CalcGospel 國際數學圖譜
返回

IAL 2024 Oct Q7

A Level / Edexcel / P1

IAL 2024 Oct Paper · Question 7

题目

Problem

Figure 3 shows a plot of part of the curve C1C_1 with equation

y=−4cos⁡x\begin{align*} y=-4\cos x \end{align*}

where xx is measured in radians.

Figure 3

Points PP and QQ lie on the curve and are shown in Figure 3.

(a) State

(i) the coordinates of PP

(ii) the coordinates of QQ

(3)

The curve C2C_2 has equation y=−4cos⁡x+ky=-4\cos x+k where xx is measured in radians and kk is a constant.

Given that C2C_2 has a maximum yy value of 1111,

(b) (i) state the value of kk

(ii) state the coordinates of the minimum point on C2C_2 with the smallest positive xx coordinate.

(3)

On the opposite page there is a copy of Figure 3 labelled Diagram 1.

Diagram 1

(c) Using Diagram 1, state the number of solutions of the equation

−4cos⁡x=5−10πx\begin{align*} -4\cos x=5-\frac{10}{\pi}x \end{align*}

giving a reason for your answer.

(2)

解答

(a)(i)

解法一

思路

展开

y=−4cos⁡xy=-4\cos x 的最小值是 −4-4，发生在 cos⁡x=1\cos x=1 的位置。图中 PP 是左边那个最低点，所以对应 x=−2πx=-2\pi。

答题过程

展开

For y=−4cos⁡xy=-4\cos x, a minimum occurs when cos⁡x=1\cos x=1.

On the left of the origin shown in the diagram, this occurs at

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

Therefore

P=(−2π,−4).\begin{align*} P=(-2\pi,-4). \end{align*}

(a)(ii)

解法一

思路

展开

QQ 是右边的 xx 轴截点。令 −4cos⁡x=0-4\cos x=0，即 cos⁡x=0\cos x=0，图中对应的是 x=3π2x=\frac{3\pi}{2}。

答题过程

展开

At an xx-axis intersection,

−4cos⁡x= 0cos⁡x= 0.\begin{align*} -4\cos x=&\,0\\ \cos x=&\,0. \end{align*}

For the point QQ shown,

x=3π2.\begin{align*} x=\frac{3\pi}{2}. \end{align*}

Therefore

Q=(3π2,0).\begin{align*} Q=\left(\frac{3\pi}{2},0\right). \end{align*}

(b)(i)

解法一

思路

展开

−4cos⁡x-4\cos x 的最大值是 44。整体向上平移 kk 后，最大值变成 4+k4+k。

答题过程

展开

The maximum value of −4cos⁡x-4\cos x is 44.

So the maximum value of −4cos⁡x+k-4\cos x+k is

4+k.\begin{align*} 4+k. \end{align*}

Given that the maximum value is 1111,

4+k= 11k= 7.\begin{align*} 4+k=&\,11\\ k=&\,7. \end{align*}

(b)(ii)

解法一

思路

展开

C2C_2 是把 C1C_1 向上平移 77。最小值从 −4-4 变成 33，最小正 xx 坐标的最低点在 x=2πx=2\pi。

答题过程

展开

The minimum value of C2C_2 is

−4+7=3.\begin{align*} -4+7=3. \end{align*}

The minimum point with the smallest positive xx coordinate occurs at

x=2π.\begin{align*} x=2\pi. \end{align*}

Therefore the point is

(2π,3).\begin{align*} (2\pi,3). \end{align*}

(c)

解法一

思路

展开

方程

−4cos⁡x=5−10πx\begin{align*} -4\cos x=5-\frac{10}{\pi}x \end{align*}

表示曲线 y=−4cos⁡xy=-4\cos x 与直线 y=5−10πxy=5-\frac{10}{\pi}x 的交点。数交点个数即可。

答题过程

展开

Draw the line

y=5−10πx\begin{align*} y=5-\frac{10}{\pi}x \end{align*}

on Diagram 1.

This line has negative gradient and passes through (0,5)(0,5).

It intersects the curve y=−4cos⁡xy=-4\cos x once.

Therefore the equation has

1\begin{align*} 1 \end{align*}

solution.