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IAL 2023 Oct Q4

A Level / Edexcel / P1

IAL 2023 Oct Paper · Question 4

题目

Problem

Figure 1 shows a sketch of part of the curve CC with equation

y=1x+2.\begin{align*} y=\frac{1}{x+2}. \end{align*}

Figure 1

(a) State the equation of the asymptote of CC that is parallel to the yy-axis.

(1)

(b) Factorise fully x3+4x2+4xx^3+4x^2+4x.

(2)

A copy of Figure 1, labelled Diagram 1, is shown on the next page.

Diagram 1

(c) On Diagram 1, add a sketch of the curve with equation

y=x3+4x2+4x.\begin{align*} y=x^3+4x^2+4x. \end{align*}

On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.

(3)

(d) Hence state the number of real solutions of the equation

(x+2)(x3+4x2+4x)=1,\begin{align*} (x+2)(x^3+4x^2+4x)=1, \end{align*}

giving a reason for your answer.

(1)

解答

(a)

解法一

思路

展开

竖直渐近线来自分母为 00 的位置。

答题过程

展开

The vertical asymptote occurs when

x+2= 0x= −2.\begin{align*} x+2=&\,0\\ x=&\,-2. \end{align*}

Therefore the asymptote is

x=−2.\begin{align*} x=-2. \end{align*}

(b)

解法一

思路

展开

先提出公因式 xx，剩下的是完全平方三项式。

答题过程

展开 x3+4x2+4x= x(x2+4x+4)= x(x+2)2.\begin{align*} x^3+4x^2+4x =&\,x(x^2+4x+4)\\ =&\,x(x+2)^2. \end{align*}

(c)

解法一

思路

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由 (b) 可知 y=x(x+2)2y=x(x+2)^2。所以图像在 x=0x=0 穿过 xx 轴，在 x=−2x=-2 与 xx 轴相切。由于最高次项系数为正，整体是正三次曲线。

答题过程

展开

From part (b),

y=x(x+2)2.\begin{align*} y=x(x+2)^2. \end{align*}

The intercepts with the xx-axis are found from

x(x+2)2=0.\begin{align*} x(x+2)^2=0. \end{align*}

So

x=0orx=−2.\begin{align*} x=0 \quad\text{or}\quad x=-2. \end{align*}

Hence the curve cuts or meets the coordinate axes at

(0,0)and(−2,0).\begin{align*} (0,0)\quad\text{and}\quad(-2,0). \end{align*}

The sketch should show a positive cubic, crossing the axes at (0,0)(0,0) and touching the xx-axis at (−2,0)(-2,0).

(d)

解法一

思路

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把方程变形为

x3+4x2+4x=1x+2.\begin{align*} x^3+4x^2+4x=\frac{1}{x+2}. \end{align*}

所以解的个数就是 (c) 的三次曲线和原图中 reciprocal curve 的交点个数。

答题过程

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The equation

(x+2)(x3+4x2+4x)=1\begin{align*} (x+2)(x^3+4x^2+4x)=1 \end{align*}

can be rewritten as

x3+4x2+4x=1x+2.\begin{align*} x^3+4x^2+4x=\frac{1}{x+2}. \end{align*}

So the real solutions correspond to intersections of the two graphs.

From the sketch, the two curves intersect twice.

Therefore the equation has

2\begin{align*} 2 \end{align*}

real solutions.