Matematika Murni 2
Pearson Edexcel · International A-Level · Mathematics Topik 2 5:12 Narasi bahasa Inggris · Subtitle bahasa Inggris + 中文 disematkan langsung
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Paper WMA12: 1 hour 30 minutes, 75 marks, and P1 may be tested too.
WMA12 试卷：1 小时 30 分钟，75 分，也可能考 P1。
Memorise the laws of logarithms, the two trigonometric identities, and the definite integral as an area.
记住对数运算法则、两个三角恒等式，以及定积分表示面积。
Factorise first: m plus 1 times m plus 2.
先因式分解：(m + 1)(m + 2)。
A number not divisible by 3 is 3q plus 1 or 3q plus 2, and those are all the cases.
不能被 3 整除的数是 3q + 1 或 3q + 2，这就是全部情况。
Each produces a factor of 3, so the claim holds every time.
每种情况都产生因数 3，所以结论总成立。
Is n squared minus n plus 1 always prime?
n² − n + 1 总是质数吗？
The first values, 3, 7, 13, are all prime, which is tempting.
前几个值 3、7、13 都是质数，很有迷惑性。
But n equals 5 gives 21, which is 3 times 7: one counterexample, and the claim is false.
但 n = 5 时得 21 = 3 × 7：一个反例，结论就不成立。
Here is the cubic with a equals 8 and b equals 26.
这是 a = 8、b = 26 的三次函数。
Because x plus 3 is a factor, it crosses the axis at minus 3.
因为 x + 3 是因式，它在 −3 处穿过 x 轴。
Because dividing by 2x minus 1 leaves 7, it has height 7 at a half.
因为除以 2x − 1 余 7，它在 ½ 处的高度是 7。
f of minus 3 equals 0 gives b equals 3a plus 2.
f(−3) = 0 得 b = 3a + 2。
f of a half equals 7 gives a plus 2b equals 60.
f(½) = 7 得 a + 2b = 60。
Together, a is 8 and b is 26.
联立得 a = 8，b = 26。
Dividing by x minus 2 leaves remainder 175, which is f of 2.
除以 x − 2 余 175，正是 f(2)。
The ends of the diameter fix everything.
直径两端确定一切。
The centre is the midpoint, 1, 10, and the radius squared is 34.
圆心是中点 (1, 10)，半径平方为 34。
The point nearest the x-axis lies straight down from the centre, root 34 below it.
离 x 轴最近的点在圆心正下方，距圆心 √34。
With ratio 0.8 each term is smaller than the last, so the partial sums climb ever more slowly and settle at 50, the sum to infinity.
公比为 0.8，每一项都比前一项小，所以部分和增长越来越慢，最终趋于 50，即无穷项和。
The k-th term is 10 times 0.8 to the k minus 1.
第 k 项是 10 × 0.8^{k−1}。
Take logs, then divide by log 0.8, which is negative, so the inequality flips: k minus 1 is above 44.38, and the smallest k is 46.
取对数，再除以 log 0.8；它是负数，所以不等号反向：k − 1 > 44.38，最小的 k 是 46。
The terms fall by 7.
每项减少 7。
The 122nd is 3, still positive; the 123rd is minus 4, which would only reduce the total.
第 122 项是 3，仍为正；第 123 项是 −4，只会让总和变小。
So the greatest sum is S 122: 52,033.
所以最大和是 S₁₂₂ = 52 033。
Move the 2 inside as a power and write 3 as log 2 of 8, so each side is a single log.
把 2 作为指数移进去，把 3 写成 log₂8，两边各成一个对数。
Equal logs mean equal arguments: x is 14 or minus 2.
对数相等则真数相等：x = 14 或 −2。
But 4 minus x must be positive, so 14 is rejected, leaving minus 2.
但 4 − x 必须为正，所以舍去 14，剩下 −2。
The curves y equals 3 to the 2x plus 1 and y equals 6 times 3 to the x meet at P.
曲线 y = 3^{2x+1} 与 y = 6 × 3ˣ 交于 P。
With u equal to 3 to the x, 3u squared equals 6u, so u is 2: x is log 3 of 2, and y is 12.
令 u = 3ˣ，3u² = 6u，所以 u = 2：x = log₃2，y = 12。
Divide by 3: tan of the bracket is 1 over root 3, so the bracket is pi over 6 plus k pi.
除以 3：括号的正切值是 1/√3，所以括号等于 π/6 + kπ。
Solve for x, then keep the values in the interval: 29 pi over 60 and 59 pi over 60.
解出 x，保留区间内的值：29π/60 和 59π/60。
Replace tan and sine squared to leave only cosine: a quadratic.
替换 tan 和 sin² 后只剩余弦：一个二次方程。
One root is below minus 1, impossible.
一个根小于 −1，不可能。
The other, 0.6385, meets the cosine graph twice in 0 to 360: at 50.3 and 309.7 degrees, by symmetry.
另一个根 0.6385 在 0 到 360° 内与余弦图像相交两次：由对称性得 50.3° 和 309.7°。
A trough holding 12 cubic metres uses S equals 6x squared plus 20 over x of metal.
容积 12 立方米的水槽用料 S = 6x² + 20/x。
Plot S against x: too narrow wastes metal on the long sides, too wide on the ends.
画出 S 关于 x 的图：太窄则长边费料，太宽则两端费料。
The lowest point is where the gradient is zero.
最低点在斜率为零处。
The volume gives L as 4 over x squared, which turns S into 6x squared plus 20 over x.
由体积得 L = 4/x²，于是 S = 6x² + 20/x。
Setting the derivative to zero gives x cubed equals five thirds, and S is 25.3.
令导数为零得 x³ = 5/3，S = 25.3。
The second derivative is positive: a minimum.
二阶导数为正：是最小值。
The line through P and Q lies above the curve between them.
过 P、Q 的直线在两点之间位于曲线上方。
The region between is the area under the line, 24, minus the area under the curve, 10.5: 13.5 square units.
两者之间的区域面积是直线下的面积 24 减去曲线下的面积 10.5：13.5 平方单位。
Four straight strips under y equals 1 over x, each of width 0.5.
在 y = 1/x 下画四条宽 0.5 的直边梯形。
The curve bends upwards, so every strip's top sits above it: the estimate 1.12 is too big.
曲线向上弯，所以每条梯形的顶边都在曲线上方：估计值 1.12 偏大。
The true value is 1.099.
真实值是 1.099。
Check yourself.
自我检测。
One counterexample disproves a claim.
一个反例就能推翻结论。
The remainder on 2x minus 1 is f of a half.
除以 2x − 1 的余数是 f(½)。
A sum to infinity needs r between minus 1 and 1.
无穷项和要求 |r| < 1。
Check log solutions against the domain.
对数方程的解要检查定义域。
And on a convex curve the trapezium rule overestimates.
在下凸曲线上，梯形法则会高估。