Sequences and Series

Column 1 | Column 2 | Column 3 | |||

I | The value of (x + y + z) is 15. If a , x , y , z , b are in AP while the value of $\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ is $\frac{5}{3}$ . If a , x , y , z , b are in HP , then product of ab is |
(i) | Even Integer | (P) | Positive Number |

II | The harmonic mean of the roots of the equation $\left(5+\sqrt{2}\right){x}^{2}-\left(4+\sqrt{5}\right)x+8+2\sqrt{5}=0$ | (ii) | Odd Integer | (Q) | Negative Number |

III | If x , y , z are in HP , then $\mathrm{log}\left(x+z\right)+\mathrm{log}\left(x+z-2y\right)=k\mathrm{log}\left(x-z\right)$ Then k is |
(iii) | Rational number | (R) | Non Positive Number |

IV | If x be the arithmetic mean and y , z be two geometric means between any two positive numbers , then $-\left(\frac{{y}^{3}+{z}^{3}}{xyz}\right)$is |
(iv) | Irrational Number | (S) | Non Negative Number |

Which of the following is the only CORRECT combination ?

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JEE Advanced 0

Column 1 | Column 2 | Column 3 | |||

I | The value of (x + y + z) is 15. If a , x , y , z , b are in AP while the value of $\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ is $\frac{5}{3}$ . If a , x , y , z , b are in HP , then product of ab is |
(i) | Even Integer | (P) | Positive Number |

II | The harmonic mean of the roots of the equation $\left(5+\sqrt{2}\right){x}^{2}-\left(4+\sqrt{5}\right)x+8+2\sqrt{5}=0$ | (ii) | Odd Integer | (Q) | Negative Number |

III | If x , y , z are in HP , then $\mathrm{log}\left(x+z\right)+\mathrm{log}\left(x+z-2y\right)=k\mathrm{log}\left(x-z\right)$ Then k is |
(iii) | Rational number | (R) | Non Positive Number |

IV | If x be the arithmetic mean and y , z be two geometric means between any two positive numbers , then $-\left(\frac{{y}^{3}+{z}^{3}}{xyz}\right)$is |
(iv) | Irrational Number | (S) | Non Negative Number |

Which of the following is the only INCORRECT combination?

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JEE Advanced 0

Column 1 | Column 2 | Column 3 | |||

I | The value of (x + y + z) is 15. If a , x , y , z , b are in AP while the value of $\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ is $\frac{5}{3}$ . If a , x , y , z , b are in HP , then product of ab is |
(i) | Even Integer | (P) | Positive Number |

II | The harmonic mean of the roots of the equation $\left(5+\sqrt{2}\right){x}^{2}-\left(4+\sqrt{5}\right)x+8+2\sqrt{5}=0$ | (ii) | Odd Integer | (Q) | Negative Number |

III | If x , y , z are in HP , then $\mathrm{log}\left(x+z\right)+\mathrm{log}\left(x+z-2y\right)=k\mathrm{log}\left(x-z\right)$ Then k is |
(iii) | Rational number | (R) | Non Positive Number |

IV | If x be the arithmetic mean and y , z be two geometric means between any two positive numbers , then $-\left(\frac{{y}^{3}+{z}^{3}}{xyz}\right)$is |
(iv) | Irrational Number | (S) | Non Negative Number |

Which of the following is the only CORRECT combination?

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JEE Advanced 0

If the roots of $10{x}^{3}-c{x}^{2}-54x-27=0$ are in H.P. then the value of

*c*must be equal to ____.
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JEE Advanced 0