IB Mathematics HL Year I—Skills Test (Take Home)

[Pages:14]IB Mathematics HL Year I--Skills Test (Take Home)

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Instructions and Comments. The problems on this skills test are problems taken almost verbatim from Mr. Corbett's IB Math HL Year 1 final exam, December, 2004 and account for the vast majority of this examination. However, as you will soon agree, these problems are really review, mostly from Algebra II/Trigonometry, but some requiring material from Precalculus. You have one week to complete these problems. Write the solutions in the spaces indicated along the right-hand margin. (There are 157 points on this skills test.)

1. (3 pts) Solve 2x + 1 > 3x - 4 and graph the solution on the number line provided.

-

2. (3 pts) Solve |6x - 1| 11 and sketch the solution on the number line provided.

-

3. (3 pts) Divide x3 + 2x2 - x - 2 by x - 1.

 4. (3 pts) Rationalize the denominator in 3 + 2 and simplify your result as much

3-2 as possible.

5x - 3

5. (3 pts) Solve for x in the equation 3x - 5 =

.

x

6. (3 pts) Give the exact value of cos 210 sin(-60).

7. (3 pts) If (3, -4) is on the terminal side of an angle in standard position, find the exact value of csc .

8. (3 pts) Find the x-intercept(s) of the graph of y = x2 - 3x + 2.

9. If f (x) = 4x - 3 and g(x) = x2, x > 0, find -3

(a)(1 pts) f 4

(b)(3 pts) all value(s) of x satisfying f (x) = g(x).

(c)(1 pts) f (g(-2)). (d)(3 pts) Let h(x) = 1 . Compute h-1(x) and state the domain and range

g(x) of h-1.

h-1(x) Domain of h-1(x) Range of h-1(x) (e)(2 pts) Determine whether or not the function k(x) = xg(x) (here allow x to take all real values) is an odd function, showing your work.

10. In ABC, CA = 8, AB = 12 and A = 25. Find to two decimal places: (a)(5 pts) BC

(b)(3 pts) the area of ABC.

11. A runner left home (point A) and travelled on a bearing of 60 for 550 m to point C, then travelled on a bearing of 120 for 700 m to point B. (a)(3 pts) Draw a rough diagram of this situation.

(b)(5 pts) How far is it from point A to point B, to two decimal places?

(c)(3 pts) What is the bearing from his home to point B, to the nearest degree?

12. (2 pts) Find the value of y if the slope of the line passing through the points (-3, -4) and (-5, y) is -2

13. (2 pts) Find the values of a and b if the midpoint of the line segment with end points A(-2, -4) and B(a, b) is M (5, -8).

a=

b=

14. (2 pts) Find the distance between the points A and M in question #13.

15. (2 pts) Find the equation of the perpendicular bisector of the line segment joining the points A(3, -2) and B(7, 6).

16. (3 pts) Find the equation of the line passing through the points of intersection of the curves with equations y = 2x2 - 3x and y = x2.

17. (2 pts) Find the maximum and minimum values of the function f (x) = 3 cos(2x- )

18. (2 pts) Use your graphing calculator to find the first-quadrant solution of the equation 1 + sin 2x = sin x. Express your answer in radians, correct to three decimal places.

3

19. (2 pts) Solve the equation cos x = - , where = - x . 2

20. (5 pts) Find all real solutions of the equation 2 cos2 x - 5 cos x + 2 = 0.

21. (3 pts) Use a trigonometric identity to show how you would solve the equation 5 - 5 cos x = 3 sin2 x. Do not solve; leave your answer as a product of two factors equal to 0.

22. (3 pts) Prove the identity csc x = cos x.

cot x + tan x

23. (3 pts) Prove the identity sin 2x = tan x.

1 + cos 2x

24. (2 pts) Find the exact value of sin 75 using the identity for sin(A + B).

25. (2 pts) Write sin 35 cos 50 + cos 35 sin 50 as a single trig function.

26.

(3

pts)

If

A

is

a

Quandrant

II

angle

with

sin A

=

3 5

and

B

is

a

Quadrant

III

angle

with

cos B

=

-4 5

,

find

the

exact

value

of

cos(A

+

B).

27. (3 pts) Solve sin 2x + sin x = 0, exactly, where - x .

28. (1 pt) Convert 25 to exact radian measure.

29. (3 pts) The height, h, of a projectile fired from ground level, varies partly as the square of t and partly as t. When t = 2, h = 80 and when t = 4, h = 120. Find the value of h when t = 10.

30. (2 pts) Write as a single logarithm: logx 12 -

1 2

logx

9

+

1 3

logx

8

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