HL Practice Test 5th October 2014

1. In how many ways can a committee of 9 can be formed from 10 men and 12 women such there are at least 3 men and not more than 5 women

2. There are two parallel lines. One has got m points and the other line has got n points. In how many ways can you make a triangle by joining three points amongst these (m+n) points

3. The first term of an arithmetic sequence is 0 and the common difference is 12. (a) Find the value of the 96th term of the sequence(b) The first term of a geometric sequence is 6. The 6th term of the geometric sequence is

equal to the 17th term of the arithmetic sequence above. What is the common ratio of the geometric sequence?

4. Consider the geometric sequence 16, 8, a, 2, b, …

What is the value of a and b?

The sum of the first n terms is 31.9375. Find the value of n.

5. The population of big cats in Africa is increasing at a rate of 5 % per year. At the beginning of 2004 the population was 10 000.

Write down the population of big cats at the beginning of 2005?

Find the population of big cats at the end of 2010?

6. A swimming pool is to be built in the shape of a letter L. The shape is formed from two squares with side dimensions x and √ x as shown. (a) Write down an expression for the area A of the swimming pool surface.(b) The area A is to be 30 m2. Write a quadratic equation that expresses this information.(c) Solve the quadratic equation to get the values of x.(d) What are the dimensions of the pool?

7. If f ( x )=3−x∧g ( x )=3x, x ≠0 ,

Find p ( x ) if p (x )=f (g ( x ) )

If q ( x )= 33− x

, x ≠3 , find p (q ( x ) )∈simplest terms.

8. Find all of the values of x that satisfy the inequality

2x|x−1|

<1.

9. The functions f and g are defined by f : x↦ex

, g : x↦ x+2 . Calculate

(a) f−1(3 )×g−1 (3) (b) ( f ∘g )

−1(3 ).

10.Find the coefficient of x11 in the expansion of (x2−2

x )10

11. Using induction prove that:

∑i=0

n

( i+2)2i=(n+1)2n+1 , for all n≥0.

12. Using induction on n, prove for all non-negative integers n, (7n+2 + 52n+1) | n13. The following diagram shows part of the graph of f (x) = x2 − x − 2.

Find the x-intercepts, the y intercept and the vertex.

14. Find the domain of the following functions.

a)

b)

c)

d)

e)

f) f ( x )=|4−x|

g) f ( x )=log8(x+5)

15. Given the function

a) Write the function in the form b) Find the vertexc) Find the x and y-intercepts

16. If the coefficients of three consecutive terms in the expansion of (1 + x)n are 120 , 210 and 252 respectively, find the value of n

17. The sum of the first three numbers in an arithmetic sequence is 24. If the first number is decreased by 1 and the second number is decreased by 2, then the third number and the two new numbers are in geometric sequence. Find all possible sets of three numbers which are in the arithmetic sequence.

18. Let a1, a2, … , ak be a finite arithmetic sequence with

a4 + a7 + a10 = 17 and ∑i=4

14

ai=77.

If ak = 13, what is the value of k?

19. The graph of the quadratic function f(x) = ax2 + bx + c has y-intercept y = 5 and has vertex at the point (−2, 1). Find the values of a,b and c.

20. How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed?

21. How many five digit positive integers that are divisible by 3 can be formed using the digits 0, 1, 2, 3, 4 and 5, without any of the digits getting repeating

22. There are 7 men and 6 women in an office. How many ways are there to make a committee of five if:

a) Gender is irrelevant?

b) The committee must be all male?

c) The committee must consist of 2 men and 3 women?

23. How many positive integers less than 700 can be made using only digits that are prime numbers?

24. How many ways are there to rearrange the letters of the word MISSPELLED?25. In how many ways can 11 books on English and 9 books on French be placed in a row on a

shelf if: a) There are no restrictionsb) All English books must be togetherc) No two books on French may be together?