*Question*

All living plants contain an isotope of carbon called carbon-14. When a plant dies, the isotope decays so that the amount of carbon-14 present in the remains of the plant decreases. The time since the death of a plant can be determined by measuring the amount of carbon-14 still present in the remains.

The amount, A , of carbon-14 present in a plant t years after its death can be modelled by A = A_{0}e^{-kt} where t ≥ 0 and A_{0} , k are positive constants.

At the time of death, a plant is defined to have 100 units of carbon-14.

(a) Show that A_{0} = 100 . [1]

The time taken for half the original amount of carbon-14 to decay is known to be 5730 years.

(b) Show that \(k=\frac{ln 2}{5730}\) [3]

(c) Find, correct to the nearest 10 years, the time taken after the plant’s death for 25 % of the carbon-14 to decay. [3]

**▶️Answer/Explanation**

Ans:

(a) When $t=0$, $A=100$, i.e., we have $$\begin{eqnarray} 100 = A_0\text{e}^{-k\left(0\right)} \nonumber \\ A_0 = 100. \end{eqnarray}$$ (b) Let half-life be $t=5730$, then $$\begin{eqnarray} \frac{1}{2} = \text{e}^{-k\left(5730\right)} \nonumber \\ -\ln 2 = -k\left(5730\right) \nonumber \\ k = \frac{\ln 2}{5730}. \end{eqnarray}$$ (c) When $25%$ of carbon-14 has decayed, we are left with $A=75$. Thus, from the graphing calculator, solving $75=100\text{e}^{-\frac{\ln 2}{5730}t}$, we have $t\approx 2380\text{ years}$.

*Question*

(a) Find the solution of the equation

\[\ln {2^{4x – 1}} = \ln {8^{x + 5}} + {\log _2}{16^{1 – 2x}},\]

expressing your answer in terms of \(\ln 2\).

(b) Using this value of *x*, find the value of *a* for which \({\log _a}x = 2\), giving your answer to three decimal places.

**▶️Answer/Explanation**

### Markscheme

(a) rewrite the equation as \((4x – 1)\ln 2 = (x + 5)\ln 8 + (1 – 2x){\log _2}16\) *(M1)*

\((4x – 1)\ln 2 = (3x + 15)\ln 2 + 4 – 8x\) *(M1)(A1)*

\(x = \frac{{4 + 16\ln 2}}{{8 + \ln 2}}\) *A1*

(b) \(x = {a^2}\) *(M1)*

\(a = 1.318\) *A1*

**Note:** Treat 1.32 as an ** AP**.

Award ** A0** for ±.

*[6 marks]*

**Question**

**Question**

(a) Let a = log_{2}x, b = log_{2}y, c = log_{2}z. Write \(log_2(\frac{x^3\sqrt{y}}{z^4})\) in terms of a, b and c.

(b) Let A = log_{2}x, B = log_{4}y, C = log_{8}z. Write \(log_2(\frac{x^3\sqrt{y}}{z^4})\) in terms of A, B and C.

**▶️Answer/Explanation**

**Ans**(a) \(log_2(\frac{x^3\sqrt{y}}{z^4})=3a+\frac{1}{2}b-4c\).

(b) \(log_2(\frac{x^3\sqrt{y}}{z^4})\)=3A+B-12C\)

**Question**

**Question**

Solve the simultaneous equations

\(log_2(y-1)=1+log_2x\)

\(2log_3y=2+log_3x\)

**▶️Answer/Explanation**

**Ans**

x = 1, y = 3 or \(x=\frac{1}{4}, y=\frac{3}{2}\)