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มาแล้วครับ...ข้อสอบชุดที่ 1 .. อาจยากไปหน่อย เพราะเป็นภาษาอังกฤษซะด้วย
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PROBLEM PAPERS 1

1. Through a given point draw a straight line which shall form with two given straight lines a triangle of given perimeter.

2. Any radius of a circle is drawn and a circle is described upon it as diameter. Prove geometrically that the locus of the centre of a circle described so as to touch the large circle internally and the small circle externally is an ellipse and find the position of its foci and centre. Find also the magnitude of its eccentricity and the lengths of its axes and latus-rectum.

3. Resolve into five factors the expression $a^5 (b - c) + b^5 (c - a) + c^5 (a - b) + abc(b - c)(c - a)(a - b)$.

4. Apply the Binomial Theorem to shew that $\left( {{\textstyle{3 \over 4}}} \right)^{{\textstyle{4 \over 5}}} = 0.7944$ approximately.

5. Shew that $\sin ^2 12^\circ + \sin ^2 21^\circ + \sin ^2 39^\circ + \sin ^2 48^\circ = 1 + \sin ^2 9^\circ + \sin ^2 18^\circ $.

6. $A_1A_2...A_n$ is a regular polygon of $n$ sides and $P$ any point between $A_1$ and $A_n$ on its circumcircle which is of radius $R$. If $PA_1$ subtend an angle $2\alpha$ at the centre, shew that the sum of the chords $PA_1, PA_2, …, PA_n$ is $2R\left( {\cos \alpha \cot {\textstyle{\pi \over {2n}}} + \sin \alpha } \right)$.

7. Shew that the equation to the circumcircle of the triangle formed by the lines $y = \pm kx$ and $x\cos \alpha + y\sin \alpha - p = 0$ is $(\cos ^2 \alpha - k^2 \sin ^2 \alpha )(x^2 + y^2 ) - p(1 + k^2 )(x\cos \alpha - y\sin \alpha ) = 0$.

8. $PQ$ is a double ordinate of a parabola, and the line joining $P$ to the foot of the directrix cuts the curve in $P’$. Shew that $P’Q$ passes through the focus.

9. A balance has its arms unequal in length and weight. A certain article appears to weigh $Q_1$ or $Q_2$ according as it is put into one scale-pan or the other. Similarly another article appears to weigh $R_1$ or $R_2$ . Shew that the true weight of an article which appears to weigh the same in whichever scale-pan it is put is ${\textstyle{{Q_1 R_1 - Q_2 R_2 } \over {Q_1 - Q_2 - R_1 + R_2 }}}$.

10. A square lamina rests in a vertical plane perpendicular to a smooth vertical wall, one corner being attached to a point in the wall by a string whose length is equal to a side of the square. Shew that in the position of equilibrium the inclination of the string to the wall is $\cot ^{ - 1} 3$.

11. A smooth wedge is placed on a smooth table, the principal vertical section of the wedge being a right-angled triangle, whose hypotenuse is inclined to the horizontal at an angle $\alpha$. A string passing over a pulley at the top of the wedge connects a mass $m$ hanging freely with a mass $m'$ on the plane. If the mass $m$ descend, prove that in order to keep the wedge from sliding a horizontal force will be required equal to ${\textstyle{{m'(m - m'\sin \alpha )\cos \alpha } \over {m + m'}}}g$.

12. Prove that the greatest range of a particle, projected with a given velocity, on a given inclined plane, is four times the greatest vertical altitude above the inclined plane.


สำหรับข้อ 9-12 เป็นการประยุกต์คณิตศาสตร์กับโจทย์กลศาสตร์พื้นฐาน ซึ่งน้องๆ บางคนอาจจะยังมองภาพไม่ออก!
แต่สำหรับข้อ 1-8 น่าจะพอทำได้ด้วยความรู้ที่เรียนกันมาแล้ว
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