CalcGospel 國際數學圖譜
返回

IAL 2023 Jan Q8

A Level / Edexcel / P1

IAL 2023 Jan Paper · Question 8

题目

Problem

Figure 2 shows a sketch of the straight line ll and the curve CC.

Figure 2

Given that ll cuts the yy-axis at −12-12 and cuts the xx-axis at 44, as shown in Figure 2,

(a) find an equation for ll, writing your answer in the form y=mx+cy=mx+c, where mm and cc are constants to be found.

(2)

Given that CC

  • has equation y=f(x)y=f(x) where f(x)f(x) is a quadratic expression
  • has a minimum point at (7,−18)(7,-18)
  • cuts the xx-axis at 44 and at kk, where kk is a constant

(b) deduce the value of kk,

(1)

(c) find f(x)f(x).

(3)

The region RR is shown shaded in Figure 2.

(d) Use inequalities to define RR.

(2)

解答

(a)

解法一

思路

展开

直线的 yy 轴截距是 −12-12，所以 c=−12c=-12。它还经过 (4,0)(4,0)，可以用两个截距求斜率。

答题过程

展开

The line passes through (0,−12)(0,-12) and (4,0)(4,0), so its gradient is

m= 0−(−12)4−0= 124= 3.\begin{align*} m =&\,\frac{0-(-12)}{4-0}\\ =&\,\frac{12}{4}\\ =&\,3. \end{align*}

Since the yy-intercept is −12-12,

y=3x−12.\begin{align*} y=3x-12. \end{align*}

(b)

解法一

思路

展开

二次函数的对称轴经过最低点，所以对称轴是 x=7x=7。两个 xx 轴截距关于 x=7x=7 对称。已知其中一个是 x=4x=4，另一个到 77 的距离也应为 33。

答题过程

展开

The minimum point is (7,−18)(7,-18), so the axis of symmetry is

x=7.\begin{align*} x=7. \end{align*}

The two roots are symmetric about x=7x=7. Since one root is x=4x=4,

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

Therefore the other root is

k=7+3=10.\begin{align*} k=7+3=10. \end{align*}

(c)

解法一

思路

展开

已知两个根是 44 和 1010，可以写成 f(x)=A(x−4)(x−10)f(x)=A(x-4)(x-10)。再把最低点 (7,−18)(7,-18) 代入求 AA。

答题过程

展开

Since the roots are 44 and 1010,

f(x)=A(x−4)(x−10).\begin{align*} f(x)=A(x-4)(x-10). \end{align*}

Use the point (7,−18)(7,-18):

−18= A(7−4)(7−10)−18= A(3)(−3)−18= −9AA= 2.\begin{align*} -18=&\,A(7-4)(7-10)\\ -18=&\,A(3)(-3)\\ -18=&\,-9A\\ A=&\,2. \end{align*}

Therefore

f(x)=2(x−4)(x−10).\begin{align*} f(x)=2(x-4)(x-10). \end{align*}

解法二

思路

展开

最低点是 (7,−18)(7,-18)，所以也可以从完成平方形式开始：f(x)=a(x−7)2−18f(x)=a(x-7)^2-18。再代入一个 xx 轴截距求 aa。

答题过程

展开

Since the minimum point is (7,−18)(7,-18), write

f(x)=a(x−7)2−18.\begin{align*} f(x)=a(x-7)^2-18. \end{align*}

Use the root x=4x=4:

0= a(4−7)2−180= 9a−18a= 2.\begin{align*} 0=&\,a(4-7)^2-18\\ 0=&\,9a-18\\ a=&\,2. \end{align*}

Therefore

f(x)=2(x−7)2−18.\begin{align*} f(x)=2(x-7)^2-18. \end{align*}

This is equivalent to

f(x)=2(x−4)(x−10).\begin{align*} f(x)=2(x-4)(x-10). \end{align*}

(d)

解法一

思路

展开

阴影区域在 yy 轴右侧、x=4x=4 左侧，也就是 0<x<40<x<4。从图像看，它在直线之上、二次曲线之下。

答题过程

展开

The region is to the right of the yy-axis and to the left of x=4x=4, so

0<x<4.\begin{align*} 0<x<4. \end{align*}

It is above the line and below the curve, so

3x−12<y<2(x−4)(x−10).\begin{align*} 3x-12<y<2(x-4)(x-10). \end{align*}

Therefore the region RR is defined by

0<x<4,3x−12<y<2(x−4)(x−10).\begin{align*} 0<x<4, \qquad 3x-12<y<2(x-4)(x-10). \end{align*}