Question 8 - A-Level Maths

Edexcel C12 2017 June — Question 8 8 marks

Exam Board	Edexcel
Module	C12 (Core Mathematics 1 & 2)
Year	2017
Session	June
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Indefinite & Definite Integrals
Type	Integration with given constant
Difficulty	Moderate -0.3 This is a straightforward multi-part integration question requiring standard techniques: finding an indefinite integral using power rule, evaluating a definite integral, and solving a cubic equation that factors simply. While it has multiple steps, each is routine for C1/C2 level with no novel problem-solving required, making it slightly easier than average.
Spec	1.07m Tangents and normals: gradient and equations 1.08b Integrate x^n: where n != -1 and sums 1.08d Evaluate definite integrals: between limits 1.08e Area between curve and x-axis: using definite integrals

8. (a) Find $\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x$, simplifying each term. Given that $b$ is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$ (b) show that $b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0$ (c) Hence find the possible values of $b$.

Show mark scheme Show mark scheme source

Question 8 (Integration):

Part (a):

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$\int(3x^2+4x-15)\,dx = x^3+2x^2-15x+c$	M1, A1, A1	M1: raises index of any term in $x$ by one; first A1: two of three algebraic terms correct (unsimplified), e.g. accept $2x^2=\frac{4}{2}x^{1+1}$; second A1: cao including $+c$

Part (b):

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$\int_b^4(3x^2+4x-15)\,dx = \left[x^3+2x^2-15x\right]_b^4 = 36$
$(64+32-60)-(b^3+2b^2-15b) = 36$	M1	Substitutes 4 and $b$ into integrated expression, subtracts either way and sets equal to 36
$b^3+2b^2-15b = 0$	A1*	Given answer; intermediate line(s) must be correct; minimum expectation: $36-(b^3+2b^2-15b)=36$ or equivalent with bracket removed

Part (c):

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$b = 0$	B1
$b^2+2b-15=0 \Rightarrow (b+5)(b-3)=0$	M1	Factorises/attempts to solve the quadratic
$b = -5,\ 3$	A1

# Question 8 (Integration):

## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int(3x^2+4x-15)\,dx = x^3+2x^2-15x+c$ | M1, A1, A1 | M1: raises index of any term in $x$ by one; first A1: two of three algebraic terms correct (unsimplified), e.g. accept $2x^2=\frac{4}{2}x^{1+1}$; second A1: cao including $+c$ | **(3)** |

## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_b^4(3x^2+4x-15)\,dx = \left[x^3+2x^2-15x\right]_b^4 = 36$ | | |
| $(64+32-60)-(b^3+2b^2-15b) = 36$ | M1 | Substitutes 4 and $b$ into integrated expression, subtracts either way and sets equal to 36 |
| $b^3+2b^2-15b = 0$ | A1* | Given answer; intermediate line(s) must be correct; minimum expectation: $36-(b^3+2b^2-15b)=36$ or equivalent with bracket removed | **(2)** |

## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b = 0$ | B1 | |
| $b^2+2b-15=0 \Rightarrow (b+5)(b-3)=0$ | M1 | Factorises/attempts to solve the quadratic |
| $b = -5,\ 3$ | A1 | | **(3)(8 marks)** |

---

Show LaTeX source

8. (a) Find $\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x$, simplifying each term.

Given that $b$ is a constant and

$$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$

(b) show that $b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0$\\
(c) Hence find the possible values of $b$.

\hfill \mbox{\textit{Edexcel C12 2017 Q8 [8]}}

This paper (15 questions)

View full paper

Q1 4 Q2 4 Q3 4 Q4 12 Q5 10 Q6 9 Q7 7 Q8 8 Q9 10 Q10 8 Q11 10 Q12 12 Q13 10 Q14 8 Q15 9