\) and \(\left(p^2-q\right)\left(p^2-3 q\right)\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118278 Let \(\theta\) be an acute angle such that the equation \(x^3+4 x^2 \cos \theta+x \cot \theta=0\) has multiple roots. Then the value of \(\theta\) (in radians) is

1 \(\frac{\pi}{3}\)
2 \(\frac{\pi}{8}\)
3 \(\frac{\pi}{12}\) or \(\frac{5 \pi}{12}\)
4 \(\frac{\pi}{6}\) or \(\frac{5 \pi}{12}\)
AP EAMCET-21.04.2019
Join With Us for More Updates
Complex Numbers and Quadratic Equation

118274 The equation \(6^x+8^x=10^x\) has

1 No real root.
2 Infinitely many rational roots.
3 Exactly one real root.
4 Two distinct real root.
WB JEE-2021
Complex Numbers and Quadratic Equation

118275 Roots of the equation \(x^3-3 x^2+3 x-9=0\) are...

1 \(3,1+2 \omega, 1+2 \omega^2\)
2 \(3,-1+2 \omega,-1-2 \omega^2\)
3 3,2 - \(\omega, 2\) - \(\omega^2\)
4 \(1,1+2 \omega, 1+2 \omega^2\)
AP EAMCET-17.09.2020
Complex Numbers and Quadratic Equation

118276 If \(\alpha, \beta\) are the roots of the equation
\(\mathbf{x}^2-\mathbf{2 x}+\mathbf{4}=\mathbf{0} \text {, then } \boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\ldots \ldots .\)
\(\cos \left(\frac{n \pi}{3}\right)\) for any \(n \in N\)

1 \(2^{\mathrm{n}}\)
2 \(2^{\mathrm{n}+1}\)
3 \(2^{\mathrm{n}-1}\)
4 \(2^{\mathrm{n}-2}\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118277 If \(\alpha, \beta\) are the roots of \(x^2+p x+q=0\), then the values of \(\alpha^3+\beta^3\) and \(\alpha^4+\alpha^2 \beta^2+\beta^4\) are respectively .... and .....

1 \(\left(3 p q-p^3\right)\) and \(\left(p^4-3 p^2 q+3 q^2\right)\)
2 \(-p\left(3 q-p^2\right)\) and \(\left(p^2-q\right)\left(p^2+3 q\right)\)
3 \((\mathrm{pq}-4)\) and \(\left(\mathrm{p}^4-\mathrm{q}^4\right)\)
4 \(\left(3 p q-p^3\right)\) and \(\left(p^2-q\right)\left(p^2-3 q\right)\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118278 Let \(\theta\) be an acute angle such that the equation \(x^3+4 x^2 \cos \theta+x \cot \theta=0\) has multiple roots. Then the value of \(\theta\) (in radians) is

1 \(\frac{\pi}{3}\)
2 \(\frac{\pi}{8}\)
3 \(\frac{\pi}{12}\) or \(\frac{5 \pi}{12}\)
4 \(\frac{\pi}{6}\) or \(\frac{5 \pi}{12}\)
AP EAMCET-21.04.2019
For More Updates join with Us
Complex Numbers and Quadratic Equation

118274 The equation \(6^x+8^x=10^x\) has

1 No real root.
2 Infinitely many rational roots.
3 Exactly one real root.
4 Two distinct real root.
WB JEE-2021
Complex Numbers and Quadratic Equation

118275 Roots of the equation \(x^3-3 x^2+3 x-9=0\) are...

1 \(3,1+2 \omega, 1+2 \omega^2\)
2 \(3,-1+2 \omega,-1-2 \omega^2\)
3 3,2 - \(\omega, 2\) - \(\omega^2\)
4 \(1,1+2 \omega, 1+2 \omega^2\)
AP EAMCET-17.09.2020
Complex Numbers and Quadratic Equation

118276 If \(\alpha, \beta\) are the roots of the equation
\(\mathbf{x}^2-\mathbf{2 x}+\mathbf{4}=\mathbf{0} \text {, then } \boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\ldots \ldots .\)
\(\cos \left(\frac{n \pi}{3}\right)\) for any \(n \in N\)

1 \(2^{\mathrm{n}}\)
2 \(2^{\mathrm{n}+1}\)
3 \(2^{\mathrm{n}-1}\)
4 \(2^{\mathrm{n}-2}\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118277 If \(\alpha, \beta\) are the roots of \(x^2+p x+q=0\), then the values of \(\alpha^3+\beta^3\) and \(\alpha^4+\alpha^2 \beta^2+\beta^4\) are respectively .... and .....

1 \(\left(3 p q-p^3\right)\) and \(\left(p^4-3 p^2 q+3 q^2\right)\)
2 \(-p\left(3 q-p^2\right)\) and \(\left(p^2-q\right)\left(p^2+3 q\right)\)
3 \((\mathrm{pq}-4)\) and \(\left(\mathrm{p}^4-\mathrm{q}^4\right)\)
4 \(\left(3 p q-p^3\right)\) and \(\left(p^2-q\right)\left(p^2-3 q\right)\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118278 Let \(\theta\) be an acute angle such that the equation \(x^3+4 x^2 \cos \theta+x \cot \theta=0\) has multiple roots. Then the value of \(\theta\) (in radians) is

1 \(\frac{\pi}{3}\)
2 \(\frac{\pi}{8}\)
3 \(\frac{\pi}{12}\) or \(\frac{5 \pi}{12}\)
4 \(\frac{\pi}{6}\) or \(\frac{5 \pi}{12}\)
AP EAMCET-21.04.2019
NEET DIGITAL LIBRARY
Complex Numbers and Quadratic Equation

118274 The equation \(6^x+8^x=10^x\) has

1 No real root.
2 Infinitely many rational roots.
3 Exactly one real root.
4 Two distinct real root.
WB JEE-2021
Complex Numbers and Quadratic Equation

118275 Roots of the equation \(x^3-3 x^2+3 x-9=0\) are...

1 \(3,1+2 \omega, 1+2 \omega^2\)
2 \(3,-1+2 \omega,-1-2 \omega^2\)
3 3,2 - \(\omega, 2\) - \(\omega^2\)
4 \(1,1+2 \omega, 1+2 \omega^2\)
AP EAMCET-17.09.2020
Complex Numbers and Quadratic Equation

118276 If \(\alpha, \beta\) are the roots of the equation
\(\mathbf{x}^2-\mathbf{2 x}+\mathbf{4}=\mathbf{0} \text {, then } \boldsymbol{\alpha}^{\mathrm{n}}+\boldsymbol{\beta}^{\mathrm{n}}=\ldots \ldots .\)
\(\cos \left(\frac{n \pi}{3}\right)\) for any \(n \in N\)

1 \(2^{\mathrm{n}}\)
2 \(2^{\mathrm{n}+1}\)
3 \(2^{\mathrm{n}-1}\)
4 \(2^{\mathrm{n}-2}\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118277 If \(\alpha, \beta\) are the roots of \(x^2+p x+q=0\), then the values of \(\alpha^3+\beta^3\) and \(\alpha^4+\alpha^2 \beta^2+\beta^4\) are respectively .... and .....

1 \(\left(3 p q-p^3\right)\) and \(\left(p^4-3 p^2 q+3 q^2\right)\)
2 \(-p\left(3 q-p^2\right)\) and \(\left(p^2-q\right)\left(p^2+3 q\right)\)
3 \((\mathrm{pq}-4)\) and \(\left(\mathrm{p}^4-\mathrm{q}^4\right)\)
4 \(\left(3 p q-p^3\right)\) and \(\left(p^2-q\right)\left(p^2-3 q\right)\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation

118278 Let \(\theta\) be an acute angle such that the equation \(x^3+4 x^2 \cos \theta+x \cot \theta=0\) has multiple roots. Then the value of \(\theta\) (in radians) is

1 \(\frac{\pi}{3}\)
2 \(\frac{\pi}{8}\)
3 \(\frac{\pi}{12}\) or \(\frac{5 \pi}{12}\)
4 \(\frac{\pi}{6}\) or \(\frac{5 \pi}{12}\)
AP EAMCET-21.04.2019