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IAL 2026 Jan Q9

A Level / Edexcel / P1

IAL 2026 Jan Paper · Question 9

题目

Problem

Figure 4

Figure 4 shows a sketch of part of the graph of the trigonometric function with equation y=f(x)y=f(x).

(a) Write down an expression for f(x)f(x).

(2)

Copies of Figure 4 (labelled Diagram 1 and Diagram 2) can be found on the following pages.

(b) (i) On Diagram 1 sketch a graph of the curve with equation

y=10−x2\begin{align*} y=10-x^2 \end{align*}

Diagram 1

(ii) Hence find the number of solutions of the equation

f(x)=10−x2\begin{align*} f(x)=10-x^2 \end{align*}

in the interval −100π⩽x⩽100π-100\pi\leqslant x\leqslant100\pi.

(3)

(c) (i) On Diagram 2 sketch a graph of the curve with equation

y=tan⁡x,−2π⩽x⩽2π\begin{align*} y=\tan x, \qquad -2\pi\leqslant x\leqslant2\pi \end{align*}

Diagram 2

(ii) Hence find the number of solutions of the equation

f(x)=tan⁡x\begin{align*} f(x)=\tan x \end{align*}

in the interval −100π⩽x⩽100π-100\pi\leqslant x\leqslant100\pi, giving a reason for your answer.

(4)

解答

(a)

解法一

思路

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图像振幅是 1010，周期是 2π2\pi。在 x=0x=0 时图像取最低值 −10-10，所以是 −10cos⁡x-10\cos x。

答题过程

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The graph has amplitude 1010 and period 2π2\pi.

Since the graph has a minimum value of −10-10 at x=0x=0,

f(x)=−10cos⁡x\begin{align*} f(x)=-10\cos x \end{align*}

(b)

解法一

思路

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y=10−x2y=10-x^2 是开口向下的抛物线，顶点在 (0,10)(0,10)，大约在 x=±10x=\pm\sqrt{10} 处过 xx 轴，接近但略大于 ±π\pm\pi。从图像交点判断方程解的个数。

答题过程

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The curve

y=10−x2\begin{align*} y=10-x^2 \end{align*}

is a downward parabola with vertex (0,10)(0,10).

On the graph, it intersects

y=f(x)=−10cos⁡x\begin{align*} y=f(x)=-10\cos x \end{align*}

twice.

Therefore the number of solutions is

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

(c)

解法一

思路

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tan⁡x\tan x 的周期是 π\pi，渐近线在 x=π2+kπx=\dfrac{\pi}{2}+k\pi。

在区间 −2π⩽x⩽2π-2\pi\leqslant x\leqslant2\pi 中，与 −10cos⁡x-10\cos x 有 44 个交点。整个区间 −100π-100\pi 到 100π100\pi 长度是 200π200\pi，也就是 5050 个长度为 4π4\pi 的区间。

答题过程

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The graph of

y=tan⁡x\begin{align*} y=\tan x \end{align*}

has vertical asymptotes at

x=π2+kπ,k∈Z\begin{align*} x=\frac{\pi}{2}+k\pi, \qquad k\in\mathbb{Z} \end{align*}

and crosses the xx-axis at multiples of π\pi.

From the graph on

−2π⩽x⩽2π,\begin{align*} -2\pi\leqslant x\leqslant2\pi, \end{align*}

there are 44 intersections of

y=−10cos⁡xandy=tan⁡x\begin{align*} y=-10\cos x \qquad\text{and}\qquad y=\tan x \end{align*}

The interval

−100π⩽x⩽100π\begin{align*} -100\pi\leqslant x\leqslant100\pi \end{align*}

has length 200π200\pi. Since 4π4\pi contains 44 solutions, there are

50×4=200\begin{align*} 50\times4=200 \end{align*}

solutions.