Oasis School Mathematics-10 297 i.e. n(A∩B) = 1 Now, probability of event A, P(A) = n(A) n(S) 7 21 1 3 = = Probability of event B, P(B) = n(B) n(S) 3 21 1 7 = = We have, P(A∪B) = P(A) + P(B) – P(A∩B) = 1 3 1 7 1 21 + – = 9 21 ∴ The probability of getting a card whose number is divisible by 3 or 7 is 3 7 . exercise 16.1 1. Identify whether the given events are mutually exclusive or mutually non-exclusive. (a) getting a king or an ace from a pack of 52 cards. (b) getting a heart or a diamond from a pack of 52 cards. (c) getting a heart or a king from the pack of 52 cards. (d) getting a face card or a red card from a pack of 52 cards. (e) getting a number which is a multiple of 5 or 7 from a stack of cards numbered 1 to 30. (f) getting a number which is a prime number or an even number from a stack of cards numbered 1 to 20. (g) getting a number which is the multiple of 3 or 4 from a stack of cards numbered 1 to 25. 2. (a) If 'A' and 'B' are two mutually exclusive events, where P(A) = 1 2 and P(B) = 1 3 , find P(A∪B). (b) Two events M and N are mutually exclusive events with P(M) = 1 8 and P(N) = 5 8 . Find; (i) P(M∪N) (ii) P(M∪N ) (c) P(A∪B) = 3 4 and P(A) = 1 2 . Find the value of P(B), if A and B are two mutually exclusive events. 3. (a) A marble is drawn from a box containing 15 black, 5 green, 10 red and 10 yellow marbles. Find the probability of getting, (i) A black marble or green marble. (ii) A green or a red marble. (iii) A yellow or black marble. (b) A card is drawn from a well shuffled pack of 52 cards. Find the probability of getting, (i) an ace or a king (ii) a black ace or a red king.
298 Oasis School Mathematics-10 (iii) a red jack or a queen. (iv) a face card or an ace. (v) a heart or a spade (vi) a diamond or a black card. (vii) neither a king nor an ace. (c) A card is drawn at random from a stack of cards numbered 5 to 30. Find the probability of getting a (i) square numbered or cube numbered card. (ii) multiple of 5 or 9. (iii) even numbered or prime numbered. (iv) multiple of 5 or multiple of 7. (d) From a set of 15 cards numbered 1, 2, 3, 4 ……. 15, one is drawn at random. What is the probability of getting card which is: (i) divisible by 3 or divisible by 7 (ii) divisible neither by 3 nor by 7 (iii) multiple of 5 or 6 4. (a) If 'A' and 'B' are mutually non-exclusive events and P(A) = 2 5 , P(B) = 6 25 and P(A∩B) = 1 25, find the value of (i) P(A∪B), (ii) P(A∪B). (b) If P(A) = 1 3 , P(B) = 1 7 and P(A∪B) = 3 7 , find the value of P(A∩B) if A and B are mutually non-exclusive events. (c) If 'A' and 'B' are mutually non-exclusive events with P(A) = 1 2 , P(B) = 1 4 , and P(A∪B) = 5 8 , find (i) P(A∪B) and (ii) P(A∩B). 5. (a) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of getting, (i) an ace or a red card. (ii) a red card or a face card. (iii) an ace or a heart. (iv) a spade or a queen. (v) a king or a red card (vi) a red queen or a heart. (b) From a pack of 52 cards, a card is drawn at random. Find the probability of getting a black or a non-faced card. (c) A card is drawn at random from a set of cards numbered 1 to 20. Find the probability of getting prime numbered or an even numbered card. (d) A card is drawn at random from a set of cards numbered 1 to 36. Find the probability of getting a card of a multiple of 5 or 7. (e) A card is drawn at random from a group of cards, numbered from 2 to 13. Find the probability of getting a prime numbered or even numbered card. .
Oasis School Mathematics-10 299 Answers 1 2 5 6 3 4 1 4 1 4 3 1 2 3 . . (a) (b) (i) (ii) ( ) .( ) (ii) Consult your teacher c a 8 5 8 2 13 1 13 3 26 4 13 1 2 3 4 11 ( ) ( )(i) ( ) ( ) ( ) ( ) ( ) ( ) iii b ii iii iv v vi vii 13 5 26 9 26 21 26 5 13 7 15 8 15 1 3 ( ) c i( ) ( ) ii ( ) iii ( ) iv ( ) d i( ) ( ) ii ( ) iii 4 3 5 2 5 1 21 3 8 1 8 5 7 13 8 13 . .(i) (ii) . ( )(i) ( ) .( )(i) ( ) ( ) a b c ii a ii iii 4 13 4 13 7 13 7 26 23 26 17 20 11 36 11 12 ( ) iv ( ) v v( )i b( ) ( ) c e (d) ( ) 16.3 Independent events Two or more events are said to be independent when the result of one event doesn't affect the result of the other. For example, when a coin is tossed twice the result of the first doesn't affect the result of the second event. So, they are called independent events. multiplicative law of probability The probability of two or more independent events occurring together or in succession is the product of their individual probabilities. Symbolically, if A and B are two independent events then the probability of events A and B occurring together or in succession is denoted by, P (A and B) or P (AB) or P(A∩B) Thus, it is written as P(A and B) = P(A) × P(B) or, P(A ∩ B) = P(A) × P(B), Where, P(A and B) = the probability of occurring two events A and B together or in succession. P(A) = Probability of occurring event A P(B) = Probability of occurring an event B If A, B and C are three independent events corresponding to a random experiment, then P(ABC) = P(A) × P(B) × P(C) i.e. P(A∩B∩C) = P(A) × P(B) × P(C) Worked out examples example: 1 Find the probability of getting a tail on a coin and 4 on a dice when a coin is tossed and dice is rolled simultaneously. Solution: Let 'T' be the event of getting a tail on a coin and F be the event of getting a '4' on a dice.
300 Oasis School Mathematics-10 When a coin is tossed, the sample space is S = {H,T}, ∴ n (S) = 2 And the number of favorable cases n(E) = 1 Then probability of getting a tail P(T) = n(E) n(S) 1 2 = Again, when a dice is thrown at once, the sample space is S = {1, 2, 3, 4, 5, 6}, ∴ n(S) = 6 Number of favorable cases n(E) = 1 Then, probability of getting a 4, P (F) = n(E) n(S) 1 6 = Since these two events are independent events, ∴ P(T and F) = P(T) × P(F) = 1 2 × 1 6 = 1 12 example 2 the probability of solving a problem by A is 2 3 and the probability of solving the problem by B is 3 4 . Find the probability of (i) solving the problem by both of them (ii) the problem being solved, if they both try. Solution: Here, we have P(A) = 2 3 P(B) = 3 4 P(A and B) = P(A∩B) = ? P (A or B) = P (A∪B) = ? We have, P (A∩B) = P(A) × P(B) [∵ A and B are independent events] = 2 3 × 3 4 = 1 2 Now, using relation, P(A∪B) = P(A) + P(B) – P(A∩B) = 2 3 + 3 4 – 1 2 = 8+9–6 12 = 11 12 example 3 A dice is thrown thrice. What is the probability that it will turn up, (i) six in each time? (ii) no sixes? (iii) at least one 6?
Oasis School Mathematics-10 301 Solution: Here the possible cases for each throw n(S) = 6 (i) Probability of getting a 6 each time is P(6,6,6) = 1 6 × 1 6 × 1 6 = 1 216 (ii) Probability of not getting 6 = 1 – 1 6 = 5 6 P(no sixes) = 5 6 × 5 6 × 5 6 = 125 216 (iii) At least one six Since the probability of not getting a 6 is P (no sixes) = 125 216 Then probability of getting at least one 6 is P (at least one 6) = 1 – 125 216 = 216-125 216 = 91 216 example: 4 A card is drawn randomly from a pack of cards and a dice is thrown once. Determine the probability of not getting a king on the card as well as 6 on the dice. Solution: Here, in a pack of cards n(S) = 52 Probability of getting a king, P(K) = 4 52 = 1 13 Probability of not getting a king, P(not K) = 1 – 1 13 = 12 13 Again, in a throw of dice, n(S) = 6 Probability of getting 6, P(6) = 1 6 The probability of not getting 6, P (not 6) = 1 – 1 6 = 5 6 Finally, probability of not getting a king as well as 6 P (not K and not 6) = P (not K) × P (not 6) = 12 13 × 5 6 = 10 13 exercise 16.2 1. (a) If P(A) = 1 3 , P(B) = 1 4 , find P(A∩B) if 'A' and 'B' are independent events. (b) If P(A) = 1 3 and P(A and B) = 1 8 , find P(B) if 'A' and 'B' are independent events. 2. (a) Find the probability of getting a 3 on a dice and head on a coin when the dice is rolled and the coin is tossed simultaneously. (b) A card is drawn randomly from a pack of 52 cards and a dice is thrown once.
302 Oasis School Mathematics-10 Determine the probability of not getting a king as well as not getting 6 on the dice. (c) A card is drawn at random from a pack of 52 cards, and at the same time a marble is drawn at random from a bag containing 2 red marbles and 3 blue marbles. Find the probability of getting: (i) a king and a blue marble. (ii) a queen and a red marble. (iii) a black card and a red marble. (iv) a face card and a blue marble. 3. (a) A and B appear in an interview for two vacancies of the same post. The probability of A's selection is 1 7 and that of B's selection is 1 5 . What is the probability that (i) both of them will be selected? (ii) at least one of them will be selected? (iii) none of them will be selected? (b) The probability of solving a mathematical problem by two students 'A' and 'B' are 1 3 and 1 4 respectively. If the problem is given to both students, find the probability of; (i) solving the problem by both of them. (ii) solving the problem by at least one of them. (iii) solving the problem by none of them. 4. (a) A coin is tossed and a dice is rolled together. Find the probability of getting. (i) a head on the coin and 4 on the dice. (ii) a tail on the coin and an odd number on the dice. (b) A bag contains 4 red and 7 blue balls. A ball is drawn and replaced. If another ball is drawn, find the probability of getting (i) two red balls. (ii) two blue balls. (iii) blue in the first draw and red in the second draw. (iv) a blue and a red ball. (c) A bag contains 6 blue, 5 white and 4 black balls. A ball is drawn randomly from the bag. At the same time, a card is also drawn from a pack of 52 cards. Find the probability of getting (i) a white ball and a jack. (ii) a black ball and a black card. (d) A ball is drawn randomly from a bag containing 5 blue, 3 red and 2 white balls and replaced. If a second ball is drawn, what is the probability of getting (i) a blue ball in the first draw and a white ball in the second draw? (ii) a red or white ball in the first draw and a blue ball in the second draw? (iii) a blue ball in both draws or a white ball in both the draws? (iv) both balls of the same color?
Oasis School Mathematics-10 303 Answers 1 2 . (a) ( ) . (a) ( ) ( ) ( ) ( ) ( ) ( ) 1 12 3 8 1 12 10 13 3 65 2 65 1 5 9 65 b b c i ii iii iv 3. (a i )( ) (ii) (iii) ( ) b i( ) ( ) ii ( ) iii 4.(a i )( 1 35 11 35 24 35 1 12 1 2 1 2 ) ( ) (b)(i) ( ) ( ) ( ) (c)(i) 1 12 1 4 16 121 49 121 28 121 28 121 1 3 ii ii iii iv 9 2 15 1 10 1 4 29 100 19 50 ( ) ii (d)(i) ( ) ii ( ) iii ( ) iv 16.4 tree diagram If we have to find the combined result of two or more events, we cannot always use a table to show all possible outcomes, so in such a case, a probability tree diagram is suitable method to show all possible outcomes. The probability tree diagram is a diagrammatical representation of all the possible outcomes in any type of random experiment. Generally, we use a tree diagram to find the probability of any combined result of two or more arrows for each event starting from a point. Again, we use the same way to represent the other event. Now, the possible outcomes will be written on the right side of each of the branches of the tree diagram. We write the probabilities of particular events on the branch along the path denoting the event. Worked out examples example 1 A coin is tossed twice. Write all the possible outcomes in a tree diagram. Solution: Let 'H' and 'T' be the events of getting a head and tail respectively. example 2 A fair coin is tossed thrice. Show all the possible outcomes is a tree diagram. Also find the probability of getting. (i) all three tails (ii) tail, head, tail in order (iii) tail, tail, head in order (iv) at least two heads (v) at most two tails Solution: HH HT TH T H T H H H T T TT outcomes 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1st toss 2nd toss
304 Oasis School Mathematics-10 Let H and T be the events of the head and tail occurring on each toss. On tossing thrice, the outcomes are shown in a tree diagram. Now, finding the probability of (i) P(all three tails) i.e., P(TTT) = P(T) × P(T) × P(T) = 1 2 × 1 2 × 1 2 = 1 8 (ii) P(tail, head, tail in order) i.e., P(THT) = P(T) × P(H) × P(T) = 1 2 × 1 2 × 1 2 = 1 8 (iii) P(tail, tail, head in order) i.e., P(TTH) = P(T) × P(T) × P(H) = 1 2 × 1 2 × 1 2 = 1 8 (iv) P (at least two heads) i.e. P(HHH or HHT or HTH or THH) = P(HHH) + P(HHT) + P(HTH) + P(THH) = 1 8 + 1 8 + 1 8 + 1 8 = 4 8 = 1 2 3 outcome rd 2 toss nd 1 toss st toss HHH 1/2 × 1/2 × 1/2 = 1 8 HHT 1/2 × 1/2 × 1/2 = 1 8 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 HTH 1/2 × 1/2 × 1/2 = 1 8 HTT 1/2 × 1/2 × 1/2 = 1 8 THH 1/2 × 1/2 × 1/2 = 1 8 THT 1/2 × 1/2 × 1/2 = 1 8 TTH 1/2 × 1/2 × 1/2 = 1 8 TTT 1/2 × 1/2 × 1/2 = 1 8 Probability H T H H H H H H H T T T T T T T Fig. Tree Diagram
Oasis School Mathematics-10 305 (v) P(at most two tails) i.e. P(TTH or HTH or HHT or THH or THT or HTT or HHH) = P(TTH) + P(HTH) + P(HHT) + P(THH) + P(THT) + P(HTT) + P(HHH) = 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 + 1 8 = 7 8 or, P(at most two tails) = 1 – P(TTT) = 1 – 1 2 . 1 2 . 1 2 = 1 – 1 8 = 7 8 example 3 A bag contains 6 blue balls and 8 green balls. Two balls are drawn in succession without replacement. By drawing a tree diagram, find the probability of getting both balls blue. Solution: Let B and G be the event of getting blue and green balls respectively. Total balls = 6 + 8 = 14 A probability tree diagram is shown alongside Now, the probability of getting both balls blue is P(BB) = P(B) × P(B) = 6 14 × 5 13 = 3 7 × 5 13 = 15 21 example 4 three children are born in a family. Calculate the probability of having two sons by drawing a tree diagram. Solution: Let S and D denote a son and a daughter respectively. Then, probability and outcomes are shown in a probability tree diagram as outcomes. outcome BB BG B B G B G G GB GG 6 Blue 8 Green Fig. Tree Diagram 6/14 5/13 8/13 6/13 7/13 8/14
306 Oasis School Mathematics-10 3 outcome rd 2 birth nd 1 birth st birth SSS 1/2 × 1/2 × 1/2 = SSD 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 1/2 1/2 1/2 SDS 1/2 × 1/2 × 1/2 = SDS 1/2 × 1/2 × 1/2 = DSS 1/2 × 1/2 × 1/2 = DSD 1/2 × 1/2 × 1/2 = DDS 1/2 × 1/2 × 1/2 = DDD 1/2 × 1/2 × 1/2 = Probability S D S S S S S S S D D D D D D D Fig. Tree Diagram 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 The probability of exactly two sons is P(SSD or SDS or DSS). P(SSD) + P(SDS) + P(DSS) = 1 8 + 1 8 + 1 8 = 3 8 example 5 From the bag containing 5 white and 4 green balls, two balls are drawn with the replacement after the first drawn. Show all the possible outcomes in tree diagram. Also find the probabilities of all the outcomes. Solution: Let, W and G be the events of getting white and green ball respectively. P(W) = 5 9 P(W) = 5 9 P(G) = 4 9 P(G) = 4 9 P(G) = 4 9 P(W) = 5 9 5W 4G W G W WW 5 9 × 5 9 = 25 81 5 9 × 4 9 = 20 81 4 9 × 5 9 = 20 81 4 9 × 4 9 = 16 81 WG GW GW G W G outcomes Probabilities
Oasis School Mathematics-10 307 exercise 16.3 1. A coin is tossed twice in succession. Calculate the probability that both are tails by drawing a tree diagram. 2. A bag contains 5 red and 7 white balls. Two balls are drawn at random one after the other without replacement. Find the probability of getting (i) both red balls (ii) both white balls (iii) one red ball and one white ball in any order 3. (a) Two cards are drawn in succession from a well shuffled pack of 52 cards without replacement. Find the probability of getting (i) both cards having the same color (ii) both cards having different colors (b) Two cards are drawn one after another from a pack of 52 cards without replacement. Using a diagram, find the probability of both getting a king. 4. (a) A coin is tossed thrice. Show all possible outcomes in a tree diagram with each of the probabilities at the side of the branches. Also find the probability of getting, (i) all three tails (ii) tails, heads, tails in order (iii) tails, tails, heads in order (iv) at least two heads (b) A dice is rolled twice. Using a tree diagram, find the probability of getting a '6' in both case. 5. A family has three children. Draw a tree diagram to show all possible combinations of boys and girls. Calculate the probability of getting (i) all boys (ii) all of the same sex (iii) exactly two girls (iv) exactly one boy 6. Two spanners are colored as shown. (a) If both spanners are spun, draw and label a tree diagram showing all the possible outcomes. (b) Using a tree diagram, calculate the probability of getting, (i) two blacks (ii) two whites (iii) a white and a grey 7. A bag contains 7 white and 5 black balls. Two balls are drawn at random, one after another without replacement. Draw a tree diagram to show all the possible outcomes. Also find the probability of getting, (i) two white balls (ii) the balls of the same color (iii) balls of different colors 8. A bag contains 6 black and 4 white marbles. A marble is drawn at random from the bag and the marble is replaced second marble is drawn. By using a tree diagram, find the probability of getting, (i) both of the same color and (ii) both marbles of different colors Black Black Gray Gray White White 'A' 'B'
308 Oasis School Mathematics-10 9. (a) There are 3 sweets, one yellow, one red and one black in a bag. A sweet is taken out randomly and not replaced. Then after, another sweet is drawn. Write the sample space using a tree diagram. Calculate the probability of selecting one yellow and one red sweet. (b) There are 4 balls, 1 red, 2 blue and 1 green in a basket. Two balls are drawn one after another without replacement. Show all the possible outcomes in a tree diagram. (ii) HHH HHT 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 HTH HTT THH THT TTH TTT H T H H H H H H H T T T T T T T 1 8 1 8 1 2 1 8 (i) (iii) (iv) 4. (a) 1 8 3 8 1 4 3 8 (i) (iii) (ii) (iv) 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 B B B B B B B G G G G G G G 5. 1/4 2/4 1/2 2/3 1/3 1/3 1/3 1/3 1/3 2/3 R G B R B B G R G B 1R 2B 1G (b) W W B W B B 7/12 6/11 5/11 7/11 4/11 5/12 7 W 5B 31 66 35 66 7 22 (ii) (iii) (i) 7. W W B W B B 4/10 4/10 6/10 4/10 6/10 6/10 4 W 6 B 8. 1/6 1/6 5/6 1/6 5/6 5/6 1S 5S S S S S S 1 36 (b) S S S S S S S S S , 1 8 (b) (i) 1 8 (ii) , 3 16 (iii) 1/4 1/4 1/2 1/4 1/2 1/4 1/4 1/4 1/2 1/4 1/4 1/2 W G W B W G G B W G B B W G B 6. W WG WB GW GG GB BW BG BB W 1 3 1/3 1/3 1/3 1/2 1/2 1/2 1/2 1/2 1/2 Y R B R Y B Y B R 1Y 1R 1B 9 (a). R R B Y R Y B B Y B R Y WW WW WB WB BW BW BB BB 1. 1 4 2. (i) 5 33 (ii) 7 22 (iii) 35 66 3. a. (i) 25 52 (ii) 26 51 b. (i) 1 221 12 25 13 25 (ii) (i) Answers
Oasis School Mathematics-10 309 miscellaneous exercise 1. From the pack of 52 playing cards, one card is drawn. If ‘A’ be the events of getting an ace and ‘F’ be the event of getting a face cards; (a) What type of events are ‘A’ and ‘F’? (b) Calculate the value of P(A) and P(F). (c) Find the probability of getting an ace card or a face card. (d) If ‘R’ be the event of getting red cards, how many events of ‘F’ is common with the events ‘R’. [Ans: (b) 4 52 , 12 52 (c) 4 13 (d) 6] 2. A card is drawn from the pack of 52 playing cards. If M and N be the event of drawing a jack and a diamond respectively. (a) Find P(M) and P(N). (b) Find P(M∪N) (c) After drawing the first card, second card is drawn without replacing the first one. Show the outcomes of getting and not getting event M in tree diagram. (d) Find the probability of getting at least one jack in two draws. [Ans: (a) 1 13 , 1 4 (b) 4 13 , (c) 396 2652] 3. If ‘A’ and ‘B’ are two mutually exclusive events. (a) Write the formula to find the probability of getting at least one of A and B. (b) ‘A’ and ‘B’ are two mutually exclusive events with respective probabilities 2 3 and 1 4 . Find the probability of getting neither A nor B. (c) If 'A' be the event of getting '1' on rolling dice and 'B' be the event of getting head on tossing the coin. Find the probability of getting '1' on dice and not a head on coin. (d) In two mutually exclusive events 'A' and 'B' P(A) = 2 3 and P(B) = 5 6 . Comment on this information. [Ans: (b) 1 12 , (c) 1 12 , (d) not possible] 4. From the number card 2 to 10. If a card is drawn (a) What is the probability of getting an even number card and what is the probability of getting prime numbered card? (b) What is probability of getting prime numbered card and even number card? What is the probability of getting prime number or even number card? (c) If another card is drawn without replacing the first, is it possible to get prime number card in both draw? Justify your answer. [Ans: (a) 5 9 and 4 9 , (b) 1 9 and 8 9 , (c) No]
310 Oasis School Mathematics-10 5. A mathematical problem is given to two candidates 'A' and 'B' . The probability of solving the problem by 'A' is 1 2 and that of 'B' is 2 3 . (a) What types of events are 'A' and 'B'? (b) What is the probability of solving problems by both of them? (c) Find the probability that only student solve the problem. (d) Probability of not solving the problem is 1 6 . Comment on this result. [Ans: (b) 1 6 , (c) 1 2 , (d) Yes] 6. There are 1 red ball, 1 blue ball and 2 white ball in the basket. (a) If a ball is drawn, find the probability of getting a blue ball or a white ball. (b) If another ball is drawn without replacing the first ball, show the outcomes by tree diagram. (c) With the help of tree diagram, find the probability of getting two balls of different colour. (d) Calculate the respective probability of (b) and find the sum of probabilities and draw out the conclusion. [Ans: (a) 3 4 , (b) consult your teacher, (c) 5 6 , (d) consult your teacher.] 7. There are 50 students in a class, 30 of them are boys. Among 50 students 15 are wearing spectacles, 5 of them are girls. If student is chosen at random, (a) What is the probability of getting a boy wearing spectacles? (b) Find the probability of getting a boy or a student wearing spectacles. (c) If another student is also chosen without replacing the first one, show the result of wearing and not wearing spectacles in tree diagram. (d) Find the probability that selected both students are wearing spectacles. [Ans: (a) 1 3 , (b) 7 10 (c) consult your teacher, (d) 3 25 ] 8. There are 80 students in class out of them 50 are girls. If 60 of them are studying optional mathematics, among them 40 are girls. (a) Find the number of (i) girls not studying optional maths (ii) boys studying optional maths. (b) If one students is selected at random, find the probability that selected student is a boy or a student studying optional maths. (c) If another student is selected without replacing the first one, show that possible outcomes of students studying or not studying optional maths. (d) What is the probability that both selected students are girls? [Ans: (a) i. 1 8 , ii. 1 4 , (b) 7 8 , (c) consult your teacher, (d) 245 582 ]
Oasis School Mathematics-10 311 9. The probability of hitting the target by 'A' and 'B' are 1 4 and 1 3 . (a) What is the probability of hitting the target by both of them? (b) Calculate the probability of hitting the target by none of them. (c) Calculate the probability that one of them hit the target. (d) What is the sum of the result of (a), (b) and (c)? Comment the result. [Ans: (a) 1 12 , (b) 1 2 , (c) 5 12 , (d) consult your teacher] 10. There are 15 red and 10 yellow balls in the basket. Two balls are drawn one after another without replacement. (a) Find the probability of getting a red ball and the probability of getting yellow ball. (b) Show the possible outcomes in tree diagram. (c) Find their respective probabilities. (d) Find the sum of probabilities. Comment on the result. [Ans: (a) 15 25 , and 10 25 , (b), (c), (d) consult your teacher] Attempt all the questions: 1. An experiment is done from the pack of numbered cards 6 to 39. (a) Find the number of total events at the experiment. (1) (b) Find the probability of getting prime number or cubed number when a number card is drawn randomly in the experiment. (2) (c) Show the entire events of getting prime number card and not getting prime number card in a tree diagram when two cards are taken out randomly one after another without the replacement of the first card. (2) (d) What is the sum of the probabilities of all outcomes? (1) 2. A bag contains 12 balls out of which x are white. (a) Find the probability of getting white ball if a ball is drawn at random. (1) (b) If 6 more white balls are put into the bag, the probability of getting white ball will be double the previous one. Find the number of white ball in the bag initially. (2) (c) 12 balls in the bag are red and white in colour. Draw the tree-diagram showing the entire outcomes when two balls are taken out randomly one after another without replacement of the first ball. (2) (d) Find the respective probabilities of all outcomes. (1) Full Marks: 12
312 Oasis School Mathematics-10 trIGoNometry Contents • Area of Triangle • Height and Distance expected Learning outcomes At the end of this unit, students will be able to develop the following competencies: • To find the area of a triangle and parallelogram using the formula of trigonometry • To solve the problems of height and distance using the formula of trigonometry materials required • Protractor, field tape, chart paper, A4 size paper. Estimated Teaching Hours 8
Oasis School Mathematics-10 313 17.2 Height and Distance Height and distance is the application of trigonometry. In height and distance, we can find the height of an object or a distance of an object from a point without measuring its length. In height and distance, we have to solve right-angled triangle making height and distance as two sides of a triangle. Here are some basic terms which are used to solve the problem of height and distance. Angle of elevation If an object A is observed from the position O, O is the position of the eye of the observer. Line OA is called the line of sight. OB is the horizontal line. Then ∠AOB is the angle of elevation of A from O. O A Line of sight angle of elevation Horizontal line B 17.1 Warm-up Activities Unit 17 trigonometry • Recall the values of the standard angle of all Trigonometric ratios. • Measure your height (In cm) Measure the length of your shadow at different time (7 : 00 A.M., 9:00 AM, 12:00, Noun, 2:00 P.M., 3 P.M.) • Find the ratio of the your height and your shadow in different time. • Discuss how to find the inclination of the Sun’s rays in different time. Discuss how to find the unknown quantity in the given right angled triangle. 40cm 30° ? 60° ? 24ft ? 60m 60 3 45° 45 2 m ? ? 30m 60m
314 Oasis School Mathematics-10 Therefore, when the position of the object lies above the eye of an observer, the angle made by the line of sight with the horizontal line through the position of the eye is called the angle of elevation. Angle of depression If an object ‘A’ is observed from position O, O is the position of the eye of the observer. Line OA is the line of sight, OB is the horizontal line through O. Then ∠AOB is the angle of depression of ‘A’ from O. When the position of the object lies below the eye of the observer, the angle made by the line of sight with the horizontal line through the position of the eye is called the angle of depression. Worked out examples example 1 The angle of elevation of the top of a house at a distance of 20 m from its foot on a horizontal plane is found to be 60°. Find the height of the house. Solution: Let, AB be a house, C the position of the observer, BC the distance between the foot of the house and the observer. Here, angle of elevation, ∠ACB = 60° BC = 20 m AB = ? In right angled triangle ABC Tan 60° = AB BC or, 3 = AB BC or, AB = 20 3m or, AB = 20 × 1.732 m or, AB = 34.64 m ∴ Height of the house = 34.64 m. example 2 From the top of the tower 100 m high, the angle of depression of an object on the ground is 30°. Find the distance between the object and the foot of the tower. Solution: Let AB be a tower and C the position of the object. BC is the distance between the object and the foot of the tower. B A O Line of sight horizontal line angle of depression A B C D 100m 300 300
Oasis School Mathematics-10 315 Here, ∠DAC = angle of depression of C from A Since, DA||CB. ∠DAC = ∠ACB = 30° AB = 100 m BC = ? In the right-angled triangle ABC, Tan 30° = AB BC or, 1 3 = 100 BC or, BC = 100 3m or, BC = 100 × 1.732 or, BC = 173.2 m Hence, distance between the tower and the object = 173.2 m. example 3 the length of the shadow of a pole 3 m high is 3 3 m. What is the altitude of the sun? At the same time, if the length of the shadow of a house is 30 3 m, find the height of the house. Solution: Let AB be a pole and BC its shadow. ∠ACB is the altitude of the sun. At the same time, QC is the length of the shadow of the house PQ Here, AB = 3m, BC = 3 3m, QC = 30 3m In ∆ ABC, Tan = AB BC or, Tan = 3 3 or, Tan = 1 3 or, Tan = Tan 300 θ θ θ θ 3 ∴ Altitude of the sun = 30° example 4 A man 1.6 m tall observes the angle of elevation of building to be 60°. If the distance between the foot of the house from the observer is 60 m, find the height of the building. Solution: Let, AB be the man, CD a building, BD the distance between observer and the building. P Q C B A 3m ? θ 30 3m Again 3 3m In PQC PQ BC PQ PQ PQ m Height of th ∆ = = = ∴ = ∴ tan tan θ 30 30 3 1 3 30 3 30 0 e house = 30m. AB QC D E B A C 600 60m 1.6m
316 Oasis School Mathematics-10 Here, ∠CAE = Angle of elevation of C from A = 60° BD = AE = 60 m ED = AB = 1.6 m In right angled triangle AEC, Tan 60° = CE AE or, 3 = CE 60 or, CE = 60 3 m. = 60 ×1.732 m = 103.92 m. ∴ Height of the building = CD = CE + ED = 103.92 +1.6 m = 105.52 m. example 5 A kite is flying in the sky. If a 300 m string is let out and if it makes an angle of 45° to the horizontal line, find the height of the kite from the ground. Solution: Let A be the kite. AC is the length of the string of the kite. AB is the vertical height of the kite. ∠ACB is the angle made by the string with the horizontal line. Here, AC = 300 m ∠ACB = 45° AB = ? In ∆ ABC, Sin 45° = AB AC or AB or AB m m m m , , . 1 2 300 300 2 300 2 2 150 2 212 13 = = = × = = ∴ Height of the kite is 212.13 m. Example 6 The top of a tree, which is broken by the wind, makes an angle of 60° with the ground at a distance of 3 3 m away from the foot of the tree. Find the height of the tree before it was broken. Solution: Let BC be the height of the tree before it was broken, AC the broken part of the tree, D a point 450 B ? A C 300m
Oasis School Mathematics-10 317 on the ground where the top of the broken part touches. Here, BD = 3 3m ∠ADB = 60° In ∆ right angled ADB, Cos60° = BD AD or, 1 2 = AD 3 or, AD = 6 3 m. ∴ Length of the broken part of the tree = AC = AD = 6 3 m. Again, In ∆ABD Tan 60° = AB BD or, 3 = or, AB = 9 m. ∴ Height of the tree before it was broken. = AB + AC = (9 + 6 3 )m = (9 + 6 × 1.732) m = 9 + 10.392 = 19.392 = 19.39 m. exercise 17.1 1. (a) The angle of elevation of the top of a building from a point 40 m away from the bottom of the building is found to be 60°. Find the height of the building. (b) From the top of a tower 90 m high, the angle of depression of an object on the ground is observed to be 30°. Find the distance between the object and the foot of the tower. (c) A tree 60 m high on the bank of a river subtends an angle of 60° on the opposite bank. Find the breadth of the river. (d) A man observes the top of a tower 80 3 m in height from a distance 240 m from the foot of the tower. Find the angle of elevation. 2. (a) Find the length of the shadow of a vertical pole of height 40 ft, if the sun's rays are inclined at an angle of 45° with the ground. 600 B A C D 3 3m AB 3 3
318 Oasis School Mathematics-10 (b) When the sun's altitude is 30°, the length of the shadow of a pillar is 5 m. Find the height of the pillar. (c) The shadow cast by a tree of height 8 3 m is 8 m long. Find the altitude of the sun. (d) The length of the shadow of a pole 3 m is 3 m. At the same time, find the length of the shadow of a tower 30 m high. 3. (a) A ladder leans against a wall, with its foot 20 m from the bottom of the wall. If it makes an angle 60° with the ground, find the length of the ladder. (b) A ladder of length 20 m is leaning against a vertical wall. The ladder makes an angle of 30° with the wall. Calculate the height up to which the ladder reaches on the wall. (c) A boy is flying a kite. The stretched part of a string, which is 60 m long makes an angle of 60° with the horizontal line. Find the vertical height of the kite. 4. (a) A man of height 1.5 m observes the angle of elevation of the top of a pole situated in front of him and finds to be 60°. If the height of the pole is 121.5 m, find the distance between the pole and the man. (b) A man stands 50 m away from the foot of a pole. He finds the angle of elevation to be 30° while observing the top of the pole. If his eyes are 1.5 m above the ground, find the height of the pole. (c) A man of height 1.5 m, standing 48 m away from the foot of a building, find the angle of elevation of the top of the building to be 30°. Find the height of the building. (d) A boy of height 1.1 m was flying his kite. When the length of a string of the kite was 33 m, it made an angle of 30° with the horizon. What is the height of the kite from the ground? (e) A man observes the top of a pole of height 52 m, situated in front of him and finds the angle of elevation to be 300 . If the distance between the man and pole is 87 m, find the height of the man. (f) A girl having height 1.54 m is 30 m away from the tower whose height is 53.5m. Find the angle of elevation of the top of the tower from her eyes. 5. (a) From the top of a building, the angle of elevation of the top of a tower is found to be 30°. If the distance between the building and the tower is 120 m, find the height of the building, if the height of the tower is 100 m. (b) The angle of elevation from the roof of a house to the top of a tree is found to be 300 . If the height of the house and tree are 8 m and 20 m respectively, find the distance between the house and the tree. (c) From the top of a tower the angle of depression of the roof of a house 20 m high and 60 m away from the tower was observed and found to be 600 . Find the height of the tower.
Oasis School Mathematics-10 319 (d) Two pillars of height 115.36 m and 150 m are at a distance of 60m. Find the angle of elevation of the top of the second from the top of the first. (e) A boy was flying his kite from the roof of a house having height 12.5 m. The kite was flying at a height of 42.5 m from the ground. If the string of the kite makes an angle of 30° with the horizon, find the length of the string. 6. (a) The upper part of a tree broken by the wind makes an angle of 60° with the horizontal line at a distance of 15 m from the foot of the tree. Find the height of the tree before it was broken. (b) A tree of height 24 m is broken by the wind such that the broken part of the tree makes an angle of 30° with the ground. Find the length of the broken part of the tree. (c) The upper part of a tree broken by the wind is 18 m long. If it touches the ground at a distance of 9 m from the bottom of the tree, find the angle made by the broken part of the tree with the ground and the height of the tree before it was broken. 7. (a) The diameter of a circular pond is 120 m. The angle of elevation of the top of a pillar situated in the middle of the pond, observed from the edge of the pond, is found to be 60°. Find the height of the pillar above the surface of water. (b) A pillar is fixed at the center of a circular meadow of diameter 60 m. The angle of elevation of the top was found to be 60°, when observed from a point on the circumference of the circular meadow. Find the height of the pillar from the ground. (c) A pole of height 80 3 m is fixed at the centre of a circular pond. The angle of elevation of the top of the pole from a point on the circumference of the pond is found to be 300 . Find the diameter of the pond. (d) At the centre of a circular pond, there is a pole of 11.62 m height above the surface of the water. From a point on the edge of the pond, a man of 1.62 m height observed the angle of elevation of the top of the pole and found it to be 30°. Find the diameter of the pond. (e) The diameter of a circular pond is 100m and a pillar is fixed on the centre of the pond. A person finds the angle of elevation of the top of the pillar is Q from the bank of the pond. If the depth of the pond is 1.5m and total height of the pillar is 51.5m, find the value of Q. 8. (a) A flagstaff stands on the top of a tower. The angles of elevation of the top and the bottom of a flagstaff as observed from a point 20 m away from the foot of the tower are found to be 60° and 45° respectively. Find the height of the flagstaff. (b) A flagstaff stands on the top of a tower. The angle subtended by the flagstaff and the tower at the point 40 m away from the bottom of the tower are 150 and 300 respectively. Find the height of the flagstaff.
320 Oasis School Mathematics-10 (c) The angle of elevation of the top of the incomplete house of height 60 m is found to be 45°. By how much should the height of the house be increased so that the elevation of the top from the same point is 60°. Answers 1. (a) 69.28 m (b) 155.88 m (c) 34.64 m (d) 300 2. (a) 40 ft. (b) 2.89 m (c) 600 (d) 17.32 m 3. (a) 40m (b) 17.32m (c) 51.96 m 4. (a) 69.28 m (b) 30.37 m (c) 29.21 m (d) 17.6 m (e) 1.77 m (f) 600 5. (a) 30.72 m (b) 20.78m (c) 123.92 m (d) 300 (e) 60 m 6. (a) 55.98 m (b) 16 m (c) 600 , 33.59 m 7. (a) 103.92 m (b) 51.96 m (c) 480 m (d) 34.64 m (e) 450 8. (a) 14.64 m (b) 16.9 m (c) 43.92 m Project Work Find the indication of the sun’s rays at different times of the day by using your height and length of your shadow. Time length of shadow tan θ = θ miscellaneous exercise 1. In the given figure, if 'A' represents position of a kite. AB, vertical height of the kite. AC, length of the string of the kite. (a) Find the length of AC. (b) What would be the vertical height of the kite, if the string of the kite makes an angle 60° with the ground? (c) At what angle the length of the string is inclined if the kite is flying at the height of 30 2m? (d) If AB = BC, what would be the value of ∠C? Ans: [(a) 84.85 m , (b) 73.48 m, (c) 300 , (d) 450 ] A B 450 60 m C
Oasis School Mathematics-10 321 2. A man of height 1.6m is observing the pole of height 21.6m. The angle of elevation so formed in 60°. (a) Make the figure to represent above question. (b) Explain the figure. (c) Find the distance between the pole and the man. (d) What will be the deviation in the angle of elevation if the distance between the man and the pole is 20m? Ans: [(a), (b) consult your teacher, (c) 11.55 m, (d) Elevation decreased by 150 ] 3. On observing the top of the building from the top of the tree, angle of elevation is formed. (a) Which one is taller, a tree or a building? (b) If the angle of elevation is 30°, the height of the building is 60m and distance between them is 100m. Show this in diagram. (c) Find the height of the tree. (d) What will be the effect on angle of elevation from the same point to the top of building if it is observed after 6 months? Ans: [(a), (b) consult your teacher, (c) 42.26 m, (d) consult your teacher.] 4. On observing the top of pole from the top of the incomplete building angle of depression of 600 is formed. (a) Which one is taller, a pole or a building? (b) If the height of the pole is 20 ft., distance between the pole and the building is 60 ft., Show this information in diagram. (c) Find the height of the building. (d) After the complete construction of the building, if the top of the pole is observed from the top of the building, what will be the effect on the angle of elevation? Ans: [(a) (b) consult your teacher, (c) 123.92 ft, (d) consult your teacher] 5. A tree of height 48m is broken by the wind, if the broken part makes an angle 30° with the ground. (a) Show this information in the diagram. (b) Explain the diagram. (c) Find the length of broken part of the tree. (d) What will be the effect on the angle made by the broken part with the ground by its length? Ans: [(a) , (b) consult your teacher. (c) 32 m, (d) consult your teacher]
322 Oasis School Mathematics-10 6. A man from the edge of the circular pond observing the top of the pole at the centre of the pond. He finds the angles of elevation be 30°, (a) If the above figure represents the given question, identify the man, pole, and diameter of the pole. (b) The height of the man is 1.5m and the diameter of the pond is 80m, find the length of CE. (c) Find the height of the pole above the water level. (d) If the depth of the pond is 5m and 2m of the pole inside the ground; what is the total length of the pole. Ans: [(a) consult your teacher, (b) 23.09 m, (c) 24.59 m, (d) 31.59m] 7. The angle of depression of the top of building of 20m from 11th and 15th floor of a tower is found to be 30° and 45° respectively. (a) Show this information by figure. (b) The distance between the tower and the building is 50m. Find the height of the 11th floor of the building. (c) Find the height of 15th of the building. (d) What would be the effect on angle of depression if it is observed from 17th floor? Ans: [(a) consult your teacher, (b) 48.86 m, (c) 70 m, (d) consult your teacher] 8. Two towers of height 115.36m and 150m are at a distance of 60m. (a) What type of angle is formed on observing the top of the first tower, from the top of second? (b) If the top of the first is observed from the top of the second, show it in diagram. (c) What is the value of angle formed, in(a)? (d) What will be the effect on angle of elevation if the distance between two tower is more than the given distance? Ans: [(a), (b) consult your teacher, (c) 320 , (d) consult your teacher] 9. The length of the shadow of the pole 20m is 20 3m. (a) Show this information is diagram. (b) Find the inclination of sun's rays. (c) At the same time what will be the length of shadow of another pole whose height is 30 3m. (d) At what angle sun's rays should incline such that the height of the pole and length of its shadow are equal. Ans: [(a) consult your teacher, (b) 300 , (c) 900 m, (d) 450 ] E C A B D E 300
Oasis School Mathematics-10 323 Full Marks: 16 1. A boy was flying a kite having length 600 m from the roof of his house 10.5 m high. The kite is flying by making angle 60° with horizontal line. (a) Show the information in diagram. (1) (b) Find the total height of kite if the height of the boy is 1.5m. (1) (c) What would be the effect on the height of the kite if the string makes angle 45° with horizon? (1) (d) Compare the result in (b) and (c) and make a conclusion. (1) 2. The diameter of a circular pond is 130 m and of pillar of height 69 m is fixed at the centre of the pond such that 1 m part of the pillar is inside the ground. The depth of pond is 3 m. A person finds the angle of elevation θο of the top of the pillar from a point on the bank of the pond. (a) Construct the figure to represent the statement. (1) (b) Find the height of the pillar above the surface of water. (1) (c) Find the value of θ. (2) 3. The angle of elevation from the roof of the house to the top of the tree is found to be 300 . The height of the house and tree are 6 m and 18 m respectively. (a) Draw the model picture of above statement. (1) (b) Find the distance between house on the tray. (2) (c) According to the scale 1: 200, how many centimeter should be the height of the house and the tree in the map? (1) 4. On observing the top of house from the top of the incomplete building angle of depression of 450 is formed. (a) Which one is taller, a house or a building? (1) (b) If the height of the house is 25 ft., distance between the house and the building is 100 ft., Show this information in diagram. (1) (c) Find the height of the building. (1) (d) After the complete construction of the building, if the top of the house is observed from the top of the building, what will be the effect on the angle of elevation? (1)
324 Oasis School Mathematics-10 Specification grid for class 9 and 10 Prescribed by CDC, Nepal. S.N. Areas Total working hours Knowledge (16%) Understanding (24%) Application (40%) Higher ability (20%) Total num ber of items Total num ber of ques tions Total Marks Num ber of items Marks Num ber of items Marks Num ber of items Marks Num ber of items Marks 1. Sets 12 1 1 1 1 1 3 1 1 4 1 6 2. Arithmetic 28 2 2 2 3 3 5 2 3 9 3 13 3. Mensuration 28 2 2 2 3 2 5 2 3 8 3 13 4. Algebra 32 2 2 2 4 3 7 1 2 8 3 15 5. Geometry 28 2 2 2 3 2 5 2 3 8 3 13 6. Statistics and Prob ability 24 2 2 2 3 2 4 2 2 8 2 11 7. Trigonometry 8 1 1 1 1 1 1 1 1 4 1 4 Total 160 12 18 30 15 49 16 75 b|:6JoM • k|Zgkq lgdf{0f ubf{ k|To]s If]qdf / ;du|df 1fg, af]w, k|of]u / pRr bIftfsf nflu tf]lsPcg';f/sf ef/ ldn]sf] x'g'kb{5 . t/ ;+1fgfTds txdf @ c+s;Dd 36a9 x'g ;Sg]5 . • ;Gbe{ lbP/ k|Zg lgdf{0f ug'{kg]{5 . k|To]s k|Zgdf PseGbf a9L ;+1fgfTds txsf pkk|Zg ;dfj]z ug{ ;lsg]5 . • Application / higher ability txsf k|Zg lgdf{0f ubf{ ;DalGwt If]qsf cnfjf cGo If]qsf ljifoj:t';+u ;DalGwt k|Zg klg /xg ;Sg]5g\ . • x/]s If]qcGtu{t /x]sf ;a} pkIf]qsf ljifoj:t' ;dfg'kflts ?kdf ;dfj]z x'g] u/L k|Zg lgdf{0f ug'{kg]{5 . Subject: Compulsory Mathematics-10 time : 3 hrs. Full Marks : 75