13 (CIE 2015, s, paper 22, question 6)
(a) Solve $6^{x-2}=\dfrac{1}{4}$. $[2]$
(b) Solve $\log _{a} 2 y^{2}+\log _{a} 8+\log _{a} 16 y-\log _{a} 64 y=2 \log _{a} 4$ $[4]$
14 (CIE $2015, \mathrm{w}$, paper 13, question 12 )
(a) Given that $2^{2 x-1} \times 4^{x+y}=128$ and $d\frac{9^{2 y-x}}{27^{y-4}}=1$, find the value of each of the integers $x$ and $y$. [4]
(b) Solve $\quad 2(5)^{2 z}+5^{z}-1=0$. [4]
15 (CIE 2016, march, paper 12, question 2)
Given that $\dfrac{p^{-2} q r^{-\frac{1}{2}}}{\sqrt{p^{\frac{1}{3}} q^{2} r^{-3}}}=p^{a} q^{b} r^{c}$, find the values of $a, b$ and $c$. $[3]$
16 (CIE 2016, s, paper 11, question 2)
(a) Solve the equation $16^{3 x-1}=8^{x+2}$. $[3]$
(b) Given that $\dfrac{\left(a^{\frac{1}{3}} b^{-\frac{1}{2}}\right)^{3}}{a^{-\frac{2}{3}} b^{\frac{1}{2}}}=a^{p} b^{q}$, find the value of each of the constants $p$ and $q$. [2]
17 (CIE $2016, \mathrm{w}$, paper 11, question 2)
Given that $\dfrac{p^{\frac{1}{3}} q^{-\frac{1}{2}} r^{\frac{1}{2}}}{p^{-\frac{2}{3}} \sqrt{(q r)^{5}}}=p^{a} q^{b} r^{c}$, find the value of each of the integers $a, b$ and $c$.
18 (CIE $2016, \mathrm{w}$, paper 13 , question 2)
Express $\dfrac{4 m \sqrt{m}-\dfrac{9}{\sqrt{m}}}{2 \sqrt{m}+\dfrac{3}{\sqrt{m}}}$ in the form $A m+B$, where $A$ and $B$ are integers to be found. $[3]$
19 (CIE $2016, \mathrm{w}$, paper 23 , question 2) Solve the equation $e^{3 x}=6 e^{x}$.
20 (CIE 2017, s, paper 11, question 3)
(a) Simplify $\sqrt{x^{8} y^{10}} \div \sqrt[3]{x^{3} y^{-6}}$, giving your answer in the form $x^{a} y^{b}$, where $a$ and $b$ are integers. [2]
(b) (i) Show that $4(t-2)^{\frac{1}{2}}+5(t-2)^{\frac{3}{2}}$ can be written in the form $(t-2)^{p}(q t+r)$, where $p, q$ and 1 are constants to be found.
(ii) Hence solve the equation $4(t-2)^{\frac{1}{2}}+5(t-2)^{\frac{3}{2}}=0$. $[1]$
21 (CIE 2017, s, paper 23, question 1)
(a) Solve the equation $7^{2 x+5}=2.5$, giving your answer correct to 2 decimal places. $[3]$
(b) Express $\dfrac{(5 \sqrt{q})^{3}}{\left(625 p^{12} q\right)^{\frac{1}{4}}}$ in the form $5^{a} p^{b} q^{c}$, where $a, b$ and $c$ are constants. $[3]$
22 (CIE $2017, \mathrm{w}$, paper 11 , question 3
(a) Given that $T=2 \pi l^{\frac{1}{2}} g^{-\frac{1}{2}}$, express $l$ in terms of $T, g$ and $\pi$. $[2]$
(b) By using the substitution $y=x^{\frac{1}{3}}$, or otherwise, solve $x^{\frac{2}{3}}-4 x^{3}+3=0$. [4]
23 (CIE $2017, \mathrm{w}$, paper 22 , question 2)
Solve the equation $\dfrac{2 x^{1.5}+6 x^{-0.5}}{x^{0.5}+5 x^{-0.5}}=x$. 24 (CIE $2018, \mathrm{~s}$, paper 12 , question 12)
Do not use a calculator in this question.
(a) Given that $\dfrac{6^{p} \times 8^{p+2} \times 3^{q}}{9^{2 q-3}}$ is equal to $2^{7} \times 3^{4}$, find the value of each of the constants $p$ and $q$. [3]
(b) Using the substitution $u=x^{\frac{1}{3}}$, or otherwise, solve $4 x^{\frac{1}{3}}+x^{\frac{2}{3}}+3=0$. [4]
Answers
13. (a) $1.226$, (b) 2
14. (a) $x=4, y=-4$
(b) $z=-0.431$
15. $a=-13 / 6, b=0, c=1$
16. (a) $10 / 9$
(b) $5 / 3,-2$
17. $a=1, b=-3, c=-1$
18. $2 m-3$
19. $x=.5 \ln 6$
20. (a) $x^{3} y^{7}$
(bi) $(t-2)^{1 / 2}(5 t-6)$
(bii) $2,\frac 65$
21. (a) $-2.26$
(b) $5^{2} p^{-3} q^{5 / 4}$
22. (a) $l=\frac{T^{2} g}{4 \pi^{2}}$
(b) $x=1, x=27$
23. $x=3,2$
24. (a) $\mathrm{p}=1 / 4, \mathrm{q}=3 / 4$
(b) $x=-1, x=-27$
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