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A-Level Pure 4 · Edexcel IAL WMA14 · Mixed mock

Predicted Paper · Pure 4 mixed mock ข้อสอบรวม Pure 4 · ชุดจำลอง

10 questions 10 ข้อ 66 marks 66 คะแนน 1 h 15 min 1 ชม. 15 นาที Edexcel IAL WMA14 format · one question per topic 1 ข้อ/หัวข้อ

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

⏱ 75:00 Scoreคะแนนรวม: –
Q1. 4 marks

Express $\dfrac{2x+7}{(x+1)(x+3)}$ in partial fractions.

Mark schemeshow ▾
M1 $2x+7=A(x+3)+B(x+1)$  A1 $x=-1$: $5=2A$, $A=\tfrac52$  A1 $x=-3$: $1=-2B$, $B=-\tfrac12$  A1 $\dfrac{5}{2(x+1)}-\dfrac{1}{2(x+3)}$.
Q2. 5 marks

Prove by contradiction that $\sqrt3$ is irrational.

Mark schemeshow ▾
B1 assume $\sqrt3=\tfrac{p}{q}$ lowest terms  M1 $3q^2=p^2$ → $3\mid p^2$ → $3\mid p$, $p=3k$  M1 $3q^2=9k^2 \Rightarrow q^2=3k^2$ → $3\mid q$  A1 common factor 3 contradicts lowest terms  A1 conclusion stated. ∎
Q3. 6 marks

(a) Expand $(1+4x)^{1/2}$ in ascending powers of $x$ up to $x^2$, stating the validity. (b) Use $x=0.01$ to find an approximation for $\sqrt{1.04}$ to 5 d.p.

Mark schemeshow ▾
(a) M1 series with $u=4x$  A1 $1+2x-2x^2$  B1 valid $|x|<\tfrac14$.
(b) M1 $1+0.02-0.0002$  A1 $\sqrt{1.04}\approx1.01980$  B1 5 d.p. quoted.
Q4. 7 marks

The curve $C$: $x=t^2-1$, $y=t^3-3t$. (a) Find the coordinates of the points where $C$ crosses the $x$-axis. (b) Find $\dfrac{dy}{dx}$ in terms of $t$, and the equation of the tangent at the point where $t=2$.

Mark schemeshow ▾
(a) M1 $y=0$: $t(t^2-3)=0$, $t=0,\pm\sqrt3$  A1 points $(-1,0)$ and $(2,0)$ (both $t=\pm\sqrt3$ give $(2,0)$).
(b) M1 $\dfrac{dy}{dx}=\dfrac{3t^2-3}{2t}$  A1 at $t=2$: $\dfrac{9}{4}$  M1 point $(3,2)$  A1 $y-2=\tfrac94(x-3)$  A1 $9x-4y-19=0$.
Q5. 6 marks

The curve $x^3+3xy+y^2=19$ passes through $(1,\,y_0)$ with $y_0>0$. (a) Show $y_0=3$. (b) Find the gradient of the curve at $(1,3)$.

Mark schemeshow ▾
(a) M1 $1+3y_0+y_0^2=19 \Rightarrow y_0^2+3y_0-18=0$  A1 $(y_0-3)(y_0+6)=0$, and $y_0>0$ gives $y_0=3$ ✓
(b) M1 $3x^2+3y+3x\frac{dy}{dx}+2y\frac{dy}{dx}=0$  A1 all four terms correct  M1 $\frac{dy}{dx}=-\dfrac{3x^2+3y}{3x+2y}$  A1 at $(1,3)$: $-\dfrac{12}{9}=-\dfrac43$.
Q6. 8 marks

(a) Find $\displaystyle\int x\cos 2x\,dx$. (b) Use the substitution $u=1+x^2$ to find $\displaystyle\int_0^1 \dfrac{2x}{1+x^2}\,dx$, giving your answer as an exact logarithm.

Mark schemeshow ▾
(a) M1 parts, $u=x$, $v=\tfrac12\sin2x$  A1 $\tfrac{x}{2}\sin2x-\int\tfrac12\sin2x\,dx$  A1 $\tfrac{x}{2}\sin2x+\tfrac14\cos2x+c$.
(b) M1 $du=2x\,dx$, limits $1\to2$  A1 $\int_1^2\frac{du}{u}$  M1 $[\ln u]_1^2$  A1 $\ln2$  B1 exact form.
Q7. 7 marks

The region bounded by $y=e^{x}$, the $x$-axis, and the lines $x=0$ and $x=1$ is rotated $360^\circ$ about the $x$-axis. (a) Find the exact volume generated. (b) Water flows into a tank so that $\dfrac{dV}{dt}=0.5$ m³/min while the volume relates to depth by $V=4h^2$. Find $\dfrac{dh}{dt}$ when $h=2$ m.

Mark schemeshow ▾
(a) M1 $V=\pi\int_0^1 e^{2x}dx$  A1 $=\pi\left[\tfrac12e^{2x}\right]_0^1$  A1 $=\dfrac{\pi}{2}(e^2-1)$.
(b) M1 $\dfrac{dV}{dh}=8h=16$  M1 $\dfrac{dh}{dt}=\dfrac{dV/dt}{dV/dh}=\dfrac{0.5}{16}$  A1 $=\dfrac{1}{32}$ m/min  B1 units.
Q8. 7 marks

Solve the differential equation $\dfrac{dy}{dx}=\dfrac{3x^2}{2y}$, given $y=4$ when $x=0$, giving $y$ in terms of $x$. State the value of $y$ when $x=2$.

Mark schemeshow ▾
M1 separate: $\int 2y\,dy=\int 3x^2dx$  A1 $y^2=x^3+c$  M1 $16=0+c$  A1 $y^2=x^3+16$  A1 $y=\sqrt{x^3+16}$ (positive root, since $y(0)=4>0$)  M1 $x=2$: $y=\sqrt{24}$  A1 $=2\sqrt6$.
Q9. 8 marks

Points $A(1,2,3)$, $B(3,1,5)$. Line $\ell: \mathbf{r}=\begin{pmatrix}2\\0\\1\end{pmatrix}+\lambda\begin{pmatrix}1\\1\\0\end{pmatrix}$. (a) Find $\overrightarrow{AB}$ and $|\overrightarrow{AB}|$. (b) Find the acute angle between $\overrightarrow{AB}$ and $\ell$, to 1 d.p.

Mark schemeshow ▾
(a) M1 $\overrightarrow{AB}=\begin{pmatrix}2\\-1\\2\end{pmatrix}$  A1 $|\overrightarrow{AB}|=\sqrt{4+1+4}=3$.
(b) M1 $\overrightarrow{AB}\cdot\mathbf{d}=2-1+0=1$  M1 $|\mathbf{d}|=\sqrt2$  M1 $\cos\theta=\dfrac{1}{3\sqrt2}$  A1 $\theta=76.4^\circ$  B2 working shown to 1 d.p.
Q10. 8 marks

(a) Express $\dfrac{1}{(1-x)(1-2x)}$ in partial fractions. (b) Hence find the expansion of $\dfrac{1}{(1-x)(1-2x)}$ in ascending powers of $x$ up to $x^2$, and state the range of validity.

Mark schemeshow ▾
(a) M1 $1=A(1-2x)+B(1-x)$  A1 $x=1$: $1=-A$, $A=-1$; $x=\tfrac12$: $1=\tfrac12B$, $B=2$. So $\dfrac{2}{1-2x}-\dfrac{1}{1-x}$.
(b) M1 $2(1+2x+4x^2)-(1+x+x^2)$  A1 $=1+3x+7x^2$  A1 coefficients $1,3,7$  B1 valid for $|x|<\tfrac12$ (tighter of the two)  B2 both expansions correct.