Surds, Indices and exponents
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$( 2018/s/12/q12)$ Do not use a calculator in this question. (a) Given that $\frac{6^p\times8^{p+2}\times3^q}{9^{2q-3}}$ is equal to $2^7\times3^4$, find the value of each of the constants $p$ and $q$. $\begin{bmatrix}3\end{bmatrix}$ (b) Using the substitution $u=x^{\frac13}$,or otherwise, solve $4x^{\frac13}+x^{\frac23}+3=0.$ [4] |
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$( 2018/$w/21/q2$) $ (a) Solve $3^{\left(\frac{x}{2}-1\right)}=10.$ [3] (b) Solve 2e$^{1-2y}=3\mathrm{e}^{3y+2}.$ [4] |
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$( 2018/$w/22/q4$) $ Solve $(\mathbf{i})\quad2^{3x-1}=6$, [3] $( $ii$) \quad \log _{3}( y+ 14) = 1+ \frac 2{\log _{y}3}.$ [5] |
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$( 2019/$s$/12/$q$4) $ (a) Given that $\frac{\left(pr^2\right)^{\frac32}\sqrt{qr}}{q^2\left(pr^2\right)^{-1}}$ can be written in the form $p^aq^br^c$, find the value of each of the constants $a$, $b$ and $c.$ [3] (b) Solve $$ \begin{array}{c}{{3x^{\frac12}-y\quad^{-\frac12}=4,}}\\{{4x^{\frac{1}{2}}+3y^{-\frac{1}{2}}=14.}}\end{array} $$ [3] |
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$( 2019/$s$/13/$q$2)$ Given that $\displaystyle \frac{\sqrt{p}\left(qr\right)^{-2}}{p^2q^{\frac13}r}=\frac{1}{p^aq^br^c}$, find the value of each of the constants $a,b$ and $c.$ [3] |
| 6 | \begin{aligned}&\text{(2019/w/11/q3)}\\\\&\text{Given that}\quad7^x\times49^y=1\quad\text{and}\quad5^{5x}\times125^{\frac{2y}{3}}=\frac{1}{25},\quad\text{calculate the value of }x\text{ and of }y.\end{aligned} |
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$( 2019/$w$/12/$q$6) $ Write $\frac{\sqrt{p}\left(\frac{qp}{r}\right)^2}{p^{-1}\sqrt[3]{qr}}$ in the form $p^aq^br^c$, where $a,b$ and $c$ are constants. |
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