มาแล้วครับ...ข้อสอบชุดที่ 2 .. หวังว่าคงถูกใจผู้รักคณิตศาสต์ทั้งหลาย
PROBLEM PAPERS 2
1. A given point $D$ lies between two given lines $AB$ and $AC$. Find a construction for a line through $D$ terminated by $AB$ and $AC$, such that $D$ is one of its points of trisection. Prove also that there are two such lines.
2. If a conic circumscribe a parallelogram, its centre must be at the intersection of the diagonals.
3. Prove the identity
$$\frac{a^2(b-c)}{b+c-a} + \frac{b^2(c-a)}{c+a-b} + \frac{c^2(a-b)}{a+b-c} + \frac{(a+b+c)^2(b-c)(c-a)(a-b)}{(b+c-a)(c+a-b)(a+b-c)} = 0.$$
4. If there be any number of quantities $a, b, c, ...,$ shew that
$$a^3+b^3+c^3+...-3(abc+abd+...)$$
is divisible by $a + b + c + . . .$ and find the quotient.
5. If in a triangle $a, c$ and $C$ are given, and $b_1, b_2$ are the two values of the third side, and $r_1, r_2$ the radii of the two inscribed circles, prove that
$(i) \left(\frac{b_1}{r_1}-\cot\frac12C\right)\left(\frac{b_2}{r_2}-\cot\frac12C\right) = 1.$
$(ii) r_1r_2 = a(a-c)\sin^2\frac12C.$
6. Prove that if $\cos A = \cos\theta \sin\phi, \cos B = \cos\phi \sin\psi, \cos C = \cos\psi \sin\theta,$ and $A + B + C = \pi,$ then $\tan\theta \tan\phi \tan\psi = 1.$
7. Prove that the locus of the poles of chords of the circle $x^2 + y^2 = a^2$ which subtend a right angle at the fixed point $(h, k)$ is the circle
$$ (h^2+k^2-a^2)(x^2+y^2)-2a^2ky+2a^4 = 0.$$
8. Chords of the parabola $y^2 = 4ax$ are drawn through the fixed point $(h, k)$. Shew that the locus of their middle points is the parabola $y(y-k) = 2a(x-h).$
9. $P$ and $Q$ are extremities of two conjugate diameters of an ellipse of minor axis $2b,$ and $S$ is a focus. Prove that $PQ^2-(SP-SQ)^2 = 2b^2.$
10. A rod of length $2a$ rests on a smooth vertical circle of radius $b,$ one end being attached to a string, to the other end of which is tied a weight which hangs down over the circle. The distance of the point of contact from this end of the rod is $na$. Prove that in the position of equilibrium the rod makes with the vertical an angle
$$\tan^{-1}\left(\frac{2abn^2}{(1-n)(b^2-a^2n^2)}\right).$$
11. The base angles of a wedge are $\alpha$ and $\beta$ and its mass is $M$. Two particles of masses $m$ and $m'$ are let fall simultaneously from the vertex down the two faces. Prove that the wedge will move on the smooth horizontal plane with which it is in contact with acceleration
$$ \frac12g \frac{m\sin2\alpha - m'\sin2\beta}{M+m\sin^2 \alpha + m'\sin^2 \beta}. $$
12. If the unit of kinetic energy be that of a train of mass $m$ tons moving with a velocity of $v$ miles an hour, the unit of power that of an engine of horse-power $h$, and the unit of force the weight of $n$ tons, prove that the unit of mass is $ \frac12m\left(\frac{448vn}{75h}\right)^2$ tons.
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