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Algebra: Basics Test4
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Question 1 
If p = 0.25, q = −0.05, r = 0.5, then the value of
$ \displaystyle \frac{{{p}^{2}}{{q}^{2}}{{r}^{2}}2pq}{{{p}^{2}}+{{q}^{2}}2qr{{r}^{2}}}$
$ \displaystyle \frac{{{p}^{2}}{{q}^{2}}{{r}^{2}}2pq}{{{p}^{2}}+{{q}^{2}}2qr{{r}^{2}}}$
$ \displaystyle \frac{7}{8}$  
$ \displaystyle \frac{8}{9}$  
$ \displaystyle \frac{1}{8}$  
$ \displaystyle \frac{5}{8}$ 
Question 1 Explanation:
$ \displaystyle \begin{array}{l}\frac{{{p}^{2}}{{q}^{2}}{{r}^{2}}2qr}{{{p}^{2}}+{{q}^{2}}2pq{{r}^{2}}}\\=\frac{{{p}^{2}}\left( {{q}^{2}}+{{r}^{2}}+2qr \right)}{\left( {{p}^{2}}+{{q}^{2}}2pq \right){{r}^{2}}}\\=\frac{{{p}^{2}}{{\left( q+r \right)}^{2}}}{{{\left( pq \right)}^{2}}{{r}^{2}}}\\=\frac{\left( p+q+r \right)\left( pqr \right)}{\left( pq+r \right)\left( pqr \right)}\\=\frac{p+q+r}{pq+r}=\frac{0.250.05+0.5}{0.25+0.05+0.5}\\=\frac{0.7}{0.8}=\frac{7}{8}\end{array}$
Question 2 
If a x b = a^{2}+ b^{2}–ab, then the value of 9 × 11 is
103  
113  
119  
129 
Question 2 Explanation:
$ \displaystyle \begin{array}{l}a\times b={{a}^{2}}+{{b}^{2}}ab\,\left( Given \right)\\\Rightarrow 9\times 11={{9}^{2}}+{{11}^{2}}3\times 11\\=81+12199\\=20299=103\end{array}$
Question 3 
If p = 999, then the value of
$ \displaystyle 3\sqrt{p\left( {{p}^{2}}+3p+3 \right)+1}$
$ \displaystyle 3\sqrt{p\left( {{p}^{2}}+3p+3 \right)+1}$
1000  
999  
998  
1002

Question 3 Explanation:
p= 999 (Given)
$ \displaystyle \begin{array}{l}Now,\,3\sqrt{p\left( {{p}^{2}}+3p+3 \right)+1}\,\\3\sqrt{{{p}^{3}}+3{{p}^{2}}+3p+1}\,\\=3\sqrt{{{\left( p+1 \right)}^{3}}}\,=p+1\\=333+1=1000\end{array}$
$ \displaystyle \begin{array}{l}Now,\,3\sqrt{p\left( {{p}^{2}}+3p+3 \right)+1}\,\\3\sqrt{{{p}^{3}}+3{{p}^{2}}+3p+1}\,\\=3\sqrt{{{\left( p+1 \right)}^{3}}}\,=p+1\\=333+1=1000\end{array}$
Question 4 
If 5^{5a + 5}=1, then a equals
1
 
2  
3/5  
4/5 
Question 4 Explanation:
$ \displaystyle \begin{array}{l}{{5}^{5a+5}}=1\\\Rightarrow {{5}^{5a}}\times {{5}^{5}}\Rightarrow {{5}^{5a}}=\frac{1}{{{5}^{5}}}\\\Rightarrow {{5}^{5a}}={{5}^{5}}\Rightarrow 5a5\\\Rightarrow a=1\end{array}$
Question 5 
If 3^{r+3}+7=250, then r is equal to
4  
6  
2  
8

Question 5 Explanation:
$ \displaystyle \begin{array}{l}{{3}^{r+3}}+7=250\\\Rightarrow {{3}^{r+3}}=243\,\,\,\,\,\,\,\,\Rightarrow {{3}^{r+3}}={{3}^{5}}\\\Rightarrow r+3=5\Rightarrow r=2\end{array}$
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