A \( Let a-d, a, a+d \text { are roots of }\) \(x^3-p x^2+q x-r=0\) \(\text { Sum of roots }=-\frac{b}{a}\) \(a-d+a+a+d=-\frac{(-p)}{1}\) \(3 a=p\) \(a=\frac{p}{3}\) \(\therefore a=\frac{p}{3} \text { should be satisfied by given equation }\) \(\text { So, put } x=\frac{p}{3} \text { in eq. (i) }\) \(\left(\frac{p}{3}\right)^3-p\left(\frac{p}{3}\right)^2+q\left(\frac{p}{3}\right)-r=0\) \(\frac{p^3}{27}-\frac{p^3}{9}+\frac{p q}{3}-r=0\) \(p^3-3 p^3+9 p q-27 r=0\) \(-2 p^3+9 p q-27 r=0\) \(2 p^3-9 p q+27 r=0\)So, put \(\mathrm{x}=\frac{\mathrm{p}}{3}\) in eq. (i)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118266
If \(\alpha, \beta, \gamma\) are the roots of the equation \(3 x^3-9 x^2+5 x-7\), then what is the value of \(\alpha+\beta\) \(+\gamma\) ?
1 3
2 -3
3 9
4 -9
Explanation:
A Given, \(\alpha, \beta, \gamma\) are roots of \(3 x^3-9 x^2+5 x-7=0\) Sum of the roots, \(\alpha+\beta+\gamma=-\frac{b}{a}=-\frac{(-9)}{3}\) \(\alpha+\beta+\gamma=3\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118267
The sum of real roots of the equation \(x^4-2 \mathrm{x}^3+\) \(\mathbf{x}-\mathbf{3 8 0}=\mathbf{0}\)
1 -1
2 0
3 1
4 2
Explanation:
C Given \(x^4-2 x^3+x-380=0\) \(\text { Now, by trail error method use find that } x=5 \text { is a }\) \(\text { solution to the equation }\) \(5^4-2 \times 5^3+5-380=0\) \(625-250+5-380=0\) \(380-380=0\) \(\text { Thus, } x=5 \text { is solution }\) \(\text { We also find that } x=-4 \text { is solution }\) \((-4)^2-2 \times(-4)^3-4-380=0\) \(=380-380=0\) \(\text { The real roots are } 5 \text { and }-4 \text {. Sum of real roots are } 5+(-4)\) \(=1\) Now, by trail error method use find that \(\mathrm{x}=5\) is a solution to the equation The real roots are 5 and -4 . Sum of real roots are \(5+(-4)\) \(=1\)
A \( Let a-d, a, a+d \text { are roots of }\) \(x^3-p x^2+q x-r=0\) \(\text { Sum of roots }=-\frac{b}{a}\) \(a-d+a+a+d=-\frac{(-p)}{1}\) \(3 a=p\) \(a=\frac{p}{3}\) \(\therefore a=\frac{p}{3} \text { should be satisfied by given equation }\) \(\text { So, put } x=\frac{p}{3} \text { in eq. (i) }\) \(\left(\frac{p}{3}\right)^3-p\left(\frac{p}{3}\right)^2+q\left(\frac{p}{3}\right)-r=0\) \(\frac{p^3}{27}-\frac{p^3}{9}+\frac{p q}{3}-r=0\) \(p^3-3 p^3+9 p q-27 r=0\) \(-2 p^3+9 p q-27 r=0\) \(2 p^3-9 p q+27 r=0\)So, put \(\mathrm{x}=\frac{\mathrm{p}}{3}\) in eq. (i)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118266
If \(\alpha, \beta, \gamma\) are the roots of the equation \(3 x^3-9 x^2+5 x-7\), then what is the value of \(\alpha+\beta\) \(+\gamma\) ?
1 3
2 -3
3 9
4 -9
Explanation:
A Given, \(\alpha, \beta, \gamma\) are roots of \(3 x^3-9 x^2+5 x-7=0\) Sum of the roots, \(\alpha+\beta+\gamma=-\frac{b}{a}=-\frac{(-9)}{3}\) \(\alpha+\beta+\gamma=3\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118267
The sum of real roots of the equation \(x^4-2 \mathrm{x}^3+\) \(\mathbf{x}-\mathbf{3 8 0}=\mathbf{0}\)
1 -1
2 0
3 1
4 2
Explanation:
C Given \(x^4-2 x^3+x-380=0\) \(\text { Now, by trail error method use find that } x=5 \text { is a }\) \(\text { solution to the equation }\) \(5^4-2 \times 5^3+5-380=0\) \(625-250+5-380=0\) \(380-380=0\) \(\text { Thus, } x=5 \text { is solution }\) \(\text { We also find that } x=-4 \text { is solution }\) \((-4)^2-2 \times(-4)^3-4-380=0\) \(=380-380=0\) \(\text { The real roots are } 5 \text { and }-4 \text {. Sum of real roots are } 5+(-4)\) \(=1\) Now, by trail error method use find that \(\mathrm{x}=5\) is a solution to the equation The real roots are 5 and -4 . Sum of real roots are \(5+(-4)\) \(=1\)
A \( Let a-d, a, a+d \text { are roots of }\) \(x^3-p x^2+q x-r=0\) \(\text { Sum of roots }=-\frac{b}{a}\) \(a-d+a+a+d=-\frac{(-p)}{1}\) \(3 a=p\) \(a=\frac{p}{3}\) \(\therefore a=\frac{p}{3} \text { should be satisfied by given equation }\) \(\text { So, put } x=\frac{p}{3} \text { in eq. (i) }\) \(\left(\frac{p}{3}\right)^3-p\left(\frac{p}{3}\right)^2+q\left(\frac{p}{3}\right)-r=0\) \(\frac{p^3}{27}-\frac{p^3}{9}+\frac{p q}{3}-r=0\) \(p^3-3 p^3+9 p q-27 r=0\) \(-2 p^3+9 p q-27 r=0\) \(2 p^3-9 p q+27 r=0\)So, put \(\mathrm{x}=\frac{\mathrm{p}}{3}\) in eq. (i)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118266
If \(\alpha, \beta, \gamma\) are the roots of the equation \(3 x^3-9 x^2+5 x-7\), then what is the value of \(\alpha+\beta\) \(+\gamma\) ?
1 3
2 -3
3 9
4 -9
Explanation:
A Given, \(\alpha, \beta, \gamma\) are roots of \(3 x^3-9 x^2+5 x-7=0\) Sum of the roots, \(\alpha+\beta+\gamma=-\frac{b}{a}=-\frac{(-9)}{3}\) \(\alpha+\beta+\gamma=3\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118267
The sum of real roots of the equation \(x^4-2 \mathrm{x}^3+\) \(\mathbf{x}-\mathbf{3 8 0}=\mathbf{0}\)
1 -1
2 0
3 1
4 2
Explanation:
C Given \(x^4-2 x^3+x-380=0\) \(\text { Now, by trail error method use find that } x=5 \text { is a }\) \(\text { solution to the equation }\) \(5^4-2 \times 5^3+5-380=0\) \(625-250+5-380=0\) \(380-380=0\) \(\text { Thus, } x=5 \text { is solution }\) \(\text { We also find that } x=-4 \text { is solution }\) \((-4)^2-2 \times(-4)^3-4-380=0\) \(=380-380=0\) \(\text { The real roots are } 5 \text { and }-4 \text {. Sum of real roots are } 5+(-4)\) \(=1\) Now, by trail error method use find that \(\mathrm{x}=5\) is a solution to the equation The real roots are 5 and -4 . Sum of real roots are \(5+(-4)\) \(=1\)
A \( Let a-d, a, a+d \text { are roots of }\) \(x^3-p x^2+q x-r=0\) \(\text { Sum of roots }=-\frac{b}{a}\) \(a-d+a+a+d=-\frac{(-p)}{1}\) \(3 a=p\) \(a=\frac{p}{3}\) \(\therefore a=\frac{p}{3} \text { should be satisfied by given equation }\) \(\text { So, put } x=\frac{p}{3} \text { in eq. (i) }\) \(\left(\frac{p}{3}\right)^3-p\left(\frac{p}{3}\right)^2+q\left(\frac{p}{3}\right)-r=0\) \(\frac{p^3}{27}-\frac{p^3}{9}+\frac{p q}{3}-r=0\) \(p^3-3 p^3+9 p q-27 r=0\) \(-2 p^3+9 p q-27 r=0\) \(2 p^3-9 p q+27 r=0\)So, put \(\mathrm{x}=\frac{\mathrm{p}}{3}\) in eq. (i)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118266
If \(\alpha, \beta, \gamma\) are the roots of the equation \(3 x^3-9 x^2+5 x-7\), then what is the value of \(\alpha+\beta\) \(+\gamma\) ?
1 3
2 -3
3 9
4 -9
Explanation:
A Given, \(\alpha, \beta, \gamma\) are roots of \(3 x^3-9 x^2+5 x-7=0\) Sum of the roots, \(\alpha+\beta+\gamma=-\frac{b}{a}=-\frac{(-9)}{3}\) \(\alpha+\beta+\gamma=3\)
AP EAMCET-22.09.2020
Complex Numbers and Quadratic Equation
118267
The sum of real roots of the equation \(x^4-2 \mathrm{x}^3+\) \(\mathbf{x}-\mathbf{3 8 0}=\mathbf{0}\)
1 -1
2 0
3 1
4 2
Explanation:
C Given \(x^4-2 x^3+x-380=0\) \(\text { Now, by trail error method use find that } x=5 \text { is a }\) \(\text { solution to the equation }\) \(5^4-2 \times 5^3+5-380=0\) \(625-250+5-380=0\) \(380-380=0\) \(\text { Thus, } x=5 \text { is solution }\) \(\text { We also find that } x=-4 \text { is solution }\) \((-4)^2-2 \times(-4)^3-4-380=0\) \(=380-380=0\) \(\text { The real roots are } 5 \text { and }-4 \text {. Sum of real roots are } 5+(-4)\) \(=1\) Now, by trail error method use find that \(\mathrm{x}=5\) is a solution to the equation The real roots are 5 and -4 . Sum of real roots are \(5+(-4)\) \(=1\)
