| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Second derivative and nature determination |
| Difficulty | Moderate -0.3 This is a straightforward chain rule application requiring two differentiations of a power function, followed by a routine geometric progression calculation. The differentiation is mechanical (standard chain rule with fractional power), and the GP condition r = f'(2)/f(2) = kf''(2)/f'(2) leads to simple algebra. Slightly easier than average due to being purely procedural with no conceptual challenges. |
| Spec | 1.04i Geometric sequences: nth term and finite series sum1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f'(x) = \left[\dfrac{3}{2}(4x+1)^{1/2}\right][4]\) | B1B1 | Expect \(6(4x+1)^{1/2}\) but can be unsimplified |
| \(f''(x) = 6 \times \frac{1}{2} \times (4x+1)^{-1/2} \times 4\) | B1\(\checkmark\) | Expect \(12(4x+1)^{-1/2}\) but can be unsimplified. Ft from their \(f'(x)\) |
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(2),\ f'(2),\ kf''(2) = 27,\ 18,\ 4k\) OR \(12\) | B1B1\(\checkmark\)B1\(\checkmark\) | Ft dependent on attempt at differentiation |
| \(27/18 = 18/4k\) oe OR \(kf''(2) = 12 \Rightarrow k = 3\) | M1A1 | |
| Total: | 5 |
## Question 7(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = \left[\dfrac{3}{2}(4x+1)^{1/2}\right][4]$ | B1B1 | Expect $6(4x+1)^{1/2}$ but can be unsimplified |
| $f''(x) = 6 \times \frac{1}{2} \times (4x+1)^{-1/2} \times 4$ | B1$\checkmark$ | Expect $12(4x+1)^{-1/2}$ but can be unsimplified. Ft from their $f'(x)$ |
| **Total:** | **3** | |
## Question 7(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(2),\ f'(2),\ kf''(2) = 27,\ 18,\ 4k$ OR $12$ | B1B1$\checkmark$B1$\checkmark$ | Ft dependent on attempt at differentiation |
| $27/18 = 18/4k$ oe OR $kf''(2) = 12 \Rightarrow k = 3$ | M1A1 | |
| **Total:** | **5** | |7 The function f is defined for $x \geqslant 0$ by $\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }$.\\
(i) Find $\mathrm { f } ^ { \prime } ( x )$ and $\mathrm { f } ^ { \prime \prime } ( x )$.\\
The first, second and third terms of a geometric progression are respectively $\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )$ and $k \mathrm { f } ^ { \prime \prime } ( 2 )$.\\
(ii) Find the value of the constant $k$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q7 [8]}}