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## Algebra: Basics Test-7

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Question 1 |

If p â‰ q, then which of the following statements is true?

A | $ \displaystyle \frac{p+q}{1}=\sqrt{pq}$ |

B | $ \displaystyle \frac{p+q}{2}<\sqrt{pq}$ |

C | $ \displaystyle \frac{p+q}{2}>\sqrt{pq}$ |

D | all of the above |

Question 1 Explanation:

$ \displaystyle \begin{array}{l}Arithmetic\,\,\,\,mean\,\,\,(AM)\,\,\,=\frac{p+q}{2}\\Geometric\,\,\,\,mean\,\,\,\,\left( GM \right)\,\,=\sqrt{pq}\\As\,\,\,\,AM>GM\\\frac{p+q}{2}>\sqrt{pq}\end{array}$

Question 2 |

$ \displaystyle \left( f+\frac{1}{f} \right)\left( f-\frac{1}{f} \right)\left( {{f}^{2}}+\frac{1}{{{f}^{2}}}-1 \right)\left( {{f}^{2}}+\frac{1}{{{f}^{2}}}+1 \right)$

is equal to the

is equal to the

A | $ \displaystyle {{f}^{6}}+\frac{1}{{{f}^{6}}}$ |

B | $ \displaystyle {{f}^{8}}+\frac{1}{{{f}^{8}}}$ |

C | $ \displaystyle {{f}^{8}}-\frac{1}{{{f}^{8}}}$ |

D | $ \displaystyle {{f}^{6}}-\frac{1}{{{f}^{6}}}$ |

Question 2 Explanation:

$ \displaystyle \begin{array}{l}=\left( f+\frac{1}{f} \right)\left( f-\frac{1}{f} \right)\left( {{f}^{2}}+\frac{1}{{{f}^{2}}}-1 \right)\left( {{f}^{2}}+\frac{1}{{{f}^{2}}}+1 \right)\\=\left( {{f}^{2}}-\frac{1}{{{f}^{2}}} \right)\left[ {{\left( {{f}^{2}}+\frac{1}{{{f}^{2}}} \right)}^{2}}-1 \right]\\=\left( {{f}^{2}}-\frac{1}{{{f}^{2}}} \right)\left( {{f}^{4}}+\frac{1}{{{f}^{4}}}+1 \right)\\={{f}^{6}}-\frac{1}{{{f}^{6}}}\end{array}$

Question 3 |

If p, q are two positive real numbers and p

^{1/3}= q^{1/3}, then which of the following relations is true?A | p ^{20}=q^{15} |

B | p ^{3}=q |

C | p =q ^{4} |

D | p ^{3}=q^{4} |

Question 3 Explanation:

$ \displaystyle \begin{array}{l}{{p}^{\frac{1}{3}}}={{q}^{\frac{1}{4}}}\\\Rightarrow {{\left( {{p}^{\frac{1}{3}}} \right)}^{12}}={{\left( {{q}^{\frac{1}{4}}} \right)}^{12}}\\\Rightarrow {{p}^{4}}={{q}^{3}}\\\Rightarrow {{\left( {{p}^{4}} \right)}^{5}}={{\left( {{q}^{3}} \right)}^{5}}\\\Rightarrow {{p}^{20}}={{q}^{15}}\end{array}$

Question 4 |

If q

^{2p+2}=1, where q is a positive real number other than 1, then p is equal toA | -4 |

B | -3 |

C | 3 |

D | -1 |

Question 4 Explanation:

We know that q

Therefore

$ \displaystyle \begin{array}{l}{{q}^{2p+2}}=1\\\Rightarrow 2p+2=0\\\Rightarrow p=\frac{-2}{2}=-1\end{array}$

^{0}=1Therefore

$ \displaystyle \begin{array}{l}{{q}^{2p+2}}=1\\\Rightarrow 2p+2=0\\\Rightarrow p=\frac{-2}{2}=-1\end{array}$

Question 5 |

If p is real, then the minimum value of (p

^{2 }-p +1 ) isA | 3/4 |

B | 3/2 |

C | 1/4 |

D | none of these |

Question 5 Explanation:

$ \displaystyle \text{For }\,\,\,\text{expression }\,\,\text{a}{{x}^{2}}\text{ +bx +c, a 0, the minimum value}$

$ \displaystyle \begin{array}{l}\frac{4ac-{{b}^{2}}}{4a}\\Here,\,\,\,for\,\,\,{{p}^{2}}-p+1\\a=1\\b=-1\\c=1\\Therefore\,\,\,Minimum\,\,\,\,value\\=\frac{4\times 1\times 1-1}{4\times 1\times 1}=\frac{3}{4}\end{array}$

$ \displaystyle \begin{array}{l}\frac{4ac-{{b}^{2}}}{4a}\\Here,\,\,\,for\,\,\,{{p}^{2}}-p+1\\a=1\\b=-1\\c=1\\Therefore\,\,\,Minimum\,\,\,\,value\\=\frac{4\times 1\times 1-1}{4\times 1\times 1}=\frac{3}{4}\end{array}$

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