加拿大安大略国际学院
Ontario International College
Mathematics Entrance Test (2nd set)
Duration: 2 hours
1. In the diagram, the tangents to the two circles intersect at 90o
as shown. If the radius of the smaller circle is 2, and the radius
of the larger circle is 5, what is the distance between the
centres of the two circles?
(如图所示，两条相互垂直的直线相切于两个圆。如果小园
半径是 2，大圆半径是 5，求两圆圆心之间的距离是多少)
2. For what value(s) of k will the graphs of the functions 12 +−= xy and kxy ++= 2
intersect?
（求 k为何值时方程式 12 +−= xy 和 kxy ++= 2 所
表
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示的曲线相交？
3. If ,
6
131 =+
x
x determine all values of .12
2
x
x +
如果 ,
6
131 =+
x
x ，求 .122 xx + 所有的解。
4. If ,log what is the value of 3log 1010 yx += ?y
x
如果 求 ,log3log 1010 yx += y
x 的解。
5. Solve the following trigonometric equation: .4tan2sec 2 =− xx
求三角方程 的解。 .4tan2sec 2 =− xx
6. A hyperbola has centre O(0,0) and one vertex A )0,6( . Hyperbola passes though B(9,5).
Determine the equation of the hyperbola.
一个双曲线的中心为 O（0,0）， 一个顶点为 A )0,6( 。双曲线穿过 B（9，5），
求双曲线的方程。
7. Give the formula for the common term tn of the following sequence: 2, 5, 10, 17,
26, 37, …
求数列 2, 5, 10, 17, 26, 37, …的公式 tn。
8. In determining the height, MN, of a tower on an island,
two points A and B, 100 m apart, are chosen on the same
horizontal plane as N. If ∠ NAB = 108o, ∠ ABN = 47o
and ∠ MBN = 32o, determine the height of the tower.
岛上有一座塔，塔高为MN，设 A点和 B点与 N在
同一水平面上，A、B相距 100米。如果∠ NAB =
108o, ∠ ABN = 47o and ∠ MBN = 32o, 求塔高MN的值
9. If a class of 10 students has five men, how many ways can the men and women be
arranged in a circle so that no two men sit next to each other?
如果一个班有 10名学生，5名男生，5名女生，如果全班同学围成一个园，求有多
少种方法可使任何两个男生都不会相邻而座？
10. Alice is taking a probability class and at the end of each week she can be either up-to-
date or she may have fallen behind. If she is up-to-date in a given week, the probability
that she will be up-to-date (or behind) in the next week is 0.8 (or 0.2, respectively). If she
is behind in a given week, the probability that she will be up-to-date (or behind) in the
next week is 0.4 (or 0.6, respectively). Alice is (by default) up-to-date when she starts the
class. What is the probability that she is up-to-date after three weeks?
Alice准备参加一个试验班，她每周末要么升级要么降级。如果她在规定的一周内
升级成功，则下一周升级的概率是 0.8 （即降级的概率是 0.2）。如果她在规定的
一周内降级，则下一周升级的概率是 0.4（即降级的概率是 0.6）。在课程开始的时
间，默认为升级。求三周后她会升级的概率是多少。
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