The Leading Maths - 10

350 Allied The Leading Mathematics-10 Probability 351 Statistics and Probability 7. A bag contains 6 red counters and 4 white counters. A counter is drawn at random, note colour and replace, then second counter is drawn, note colour. Calculate the probability of getting; (a) two white counters (b) two red counters (c) two counter of same colour (d) first red and second white (e) one red and other white 8. A die is thrown twice. Find; (a) P(5 on both throws) (b) P(4 on both throws) (c) P(same numbers on both throws) 9. One box contains 3 white balls and 5 black balls, another contains 9 white balls and 3 black balls, one ball is selected at random from each bag. Calculate the probability of selecting; (a) two black balls. (b) one of each colour. (c) two white colour balls. 10. A special deck of cards consists of only the aces and face cards, what is the probability of (drawing without replacement). (a) a King followed by an Ace? (b) a Queen followed by a Queen? (c) an Ace followed by a King or a King followed by a Queen? 11. A card is picked at random from a pack of nine numbered cards. It is not replaced and a second card is picked. The cards are numbered; 2 3 4 5 6 7 8 9 10 (a) Calculate the probability that both cards will show prime numbers. (b) What is the probability that only one card will have prime number? (c) Find the probability that the product of the two numbers will be odd. 12. From the deck of number cards numbered from 9 to 35, a card is drawn at random. Calculate the probability of getting; (a) divisible by 7 and divisible by 6 (b) prime number and multiple of 5 13. From the deck of letter cards of the word the given word, a card is drawn at random. (i) COMMUNICATION (ii) QUANTITATIVE Calculate the probability of occurring the letter one after another with replacement; (a) N and I (b) T and N (c) U and N and I

352 Allied The Leading Mathematics-10 Probability 353 Statistics and Probability 14. (i) When a coin and a die are thrown together, calculate the probability of getting; (a) Tail on coin and composite number on die (b) Head on coin or odd number on die (ii) When a die is rolled and a spinner having 5 equal sectors 1, 2, 3, 4, 5 is rotated together. Calculate the probability of getting; (a) 1 on die or prime number on spinner (b) even on die and odd number on spinner 15. An unbiased die is rolled and the obtained results are shown in the table below: Result 1 2 3 4 5 6 Frequency 10 15 20 30 25 25 (a) How many times was the die rolled ? (b) What is the probability of occurring 1 and 6 ? Find it. (c) Calculate the probability of occurring the frequency less than 4 at once. 3. (a) 1 2 (b) 1 6 , 1 6 4. (a) 1 9 (b) 2 9 5. (a) 1 16 (b) 3 8 (c) 9 16 6. (a) 4 25 (b) 12 25 (c) 13 25 7. (a) 4 25 (b) 9 25 (c) 13 25 (d) 6 25 (e) 12 25 8. (a) 1 36 (b) 1 36 (c) 1 6 9. (a) 5 32 (b) 7 8 (c) 9 32 10. (a) 1 15 (b) 1 20 (c) 2 15 11. (a) 1 6 (b) 5 18 (c) 1 6 12. (a) 16 729 (b) 14 243 13. (i) (a) 4 169 (b) 2 169 (c) 4 2197 (ii) (a) 1 27 (b) 1 48 (c) 1 864 14. (i) (a) 1 6 (b) 1 4 (ii) (a) 1 10 (b) 3 10 15. (a) 125 (b) 2 125 (c) 9 25 Answers Project Work Three friends A, B and C are playing the game with playing cards. Two cards are drawn from a deck of playing cards without replacement by each person. Find the probabilities of getting both King cards for A, first Ace and second spade cards for B and both Jack cards or Queen cards.

352 Allied The Leading Mathematics-10 Probability 353 Statistics and Probability 14.3 Probability By Tree Diagram At the end of this topic, the students will be able to: ¾ construct probability tree diagram and calculate the probability of events using tree diagram. Learning Objectives I Tree Diagram for Independent Events When attempting to determine a sample space, the possible outcomes from an experiment of two or more simultaneous events, it is often helpful to draw a diagram, such diagram is a tree diagram. The tree diagram can be used to determine the probability of individual outcomes within the sample space. Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ..... tree diagrams to the rescue! The probability of any outcome in the sample space is the product (multiply) of all possibilities along the path or branch that represents the corresponding probability of the events written besides the branches of the tree diagram. This is called probability tree diagram. II Tree Diagram for Independent Events An experiment that has constant probability in each step of trail is the probability for independent events. Here is a tree diagram for the toss of a coin: There are two “branches” {Head (H) and Tail (T)}. The probability of each branch is written on the branch. The outcome is written at the end of the branch. We can extend the tree diagram to two tosses of a coin: How do we calculate the overall probabilities? (a) We multiply probabilities along the branches. (b) We add probabilities down columns. 1 2 1 2 1 2 1 2 1 2 1 2 HH 1 2 × 1 2 = 1 4 1 2 × 1 2 = 1 4 1 2 × 1 2 = 1 4 1 2 × 1 2 = 1 4 HT P(T) = P(T) = P(T) = TH TT Add Multiply Outcomes 1.00 H T H H H T T T P(H) = P(H) = P(H) = 0.5 0.5 Outcome Probability "Branch" Head Tail H T

354 Allied The Leading Mathematics-10 Probability 355 Statistics and Probability Example-1 Show the sample space for tossing one coin and rolling one die. Also, find the probability of each outcome. (H = heads, T = tails) Solution: By following the different paths in the tree diagram, we can arrive at the sample space. 1 H1 1 T1 4 H4 4 T4 2 H2 2 T2 5 H5 5 T5 3 H3 3 T3 6 H6 6 T6 Outcomes H T Start 1 2 1 2 1 6 1 6 Sample space: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} The probability of each of these outcomes is 1 2 × 1 6 = 1 12. [The Counting Principle could also verify that this answer yields the correct number of outcomes: 2 × 6 = 12 outcomes.] Example-2 A family has three children. How many outcomes are in the sample space that indicates the sex of the children? Assume that the probability of male (M) and the probability of female (F) are each 1 2 . Solution: Construction of the diagram, M MMM 3rd Child 2nd 1 Child st Child Start F MFF F FMF F MMF M FFM M MFM M FMM F FFF Sample space 1 2 1 2 1 2 1 2 1 2 1 2 M F M F M F There are 8 outcomes in the sample space. The probability of each outcome is 1 2 × 1 2 × 1 2 = 1 8 .

354 Allied The Leading Mathematics-10 Probability 355 Statistics and Probability Example-3 A bag contains 3 red balls and 5 blue balls. Draw a ball at random and observe and replace it. Another ball is drawn in random. (a) Prepare the sample space of all possible outcomes in the table. (b) Construct probability tree. (c) Calculate the probability of drawing both balls red. Solution: (a) Let us number the balls as follows R1 , R2, R3 for red balls and B1 , B2, B3, B4, and B5 for blue balls. The sample space is shown in the following two–way table. R1 R2 R3 B1 B2 B3 B4 B5 R1 R2 R3 R1 R2 R3 R4 R5 R1 R1 R2 R1 R3 R1 B1 R1 B2 R1 B3 R1 B4 R1 B5 R1 R1 R2 R2 R2 R3 R2 B1 R2 B2 R2 B3 R2 B4 R2 B5 R2 R1 R3 R2 R3 R3 R3 B1 R3 B2 R3 B3 R3 B4 R3 B5 R3 R1 B1 R2 B1 R3 B1 B1 B1 B2 B1 B3 B1 B4 B1 B5 B1 R1 B2 R2 B2 R3 B2 B1 B2 B2 B2 B3 B2 B4 B2 B5 B2 R1 B3 R2 B3 R3 B3 B1 B3 B2 B3 B3 B3 B4 B3 B5 B3 R1 B4 R2 B4 R3 B4 B1 B4 B2 B4 B3 B4 B4 B4 B5 B4 R1 B5 R2 B5 R3 B5 B1 B5 B2 B5 B3 B5 B4 B5 B5 B5 (b) The following diagram is the probability tree of the above experiment. = RR Outcomes = RB = BR = BB 3 8 5 8 3 8 5 8 3 8 5 8 3R 5B Start R B B R B R (c) By using probability tree considering the events of separate experiments and multiplying their probabilities which is as follows; The probability of drawing both balls red, P(RR) = 3 8 × 3 8 = 9 64.

356 Allied The Leading Mathematics-10 Probability 357 Statistics and Probability Example-4 Devi has a pentagonal spinner labelled A, B, C, D and E and a die labelled 1, 2, 3, 4, 5 and 6. Find the probability that first getting A in the spinner and then a prime number (P) on the die by using probability tree. Solution: Here, ocurring event in spinner = A Non-ocurring event in spinner = A Ocurring event in dice, Prime number = P Not ocurrng event in dice, not prime = P Now, drawing probability tree diagram, Die Start AP AP AP A P Spinner Event 3 6 3 6 3 6 3 6 4 5 1 5 A not A not P P not P P The probability that first getting A in the spinner and then a prime number (P) on the die, P(A and P) = P(A) × P(P) = 1 5 × 3 6 = 3 30 = 1 10. Example-5 A bag contains 2 red, 5 blue and 3 green counters. Angdaba selects a counter, notes its colour and replaces the counter. He draws a second counter and notes its colour. Calculate the probability that Angdaba selects: (a) two blue counters and (b) two counters of different colour. Solution: Here, No. of red counter (R) = 2, No. of blue counter (B) = 5 No. of greel counter (G) = 3 Now, drawing probability tree diagram, 2nd draw Outcomes 1st draw = RR = RB = BG = BR = BB = BG = GR = GB = GG 2 10 2 10 2 10 2 10 5 10 5 10 5 10 5 10 3 10 3 10 3 10 3 10 2R 5B 3G Start G G G G B B B B R R R R (a) P(BB) = P(B) × P(B) = 5 10 × 5 10 = 1 4 . E D B C A

356 Allied The Leading Mathematics-10 Probability 357 Statistics and Probability (b) P(different colour) = P(RB) + P(RG) + P(BR) + P(BG) + P(GR) + P(GB) = P(R).P(B) + P(R).P(G) + P(B).P(R) +P(B).P(G) + P(G).P(R) + P(G). P(B) = 2 10 × 5 10 + 2 10 × 3 10 + 5 10 × 2 10 + 5 10 × 3 10 + 3 10 × 2 10 + 3 10 × 5 10 = 10 + 6 + 10 + 15 + 6 + 15 10 × 10 = 62 100 = 31 50 Example-6 Three children are born in a family. (a) Draw a tree diagram to represent the possible order of boys (B) and girls (G). (b) Calculate the probability that; (i) all children are girls. (ii) at least a child is a girl. (iii) two children are boys. Solution: (a) Here, Event of borning boy = B, event of borning girl = G Now, drawing probability tree diagram, 3 Outcomes rd birth 2nd birth 1st birth BBB BBG√ BGB√ BGG GBB√ GBG GGB GGG 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 B G Start G G G G G G G B B B B B B B (b) (i) P(GGG) = P(G) × P(G) × P(G) = 1 2 × 1 2 × 1 2 = 1 8 (ii) P(GGG) + P(GGB) + P(GBG) + P(GBB) + P(B G B) + P(BGB) + P(BBG) = 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 = 7 8 (iii) P(B B G) + P(B G B) + P(G B B) = 1 8 + 1 8 + 1 8 = 3 8 . Example-7 A box contains 6 white balls and 4 black balls. If 2 balls are drawn in succession, what is the probability that both are black if the first ball is replaced before the second is drawn? Solution: Here, No. of white balls (W) = 6, No. of black balls (B) = 4

358 Allied The Leading Mathematics-10 Probability 359 Statistics and Probability Now, drawing probability tree diagram for without replacement, = WW Outcomes = WB = BW = BB 6 10 4 10 6 10 4 10 6 10 4 10 6W 4B Start 1st draw 2nd draw W W B W B B Now, we have ∴ The probability that both are black, P(BB) = 4 10 × 4 10 = 16 100 = 4 25. III Tree Diagram for Dependent Events An experiment that has decreasing probability of the outcome event in each step of trail is the probability for dependent events. Here is a tree diagram for the drawing a ball: There are two “branches” (blue ball and black ball). The probability of each branch is written on the branch. The outcome is written at the end of the branch. We can extend the tree diagram to drawing two balls: But events can also be “dependent” ... which means they can be affected by previous events ... 2 blue and 3 red balls are in a bag. What are the chances of getting a blue ball? The chance is 2 in 5 as shown in the figure above. But after taking one out the chances change! So the next time: (a) if we got a black ball before, then the chance of a blue ball next is 2 in 4. (b) if we got a blue ball before, then the chance of a blue ball next is 1 in 4. See how the chances change each time? Each event depends on what happened in the previous event, and is called dependent. That is the kind of thing we look at here. A tree diagram is a wonderful way to picture what is going on, so let’s build one for our balls example. 2 in 5 2 in 4 1 in 4

358 Allied The Leading Mathematics-10 Probability 359 Statistics and Probability There is a 2 5 chance of pulling out a blue ball and a 3 5 chance for black: We can even go one step further and see what happens when we select a second ball: If a blue ball was selected first there is now a 1 4 chance of getting a blue ball and a 3 4 chance of getting a black ball. If a black ball was selected first there is now a 2 4 chance of getting a blue ball and a 2 4 chance of getting a black ball. Now we can answer questions like “What are the chances of drawing 2 blue balls?” It is a 2 5 chance followed by a 1 4 chance: 2 5 3 5 1 4 3 4 2 4 2 4 2 5 × 1 4 = 2 20 = 1 10 Did you see how we multiplied the chances, and got 1 10 as a result. The chances of drawing 2 blue balls is 1 10. Example-8 Find the probability of drawing 2 Kings from a deck of playing cards by using tree diagram. Solution: The events K1 and K1 are drawn a King and non-King at first respectively, and Event K2 and K2 are drawn a King and non-King at second respectively. Drawing probability tree diagram below, 2 5 3 5 2 5 3 5 1 4 3 4 2 4 2 4

360 Allied The Leading Mathematics-10 Probability 361 Statistics and Probability 1st drawn 2nd drawn 4K1 48K1 P(K2) = 3 51 P(K1) = 4 52 P(K2) = 48 51 P(K2) = 4 51 P(K2) = 47 51 P(K1) = 48 52 3K2 48K2 4K2 47K2 4K3 46K3 3K3 47K3 3K3 47K3 2K3 48K3 For the first card the chance of drawing a King is 4 out of 52 (there are 4 Kings in a deck of 52 cards): P(K1) = 4 52 But after removing a King from the deck the probability of the 2nd card drawn is less likely to be a King (only 3 of the 51 cards left are Kings): P(K2) = 3 51 And so, P(K1 and K2) = P(K1) × P(K2) = 4 52 × 3 51 = 12 2652 = 1 221 So, the chance of getting two Kings is 1 in 221, or about 0.0045%. Example-9 A box contains 6 white balls and 4 black balls. If 2 balls are drawn in succession, what is the probability that both are black if the first is not replaced before the second is drawn? Solution: Here, No. of white balls (W) = 6, No. of black balls (B) = 4 Now, drawing probability tree diagram for without replacement, = RR Outcomes = WB = BW = BB 5 9 4 9 6 9 3 9 6 10 4 10 6W 4B 5W 4B 4W 4B 5W 3B 5W 3B 6W 2B 6W 3B Now, we have, The probability of getting both black balls, P(BB) = 4 10 × 3 9 = 12 90 = 2 15.

360 Allied The Leading Mathematics-10 Probability 361 Statistics and Probability Example-10 A bag contains 2 red balls and 3 green balls. Draw a ball and observe its colour and do not replace it, and again draw a ball. Find sample space by using probability tree diagram, and find the probability of the following events; (a) Two red balls. (b) Two green balls. (c) The first ball is red and the second ball is green. (d) The first ball is green and the second ball is red. Solution: Here, No. of red balls (R) = 2, No. of green balls (G) = 3 Now, drawing probability tree diagram, 1 Outcomes st draw 2nd draw P(G) = 2 4 P(R) = 2 4 P(G) = 3 4 P(R) = 1 4 P(R) = 2 5 P(G) = 3 5 2R 2G 1R 3G 2R 3G 0R 3G 1R 2G 1R 2G 2R 1G = RR = RG = GR = GG Sample space, S = {RR, RG, GR GG} (a) P(RR) = 2 20 = 1 10 ; (b) P(GG) = 6 20 = 3 10 ; (c) P(RG) = 6 20 = 3 10 ; (d) P(GR) = 6 20 = 3 10. PRACTICE 14.3 Read Think Understand Do Keeping Skill Sharp 1. Circle ( ) the correct answer. (a) What is the probability of getting two heads from the given tree diagram ? (i) 1 2 (ii) 1 3 (iii) 1 4 (iv) 1 5 H H H T T T

362 Allied The Leading Mathematics-10 Probability 363 Statistics and Probability (b) What is the probability of getting green ball at first and yellow ball at second times without replacement from the given tree diagram ? (i) 2 3 (ii) 3 10 (iii) 1 10 (iv) 2 3 (c) What is the probability of getting two ace cards without replacement from the given tree diagram ? (i) 1 221 (ii) 5 2652 (iii) 3 221 (iv) 7 2652 (d) What is the probability of birthing different genders from the given tree diagram ? (i) 1 4 (ii) 3 4 (iii) 1 2 (iv) 1 3 (e) What is the probability of birthing the same sex from the given tree diagram ? (i) 1 8 (ii) 3 4 (iii) 3 8 (iv) 1 4 Check Your Performance 2. An urn contains 10 white TT balls and 5 blue TT balls. A ball is selected and then replaced. A second ball is selected. (a) Copy the adjoining tree diagram and complete with filling probabilities on each branch. (b) Calculate the probabilities of the following: (i) both white TT balls (ii) one white and black TT balls green green Start green yellow 1/4 yellow 3/4 2/4 2/4 3/5 2/6 yellow A A A A A A 4A 48A P2(A)=48/51 P2(A)=47/51 P2(A)=3/51 P1(A)=4/52 P1(A)=48/52 P2(A)=4/51 Girl Girl Girl Boy Boy Boy First Child Second Child Girl Girl Girl Girl Girl Girl Girl Boy Boy Boy Boy Boy Boy Boy 2nd 1 Child st Child 3rd Child Couple 10 W 5 B 10 W 5 B 10 W 5 B 10 W 5 B 10 W 5 B 10 W 5 B 10 W 5 B 1st drawn 2nd drawn P(W) = ... P(W) = ... P(B) = ... P(B) = ... P(B) = ... P(W) = 10 15

362 Allied The Leading Mathematics-10 Probability 363 Statistics and Probability 3. A ludo die is thrown three times and their results are shown in the given tree diagram. (a) Copy the tree diagram and complete with filling probabilities on each branch. (b) Write its outcomes. (c) Calculate the probabilities of the following; (i) all 1's (ii) Two 1's and third other 4. 2 white sweets (W) and 2 yellow sweets (Y) are mixed in a box. Three sweets are taken without replacement by closing their eyes. (a) Draw a tree diagram and complete with the probabilities on each branch. (b) Write its sample space S. (b) Calculate the probabilities of the following: (i) WWY (ii) at most one white sweet 5. A pack of 52 cards is well shuffled and drawn three cards for face or non-faced cards without replacement as shown in the adjoining probability tree diagram. (a) Draw a tree diagram and complete with the probabilities on each branch. (b) Write its outcomes. (c) Calculate the probabilities of the following; (i) 1st one faced card and others non-faced cards (ii) 1st two faced cards and next non-faced card 1st thrown 2nd thrown 3rd thrown 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O 1O 5O P(O) = ... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... P(O) = ... 1st selection 2nd selection 3rd selection 0W 2Y 1W 2Y 2W 2Y 1W 1Y 1W 1Y 2W 1Y 2W 0Y 0W 1Y 0W 1Y 1W 1Y 1W 0Y 1W 0Y P(W) = ... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... P(Y) = ... 1st drawn 2nd drawn 3rd drawn Outcomes 12F 48F P(F) = ... ..... ................ ................ ................ ................ ................ ................ ................ ................ ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... P(F) = ... 11F 48F 12F 47F 10F 48F 11F 47F 11F 47F 12F 46F 9F 48F 10F 47F 10F 47F 10F 47F 11F 46F 11F 46F 11F 46F 12F 45F

364 Allied The Leading Mathematics-10 Probability 365 Statistics and Probability 6. At first, a die is thrown and then a coin is tossed. The possible results are shown in the given tree diagram aside. (a) Copy the given tree diagram and complete with filling probabilities on each branch. (b) Write its sample space. (c) Find the probabilities of the following; (i) 1 on die and head on coin (ii) prime number on die and tail on coin 7. By drawing tree diagram, find the probability of getting; (a) three heads while tossing a coin three times, (b) two heads and one tail while tossing a coin three times, (c) one head and two tails while tossing a coin three times. 8. (a) Two cards are drawn from a deck of 52 cards. Find the probability that the two cards will be two Kings if the first card is returned to the deck and reshuffled before the second is drawn by using tree diagram. (b) Two cards are drawn from a deck of 52 cards. Find the probability that the two cards will be Club if the first card is returned to the deck and reshuffled before the second is drawn by using tree diagram. 9. A bag contains 7 red balls and 4 blue balls. Two balls are drawn one at a time at random and with replacement. By using tree diagram, find the probability of getting; (a) 1 red ball and 1 blue ball (b) a blue ball followed by a red ball (c) a red ball followed by a blue ball. 10. A bag contains 3 red balls and 5 blue balls. Two balls are drawn one at a time at random and with replacement. By drawing tree diagram, find the probability of getting; (a) 1st red and 2nd blue (b) 1st blue and 2nd red (c) 1 red and 1 blue. 11. A committee consists of 4 men and 6 women. A subcommittee is formed from it made up of 3 people. Using tree diagram, what is the probability that (a) they are all women, (b) 2 of them are women? 12. (a) Two cards are drawn from a deck of 52 cards. Find the probability by using tree diagram that the two cards will be two Kings if the first card is not returned to the deck before the second is drawn. H T H T H T H T H T H T 1 2 3 4 5 6

364 Allied The Leading Mathematics-10 Probability 365 Statistics and Probability (b) Two cards are drawn from a deck of 52 cards. Find the probability by using tree diagram that the two cards will be two Diamonds if the first card is not returned to the deck before the second is drawn. 13. If the chance of a fine day is 1 5 , find, by using a probability tree diagram, the probability of; (a) two fine days in succession (b) one fine day out of two days. (c) two consecutive days that are not fine. 14. A coin containing Nepal map (N) - Sagarmatha (S) is tossed and then a spinner with 3 sectors 1, 2, 3 are thrown. (a) Show the all possible outcomes in the same tree diagram. (b) Write its sample space. (c) Find the probabilities of the following; (i) Nepal map (M) on coin and odd number on spinner. (ii) Sagarmatha (S) on coin and prime number on spinner. 15. A coin containing head (H) and tail (T) is tossed and then a die is thrown. (a) Show the all possible outcomes in the same tree diagram. (b) Write its sample space. (c) Find the probabilities of the following; (i) Head on coin and even number on die (ii) Tail on coin and greater than 4 on die 7. (a) 1 8 (b) 3 8 (c) 3 8 8. (a) 1 169 (b) 1 16 9. (a) 56 121 (b) 28 121 (c) 28 121 10. (a) 15 64 (b) 15 64 (c) 15 32 11. (a) 1 6 (b) 1 2 12. (a) 1 221 (b) 1 17 13. (a) 1 25 (b) 8 25 (c) 16 25 14. (c) (i) 1 3 (ii) 1 3 15. (c) (i) 1 4 (ii) 1 6 Answers

366 Allied The Leading Mathematics-10 Probability 367 Statistics and Probability CONFIDENCE LEVEL TEST - VI Unit VI : Statistics and Probability Class: 10, The Leading Maths Time: 45 mins. FM: 27 Attempt all questions. 1. In the grouped data, Σf = 10 + 3p, Σfx = 30p + 100. (a) Write the formula for finding mean in the continuous series. [1] (b) Find the arithmetic mean. [2] (c) If median = 9, what would be the value of mode? Calculate it. [2] 2. The median of the given data is 24. Marks 0 - 10 10 - 20 30 - 40 40 - 50 20 - 30 No. of students 4 12 9 5 x (a) Write the formula for finding median in the continuous series. [1] (b) In which class does median lie ? [1] (c) Find the value of x. [2] (d) If the mean of the data is 25, what would be the value of mode ? [1] 3. If the third quartile (Q3) of the given data is 44. Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 No. of students 6 3 9 7 m 6 (a) In which class does Q3 lie? [1] (b) Make a cumulative frequency table. [1] (c) Find the value of m. [2] (d) How many maximum number of students who got more than 75% marks. [1] 4. From the deck of number cards numbered from 5 to 30, a card is drawn at random. (a) Which of the following scale is used in probability? [1] (i) 0 < P(A) < 1 (ii) 0 ≤ P(A) < 1 (iii) 0 < P(A) ≤ 1 (iv) 0 ≤ P(A) ≤ 1 (b) Is the event of the number card which is divisible by 5 and 7 mutually exclusive respectively? Give reason. [1] (c) Find the probability of getting a number card which is divisible by either 5 or 7. [2] (d) Find the probability of getting a number card which is divisible by 5 and 7. [2] 4. A coin and a die are thrown together. (a) Write the sample space of rolling a normal dice. [1] (b) Calculate the probability of getting a head on coin and even number on die by using a tree diagram. [2] (c) Are the events independent? Give reason. [1] (d) Is the probability of getting a prime number and probability of not getting a prime number equal when a single die is thrown? Justify with reason. [2] Best of Luck

366 Allied The Leading Mathematics-10 Probability 367 Statistics and Probability Additional Practice – VI Mean of Grouped Data 1. (a) If Σfx = 250 and N = 25 in a continuous data what is the mean? (b) If Σfm = 375 and mean (x) = 15 in a continuous data , what is the number of observations? 2. If the mean of a grouped data having Σfm = 455 is 35, find the numbers of terms. 3. In the grouped data, Σf =30 + 2p, Σfx = 100p + 280, and x = 40. Calculate the value of n. 4. Find the arithmetic mean for the data given below: Age in Years 30-35 35-40 40-45 45-50 50-55 55-60 No. of teachers 65 80 120 135 90 10 5. If the arithmetic mean of the following cases is 34.87, find the value of a. Age in yrs. 4.5-14.5 14.5-24.5 24.5-34.5 34.5-44.5 44.5-54.5 54.5-64.5 No. of cakes 6 11 21 a 14 5 Median of Grouped Data 6. In the given continuous data, the median lies in the 28th position. Which is of taken for computing the value of median ? Marks (x) 10-20 20-30 30-40 40-50 No. of students (f) 135 150 200 160 7. Find the class interval in the given data where the median lies. Marks 12-16 16-20 20-24 24-28 28-32 No. of Students 8 2 5 6 3 8. Find median of the data given below: x 10–20 20–30 30–40 40–50 50–60 60–70 70–80 80–90 f 78 160 121 85 63 20 12 11 9. Compute the median from the given data: x 0–10 10–20 20–30 30–40 40–50 cf 2 7 20 26 35 10. If the median of the given data is 24, find the value of p: x 0–10 40–50 20–30 10–20 30–40 cf 9 10 p 21 15 Quartiles of Grouped Data 11. (a) Write the formula to calculate the value of Q3 in the grouped data. (b) In which percent does the third quartile divide the given data?

368 Allied The Leading Mathematics-10 Probability 369 Statistics and Probability 12. In the given grouped data the first quartile lies in the 7th position. Which is cf taken to calculation of the value of the first quartile? 13. Find the lower limit of the first quartile class in the given data where the first quartile lies. Ages (in Months) 0 – 5 5 – 10 10 – 15 15 – 20 20 – 25 No. of Children 7 8 6 3 4 14. Calculate the lower quartile and upper quartile for the data given below: Marks obtained 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100 No. of students 10 16 20 30 10 15. The speed of vehicles running in the ring road is given below: Speed (km) 30-40 40-50 50-60 60-70 70-80 No. of vehicles 50 ? 85 45 30 If the maximum number of vehicles of upper 25% is 62, find the number of vehicles whose speed is 40-50. Probability on Mutually Exclusive Events 16. (a) When a grouped data has 60 observations, write the position of the median class of the data (b) When a grouped data has 75 observations, write the position of the third quartile of the data. 17. Find the third quartile class from the given cumulative frequency curves: 2 6 10 14 4 8 12 18 0 10 20 30 40 50 60 70 80 5 15 25 35 10 20 30 40 O 4 8 12 14 16 20 24 28 10 30 50 70 20 40 60 80 O 20 40 60 80 100 120 140 180 (a) (b) (c) 18. For each of these activities say whether or not the two given events are mutually exclusive: (a) Throwing a die: obtaining multiple of 3, obtaining an even number. (b) Throwing a die: ‘obtaining an odd number, obtaining a 6.’ 19. A letter is chosen at random from the world MATHEMATICS. Calculate the probability of getting ; (a) A or M (b) Vowel

368 Allied The Leading Mathematics-10 Probability 369 Statistics and Probability 20. Experiment that two coins are tossed simultaneously. Sample space S = {HH, HT, TH, TT} Calculate the probability of; the event B is that the first coin is a tail. 21. A single ball is drawn from a bag containing 3 red, 5 white and 2 blue balls. Find the probability of each event. Drawing a red or a blue ball in a draw. 22. Prasana chooses a date for a party at random from the calendar below. Calculate the probability that she chooses: Baisakh 2080 Sunday Monday Tuesday Wednesday Thursday Friday Saturday 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (a) a Saturday (b) a Friday (c) a Saturday or Friday Probability on Independent and Dependent Events 23. Maya has two spinners as given in the figure. Calculate the probability of; (a) getting even number and A. (b) getting number less than 5 and B. 24. If the chance of a fine day is 1 5 , find, by using a probability tree, the probability of ; (a) two fine days in succession. (b) one fine day out of two days. (c) two consecutive days that are not fine. 25. A soccer player rates his chance of scoring a goal in a match as 2 5 . Find the probability that in his nest two matches; (a) he scores a goal in each. (b) he scores a goal in only one. (c) he does not score in either match. R R R B B W W W W W C B A 1 5 2 4 3

370 Allied The Leading Mathematics-10 Probability PB Statistics and Probability 26. A bag contains 3 white and 4 red marbles. What is the probability of drawing (without replacements); a red marble followed by a white marble? Probability by Tree Diagram 27. The probability that a newly born baby being a boy or a girl is equal. Of 3 babies in a family, using tree diagram, find the probability of being; all of the same sex, 28. A bag contains 5 black ball and 3 white balls. Two balls are drawn one after another with replacement. Draw the probability tree diagram representing the experiment and find the probability of getting both black. 29. Three red and two white pencils of the same size are kept in a box. Two draws are made one after another without replacement. Show the outcomes in a tree diagram. 30. (a) Three red and two white balls of the same size are kept in a box. Two draws are made one after the other without replacement. Show the probability of getting red and white balls in the tree diagrams. (b) There is one red, one white and one yellow sweet in a pot. A sweet is taken out randomly and not replaced. Then another sweet is drawn. Write the sample space of the experiment using a tree diagram. 31. An archer shoots twice. If the probability of hitting a target is 0.7, draw the probability tree showing the outcomes and then find the probability of hitting once and missing once. 2. 13 3. 46 4. 43.85 5. 23 7. 20-24 8. 33.05 9. 36.15 10. 25 13. 0 14. 34.37, 72.3 15. 54 19. (a) 4 11 (b) 4 11 20. 1 2 21. 1 2 22. (a) 5 31 (b) 5 31 (c) 10 31 23. (a) 2 15 (b) 4 15 24. (a) 1 25 (b) 1 10 (c) 16 25 25. (a) 25 64 (b) 1 16 (c) 7 16 26. 2 7 27. 1 4 28. 25 64 31. 0.21 Answers

Trigonometry PB Allied The Leading Mathematics-10 Trigonometry 371 To solve the simple problems related to height and distance by using trigonometric ratios. Marks Weightage : 4 Estimated Working Hours : Competency Learning Outcomes 1. To develop the concept of angle of depression and angle of elevation. 2. To solve the behaviour problems related to height and distance by using the trigonometric ratios. Chapters / Lessons 15 Trigonometry - Angles of Elevation and Depression - Altitude of the Sun - Solving the Problems of Height and Distance TRIGONOMETRY VII UNIT 8 (Th. + Pr.) Specification Grid Unit Areas Total working hour Knowledge Understanding Application Higher ability Total number of Items Total number of questions Total Marks No. of items Marks No. of items Marks No. of item Marks No. of item Marks VII. Trigonometry 8 1 1 1 1 1 1 1 1 4 1 4 FEE Concept: "Maths is Fun, Maths is Easy and Maths is Everywhere"

Trigonometry 372 Allied The Leading Mathematics-10 Trigonometry 373 WARM-UP 1) Trigonometric Ratios: The ratios any two sides of a right-angled triangle with a acute angle θ is known as trigonometric ratios for the acute angle θ. 2) In practice, for the acute angle OAB = θ in the right-angled triangle OAB alongside, we write the trigonometric ratios as; sine of the angle θ, sin θ = Perpendicular Hypotenuse = p h cosine of the angle θ, cos θ = Base Hypotenuse = b h tangent of the angle θ, tan θ = Perpendicular Base = p b cotangent of the angle θ, cot θ = Base Perpendicular = b p secant of the angle θ, sec θ = Hypotenuse Base = h b cosecant of the angle θ, cosec θ = Hypotenuse Perpendicular = h p , where sin θ, cos θ and tan θ are called the Fundamental Trigonometric Ratios and cot θ, sec θ and cosec θ are called Reciprocal Trigonometric Ratios. So, (i) sin θ and cosec θ are reciprocal to each other. Then, sin θ × cosec θ = 1 (ii) cos θ and sec θ are reciprocal to each other. Then, cos θ × sec θ = 1 (iii) tan θ and cot θ are reciprocal to each other. Then, tan θ × cot θ = 1 3) Trigonometric Ratios of Standard Angles: The table of the values trigonometric ratios of standard angles 0°, 30°, 45°, 60° and 90° are given below. O θ B A Perpendicular (p) Hypotenuse (h) Base (b) sin cos tan cot sec cosec 1 The Magic Hexagon THE HAND TRICK R E V I S I T I N G T R I G O N O M E T R Y

Trigonometry 372 Allied The Leading Mathematics-10 Trigonometry 373 15.1 Height and Distance At the end of this topic, the students will be able to: ¾ solve the problems related to height and distance by using trigonometric ratios. Learning Objectives I Introduction This chapter helps us to calculate the area height of erect objects like a house, building, tower, pole, tree, hill, flying kite, flying bird, etc, and find easily the distance between the observer and the erect object. Also, we find out the slope of the inclined objects and its length. We can measure the angle by using Theodolite or big protractor or any other angle measuring device. In the adjoining figure, if we know any two parts among the height of wall, distance between the base of ladder and foot of wall and angle and inclined ladder, we easily compute the value of the next part. II Angle of Elevation and Angle of Depression When an observer observes any object above the higher level than him/her, the eye’s ray makes some angle with the ground or horizontal line. This angle is called the angle of elevation. In the adjoining figure, ∠EAT is the angle of elevation. Likewise, when the observer observes any object below the lower level than him/her, the eye’s ray makes some angle with the horizontal line. This angle is called the angle of depression. In the adjoining figure, ∠EAB is the angle of elevation. ∠DTA is also the angle of depression. We know that ∠EAB = ∠ABC and ∠DTA = ∠TAE because of alternate angles. Measure of Angle of Elevation and Angle of Depression and Distance We can measure the angles of elevation and depression by using simple protractor or big protractor or any other device, etc. by the following ways: Angle of depression Horizontal line T B E D A C Angle of elevation Eye's ray Eye's ray α β Ground Level CHAPTER 15 TRIGONOMETRY

Trigonometry 374 Allied The Leading Mathematics-10 Trigonometry 375 Similarly, we measure the distance between the observer and the foot of the erect objects by using measuring tape, ray measure, any other measuring device. Altitude of Sun or Sun’s Elevation An angle made by a ray of the sun light at the ground or horizontal line, is called the altitude of the sun. It is also called the height of the sun. In the figure, ∠SOB = θ is the altitude of the sun S. Do you know ? The altitude of the sun at noon of day is 90°. Points to be Remembered 1. An angle defined between the horizontal plane and oblique line from the observer’s eyesight to some object above eye level, is called angle of elevation. 2. An angle defined between the horizontal plane and oblique line from the observer’s eyesight to some object below eye level, is called angle of depression. 3. An angle made by a ray of the sun-light at the ground or horizontal line, is called the altitude of the sun or height of the sun. Tricks for Height and Distance 1. When the angle of elevation θ = 45°; the relation between the perpendicular (p) and base (b) sides of a right triangle when the measure of the angle of elevation is 45°, is p = b and h = 2 p or h = 2 b. 2. When the angle of elevation θ = 60°; the relation between the perpendicular (p) and base (b) sides of a right triangle when the measure of the angle of elevation is 60°, is p = 3 b and h = 2b or 2h = 3 p. 3. When the angle of elevation θ = 30°; the relation between the perpendicular (p) and base (b) sides of a right triangle when the measure of the angle of elevation is 30°, is b = 3 p and h = 2p or 2h = 3 b. Example-1 A ladder rests on a wall. The lower end of the ladder makes an angle of 45° with the horizon and the lower end lies at the same horizon at a distance of 15 m from the foot of the wall. (a) Which trigonometric ratio for the acute angle θ is used to find the height of the wall ? Write its ratio. O A B S 3 60° 1 2 2 30° 45° 45° 1 1 3 60° 30° 2 1 A C B 45°

Trigonometry 374 Allied The Leading Mathematics-10 Trigonometry 375 (b) Find the height of the wall. (c) When the measure of one acute angle of the right triangle is 45°, what is the relation among its perpendicular and base sides? Solution: (a) tan θ is used to find the height of the wall and tan θ = p b. (b) In the given figure, AB = Height of wall, AC = length of ladder, ∠ACB = 45°, BC = 15 m. Now, in right triangle ABC; tan 45° = AB BC or, 1 = AB 15 m or, AB = 15 m. Hence, the height of the wall is 15m. (b) When the measure of one acute angle of the right triangle is 45°, its perpendicular and base sides are equal. Example-2 A 1.5 m tall person observes a tower of the height 79.44 m and finds the angle of elevation to be 60°. (a) Define the angle of elevation. (b) Find the distance of the man from the tower. (c) If the height of the person is 'a', that of the tower, 'b' and the distance between them, 'd', then write the relation between a, b and d in the above situation. Solution: (a) An angle made by the observer’s eye-sight with the horizontal line when observed an object above eye, is called angle of elevation. (b) In the given figure, CD = Height of a man, AE = Height of the tower DE = Distance between the man and the tower Then, ∠ACB = 60°, CD = 1.5 m, AE = 79.44 m, AB = 79.44 - 1.5 = 77.94 m, DE = BC = ? Now, in right triangle ABC, tan 60° = AB BC or, 3= 77.94 DE or, DE = 77.94 3 = 45 m. Hence, the distance of the man from the tower is 45 m. (c) If the height of the house is 'a', that of the building, 'b' and the distance between them, 'd', then write the relation between a, b and d in the above situation, is b = a + 3d. A C B 45° 60° A B E C D 1.5 m 1.5 m 79.44 m

Trigonometry 376 Allied The Leading Mathematics-10 Trigonometry 377 Example-3 From the top of a vertical column, a girl observes the angle of depression of a car on the road to be 30°. The height of the column is 15 m and the girl is 1.5 m tall. (a) What is the angle of elevation ? Define it. (b) Find the distance of the car from the foot of the column. (c) If the height of the girl is 'a', that of the column, 'b' and the distance between them, 'd', then write the relation between a, b and d in the above situation. Solution: (a) An angle made by the observer’s eyesight with the horizontal line when observed an object below eye, is called angle of depression. (b) In the figure, AB = 1.5 m (height of a girl), BC = 15 m (height of column) ∠EAD = 30°, angle of depression to the car at D. CD = distance of the car from the foot of the column = ? Since AE//CD; ∠ADC = 30°, and AC= AB + BC = 1.5 + 15 = 16.5 m Now, in right-angled triangle ACD; tan 30° = AC CD or, 1 3 = 16.5 CD or, CD = 16.5 3 = 16.5 × 1.732 = 28.58 m. Hence, the distance of the car from the foot of the column is 28.58 m. (c) If the height of the girl is 'a', that of the column, 'b' and the distance between them, 'd', the relation between a, b and d in the above situation, is d = 3 (a + b). Example-4 An 8 feet long stick casts a shadow of 6 feet long on the ground in front of a building when the sun rises from the back of the building. (a) What is the altitude of the sun ? Define it. (b) Find the length of the shadow of the hill of 360 ft high at the same time. (c) Write the relation between the height of the stick and building at the same time. Solution: (a) An angle made by of the sun light with the ground level or horizontal line is called the altitude of the sun. (b) In the figure, OB be the shadow of a stick AB and OD be the shadow of a building CD. Then AB = 8 ft, OB = 6 ft, CD = 360 ft, ∠COD = θ = ? and OD = ? Now, in right ∆OBA, A E B C D 30° A E B C D 30° 1.5m 30° 15m

Trigonometry 376 Allied The Leading Mathematics-10 Trigonometry 377 tan θ = AB OB = 8 ft. 6 ft. = 4 3 Again, in rt.∠d ∆COD, tan θ = CD OD or, 4 3 = 360 OD or, OD = 3 × 360 m 4 = 270 ft Thus, the length of the shadow of the building is 270 ft. (c) The height of the stick and building at the same time has equal tangent ratios and their height is proportional to their shadows. Example-5 A 24 meter high tree is broken by the storm so that its top touches the ground and makes an angle of 45o . (a) What is hypotenuse in a right triangle ? In which ratios are the hypotenuse? (b) Find the length of the broken part of the tree. (c) How much percentage of part of the tree is remained on the ground? Solution: (a) The opposite side of the right angle of a right triangle is called hypotenuse. The trigonometric ratios sine and cosines have hypotenuse. (b) The situation may be illustrated diagrammatically as where, TR = 24 m (height of the tree), ∠BAR = 45°, AB = TB = x (broken part of the tree), then BR = (24 – x) m Now, in right triangle ABR; sin45o = BR AB or, 1 2 = 24 – x x or, x = 24 2– 2 x or, x + 2 x = 24 2 or, x(1 + 2) = 24 2 or, x = 24 2 1 + 2 = 14.06 m. Hence, the length of the broken part of the tree is 14.06m. (c) The remaining part of the tree on the ground = 24 – 14.06 = 9.94. ⸫ The percentage of the remaining part of the tree on the ground = 9.94 24 ×100% = 41.42% C O B D A 8 ft 6' θ 360 ft T B A R 45°

Trigonometry 378 Allied The Leading Mathematics-10 Trigonometry 379 PRACTICE 15 Read Think Understand Do Keeping Skill Sharp 1. (a) Define angle of elevation. (b) Why is the angle of depression different from the angle of elevation ? Give reason. (c) If a man observes a bird on the top of the pole from the ground, which angle is formed with the horizontal line? (d) What are the values of tan 60° and cos 45° ? (e) What are the values of sin 30° and tan 45° ? (f) If tan θ = 1 3 , what is the value of the angle θ? (g) If a ladder rests on a wall and makes the angle of 45° to the ground 20 ft far from the wall, what is the height of the wall where the ladder is touched? 2. Circle ( ) the correct answer. (a) When a lady observes a car on the road from the top of the house, which angle is formed ? (i) Obtuse angle (ii) Angle of elevation (iii) Angle of depression (iv) Right angle (b) When a boy observes a parrot on the tree from the ground, which angle is formed ? (i) Obtuse angle (ii) Angle of elevation (iii) Right angle (iv) Angle of depression (c) What is the altitude of the sun at noon of the day is ........................ . (i) Acute angle (ii) Angle of depression (iii) Angle of elevation (iv) Right angle (d) A wire is tied to a pole of the height 30 ft to erect it. If the wire makes the angle of 45° to ground, how far is the wire tied from the pole? (i) 30 ft (ii) 30 3ft (iii) 10 3ft (iv) 20 3ft

Trigonometry 378 Allied The Leading Mathematics-10 Trigonometry 379 (e) The height of the erected tower is 150 ft. If an observer finds the angle of elevation as 30°, what is the distance between the tower and the observer ? (i) 150 ft (ii) 150 3ft (iii) 75 3ft (iv) 50 3ft (f) If a ladder rests on a wall and makes the angle of 60° to the ground 12 ft far from the wall, what is the length of the ladder? (i) 12 ft (ii) 6 ft (iii) 24 ft (iv) 6 3ft Check Your Performance 3. Observe the following right-angled triangle and answer the following questions. (i) (ii) (iii) (iv) (a) Write the relation between x and y in each right-angled triangle. (b) Find the values of x and y in each of the above right-angled triangle. Answer the given questions for each problem. 4. (i) An iron pillar supports a wall. The lower end of the pillar makes an angle of 45° with the ground and the lower end lies at the same horizon at a distance of 23 ft from the foot of the wall. (a) Which trigonometric ratio for the acute angle θ is used to find the height of the wall ? Write its ratio. (b) Find the height of the wall. (c) When the angle of elevation is 45° in a right triangle, what is the relation among its perpendicular and base sides? (ii) A lamp hanging on the lamppost of the height 40 ft is observed by an observer and finds the angle of elevation of the lamp post to be 45°. (a) Which trigonometric ratio for the acute angle θ is used to find the distance between the lamppost and observer? Write its ratio. (b) Find the distance between the lamppost and observer. (iii) From a point 100 m away to the base of a building; the angle of depression from its top is found to be 30°. (a) Find the height of the building. A 30° y 5m x B C 45° y 15m G x I H 30° 12m y J x K L 30° y 13m x D E F

Trigonometry 380 Allied The Leading Mathematics-10 Trigonometry 381 (b) When the angle of elevation is 30° in a right triangle, what is the relation among its perpendicular and base sides? (iv) A man observes a car on the road from the top of the tower of the height 250 m and finds the angle of 60°. (a) Find the distance between the tower and the man. (b) When the angle of elevation is 60° in a right triangle, what is the relation among its perpendicular and base sides? 5. (i) An observer 1.5 m tall observes the 60 m tall building at an angle of elevation of 45°. (a) What is the angle of elevation? Define it. (b) How far is the building from the point of observation? Find it. (c) If the height of the observer is 'a', that of the building, 'b' and the distance between them, 'd', then establish the relation between a, b and d in the above situation. (ii) An observer observes the 60 m tall building from the 15 m high house and finds an angle of elevation of 30°. (a) What kind of angle is formed when the observer observes the top of the building from the top of the house ? (b) How far is the building from the point of observation? Find it. (c) If the height of the house is 'a', that of the building, 'b' and the distance between them, 'd', then write the relation between a, b and d in the above situation. 6. (i) From the top of a house 15 m high, a man of 1.5 m tall observes the angle of depression of a motorbike to be 60°. (a) What is the angle of depression ? Define it. (b) How far is the motorbike from the bottom of the house? Find it. (c) If the height of the man is 'a', that of the house, 'b' and the distance between them, 'd', write the relation between a, b and d in the above situation. (ii) From the top of a tower 150 m high, a man of 1.5 m tall observes the angle of depression of a car on the road to be 30°. (a) How far is the car from the bottom of the tower? Find it. (b) If the height of the man is 'a', that of the tower, 'b' and the distance between them, 'd', write the relation between a, b and d in the above situation. 7. (i) A 42 feet high Nepal’s national flag casts a shadow of 24.25 feet in an afternoon. (a) Define the altitude of the sun. (b) Find the altitude of the sun from the ground. 24.25 ft 42 ft ft

Trigonometry 380 Allied The Leading Mathematics-10 Trigonometry 381 (c) How long shadow of a building of the height 250 ft does the sun cast at the same time? Find it. (d) Write the relation between the height of the flag and building at the same time. (ii) A 2 m high stick casts a shadow of 6 m long on the ground. (a) Find the height of the sun. (b) What is the length of the shadow of a 360 ft high tower at the same time ? Find. (c) Write the relation between the height of the stick and tower at the same time. 8. (i) A 10.5 m long ladder rests against a wall making an angle of 45° to the ground. (a) What is hypotenuse in a right triangle ? In which ratios have the hypotenuse ? (b) Find the height of the wall. (c) Write the relation between the hypotenuse and perpendicular of a right triangle when its reference angle is 45°. (ii) A 24 m long ladder rests against a wall and the upper end of the ladder reaches just 12 m above the ground on the wall. (a) Find the angle that the ladder makes with the ground. (b) How far the bottom of the ladder is from the wall ? Find. (c) Write the relation between the hypotenuse and perpendicular of a right triangle when its reference angle is 30°. 9. (i) From the balcony 23 m below the top of a tower 100 m high, the angle of depression of top of a lamp post 15 m tall is 45°. How far is the lamp post from the foot of the tower? (ii) From the top of the Dharahara of 203 feet high, a man observes the roof of the building of Nepal Telecom of the height 55 feet and finds the angle of depression of 30°. How far is the building from the Dharahara? 10. (i) A flag of 30m high is fixed at the center of the cricket ground. A cricketer observes its top from the edge of the ground and finds to be an angle of 30o . (a) Find the diameter of the ground. (b) Find the circumference of the ground. (c) Establish the relation between its diameter and circumference for the above situation.

Trigonometry 382 Allied The Leading Mathematics-10 Trigonometry 383 (ii) A girl of 5.3ft high observes the fixed pole of 30 ft high at the center of the circular pole and she finds to be an angle of 60o from the edge of its circumference. (a) Find the length of its radius. (b) Find the circumference of the ground. (c) Establish the relation between its radius and circumference. 11. (i) A tall tree is broken by the storm and has made an angle of 60o at 24 meter apart from its base. (a) Find the length of the broken part of the tree. (b) Find the height of the tree before broken. (c) How much percentage of part of the tree is remained on the ground? (ii) A 88m tall tree is broken by the storm and its top makes an angle of 30o with the ground. (a) Find the length of the broken part of the tree. (b) How much percentage of part of the tree is remained on the ground? 3. (i) 10 m, 5 3m (ii) 5 3m, 15 m (iii) 13 3 2 m, 6.5 m (iv) 4 3m, 8 3m 4. (i) (a) tan q, p b (b) 23 ft (c) Equal (ii) (a) tan q, p b (b) 40 ft (iii) (a) 57.74 m (b) b = 3 p (iv) (a) 144.34 m (b) p = 3 b 5. (i) (b) 58.5 m (c) b = a + b (ii) (a) Angle of elevation (b) 45 3m (c) 3 b = 3 a + d 6. (i) (b) 9.53 m (c) 3 d = a + b (ii) (a) 262.41 m (b) d = 3 (a + b) 7. (i) (b) 60° (c) 144.34 ft (d) Proportional (ii) (a) 18.43° (b) 1080 ft (c) Proportional 8. (i) (b) 7.42 m (c) h = 2 p (ii) (a) 30° (b) 12 3m (c) h = 2p 9. (i) 62 m (ii) 148 3ft 10. (i) (a) 103.92 (b) 326.62 m (c) 7c = 22d (ii) (a) 14.26 ft (b) 89.63 ft (c) 7c = 44r 11. (i) (a) 48 m (b) 89.57 m (c) 46.41% (ii) (a) 58.67 m (b) 29.33m, 33.33% Answers Project Work Measure the distance between you and a tree of your school or around you. Find the angle made by your eyesight with the horizontal line to the top of the tree by using a protractor or other device. Find the height of the tree by using trigonometric ratios.

Trigonometry 382 Allied The Leading Mathematics-10 Trigonometry 383 CONFIDENCE LEVEL TEST - VII Unit VII : Trigonometry Class: 10, The Leading Maths Time: 45 mins. FM: 20 Attempt all questions. 1. A man 1.5m tall standing at a distance of 5m from a wall observes the angle of elevation of the top of the wall is 45°. (a) Write the definition of the angle of elevation. [1] (b) Find the height of the wall. [1] (c) If a man wants to adjust a ladder on the wall from the foot of the man, what would be the length of the ladder? Calculate it. [1] (d) If the angle of elevation is more than 45, what will be the height of the wall and the length of the ladder comparatively? [1] 2. A boy, 1.2m tall, is flying a kite. When the length of the string of the kite is 90 m, it makes an angle of 30° with the horizontal line. (a) Construct a figure with given measurements. [1] (b) At what height is the kite from the ground? [1] (c) If the angle of elevation is 45°, what may be the differences between height of kite comparatively? [2] 3. The circumference of a circular pond is 88 m and a pillar is fixed at the center of the pond. A person finds the angle of elevation of 60° of the top of the pillar from any point on the bank of the pond. (a) Construct a figure with given measurements. [1] (b) Find the diameter of the pond. [1] (c) Calculate the height of the pillar above the water level. [1] (d) If the angle of elevation is 45°, what may be the height of the pole? [1] 4. The top of a tree which is broken by the wind makes an angle of 60° with the ground at a distance of 5√3 m from the foot of the tree. (a) Find the height of the tree before broken. [2] (b) Calculate the height of the remaining tree after broken. [1] (c) Compare the remaining part and broken part of the tree. [1] 5. In the given figure, two vertical poles AB and CD are fixed 60 m apart. The angle of depression of the top of the first pole from the top of the second pole which is 150 m high is 30°. (a) Define the angle of depression. [1] (b) Find the height of CE. [1] (c) Calculate the height of first pole AB. [1] (d) If the angle of depression is 45o, what will be the height of first pole AB? [1] Best of Luck A C B D E 60 m 150 m 30° 30°

Trigonometry 384 Allied The Leading Mathematics-10 Trigonometry PB Additional Practice – VII 1. Find the value of x and y in each of the right angled triangle given below: (a) (b) (c) 2. A railway uphill makes the angle of elevation of 15° with horizon. A train travels 6 km on the railway track. How high is the train from the horizon? [Use sin15o = 0.26] 3. An aeroplane flies at a constant height of 1200 m above the ground. At some distant the radar at the airport reads the angle of elevation of 30°. How far is the aeroplane from the airport? 4. The Red Machhindranath chariot, the longest festival of the world, is supported by a 40 feet long rope at 20 feet below the top. If the rope makes an angle of 75o when the men of the average height of 5feet are pulling tightly, what is the height of the chariot? [Use sin75o = 0.966] 5. A 15 cm long pencil casts a shadow of 15 3cm in an afternoon. A rod casts a shadow of 30 m long at the same time. How long was the rod? 6. How long the shadow of a 25 ft high post when the altitude of the sun changes from 45o to 30°? 1. (a) 12 m, 6 2m (b) 16 cm, 8 3cm (iii) 2 m, 2 m 2. 1.56 km 3. 2078.46 m 4. 63.64 ft 5. 10 3m 6. 18.30 ft Answers 30° 30° y 1m x Q T R S 30° 8cm y x B C A D E 45° 45° y x 12m M P N O

PB Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 385 I. PRACTICE QUESTION SET FM: 75 Class: 10 Subject: The Leading Maths Time: 3 hrs. Attempt all questions. 1. Out of group of 3500 Chinese tourists who visited Nepal, 40% have been already to Pokhara and 43% to Lumbini. 8% of them have been to both places. (a) Write the above information in set notation. [1] (b) Show the above information in a Venn diagram. [1] (c) How many Chinese tourists have visited at most one place. [3] (d) How many more or less percentage are there among the Chinese who visited both places or visited neither of both places? [1] 2. Tilak borrowed a sum of Rs. 75000 at 11% p.a. compounded interest annually. On the same day, Tilak lent out this money to Samir at 10.5% compound interest quarterly. (a) Write the formula for finding compound interest compounded semi-annual. [1] (b) How much amount would be returned by Tilak? Find it. [2] (c) How much amount will Tilak get as a profit or loss on his transaction? Find it. [2] 3. Aamir sold a plot of land for Rs. 12500000. He bought a car for Rs. 4450000 from that amount and invested the remaining amount in a company. The value of the car decreased by 10% p.a. and the company gave 8% profit in each year. (a) Write any one difference between compound growth and compound depreciation. [1] (b) How much amount did Aamir invest in the company? Find it. [1] (c) Find the difference between the compound growth amount on investment in the company and the compound depreciated amount on selling the car. [2] 4. It is given that EUR 120 = Rs. 17044.80 and $ 90 = Rs. 11939.40. A boy needs to exchange $ 100 into Euro (EUR). (a) Find how much Euro does he get from $ 100 by using the chain rule. [2] (b) How much US dollar is needed a watch of the cost EUR 100 ? Calculate it. [1] (c) Which currency is expensive among the US dollar and Euro by how many percent? Find it. [1] 5. A squared pyramid made by bricks on the ground has the length of the base side 120 ft. with slanting edge 65 ft. (a) What is the volume of a pyramid of the base area a ft2 and vertical height b ft? [1] (b) Find the slant height of the given pyramid. [1] (c) Calculate the lateral surface area of the pyramid. [1] (d) Estimate the cost of plastering on its faces at the rate of Rs. 65 per sq. ft. [1] 65 m 120 m 120 m

386 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 387 6. A hollow cone of height 25 cm and radius 14 cm is completely filled with water. The water is poured in a cylinder of the height 30 cm and base diameter 21 cm. (a) Calculate the quantity of the water in the cone. [2] (b) Find the height of water in the cylinder. [2] 7. Observe the given net of solid alongside. (a) Which solids form the given combined solid? [1] (b) If the slant height of the upper solid and the radius of the lower circle are 14 cm and 7 cm respectively, and the breadth of the rectangle is 21 cm, find what quantity of paper is used for of the solid formed by the given net. [2] (c) Establish the relation between the capacities of the cylindrical part and the conical part of the solid formed by using the dimension of the same given net. [2] 8. The given numbers 20, 16, 12, 8, … are in arithmetic sequence. (a) What is the geometric mean between two numbers p and q ? Write it. [1] (b) Find the nth term of the above given sequence. [2] (c) Which term will be – 20 ? Calculate it. [2] 9. (a) Observe the equation 3x2 + 2x – 1 = 0. Answer the following questions. (i) How many roots does the given equation give? [1] (ii) Solve the given equation. [2] (b) Simplify: 1 a2 – 5a + 6 + 1 (a – 1) (3 – a) + 1 a2 – 3a + 2 [3] 10. Observe the equation 4 × 3x + 1 – 9x = 27. (a) Find the value of the given equation. [2] (b) Is the value of x also satisfied in the equation 32x – 4 × 3x + 1 + 27 = 0? Check it. [2] 11. In the given figure, parallelogram ABCD and ΔAXB are standing on the same base AB and between the same parallel lines AB and DC. (a) If the area of parallelogram ABCD = 32 cm2 , find the area of ∆AXB. [1] (b) Prove that: Ar(∆AXB) = 1 2 of Ar( ABCD) [3] (c) Is there any relationship between Ar(∆ADX) and Ar(∆BXC) with Ar(∆AXB)? Give reason. [1] 12. (a) Construct a parallelogram PQRS in which PQ = 4 cm, QR = 5 cm, and ∠PQR = 60°. Also, construct a triangle POT having ∠POT = 60° equal in area to PQUS. [2] (b) How is Ar(∠POT) equal to ( PQUS) ? Justify. [2] A X C B D

386 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 387 13. In the given figure, O is the center of circle. PQ = PR with ∠QPR = 52°. (a) Write the relation of ∠PQR and ∠PSR. [1] (b) What will be the value of ∠ORP when joining OR and OP. Calculate with reason. [1] (c) Verify experimentally that the opposite angles of the cyclic quadrilaterals ABCD of the radii more than 3 cm are supplementary. [2] 14. If the third quartile (Q3) of the given data is 44. Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60 No. of students 6 3 9 7 m 6 (a) In which class does Q3 lie? [1] (b) Make a cumulative frequency table. [1] (c) Find the value of m. [2] (d) How many maximum number of students got more than 75% marks. [1] 15. A bag contains 3 red balls and 5 blue balls. Two balls are drawn one at a time at random with replacement. (a) Define independent events with example. [1] (b) Construct a tree diagram. [2] (c) Find the probability of getting 1 red ball and 1 blue ball. [2] (d) If two balls are taken from the bag one after another with and without replacement, what would be the differences between probability of getting both balls are blue. [1] 16. From the top of a house of the height 50 ft., a girl observes the angle of depression of an electric pole of the height 20 ft. in front of the house to be 60°. (a) What is the angle of elevation? Define it. [1] (b) Which trigonometric ratio is used to find the distance between the house and the pole? [1] (c) Find the distance of the pole from the foot of the house. [1] (d) What would be the distance between the house and the pole when the angle of depression is decreasing? Write your conclusion. [1] Best of Luck Q R S T 52° P O

388 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 389 III. Project and Practical Work Evaluation Sheet Practical Recorded Form School’s Name and Address: ………………………………………………………. Academic Year: 208 …. Subject: The Leading Mathematics Class: 10 RN Student’s Name Conducting experimental and project work based on main learning achievements: (8) Conducting and documenting experimental and project work: (5) Drawing - Naming - Character Description - Spotting and Oral Examination: (3) Total Marks Remarks 123456789101112 …………………… …………………….. ………………………. Signature of student Signature of parents Signature of class teacher II. Classroom Individual Form Student’s Activities Form Subject: The Leading Mathematics Class: 10 Student’s Name: …………………………............................................ Roll No. ……. SN Unit Chapter/ Lesson Practice/ Exercise Participation (3 Marks) (1.5) (1.5) Attendance Class work Home work Classroom Activities 12345678910111213 …………………… …………………….. ………………………. Signature of student Signature of parents Signature of class teacher

388 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 389 III. Project and Practical Work Evaluation Sheet Practical Recorded Form School’s Name and Address: ………………………………………………………. Academic Year: 208 …. Subject: The Leading Mathematics Class: 10 RN Student’s Name Conducting experimental and project work based on main learning achievements: (8) Conducting and documenting experimental and project work: (5) Drawing - Naming - Character Description - Spotting and Oral Examination: (3) Total Marks Remarks 123456789101112 …………………… …………………….. ………………………. Signature of student Signature of parents Signature of class teacher

390 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 391 IV. Internal Evaluation Form Practical Recorded Form for 25 Marks School’s Name and Address: ………………………………………………………. Academic Year: 208 …. Subject: The Leading Mathematics Class: 10 RN Student’s Name Participation (Attendence + Class work): (3) Project and Practical Work: (16) Terminal Exams: (6) Total Marks Percent % Grade Grade point Result Remarks First (3) Second (3) 123456789101112 …………………… …………………….. ………………………. Signature of student Signature of parents Signature of class teacher

390 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 391 V. Format of Project Work For Project Work -1.1 Make a group of students at least 15 students and ask the following questions to the students about their favourite mobiles as iPhone-Apple and Samsung-Android: (i) How many students who like iPhone-Apple mobile? (ii) How many students who like Samsung-Android mobile? (iv) How many students who do not like both mobiles? Show the above information in a Venn diagram. Find the following operations of the sets with the separate Venn diagram: (a) How many students who like iPhone-Apple mobile or Samsung-Android mobile or both? (b) How many students who like both iPhone-Apple and Samsung-Android mobiles? (c) How many students who like only iPhone-Apple mobile? (d) How many students who like Samsung-Android mobile but not iPhone-Apple mobile? (e) How many students who like at least one type of mobiles ? (f) How many students who like at most one type of mobiles ? (g) How many students who like at most both types of mobiles ? (h) How many percentage of students who like iPhone-Apple mobile? (i) How many percentage of students who like only Samsung-Android mobile ? 1. Objective: l To find the cardinality of the operations of two sets with the Venn diagrams. 2. Procedure: 1. Let them a group of at least 15 students. 2. Ask the questions to all of the group. 3. Record all the information and represent in set notation 3. Methods: A) Set Notation: l Ask the above questions and represent in set notation. Suppose, Set of all students who involve in the survey = U, Set of students who like Samsung-Android mobile = S. Set of students who like iPhone-Apple mobile = I. Then, n(U) = ……. n(I) = …… n(S) = ……. n(I ∪ S) = ……

392 Allied The Leading Mathematics-10 Allied The Leading Mathematics-10 PB B) Representation: l Represent the above information in a Venn diagram, C) Computation: (a) No. of students who like iPhone-Apple mobile or Samsung-Android mobile or both, n(I ∪ S) = n(U) – n(I ∪ S) = ……… – ……… = ……… Showing n(I ∪ S) in the same Venn diagram below, (b) For no. of students both iPhone-Apple and Samsung-Android mobiles n(I ∩ S), n(I ∪ S) = ……… + ……… – ……… or, ……… ……… ……… or, ……… ……… ……… or, n(I ∩ S) = ……… ……… Showing n(I ∩ S) in the same Venn diagram below, I S U n(I ∪ S) I S U n(I ∩ S) You can do the remaining questions from (c) to (i) as the similar process as above. 4. Demonstration or Presentation: l Introduce about me or every members of a group. l Present the whole work step by step and turn by turn. 5. Application: l Tell and show if possible, how do we use Set Operations and Venn diagram in Real Life. Most of us have collections of our favorite things, groups of objects, like our favorite clothes, favorite foods, favorite people and places, arrangement of books, kitchen utensils and groceries etc. These are all parts of sets, and we use them every day and in our daily lives. In addition, applications of the operations of sets are frequently used in science and mathematics fields like physics, biology, chemistry, as well as in electrical engineering and computer engineering etc. 6. Conclusion: l From the project work, i/we conclude that I learned …………………………………… …… ……………………………………………………………………………………… . Thank you for listening to me/us. I ∪ S n(U) = ... n(I) = ... n(S) = ... n(I ∪ S) = ... I S U