Moderate -0.3 This is a straightforward application of the product rule to find dy/dx, followed by evaluating at x=-1 to find the gradient, then using y-y₁=m(x-x₁). While it requires careful algebraic manipulation of the product rule with a chain rule component, it's a standard textbook exercise with no conceptual challenges—slightly easier than the average A-level question due to its routine nature.
1 Find the equation of the tangent to the curve
$$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$
at the point \(( - 1,3 )\), giving your answer in the form \(y = m x + c\).
Differentiate to produce form \(k_1x(x+2)^m + k_2x^2(x+2)^n\)
*M1
For positive integers \(k_1, k_2, m, n\); allow M1 if slip to, for example, \((x+3)\) in both brackets
Obtain \(6x(x+2)^6 + 18x^2(x+2)^5\)
A1
Or unsimplified equiv
Substitute \(x=-1\) to obtain value 12
A1
From correct work only
Attempt equation of tangent (not normal) through point \((-1,3)\)
M1
Dep *M; using non-zero numerical value of gradient; condone slip in use of coordinates
Obtain \(y=12x+15\)
A1 [5]
Answer required in \(y=mx+c\) form
Question 2i:
Answer
Marks
Guidance
Answer
Marks
Guidance
Expand to produce form \(k_1 + \frac{k_2}{x} + \frac{k_3}{x^2}\)
M1
For non-zero constants \(k_1, k_2, k_3\); allow if middle term appears as two, so far, unsimplified terms
Obtain \(4x - 4\ln x - \frac{1}{x}\) or \(4x - 4\ln x - x^{-1}\)
A1
Condoning absence of modulus signs but A0 if expression involves \(
Question 2ii:
Answer
Marks
Guidance
Answer
Marks
Guidance
Integrate to obtain form \(k(4x+1)^{\frac{4}{3}}\)
M1
Any non-zero constant \(k\)
Obtain \(\frac{3}{16}(4x+1)^{\frac{4}{3}}\)
A1
With coefficient simplified
Include \(\ldots + c\) or \(\ldots + k\) at least once anywhere in answer to question 2
B1 [5]
Even if associated with incorrect integral
Question 3i:
Answer
Marks
Guidance
Answer
Marks
Guidance
Obtain 128 for value corresponding to 10
B1
Allow any value rounding to 128
Obtain 65.5 for value corresponding to 25
B1 [2]
Allow any value rounding to 65 or 66; whether obtained using powers of 0.8 or by use of formula
Question 3ii:
Answer
Marks
Guidance
Answer
Marks
Guidance
Attempt to find formula for \(m\) of form \(200e^{kt}\) or \(200 \times r^{\lambda t}\)
M1
Whether attempted in part (i) or (ii)
Obtain \(200e^{(0.2\ln 0.8)t}\) or \(200e^{-0.0446t}\) or \(200\times 0.8^{0.2t}\) or \(200\times 0.956^t\)
A1
Or equiv
Show correct process for solving equation of form \(200e^{kt}=50\) or \(200r^{\lambda t}=50\)
M1
Obtain 31
A1 [4]
Or greater accuracy rounding to 31; ignore any units given; second M1 is implied by correct answer
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to produce form $k_1x(x+2)^m + k_2x^2(x+2)^n$ | *M1 | For positive integers $k_1, k_2, m, n$; allow M1 if slip to, for example, $(x+3)$ in both brackets |
| Obtain $6x(x+2)^6 + 18x^2(x+2)^5$ | A1 | Or unsimplified equiv |
| Substitute $x=-1$ to obtain value 12 | A1 | From correct work only |
| Attempt equation of tangent (not normal) through point $(-1,3)$ | M1 | Dep *M; using non-zero numerical value of gradient; condone slip in use of coordinates |
| Obtain $y=12x+15$ | A1 [5] | Answer required in $y=mx+c$ form |
## Question 2i:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Expand to produce form $k_1 + \frac{k_2}{x} + \frac{k_3}{x^2}$ | M1 | For non-zero constants $k_1, k_2, k_3$; allow if middle term appears as two, so far, unsimplified terms |
| Obtain $4x - 4\ln x - \frac{1}{x}$ or $4x - 4\ln x - x^{-1}$ | A1 | Condoning absence of modulus signs but A0 if expression involves $|\ln x|$ or $|4\ln x|$ |
## Question 2ii:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain form $k(4x+1)^{\frac{4}{3}}$ | M1 | Any non-zero constant $k$ |
| Obtain $\frac{3}{16}(4x+1)^{\frac{4}{3}}$ | A1 | With coefficient simplified |
| Include $\ldots + c$ or $\ldots + k$ at least once anywhere in answer to question 2 | B1 [5] | Even if associated with incorrect integral |
## Question 3i:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain 128 for value corresponding to 10 | B1 | Allow any value rounding to 128 |
| Obtain 65.5 for value corresponding to 25 | B1 [2] | Allow any value rounding to 65 or 66; whether obtained using powers of 0.8 or by use of formula |
## Question 3ii:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to find formula for $m$ of form $200e^{kt}$ or $200 \times r^{\lambda t}$ | M1 | Whether attempted in part (i) or (ii) | If formula attempted in part (i), marks earned must be recorded in part (ii) |
| Obtain $200e^{(0.2\ln 0.8)t}$ or $200e^{-0.0446t}$ or $200\times 0.8^{0.2t}$ or $200\times 0.956^t$ | A1 | Or equiv |
| Show correct process for solving equation of form $200e^{kt}=50$ or $200r^{\lambda t}=50$ | M1 | |
| Obtain 31 | A1 [4] | Or greater accuracy rounding to 31; ignore any units given; second M1 is implied by correct answer | Special case: no formula anywhere and answer 31 (or greater accuracy) given, award B2 (i.e. 2/4 for part (ii)) |
1 Find the equation of the tangent to the curve
$$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$
at the point $( - 1,3 )$, giving your answer in the form $y = m x + c$.
\hfill \mbox{\textit{OCR C3 2016 Q1 [5]}}