高二国际班试卷2.doc

学校 班级 姓名 学号 A
密 封 线 内 请 不 要 答 题
2013——2014学年第二学期期中考试试卷
高二国际班 数学
命题人：王瑞
1、[12] (1) In the diagram，O is the centre of a circle ABCD，
and BCis parallel to OD.Given that AOB is a
straight line and AD is produced to E，
prove that
(a) AD=CD (b)
(2) In the diagram，AB is a chord and P is the point
of intersection of BP and the tangent ai A. If AC
bisects , prove that
(a) CA=CB
(b)

2、[15] (1) Given that， and that A and B are in the same
quadrant，evaluate
(a) (b) (c)
(2) The curveintegers and
an amplitude of 4 and a period of The maximum value of y
is 6， (a) State the value of a，b and c (b) Sketch this curve.

3、[20] (1) Given that，find the exact value of .
(2) Without using a calculator，find the values of
,if is obtuse.
(3) Find the maximum and minimum values of and the
corresponding values of ,where .
(4) Without using a calculator，express in the form .

4、(1). Given that ，find and hence，obtain the equation of the
tangent to the curve at the point where the curve crosses the line .[6]
(2) The tangent and the normal to the curve at the point P(7,12) cut the x-axis at points M and N respectively .Calculate the area of the triangle PMN. [7]

5、A curve has the equation
( a ) Find the gradient of the curve when . [4]
( b ) Given that x is increasing at a constant rate of 0.06 units per second, find
the rate of change of y when . [6]

`

6、[10] A piece of wire of length 2400cm is used to make the edges of a rectangular
box ,as shown. The box has breadth cm, length cm and height cm .
( a ) Express in terms of .
( b ) Show that the volume of the box , ,is given by.
( c ) Find the value of for which the volume is a maximum.

7、(a) Given the curve Calculate the value of
for which the curve (1)meets the x-axis,(2) has turning points. [6]
(b) Find the coordinates of the turning point of the curve
and determine the nature of this turning point . [5]
(c) Find the coordinates of the stationary point of the curve.[5]

8、(a) Prove that ： [4]
(b) Differentiation ： [6]
学校 班级 姓名 学号 A
密 封 线 内 请 不 要 答 题
(c) Integration ： (1) (2) [6]

9、[18] (a) Given that evaluate
(b) Show that ，hence，evaluate
(c) Given that ，find
(1) (2) .
Find the value of the constant for which

10、[20] A curve is such that .
( a ) Given that the curve passes through the point (1, 5), find the equation of the
curve.
( b) Find the x-coordinates of the stationary points of the curve.
( c) Obtain an expression for and hence, or otherwise, determine the nature of each stationary point.
高二年级 数学 第3页 共3页