Information Technology Department - Math Section General Foundation Programme FPMP0003 Pure Mathematics – Full book worksheet Student Name Student Id Group No. Chapter – 1: f (3) g (0) . 2h(9) ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 2. Find the following for the function f ( x) 3x 2. 1 1. Let f ( x) x 2 1, h( y) y 2 and g (m) 2m 3 . Find the value of a) f ( x h) f ( x ) h b) f (1 h) f (1) h c) f ( x) f (a) xa ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 3. Find the domain of the following functions: z f ( z) 2 a) f ( x) 3x 2 b) c) f ( x) 3 2 x 1 z 16 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 1 4. Test if the given graph represents a function by using the vertical line test. 5. Test if the given graph represents a one to one function by using the horizontal line test. 6. Which of the following are functions? For functions, write the domain, co-domain and range. 7. Classify the following functions as one to one, onto, bijective or none. 2 8. Which of the following are functions? Specify the domain and range for each function: (a) S 1,1, 2,1, 3,1, 4,1 (b) F 3,1, 1,3, 1,1, 1,4, 1,2 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 9. Determine which of the following equations define a function. (a) 3x 2 y 3 1 (b) y 2 2 x 1 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 10. Let A a, b, c, d and B 1,2,3,4. If the following arrow diagram defines a function f then find the following: t and g (t ) 2 t , then find f g 3 and g f 3 t 3 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 11. If f (t ) 3 12. Find inverse function f 1 ( x) for the following functions: 1 (a) f ( x) x 7 (b) f ( x) 2 x 3 2 ……………………………………………………………………………………………… ……………………………………………………………………………………………… …………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 13. Solve the following quadratic equations using the quadratic formula : 2 y 2 6 y 7 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 14. Find the slope and intercepts, and then sketch the graph of the linear function 3x 2 y 6 ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. ………………………………………….. 4 15. Graph the function f ( x) x 2 4 and find the following: 2 16. Graph the function f ( x) x 2 4 x 5 and find the following: 17. The height of a ball at t seconds after it is thrown is modeled by the function h(t ) t 2 2t 8 , where h(t ) is the height of the ball in meters? (a. When does the ball reach the maximum height? (b) Find the maximum height reached by the ball. ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… 5 Chapter – 2: 1. Sketch the graph of the following exponential and logarithmic functions: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 2. Solve for 𝑥 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 6 3. Rewrite in equivalent logarithmic form. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 4. Rewrite in equivalent exponential form. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 5. Find 𝑥, 𝑦 or 𝑏 for the following logarithmic equations. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 6. Write the below expression in terms of single logarithm. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 7. If log 𝑥 = 2.5 𝑎𝑛𝑑 log 𝑦 = 4, then find the value of log 𝑦 2 𝑥. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 8. Solve the logarithmic equation for x ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 7 9. Simplify: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 10. Use Calculator to find the value to 3 decimal places ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 11. At the start of the experiment, the population of the bacteria was 3000. Two hours later, the population is 4200. a) Determine the growth constant k . ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… b) Determine the population 6 hours after the start of experiment. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 12. The half – life of Plutonium – 241 is 10 years. How much of an initial 5 g sample will remain after 5 years? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 8 Chapter – 3: 1. Classify each type of angle below and draw an approximate diagram to represent them. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 2. Find the measures of angles a, b, c and identify the type of each angle: 3. Find the complementary angle of each of the angle given below: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 4. Find the radian measure of each angle below: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 5. Find the degree measure of each angles below: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 8 6. Given tan , is in the second quadrant, find all other trigonometric ratios . 6 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 9 7. Find the coordinates of the point on a unit circle corresponding to the angle 130°. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 8. Verify the following identities: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 9. Find the exact value: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 10. Simplify using sum and difference properties: ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 10 Chapter – 4: 1. Find the exact value for each of the following: √3 a) 𝑠𝑖𝑛−1 (− 2 ) √3 b) 𝑐𝑜𝑠 −1 ( 2 ) ……………………………………………………………………………………………… ……………………………………………………………………………………………… 2. Evaluate exact values for each of the following: a) 𝑠𝑖𝑛−1 (−0.35) 4 b) 𝑡𝑎𝑛 (𝑐𝑜𝑠 −1 (− 5)) ……………………………………………………………………………………………… ……………………………………………………………………………………………… 3. Find all possible solutions exactly on the interval 0 ≤ 𝜃 ≤ 2𝜋. a) 2𝑐𝑜𝑠𝜃 = √3 b) 𝑐𝑠𝑐 2 𝑥 − 2 = 0 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 4. Solve the triangle ABC, given 𝑐 = 25, ∠𝐴 = 35° , ∠𝐵 = 68° . ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 5. Determine the number of triangles possible. Given 𝑏 = 100𝑚, 𝑎 = 60𝑚, ∠𝐴 = 28° . ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 11 6. Find the the shown below the altitude of aircraft from ground level in the figure ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 7. Solve the triangle, given ∠𝐴 = 35° , 𝑏 = 15𝑐𝑚, 𝑐 = 20𝑐𝑚. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 8. Solve the triangle, given 𝑎 = 12𝑐𝑚, 𝑏 = 15𝑐𝑚, 𝑐 = 20𝑐𝑚. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 12 9. Suppose a boat leaves port, travels 15 miles, turns 20 degrees and travels another 7 miles. How far from port is the boat? The following figure illustrates the distance of the boat from the terminal point. ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… ……………………………………………… Chapter – 5: 1. The marks of 10 students in a class are given below. Find the mean, median and mode. 20 23 15 20 10 12 20 9 23 25 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 2. Find the value of p if the mean if the following distribution is 3.5 x 1 2 3 4 5 f 5 1 p 7 8 ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 13 3. Find the mean of the following frequency distribution: Class 5-10 10-15 15-20 20-25 Frequency 1 3 2 4 4. The following data represent the colour of hair of 20 students in a class. Construct a categorical frequency distribution for the given below data. (W-White, B-Black, BR-Brown, R-Red, G-Grey) B W BR B R B BR W B R W BR B R W R B G G B 5. The following data represents the total number of customers visited to a mobile shop on each of 24 days. Construct a grouped frequency distribution for the data using 5 classes. 25 40 35 41 30 35 41 46 47 27 36 48 48 30 46 45 29 45 39 38 33 34 42 44 14 6. The table given below shows the favorite color of students in a class. Draw the bar graph. Favorite color Blue Green Red Purple No. of Students 15 10 20 30 7. Draw a Histogram for the following frequency distribution. Class Interval 0-20 20-40 40-60 60-80 80-100 Frequency 5 10 12 20 15 8. A software company ordered computers for 1500 OMR, Air conditioners for 1000 OMR, Chairs for 200 OMR, Cupboards for 200 OMR and Printers for 800 OMR for its new office. Construct a pie chart. 15 Chapter – 6: 1. Two coins are tossed simultaneously. Write the sample space and find the probability of getting a) 2 heads b) exactly 1 head …………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 2. A die is rolled. Find the probability of getting a) a multiple of 3 b) a number less than 5 or an even number …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 3. Two dice are rolled. Find the probability of getting a) a sum 9 b) both are even numbers ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 4. One ball is selected at random from a bag containing 5 red balls, 2 yellow balls and 4 white balls. Find the probability of selecting a red ball or a white ball. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ………………………………………………………………………………………… ……………………………………………………………………………………………… 5. A die is rolled. Let A be the event of getting an even number and B be an event of getting a number less than 6, find 𝑃(𝐴 ∪ 𝐵) and 𝑃(𝐴 ∩ 𝐵). …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 6. If a card is drawn from a deck, then find the probability of getting an ace or a diamond. …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 16 7. A die is rolled two times, and observe the numbers that faces up. Find the probability that the sum of the number is 8, given that the first number is 3. …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 8. Two coins are tossed together. Find the following probabilities using a tree diagram. a) Probability of getting two tails b) Probability of getting no tails c) Probability of getting at least one head …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 9. Ali has a bag with seven black marbles and three white marbles in it. He picks up a marble from the bag at random, and then picks another at random after replacing the first marble. With the help of a tree diagram calculate the probabilities that he picks: a) Two black marbles b) Two white marbles …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 10. Ali has a bag with five red balls and three green balls in it. He picks up a ball from the bag at random, and then picks another at random without replacing the first ball. With the help of a tree diagram calculate the probabilities that he picks: a) at least one red ball b) one red and one green ball …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 17 11. Find the number of words, with or without meaning, that can be formed with the letters of the word PROBABILITY. ……………………………………………………………………………………………… ……………………………………………………………………………………………… 12. How many ways are there to arrange 7 books on a shelf, taking all books at a time? ……………………………………………………………………………………………… ……………………………………………………………………………………………… 13. How many 2 digit numbers can we make using the digits 5,2 and 7 without repetitions? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 14. In how many ways can we select 11 members in a football team from 20 members? ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 15. How many 3 digit numbers can we make using the digits 1,2,3,4 and 5. Assuming that a) repetition of the digits is allowed. b) repetition of the digits is not allowed. ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… ……………………………………………………………………………………………… 16. A 3-digit password is selected. What is the probability that there are no repeated digits? …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… 17. A Math class consists of 14 girls and 15 boys. During the lesson, the teacher randomly selects 6 of the students to demonstrate their work on the board. a) What is the probability that all students selected are girls? b) What is the probability that 3 boys and 3 girls are selected? ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… 18
