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A-Level Pure 2 · Edexcel IAL WMA12 & CAIE 9709 · Full paper

Full Paper B · Pure 2 ข้อสอบ Pure 2 เต็มฉบับ · ชุด B

11 questions 11 ข้อ 75 marks 75 คะแนน 1 h 50 min 1 ชม. 50 นาที Edexcel IAL WMA12 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

Find, in ascending powers of $x$, the first three terms of the expansion of $(1+3x)^8$.

Mark schemeshow ▾
M1 $1+\binom81(3x)+\binom82(3x)^2$  A1 $1$  A1 $+24x$  A1 $+252x^2$.
Q2. 5 marks

$f(x)=3x^3+2x^2-19x+6$. (a) Show that $(x-2)$ is a factor of $f(x)$. (b) Factorise $f(x)$ completely.

Mark schemeshow ▾
(a) M1 $f(2)=24+8-38+6$  A1 $=0$, conclusion stated.
(b) M1 $(x-2)(3x^2+8x-3)$  A1 quotient correct  A1 $(x-2)(3x-1)(x+3)$.
Q3. 6 marks

(a) Express $2\log_2 a+\log_2 b-3\log_2 c$ as a single logarithm. (b) Solve $\log_3(2x+3)-\log_3 x=2$.

Mark schemeshow ▾
(a) M1 power law on all three terms  A1 $\log_2 a^2b-\log_2 c^3$  A1 $\log_2\dfrac{a^2b}{c^3}$.
(b) M1 $\log_3\dfrac{2x+3}{x}=2 \Rightarrow \dfrac{2x+3}{x}=9$  A1 $7x=3$  A1 $x=\dfrac37$ (valid: $x>0$).
Q4. 6 marks

$A(2,5)$ and $B(8,-3)$ are the ends of a diameter of a circle $C$. (a) Find the equation of $C$. (b) Show that the point $D(1,4)$ lies on $C$, and state the size of angle $ADB$, giving a reason.

Mark schemeshow ▾
(a) M1 centre $=$ midpoint $=(5,1)$  M1 $r=\tfrac12 AB=\tfrac12\sqrt{36+64}=5$  A1 $(x-5)^2+(y-1)^2=25$.
(b) M1 $(1-5)^2+(4-1)^2=16+9=25$ ✓  A1 $\angle ADB=90^\circ$  B1 angle in a semicircle.
Q5. 6 marks

A geometric series has second term 12 and fifth term $\dfrac32$. (a) Find the common ratio and first term. (b) Find the sum to infinity.

Mark schemeshow ▾
(a) M1 $r^3=\dfrac{3/2}{12}=\dfrac18$  A1 $r=\dfrac12$  A1 $a=\dfrac{12}{1/2}=24$.
(b) M1 $|r|<1$ so $S_\infty=\dfrac{24}{1-\frac12}$  A2 $=48$.
Q6. 7 marks

A sector $OPQ$ of a circle, centre $O$, has radius 12 cm and angle $0.9$ radians. (a) Find the arc length $PQ$. (b) Find the area of the sector. (c) Find the area of the segment cut off by the chord $PQ$, to 3 significant figures.

Mark schemeshow ▾
(a) M1 $12\times0.9$  A1 $=10.8$ cm.
(b) M1 $\tfrac12(144)(0.9)$  A1 $=64.8$ cm².
(c) M1 triangle $=\tfrac12(144)\sin 0.9=56.40\ldots$  A1 segment $=64.8-56.40$  A1 $=8.40$ cm² (3 s.f.).
Q7. 7 marks

Solve, for $0\le x<360^\circ$: (a) $\cos(x-30^\circ)=\dfrac12$; (b) $3\tan^2 x-1=0$.

Mark schemeshow ▾
(a) M1 $u=x-30^\circ\in[-30^\circ,330^\circ)$: $u=60^\circ,\ 300^\circ$  A1 $x=90^\circ$  A1 $x=330^\circ$.
(b) M1 $\tan x=\pm\dfrac{1}{\sqrt3}$  A1 $x=30^\circ,210^\circ$  A1 $x=150^\circ,330^\circ$  B1 all four, none extra.
Q8. 7 marks

The curve $y=2x^3-9x^2+12x-3$ has two stationary points. (a) Find their coordinates and determine the nature of each. (b) State the set of values of $x$ for which $y$ is increasing.

Mark schemeshow ▾
(a) M1 $y'=6x^2-18x+12=6(x-1)(x-2)$  A1 $x=1,2$  A1 $(1,2)$ and $(2,1)$  M1 $y''=12x-18$  A1 $(1,2)$ max ($y''=-6$), $(2,1)$ min ($y''=6$).
(b) M1 $y'>0$  A1 $x<1$ or $x>2$.
Q9. 8 marks

The mass of a radioactive sample is $M=120e^{-0.05t}$ grams after $t$ years. (a) Write down the initial mass. (b) Find the half-life of the sample, to 3 significant figures. (c) Find the mass after 20 years, to 3 significant figures. (d) Find how long until the mass first falls below 30 g, to 3 significant figures.

Mark schemeshow ▾
(a) B1 $120$ g.
(b) M1 $e^{-0.05t}=\tfrac12 \Rightarrow t=\dfrac{\ln 2}{0.05}$  A1 $=13.9$ years.
(c) M1 $120e^{-1}$  A1 $=44.1$ g.
(d) M1 $e^{-0.05t}=\tfrac14 \Rightarrow t=\dfrac{\ln 4}{0.05}$  A1 $=27.7$ years  B1 (two half-lives — consistent with (b)).
Q10. 9 marks

(a) Use the trapezium rule with 4 strips to estimate $\displaystyle\int_0^1\frac{1}{1+x^2}\,dx$, to 4 decimal places. (b) The exact value of this integral is $\dfrac{\pi}{4}$. Find the percentage error of your estimate, to 2 significant figures. (c) Find the exact value of $\displaystyle\int_0^2\left(x^3-2x+3\right)dx$.

Mark schemeshow ▾
(a) B2 $h=0.25$; $y$-values $1,\ 0.9412,\ 0.8,\ 0.64,\ 0.5$  M1 $\tfrac{0.25}{2}\big[1.5+2(2.3812)\big]$  A1 $=0.7828$.
(b) M1 $\dfrac{\pi/4-0.7828}{\pi/4}\times100$  A1 $\approx 0.33\%$ (underestimate).
(c) M1 $\left[\dfrac{x^4}{4}-x^2+3x\right]_0^2$  A1 $=4-4+6$  A1 $=6$.
Q11. 10 marks

The curve $y=x^2-2x+3$ and the line $y=2x+3$ enclose a region $R$. (a) Find the $x$-coordinates of the points of intersection. (b) Show that the area of $R$ is $\dfrac{32}{3}$. (c) The line $y=2x+3$ meets the $y$-axis at $P$ and the curve meets the $y$-axis at $Q$. Explain why $P$ and $Q$ are the same point, and state its coordinates.

Mark schemeshow ▾
(a) M1 $x^2-2x+3=2x+3 \Rightarrow x^2-4x=0$  A1 $x=0,\ 4$.
(b) M1 $\displaystyle\int_0^4\big[(2x+3)-(x^2-2x+3)\big]dx=\int_0^4(4x-x^2)dx$  A1 $=\left[2x^2-\dfrac{x^3}{3}\right]_0^4$  M1 $=32-\dfrac{64}{3}$  A1 $=\dfrac{32}{3}$ ∎  B1 exact working shown throughout.
(c) M1 both give $y=3$ at $x=0$  A1 $x=0$ is an intersection point from (a)  B1 $(0,3)$.