NEET Test Series from KOTA - 10 Papers In MS WORD
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Complex Numbers and Quadratic Equation
118248
If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-\) \(7 x+5=0\), then the value of \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\) is
1 \(\frac{-7}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{-3}{5}\)
4 \(\frac{3}{5}\)
Explanation:
B Given, \(\alpha, \beta, \gamma\) are the roots of equation \(x^3+3 x^2-7 x+\) \(5=0\) \(\therefore \quad \alpha+\beta+\gamma=-3\) \(\alpha \beta+\beta \gamma+\alpha \gamma=-7\) \(\alpha \beta \gamma=-5\) Now, \(\frac{\alpha \beta \gamma}{\alpha \beta \gamma}\left[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right]\) \(=\frac{1}{\alpha \beta \gamma}[\beta \gamma+\alpha \gamma+\alpha \beta]=\frac{-7}{-5}=\frac{7}{5}\)
APEAPCET-20.08.2021
Complex Numbers and Quadratic Equation
118249
The condition that \(\mathbf{x}^3-\mathbf{p x}^2+\mathbf{q x}-\mathbf{r}=\mathbf{0}\) may have two of its roots equal to each other but of opposite sign is
118250
If \(a, b, c, d\) are real numbers such that \(a\lt b\lt c\) \(\lt d\), then the roots of the equation \((x-a)(x-c)\) \(+2(x-d)=0\) are
1 Real \& need not be distinct
2 Real and distinct
3 Non-real and distinct
4 Non-real need not be distinct
Explanation:
B Given, \(\mathrm{a}\lt \mathrm{b}\lt \mathrm{c}\lt \mathrm{d}\). and the equation \((\mathrm{x}-\mathrm{a})\) \((\mathrm{x}-\mathrm{c})+2(\mathrm{x}-\mathrm{d})=0\) \(\mathrm{x}^2-(\mathrm{a}+\mathrm{c}) \mathrm{x}+\mathrm{ac}+2 \mathrm{x}-2 \mathrm{~d}=0\) or \(x^2+x(2-a-c)+a c-2 d=0\) Now, discriminant (D) \(=[2-(a+c)]^2-4(\mathrm{ac}-2 \mathrm{~d})\) \(=4+(a+c)^2-4(a+c)-4 a c+8 d\) \(=4+(a-c)^2-4 a-4 c+8 d\) \(=(a-c)^2+4(1-a-c+2 d)\) \((a-c)^2+4[(d-c)+(d-a)+1]\) \(\because \quad(a-c)^2>0\) and \(\mathrm{d}-\mathrm{c}>0(\because \mathrm{d}>\mathrm{c})\) \(\mathrm{d}-\mathrm{a}>0(\because \mathrm{d}>\mathrm{a})\) \(\therefore\) Discriminant is \(+\mathrm{ve}\). So, the equation has real and distinct root.
AP EAMCET-06.07.2022
Complex Numbers and Quadratic Equation
118251
If the product of the roots of the equation \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) is 31 , then the roots of the equation are real for \(k=\)
1 -4
2 1
3 4
4 0
Explanation:
C Given, \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) Let \(\alpha, \beta\) be the roots of equation \(\alpha \beta=31=2 \mathrm{e}^{2 \log \mathrm{k}}-1\) \(2 \mathrm{e}^{2 \log \mathrm{k}}=32\) \(\mathrm{e}^{2 \log \mathrm{k}}=16\) \(\mathrm{e}^{\log \mathrm{k}^2}=16\) \(\mathrm{k}^2=16\) \(\mathrm{k}=4\)
118248
If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-\) \(7 x+5=0\), then the value of \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\) is
1 \(\frac{-7}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{-3}{5}\)
4 \(\frac{3}{5}\)
Explanation:
B Given, \(\alpha, \beta, \gamma\) are the roots of equation \(x^3+3 x^2-7 x+\) \(5=0\) \(\therefore \quad \alpha+\beta+\gamma=-3\) \(\alpha \beta+\beta \gamma+\alpha \gamma=-7\) \(\alpha \beta \gamma=-5\) Now, \(\frac{\alpha \beta \gamma}{\alpha \beta \gamma}\left[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right]\) \(=\frac{1}{\alpha \beta \gamma}[\beta \gamma+\alpha \gamma+\alpha \beta]=\frac{-7}{-5}=\frac{7}{5}\)
APEAPCET-20.08.2021
Complex Numbers and Quadratic Equation
118249
The condition that \(\mathbf{x}^3-\mathbf{p x}^2+\mathbf{q x}-\mathbf{r}=\mathbf{0}\) may have two of its roots equal to each other but of opposite sign is
118250
If \(a, b, c, d\) are real numbers such that \(a\lt b\lt c\) \(\lt d\), then the roots of the equation \((x-a)(x-c)\) \(+2(x-d)=0\) are
1 Real \& need not be distinct
2 Real and distinct
3 Non-real and distinct
4 Non-real need not be distinct
Explanation:
B Given, \(\mathrm{a}\lt \mathrm{b}\lt \mathrm{c}\lt \mathrm{d}\). and the equation \((\mathrm{x}-\mathrm{a})\) \((\mathrm{x}-\mathrm{c})+2(\mathrm{x}-\mathrm{d})=0\) \(\mathrm{x}^2-(\mathrm{a}+\mathrm{c}) \mathrm{x}+\mathrm{ac}+2 \mathrm{x}-2 \mathrm{~d}=0\) or \(x^2+x(2-a-c)+a c-2 d=0\) Now, discriminant (D) \(=[2-(a+c)]^2-4(\mathrm{ac}-2 \mathrm{~d})\) \(=4+(a+c)^2-4(a+c)-4 a c+8 d\) \(=4+(a-c)^2-4 a-4 c+8 d\) \(=(a-c)^2+4(1-a-c+2 d)\) \((a-c)^2+4[(d-c)+(d-a)+1]\) \(\because \quad(a-c)^2>0\) and \(\mathrm{d}-\mathrm{c}>0(\because \mathrm{d}>\mathrm{c})\) \(\mathrm{d}-\mathrm{a}>0(\because \mathrm{d}>\mathrm{a})\) \(\therefore\) Discriminant is \(+\mathrm{ve}\). So, the equation has real and distinct root.
AP EAMCET-06.07.2022
Complex Numbers and Quadratic Equation
118251
If the product of the roots of the equation \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) is 31 , then the roots of the equation are real for \(k=\)
1 -4
2 1
3 4
4 0
Explanation:
C Given, \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) Let \(\alpha, \beta\) be the roots of equation \(\alpha \beta=31=2 \mathrm{e}^{2 \log \mathrm{k}}-1\) \(2 \mathrm{e}^{2 \log \mathrm{k}}=32\) \(\mathrm{e}^{2 \log \mathrm{k}}=16\) \(\mathrm{e}^{\log \mathrm{k}^2}=16\) \(\mathrm{k}^2=16\) \(\mathrm{k}=4\)
118248
If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-\) \(7 x+5=0\), then the value of \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\) is
1 \(\frac{-7}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{-3}{5}\)
4 \(\frac{3}{5}\)
Explanation:
B Given, \(\alpha, \beta, \gamma\) are the roots of equation \(x^3+3 x^2-7 x+\) \(5=0\) \(\therefore \quad \alpha+\beta+\gamma=-3\) \(\alpha \beta+\beta \gamma+\alpha \gamma=-7\) \(\alpha \beta \gamma=-5\) Now, \(\frac{\alpha \beta \gamma}{\alpha \beta \gamma}\left[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right]\) \(=\frac{1}{\alpha \beta \gamma}[\beta \gamma+\alpha \gamma+\alpha \beta]=\frac{-7}{-5}=\frac{7}{5}\)
APEAPCET-20.08.2021
Complex Numbers and Quadratic Equation
118249
The condition that \(\mathbf{x}^3-\mathbf{p x}^2+\mathbf{q x}-\mathbf{r}=\mathbf{0}\) may have two of its roots equal to each other but of opposite sign is
118250
If \(a, b, c, d\) are real numbers such that \(a\lt b\lt c\) \(\lt d\), then the roots of the equation \((x-a)(x-c)\) \(+2(x-d)=0\) are
1 Real \& need not be distinct
2 Real and distinct
3 Non-real and distinct
4 Non-real need not be distinct
Explanation:
B Given, \(\mathrm{a}\lt \mathrm{b}\lt \mathrm{c}\lt \mathrm{d}\). and the equation \((\mathrm{x}-\mathrm{a})\) \((\mathrm{x}-\mathrm{c})+2(\mathrm{x}-\mathrm{d})=0\) \(\mathrm{x}^2-(\mathrm{a}+\mathrm{c}) \mathrm{x}+\mathrm{ac}+2 \mathrm{x}-2 \mathrm{~d}=0\) or \(x^2+x(2-a-c)+a c-2 d=0\) Now, discriminant (D) \(=[2-(a+c)]^2-4(\mathrm{ac}-2 \mathrm{~d})\) \(=4+(a+c)^2-4(a+c)-4 a c+8 d\) \(=4+(a-c)^2-4 a-4 c+8 d\) \(=(a-c)^2+4(1-a-c+2 d)\) \((a-c)^2+4[(d-c)+(d-a)+1]\) \(\because \quad(a-c)^2>0\) and \(\mathrm{d}-\mathrm{c}>0(\because \mathrm{d}>\mathrm{c})\) \(\mathrm{d}-\mathrm{a}>0(\because \mathrm{d}>\mathrm{a})\) \(\therefore\) Discriminant is \(+\mathrm{ve}\). So, the equation has real and distinct root.
AP EAMCET-06.07.2022
Complex Numbers and Quadratic Equation
118251
If the product of the roots of the equation \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) is 31 , then the roots of the equation are real for \(k=\)
1 -4
2 1
3 4
4 0
Explanation:
C Given, \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) Let \(\alpha, \beta\) be the roots of equation \(\alpha \beta=31=2 \mathrm{e}^{2 \log \mathrm{k}}-1\) \(2 \mathrm{e}^{2 \log \mathrm{k}}=32\) \(\mathrm{e}^{2 \log \mathrm{k}}=16\) \(\mathrm{e}^{\log \mathrm{k}^2}=16\) \(\mathrm{k}^2=16\) \(\mathrm{k}=4\)
118248
If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+3 x^2-\) \(7 x+5=0\), then the value of \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\) is
1 \(\frac{-7}{5}\)
2 \(\frac{7}{5}\)
3 \(\frac{-3}{5}\)
4 \(\frac{3}{5}\)
Explanation:
B Given, \(\alpha, \beta, \gamma\) are the roots of equation \(x^3+3 x^2-7 x+\) \(5=0\) \(\therefore \quad \alpha+\beta+\gamma=-3\) \(\alpha \beta+\beta \gamma+\alpha \gamma=-7\) \(\alpha \beta \gamma=-5\) Now, \(\frac{\alpha \beta \gamma}{\alpha \beta \gamma}\left[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}\right]\) \(=\frac{1}{\alpha \beta \gamma}[\beta \gamma+\alpha \gamma+\alpha \beta]=\frac{-7}{-5}=\frac{7}{5}\)
APEAPCET-20.08.2021
Complex Numbers and Quadratic Equation
118249
The condition that \(\mathbf{x}^3-\mathbf{p x}^2+\mathbf{q x}-\mathbf{r}=\mathbf{0}\) may have two of its roots equal to each other but of opposite sign is
118250
If \(a, b, c, d\) are real numbers such that \(a\lt b\lt c\) \(\lt d\), then the roots of the equation \((x-a)(x-c)\) \(+2(x-d)=0\) are
1 Real \& need not be distinct
2 Real and distinct
3 Non-real and distinct
4 Non-real need not be distinct
Explanation:
B Given, \(\mathrm{a}\lt \mathrm{b}\lt \mathrm{c}\lt \mathrm{d}\). and the equation \((\mathrm{x}-\mathrm{a})\) \((\mathrm{x}-\mathrm{c})+2(\mathrm{x}-\mathrm{d})=0\) \(\mathrm{x}^2-(\mathrm{a}+\mathrm{c}) \mathrm{x}+\mathrm{ac}+2 \mathrm{x}-2 \mathrm{~d}=0\) or \(x^2+x(2-a-c)+a c-2 d=0\) Now, discriminant (D) \(=[2-(a+c)]^2-4(\mathrm{ac}-2 \mathrm{~d})\) \(=4+(a+c)^2-4(a+c)-4 a c+8 d\) \(=4+(a-c)^2-4 a-4 c+8 d\) \(=(a-c)^2+4(1-a-c+2 d)\) \((a-c)^2+4[(d-c)+(d-a)+1]\) \(\because \quad(a-c)^2>0\) and \(\mathrm{d}-\mathrm{c}>0(\because \mathrm{d}>\mathrm{c})\) \(\mathrm{d}-\mathrm{a}>0(\because \mathrm{d}>\mathrm{a})\) \(\therefore\) Discriminant is \(+\mathrm{ve}\). So, the equation has real and distinct root.
AP EAMCET-06.07.2022
Complex Numbers and Quadratic Equation
118251
If the product of the roots of the equation \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) is 31 , then the roots of the equation are real for \(k=\)
1 -4
2 1
3 4
4 0
Explanation:
C Given, \(x^2-2 \sqrt{2} k x+2 e^{2 \log k}-1=0\) Let \(\alpha, \beta\) be the roots of equation \(\alpha \beta=31=2 \mathrm{e}^{2 \log \mathrm{k}}-1\) \(2 \mathrm{e}^{2 \log \mathrm{k}}=32\) \(\mathrm{e}^{2 \log \mathrm{k}}=16\) \(\mathrm{e}^{\log \mathrm{k}^2}=16\) \(\mathrm{k}^2=16\) \(\mathrm{k}=4\)
