| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Quadratic in exponential form |
| Difficulty | Moderate -0.8 This is a standard textbook exercise on solving exponential equations by substitution. Part (i) is routine algebraic manipulation (multiply through by y), and part (ii) requires factoring a simple quadratic and taking logarithms. The technique is well-practiced in P2/C3 courses with no novel problem-solving required, making it easier than average but not trivial. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(2^{-x} = \frac{1}{y}\), or \(2^x = y^{-1}\) | B1 | |
| Substitute and obtain a 3-term quadratic in \(y\) | M1 | |
| Obtain the given answer correctly | A1 | [3] |
| (ii) Solve the given quadratic and carry out correct method for solving an equation of the form \(2^x = a\), where \(a > 0\) | M1 | |
| Obtain answer \(x = 1.58\) or \(1.585\) | A1 | |
| Obtain answer \(x = 0\) | B1 | [3] |
**(i)** State or imply $2^{-x} = \frac{1}{y}$, or $2^x = y^{-1}$ | B1 |
Substitute and obtain a 3-term quadratic in $y$ | M1 |
Obtain the given answer correctly | A1 | [3]
**(ii)** Solve the given quadratic and carry out correct method for solving an equation of the form $2^x = a$, where $a > 0$ | M1 |
Obtain answer $x = 1.58$ or $1.585$ | A1 |
Obtain answer $x = 0$ | B1 | [3]5 (i) Given that $y = 2 ^ { x }$, show that the equation
$$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$
can be written in the form
$$y ^ { 2 } - 4 y + 3 = 0$$
(ii) Hence solve the equation
$$2 ^ { x } + 3 \left( 2 ^ { - x } \right) = 4$$
giving the values of $x$ correct to 3 significant figures where appropriate.
\hfill \mbox{\textit{CAIE P2 2010 Q5 [6]}}