I. Two Were Flat Backs Ii. at Least 3 Were Flat Backs

Corrected 5/1/2015 HL Mathematics Summer Packet

Statistics

1. a. Expand

b. In New Zealand in 1946 there were two different coins of value two shellings. These were ‘normal’ kiwis and ‘flat back’ kiwis, in the ration 3:1. From a very large batch of 1946 two shelling coins, five were selected at random. Find the probability that:

i. two were ‘flat backs’ ii. At least 3 were ‘flat backs’

iii. at most 3 were ‘normal’ kiwis

2. An infectious flu virus is spreading through a school. The probability of a randomly selected student having the flu next week is 0.3. a. Mr C has a class of 25 students,

i. Calculate the probability that 2 or more students will have the flu next week.

ii. If more than 20% of the students have the flu next week, a class test will have to be cancelled. What is the probability that the test will be cancelled. b. If the school has 350 students, find the expected number that will have the flu next week.

3. Apples from a grower’s crop were normally distributed with a mean 173 grams and standard deviation 34 grams. Apples weighing less than 130 grams were too small to sell. a. Find the proportion of apples from this crop which were too small to sell. b. find the probability that in a picker’s basket of 100 apples, up to 10 apples were too small to sell

4. A university professor determines that no more than 80% of this year’s History candidates should pass the final examination. The examination results were approximately normally distributed with a mean 62 and standard deviation 13find the lowest score necessary to pass the examination.

5. The length of steel rods produced by a machine is normally distributed with a standard deviation of 3 mm. It is found that 2% of all rods are less than 25 mm long. Find the mean length of rods produced by the machine.

Logarithms and Exponential functions

Simplify:

1. 2. 3. 4. 5. 6.

Solve for x

7. 8. 9.

Solve exactly for x

10. 11.

Complex Numbers

1. If and Find in simplest rectangular form (a + bi)

a. b. c.

d. e. f.

3 2. Simplify: a.

3. Given Re means the real part and Immeans the imaginary part of a complex number. If and find:

2 a. Im(4z-3w) b. Re(zw) c.Im(iz )

4. For polar form, a complex number is or . Where r is the magnitude and  is (much like a vector)

Find the polar form of a) b) 1 c) d)

Trigonometric Identities

Prove each identity by transforming the left side into one or the expressions listed on the right. SHOW ALL YOUR WORK 1. = a. b.

2. = a. b. 1

3. + 3 = a. b.

4. = a. b.

5. = a. b.

6. Show that

7. Solve for where a). b)

8. Use the double angle formula to show that a) b)

5 Derivatives: Find the derivative of the following functions

9. If Show that

10. If Show that . Integration

dy  10e 2x  5 dx 1. A gradient function is given by . When x = 0, y = 8. Find the value of y when x = 1. 2. 3.

6. 7. s(0)=100

APPLICATIONS OF INTEGRATION AND DIFFERENTIATION

1 1. It is given that f (x)dx = 5.

1 (a) Write down 2 f (x)dx.

7 3 1 2 (b) Find the value of (3x + f (x))dx.

2 2. A part of the graph of y = 2x – x is given in the diagram below.

The shaded region is revolved through 360 about the x-axis.

(a) Write down an expression for this volume of revolution.

(b) Calculate this volume. 3. You are designing a poster to contain 50 in2 of printing with margins of 4 in each at the top and bottom and 2 in at each side. What overall dimensions will minimize the amount of paper used?

–1 4. A car starts by moving from a fixed point A. Its velocity, v m s –t after t seconds is given by v = 4t + 5 – 5e . Let d be the displacement from A when t = 4.

(a) Write down an integral which represents d.

(b) Calculate the value of d.

9 VECTORS

1. Find the vector equation of the line that passes through

a) A(1,2,1) and B(-1,3,2) b) C(0,1,2) and D(3,1,-1)

2. Find the parametric equation of the line which passes through (0,1,2) in the direction of i+ j – 2k.

3. A particle at P(x(t),y(t)) moves such that x(t)=1+2t and y(t) =2-5t with . The distances are in cm and the time is in seconds.

a) Find the initial position of P

b) Illustrate the initial part of the motion of P where t=0,1,2,3

c) Find the velocity vector of P

d) Find the speed of P