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A-Level Pure 4 · Edexcel IAL WMA14 · Full paper

Full Paper A · Pure 4 ข้อสอบ Pure 4 เต็มฉบับ · ชุด A

11 questions 11 ข้อ 75 marks 75 คะแนน 1 h 50 min 1 ชม. 50 นาที Edexcel IAL WMA14 format

Sit it like the real thing: timer on, no notes, calculator allowed. Open the mark schemes only when you finish, then self-mark honestly. ทำเหมือนสอบจริง: เปิดตัวจับเวลา ไม่เปิดโน้ต ใช้เครื่องคิดเลขได้ ทำเสร็จค่อยเปิดมาร์คสกีมแล้วให้คะแนนตัวเองตามจริง

⏱ 110:00 Scoreคะแนนรวม: –
Q1. 4 marks

Express $\dfrac{4x+6}{x(x+3)}$ in partial fractions.

Mark schemeshow ▾
M1 $4x+6=A(x+3)+Bx$  A1 $x=0$: $6=3A$, $A=2$  A1 $x=-3$: $-6=-3B$, $B=2$  A1 $\dfrac{2}{x}+\dfrac{2}{x+3}$.
Q2. 5 marks

Prove by contradiction that there are infinitely many prime numbers.

Mark schemeshow ▾
B1 Assume there are finitely many primes $p_1,p_2,\ldots,p_n$  M1 consider $N=p_1p_2\cdots p_n+1$  M1 dividing $N$ by any $p_i$ leaves remainder 1, so no $p_i$ divides $N$  A1 but $N>1$ must have a prime factor — one not in the list, contradiction  A1 so the assumption is false: there are infinitely many primes ∎.
Q3. 6 marks

(a) Expand $(1+x)^{-3}$ in ascending powers of $x$ up to and including the $x^3$ term, stating the range of validity. (b) Use your expansion with $x=0.02$ to estimate $\dfrac{1}{1.02^3}$, giving your answer to 5 decimal places.

Mark schemeshow ▾
(a) M1 series with $n=-3$  A1 $1-3x+6x^2-10x^3$  B1 valid $|x|<1$.
(b) M1 $1-0.06+0.0024-0.00008$  A1 $=0.94232$  B1 5 d.p. quoted.
Q4. 6 marks

A curve has parametric equations $x=3t^2$, $y=6t$. (a) Find $\dfrac{dy}{dx}$ in terms of $t$. (b) Find the equation of the tangent at the point where $t=2$. (c) Find a cartesian equation of the curve.

Mark schemeshow ▾
(a) M1 $\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{6}{6t}$  A1 $=\dfrac1t$.
(b) M1 point $(12,12)$, gradient $\tfrac12$  A1 $y-12=\tfrac12(x-12)$, i.e. $x-2y+12=0$.
(c) M1 $t=\dfrac{y}{6}$ substituted  A1 $y^2=12x$.
Q5. 6 marks

The curve $C$ has equation $2x^2-3xy+y^2=12$. Verify that the point $(1,-2)$ lies on $C$, and find the gradient of $C$ at this point.

Mark schemeshow ▾
B1 $2+6+4=12$ ✓  M1 $4x-3y-3x\dfrac{dy}{dx}+2y\dfrac{dy}{dx}=0$ (product rule on $-3xy$)  A1 all terms correct  M1 $\dfrac{dy}{dx}=\dfrac{3y-4x}{2y-3x}$  M1 substitute $(1,-2)$: $\dfrac{-6-4}{-4-3}$  A1 $=\dfrac{10}{7}$.
Q6. 7 marks

(a) Show that $\displaystyle\int_0^{\pi/2}x\sin x\,dx=1$. (b) Find $\displaystyle\int \ln x\,dx$.

Mark schemeshow ▾
(a) M1 parts, $u=x$, $\frac{dv}{dx}=\sin x$  A1 $\big[-x\cos x\big]_0^{\pi/2}+\int_0^{\pi/2}\cos x\,dx$  A1 $=0+\big[\sin x\big]_0^{\pi/2}$  A1 $=1$ ∎.
(b) M1 parts with $u=\ln x$, $\frac{dv}{dx}=1$  A1 $x\ln x-\int 1\,dx$  A1 $=x\ln x-x+c$.
Q7. 7 marks

(a) Find the exact value of $\displaystyle\int_0^2\frac{x}{x^2+4}\,dx$. (b) The region under $y=\dfrac{1}{\sqrt{1+2x}}$ between $x=0$ and $x=3$ is rotated $360^\circ$ about the $x$-axis. Find the exact volume generated.

Mark schemeshow ▾
(a) M1 $\tfrac12\ln(x^2+4)$ (spot $\tfrac{f'}{f}$, or $u=x^2+4$)  A1 $\tfrac12(\ln 8-\ln 4)$  A1 $=\tfrac12\ln 2$.
(b) M1 $V=\pi\displaystyle\int_0^3\frac{1}{1+2x}\,dx$  A1 $=\pi\left[\tfrac12\ln(1+2x)\right]_0^3$  A1 $=\tfrac{\pi}{2}\ln 7$  B1 exact form.
Q8. 7 marks

The population of a colony grows at a rate proportional to its size: $\dfrac{dP}{dt}=kP$. The population doubles in 5 years. (a) Find the exact value of $k$. (b) Find how long the population takes to triple, to 3 significant figures.

Mark schemeshow ▾
(a) B1 $P=P_0e^{kt}$ (from separating variables)  M1 $e^{5k}=2$  A1 $k=\dfrac{\ln 2}{5}$.
(b) M1 $e^{kt}=3$  A1 $t=\dfrac{\ln 3}{k}$  A1 $=\dfrac{5\ln 3}{\ln 2}$  A1 $=7.92$ years (3 s.f.).
Q9. 8 marks

Points $A(2,-1,3)$ and $B(4,1,2)$. The line $\ell$ has equation $\mathbf{r}=\begin{pmatrix}1\\0\\5\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}$. (a) Find $\overrightarrow{AB}$ and $|\overrightarrow{AB}|$. (b) Find the acute angle between $\overrightarrow{AB}$ and $\ell$, to 1 decimal place. (c) Determine whether $\ell$ passes through the point $(3,1,4)$.

Mark schemeshow ▾
(a) M1 $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}=\begin{pmatrix}2\\2\\-1\end{pmatrix}$  A1 $|\overrightarrow{AB}|=\sqrt{4+4+1}=3$.
(b) M1 $\overrightarrow{AB}\cdot\mathbf{d}=4+2+1=7$  M1 $|\mathbf{d}|=\sqrt6$, $\cos\theta=\dfrac{7}{3\sqrt6}$  A1 $\cos\theta=0.9526$  A1 $\theta=17.7^\circ$.
(c) M1 $(1+2\lambda,\ \lambda,\ 5-\lambda)=(3,1,4)$: first gives $\lambda=1$, second $\lambda=1$, third $\lambda=1$  A1 consistent — yes, $\ell$ passes through $(3,1,4)$.
Q10. 9 marks

$f(x)=\dfrac{5x+4}{(1-x)(2+x)}$. (a) Express $f(x)$ in partial fractions. (b) Hence find the expansion of $f(x)$ in ascending powers of $x$ up to and including the $x^2$ term, and state the range of values of $x$ for which it is valid.

Mark schemeshow ▾
(a) M1 $5x+4=A(2+x)+B(1-x)$  A1 $x=1$: $9=3A$, $A=3$  A1 $x=-2$: $-6=3B$, $B=-2$. So $f(x)=\dfrac{3}{1-x}-\dfrac{2}{2+x}$.
(b) M1 $3(1-x)^{-1}=3(1+x+x^2+\cdots)$  A1 $3+3x+3x^2$  M1 $\dfrac{2}{2+x}=\left(1+\tfrac{x}{2}\right)^{-1}=1-\tfrac{x}{2}+\tfrac{x^2}{4}-\cdots$  A1 correct  A1 $f(x)=2+\tfrac{7}{2}x+\tfrac{11}{4}x^2$  B1 valid for $|x|<1$ (the tighter of $|x|<1$ and $|x|<2$).
Q11. 10 marks

A cup of coffee cools according to $\dfrac{d\theta}{dt}=-k(\theta-25)$, where $\theta$ °C is its temperature after $t$ minutes and $k$ is a positive constant. Initially $\theta=85$, and after 10 minutes $\theta=65$. (a) Show that $\theta=25+60e^{-kt}$. (b) Find the exact value of $k$. (c) Find how long it takes the coffee to cool to $45$ °C, to 3 significant figures.

Mark schemeshow ▾
(a) M1 separate: $\displaystyle\int\frac{d\theta}{\theta-25}=-\int k\,dt$  A1 $\ln|\theta-25|=-kt+c$  M1 $\theta=25+Ae^{-kt}$  A1 $\theta(0)=85 \Rightarrow A=60$ ∎.
(b) M1 $65=25+60e^{-10k} \Rightarrow e^{-10k}=\tfrac23$  A1 $k=\dfrac{1}{10}\ln\dfrac32$.
(c) M1 $45=25+60e^{-kt} \Rightarrow e^{-kt}=\tfrac13$  A1 $t=\dfrac{\ln 3}{k}=\dfrac{10\ln 3}{\ln(3/2)}$  A1 $=27.1$ minutes (3 s.f.)  B1 units stated.