Math All chapters MCQ with answers

Solutions 1. (c) The general term of an arithmetic  n(10  n) 10n  n2 progression is given by a  (n 1)d where 'a' is the first term and 'd' 10. (d) Not available is the common difference. 11. (c) Not available Here, a = 4 and d 1310 10  7 12. (b) Not available 13. (d) Not available 743. Then nth term 14. (b) Not available  4  (n 1)3  3n 1 2. (c) The given arithmetic progression is 35, 15. (b) Not available 30, 25, ..... Here, a = 35 and d = (-5). 16. (b) Not available nth term, tn  a  (n 1)d 17. (c) Not available  40  5n For the first negative term, tn  0 18. (d) Not available  40  5n  0 19. (a) Not avJaOilabINle @iitwale in Telegram  8n  n8  n9 3. (a) Not available 20. (c) T6  T7  ......  T12  S12  S5 4. (b) Not available  12[2 6  (12 1) 4] 2 5. (d) Not available  5 [2 6  (5 1) 4]  266 2 Thus, the required sum is 266. 6. (d) Not available 21. (c) t6 13  5(5) 13  25  38 7. (a) Not available t20 13 19(5) 13  95 108 8. (d) The first 'n' even numbers are 2, 4, 6,… S15  15 [38 108]  1095.0 2n. 2  Sn  n (2  2n)  n(n 1) 22. (c) Sn  n [2a  (n 1)d] 2 2  [4n  59][n 12]  0 9. (a) Odd numbers starting from 11 are 11,  n  59 or n 12 13, 15, 17, 19, .... 4 Sum to 'n' odd numbers, where a = 11, d = 2 is Hence, the number of terms is n = 12. Sn  n [2a  (n  1)d ] 23. (c) The circumferences of the successive 2 circles (in cm) are 8

2 r,2 (r 1)  2 (r  2),..... T2 T1  2 (r 1)  2 r  2 36. (d) Since S10  360, a 115 and d = 1. T3 T2  2 (r  2)  2 (r 1)  2 tn  a  (n 1)d t15 11.5  (18 1)(1)  28.5o S5  5 (4 r  4(2 )) The angle of the biggest sector is 28.5o 2 10 (r  2)cm 37. (d) Not available 24. (a) Not available 38. (b) Not available 25. (a) Not available 39. (d) Not available 26. (b) Not available 40. (d) Not available 27. (c) Not available 28. (d) Not available 41. (b) Not available 29. (b) Not available 42. (a) a 15d  8 a = 5 (Given) 30. (d) Not available  d1 31. (b) Not available JO5 IN @iitwale in Telegram 5, 26 , 27 ,........ 39 ,8 55 5 32. (c) Not available 14 2  26  13 1  S14  2  5  5  33. (c) Not available 14  52 13  2  5    91 34. (b) Given, t4  8; S12 156 43. (b) Sum of cubes of 1st 'm' natural numbers  a = 2 and d = 2 tn  a  ( p 1)d   m(m 1) 2  3025  tn  a  ( p 1)d  2   2  2( p 1) 1000 Hence, m = 10.  p  500 35. (a) 44. (a) tm  a  (m 1)d ....(1) tn  a  (n 1)d ....(2) Subtracting (2) from (1), we get Solving the two equations, we get Circumference of 5 circles d  1and a  n  m 1  2 r  2 (r 1)  2 (r  2)  t(mn)  a  (m  n 1)d 2 (r  3)  2 (r  4)  90  n  m1 m n 1 0  10 r  70 Radius of the smallest circle, r = 7 cm. 9

45. (b) pS22  9  n(n 1)(2n 1) 2   1  2  ........' n 'terms   6   n n   9 n2 (n 1)2 (2n 1)2  n   1 2  3  .....  ' n 'terms  36  n   n2 (n 1)2 [4n2  4n 1]  n  n(n 1) 4 2n  S3[4n(n 1) 1]  n  n 1  n 1  S3(8S1 1) 22 46. (b) 7t7  11t11 51. (a) Not available  7[a  6d] 11[a 10d]  a 17(d)  0 52. (c) Not available  t18  0 53. (a) Not available 47. (a) 9, 18, 27, ......999 tn  a  (n 1)d 54. (b) Not available  999  9  (n 1)9  111  n 55. (b) Not available 48. (a) t1  a;tn2  b JOIN @iitwale in Telegram tn2  a  (n  2 1)d  b  a  (n  2 1)d  ba d n 1 49. (b) 1st term of 'n' AM.s between 'a' & 'b'  a  b  a  an  b n1 n1 Last term of 'n' A.M.s between 'a' & V  b  b  a  bn  a n1 n1 Sum of the terms  n  na  b  bn  a  2  n 1 n 1   n  (n 1)(a  b)  2  n 1   n (a  b) 2 50. (b) (1 + 1 + ....... 'n' terms) 10

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Probability 1. What is the probability of getting a prime 7. From a normal pack of cards, a card is number in a throw of a die? drawn at random. Find the probability of getting a jack or a king. (a) 2 (b) 1 2 (a) 2 (b) 1 52 52 (c) 3 (d) 6 2 (c) 2 (d) 1 13 26 2. What is, the probability that a vowel selected at random in English alphabet is an 8. Two numbers are chosen from 1 to 5. What \"i\"? is the probability for the two numbers to be consecutive? (a) 1 (b) 1 5 26 (a) 1 (b) 2 5 5 (c) 1 (d) 1 6 (c) 1 (d) 2 10 10 3. When two dice are thrown, what is the probability of always getting a number 9. Two dice are thrown at a time. What is the greater than 4 on the second dice? probability that the difference of the numbers shown on the dice is 1? (a) 1 (b) 1 6 3 (a) 5 JOIN @iit(wb)a3l16e in Telegram 18 (c) 1 (d) 1 36 2 (c) 1 (d) 1 6 12 4. What is the probability for a leap year to have 52 Mondays and 53 Sundays? 10. A bag contains 3 white and 5 red balls. If a ball is drawn at random, what is the (a) 1 (b) 1 probability that it is red? 366 52 (a) 3 8 (c) 2 (d) 1 (b) 5 7 7 8 (c) 3 5. Cm In a single throw of two dice, what is the 15 probability of getting a sum of 10? (d) 5 15 (a) 1 (b) 1 12 36 (c) 1 (d) 1 11. What is the probability of getting an even 6 8 number when a die is rolled? (a) 1 6. Three letters, to each of which corresponds 6 an addressed envelope are placed in the (b) 1 envelopes at random. What is the 36 probability that all letters are placed in the (c) 1 right envelopes? 2 (d) 1 (a) 1 (b) 1 12 3 (c) 1 (d) 0 6

12. OA card is drawn from a packet of 100 17. What is the chance that a non-leap year cards numbered 1 to 100. Find the contains 53 Saturdays? probability of drawing a number which is a square. (a) 2 (b) 1 7 7 (a) 1 (b) 9 (c) 2 (d) 1 10 100 365 365 (c) 1 (d) 2 18. From a well shuffled pack of cards, one card 100 100 is drawn at random. What is the probability that the card drawn is a king? 13. A book containing 100 pages is opened at random. What is the probability that a (a) 12 (b) 1 doublet page is found? 13 13 (a) 9 100 (c) 3 (d) 1 (b) 90 13 2 100 (c) 10 19. An unbiased coin is tossed. What is the 100 probability that neither head nor tail turns (d) 20 up? 100 (a) 1 (b) 1 2 14. If a coin is tossed twice, find the probability (c) 0 (d) 1 of getting at least one head. 3 (a) 1 (b) 1 20. A box conJtOainIsN7@redii,tw3 awlheiteinanTd e2leblgacrkam 2 4 balls, when a ball is picked at random from (c) 3 (d) 1 the box what is the probability that it is not 4 8 red? 15. Find the probability of getting a number (a) 1 (b) 11 greater than 2 or an even number in a single 12 12 throw of a fair die. (c) 7 (d) 5 12 12 (a) 1 (b) 2 21. An unbiased coin is tossed 5 times. What is 3 3 the odds in favour of getting at least one (c) 5 (d) 3 tail? (b) 1 :31 6 5 (a) 31 :1 16. Find the probability that in a family of 3 (c) 32:32 (d) 31 :32 children, there is at least one boy. (a) 3 22. A coin is tossed successively three times. 4 What is the probability of getting exactly one (b) 1 head or two heads? 8 (c) 4 (a) 3 (b) 1 8 4 4 (d) 5 8 (c) 1 (d) 2 3 3 3

23. Three of the six vertices of a regular 28. A month is randomly selected from a year. hexagon are chosen at random. What is the An event X is defined as 'the month with 30 probability that the triangle with these days'. Identify the number of outcomes of vertices is equilateral? event X. (a) 1 (a) 1 (b) 2 (b) B 5 5 (c) 3 (d) 4 (c) 1 (d) 3 10 10 29. A spinner is spun. What is the number of possible outcomes of the event that the 24. A coin is tossed successively three times. arrow will stop in the sector with an odd What is the probability of getting exactly one number. head or two heads? (a) 3 (b) 1 4 4 (c) 1 (d) 2 (a) 1 (b) 2 3 3 (c) 3 (d) 4 25. If P(A)  1 , P(B)  1 and A and B are 30. Turn the given cards facing down and 32 shuffle. mutually exclusive, find P(A' B') . (a) 5 (b) 1 6 6 (c) 1 (d) 2 Turn oneJOcaIrNd @faciinitgwauple. iWnhTatelisegthream 5 5 probability that it shows a circle? 26. The probability that A can win a race is 3 (a) 2 (b) 5 8 7 7 and the probability that B can win it is 1 . If (c) 3 (d) 1 6 7 7 both run in a race, what is the probability 31. A restaurant operator checked a sample of that one of them will win the race, assuming 200 plates and found that 10 of them were that both cannot win together? (a) 17 (b) 15 defective. The chef of the restaurant picks a 24 24 plate from this sample. What is the (c) 13 (d) 11 probability that he will get a defective plate? 24 24 (a) 0.5 (b) 0.05 (c) 0.2 (d) 20 27. A chess piece is randomly selected from a 32. Two dice are rolled at once and the box that contains all the pieces used in the numbers shown are added up. What is the game of chess. Identify the sample space of probability of getting a total of 14? this experiment. (a) {King, Queen, Bishop, Knight} (a) 1 (b) 1,2,3,4,5,6,7} 2 (c) {Bishop, Castle, King, Pawn, Queen, Knight} (b) 1 (d) {King, Knight, Pawn, Ace, Queen, (c) 0 Castle, Bishop} (d) 2 3 4

33. The given figure shows 10 alphabet cards. 38. 7 marbles shown are kept in a tin. What is the probability of getting a card If a marble is taken out randomly from the labelled 'S' when the card is chosen at tin, state the probability that the marble has random? the number 2. (a) 1 (b) 2 (a) 2 (b) 3 5 5 7 7 (c) 1 (d) 1 (c) 5 (d) 4 10 6 7 7 34. A weather forecast center predicts that it will 39. Two fair dice are thrown. Find the rain for 3 days in a duration of 20 days. probability that both dice show different Find the probability of rain on a particular numbers. day. (a) 1 (b) 5 (a) 17 (b) 3 6 6 20 17 (c) 32 (d) 29 (c) 20 (d) 3 36 36 17 20 40. A box contains 24 coloured marbles. 35. 90% of the mangoes in a bag are good. If a Eighteen of then are yellow and the rest are mango is chosen randomly from the box, either red or blue. A marble is picked at find the probability of getting a bad mango. random. Find the probability of picking an (a) 9 (b) 1 yellow maJrbOleI.N @iitwale in Telegram 100 100 (a) 1 (c) 9 (d) 1 4 10 10 (b) 3 36. A fair coin is tossed thrice. Identify the 4 probability of getting 3 tails as a fraction. (c) 3 (a) 1 (b) 3 8 8 8 (d) 1 8 (c) 7 (d) 1 41. A certain class has 's' students. If a student is 8 4 picked at random, the probability of picking 37. The given figure shows two circles such that a boy is 8 . If the class has 24 boys, what is the radius of the small shaded circle is 1 . 13 3 times the radius of the big circle. A dart is the value of 's'? thrown randomly towards the circle. Find (a) 26 (b) 39 the probability that the dart hits the shaded (c) 52 (d) 60 target. 42. A box contains 60 pens which are blue inked or black-inked. If a pen is picked at random, the probability of picking a blue- inked pen is .What is the number of blue- inked pens in the box? (a) 1 (b) 3 (a) 32 (b) 48 8 8 (c) 30 (d) 24 (c) 7 (d) 1 8 4 5

43. A certain class has 45 students. If a student 47. A factory has 120 workers in January. 90 of is picked at random, the probability of them are female workers. In February, another 15 male workers were employed. A picking a prefect is 1 . How many students in worker is then picked at random. Calculate 3 the probability of picking a female worker. the class are not prefects? (a) 60 (b) 15 (c) 30 (d) 70 (a) 3 (b) 4 4 9 44. A bag contains several coloured balls. 28 of (c) 2 (d) 1 them are red. If a ball is drawn at random, 3 3 the probability of drawing a red ball is 4 . x 48. A box contains a number of marbles with 9 serial number 18 to 38.A marble is picked at a random. Find the probability that it is a balls are added into the box. A ball is then multiple of 3. drawn at random. If the probability of drawing a red ball is now 1 ,find the value of (a) 3 2 5 x. (b) 7 20 (a) 4 (b) 6 (c) 5 (d) 7 (c) 3 45. A bag contains 40 coins, consisting of ` 2, ` 4 5 and `10 denominations. If a coin is drawn at random, the probability of drawing a ` 2 (d) 1 3 coin is 5 .lf x ` 2 coins are removed from the 49. A box conJtOainIsN40@miaitrwbleasleof irnedTaendlebgluream 8 colour. If a marble is picked at random, the bag and then a coin is drawn at random, the probability of picking a blue marble is 3 . probability of drawing a ` 2 coin is 1 . Find 8 2 Rana takes out one red marble and nine the value of x . blue marbles and then picks a marble at random. Find the probability that it is a blue (a) 5 (b) 2 marble. (c)10 (d) 8 46. The given incomplete table shows the (a) 4 (b) 2 number of coins in a box. 5 5 Coins ` 1 ` 2 ` 5 10 50 (c) 7 (d) 1 p 40 3 Amount 6 12 8 10 ? If a coin is drawn at random. the probability 50. Set P {x : 5  x  22, xis an integer} . If an of drawing a ` 2 coin is 3 . Find the element from set P is picked at random, 10 calculate the probability that it is a prime number. probability of drawing a 50 p coin. (a) 1 (a) 5 (b) 1 10 18 3 (b) 2 (c) 7 (d) 5 10 9 6 (c) 1 5 (d) 2 15 6

51. A box contains 32 coloured marbles. Eight 56. A box contains 20 balls bearing numbers of them are red marbles and the rest are 1,2,3,...,20.A ball is drawn at random from either blue or green marbles. A marble is the box. What is the probability that the drawn at random. Calculate the probability number on the balls is not divisible by 10? of drawing a marble which is not red in colour. (a) 9 (b) 1 10 10 (a) 2 (b) 5 (c) 9 (d) 1 3 8 5 5 (c) 3 (d) 7 57. A coin is tossed two times. Find the 4 16 probability of getting a tail at least once. 52. A Suppose a die is dropped at random on (a) 3 (b) 2 the rectangular region as shown in the 4 3 figure. (c) 3 (d) 1 5 5 What is the probability that it will land inside 58. If a leap year is selected at random what is the probability that it will contain 53 Tuesdays? the circle with diameter 2 m? (a) 1 (b) 2 7 7 (a)  (b)  8 28 (c) 3 (d) 4 (c) 1 (d) 1 7 JOIN @iitwa7le in Telegram 12 18 53. A lottery has a 0.00002 probability of 59. If P(A B)  0.65, P(A B)  0.15, find winning first prize. How many tickets have P(A)  P(B) . been sold for the lottery? (a) 2000 (b) 50000 (a) 1.5 (b) 1.4 (c) 2000 (d) 100000 (c) 1.3 (d) 1.2 54. 250 tickets are sold for a raffle. A girl 60. If the odds in favour of winning a race by calculates that the tickets bought by her three horses are respectively 1 : 4, 1 : 5 and family give them a 0.032 probability of 1 :6, find the probability that one of these winning first prize. How many tickets did the horses will win. family buy? (b) 9 (a) 37 (b) 39 (a) 60 60 60 (c) 50 (d) 8 (c) 41 (d) 51 60 60 55. All the three cards of spades are removed 61. Two dice are rolled at once. What is the from a well-shuffled pack of 52 cards. A probability of getting an even number on card is drawn at random from the remaining the first die or a total of 8? pack. Find the probability of getting a queen? (a) 4 (b) 5 9 9 (a) 3 (b) 3 52 49 (c) 7 (d) 2 9 9 (c) 1 (d) 1 26 52 7

Answer - Keys 1. B 2. A 3. B 4. D 5. A 6. C 7. C 8. B 13. A 14. C 9. A 10. B 11. C 12. A 19. C 20. D 25. B 26. C 15. C 16. A 17. B 18. B 31. B 32. C 37. C 38. B 21. A 22. A 23. C 24. A 43. C 44. D 49. D 50. B 27. C 28. D 29. B 30. A 55. B 56. A 33. A 34. D 61. B JOIN @iitwale in Telegram 35. D 36. A 39. B 40. B 41. B 42. D 45. C 46. A 47. C 48. D 51. C 52. B 53. B 54. D 57. A 58. B 59. D 60. A 8

Solutions 1. (b) Not available  p(E)  n(E)  3 n(S) 4 2. (a) Not available 15. (c) E {2, 3, 4, 5, 6}  n(E)  5 3. (b) Not available S {1, 2, 3, 4, 5, 6}  n(S)  6  p(E)  n(E)  5 4. (d) Not available n(S) 6 5. (a) Not available 16. (a) Not available 6. (c) Not available 17. (b) Not available 7. (c) Not available 18. (b) Not available 8. (b) Not available 19. (c) Not available 9. (a) Not available 20. (d) Not available 10. (b) Not available 21. (a) Not avJaOilabINle @iitwale in Telegram 11. (c) E {2,4,6}  n(E)  3 22. (a) Not available S {1,2,3,4,5,6} n(S)  6  p(E)  n(E)  3  1 23. (c) Not available n(S) 6 2 24. (a) Not available 12. (a) E {1,4,9,16,25,36,49,64,81,100} n(E)=10 25. (b) Not available S {1, 2, 3, ...... 100}  n(S) 100  p(E)  n(E)  10  1 26. (c) Let A be the event that A wins and B be n(S) 100 10 the event that B wins. P (one of them will win)  P(A B) 13. (a) E {11,22,33,44, 55,66,77,88,99}  P(A)  P(B) (Since, A and B are mutually  n(E)  9 exclusive.) S {1, 2, 3, ....... 100}  n(s) 100  3  1  13  p(E)  9 8 6 24 100 27. (c) The sample space is S = {Bishop, 14. (c) S {HH, HT, TH, TT}  n(s)  4 Castle, King, Pawn, Queen, Knight}. E {HT, TH, HH}  n(E)  3 28. (d) X = {April, June, September, November} 9

Hence, n(X)  4 41. (b) Not available 29. (b) X  {1,3} 42. (d) Not available Hence, n(X)=2 . 43. (c) Not available 30. (a) Since there are 7 cards, n(S) =7 . Let A be the event of picking a card having 44. (d) Let n(S)  s,n(R)  28 a circle, n(A) = 2 . P(R)  n(R)  4  28 or S  63 The probability of picking a card having a n(S) 9 S Now, 28  x  1 circle P(A) = n(A)  2 63  x 2 n(S) 7  x7 31. (b) Since there are 200 plates, n(S)  200 . 45. (c) n(S)  40, let n(C)  C Let B be the event of picking a defective P(C)  5  C  5 or C  25 plate. Then, n(B) 10 . 8 40 8 The probability of picking a defective plate Now, 25  x  1  x  10 40  x 2  P(B)  n(B)  10  0.05 n(S) 200 46. (a) Not available 32. (c) Not available JOIN @iitwale in Telegram 33. (a) Not available 47. (c) Not available 34. (d) Not available 48. (d) Not available 35. (d) Not available 49. (d) Not available 36. (a) Not available 50. (b) Not available 37. (c) Area of big circle  S   r2 51. (c) n(S)  32,n(R)  8 Area of shaded circle, p(R)  8  1 32 4 A    1 r 2  r2  The probability of drawing a marble  3  9 which is not red in colour 1 1  3  P( A)  n( A)   r2  1 44 n(S ) 9 9 r2 52. (b) n(S)  Area of rectangle  7 4  28 m2 38. (b) Not available n(A)  Area of circle   r2   (1)2   Probability  n( A)   39. (b) Not available n(S) 28 40. (b) Not available 10

53. (b) P(A)  n(A) JOIN @iitwale in Telegram n(S ) 0.00002  1 n(S ) n(S)  1  50000 0.00002 54. (d) 0.032  n(A) 250 n(A)  2500.032 n(A)  8 55. (b) Total number of cards = 52 3 face cards of spades are removed. Then remaining cards = 49 P (a queen)  3 49 56. (a) Numbers divisible by 10 from 1 to 20 are 10 and 20 i.e., 18 numbers are not divisible by 10.  P(A)  n(A)  18  9 n(S) 20 10 57. (a) Not available 58. (b) Not available 59. (d) Not available 60. (a) Not available 61. (b) Not available 11

Real Numbers 1. The L.C.M. and H.C.F. of marks scored by 6. If 4 divides 1728, which of the following Ajit and Amar in a math test are 5040 and statements is true? 12 respectively. If Amar's score is 144, what (a) 4 divides 12. (b) 6 divides 1728. is Ajit's score? (c) 2 divides 1728. (d) 4 divides 144. (a) 288 (b) 132 (c) 564 (d) 420 7. Dimensions of a rectangle are (25  7)cm and 2. 'p' is the remainder obtained when a perfect (252 73)cm . Express the area of the square is divided by 3.What is the value of 'p'? rectangle in prime factorization form. (a) 1 (b) 0 (a) 25 7cm2 (b) 2 x 73 cm2 (c) Either (a) or (b). (c) 265274 cm2 (d) 255273cm2 (d) Neither (a) nor (b). 8. Choose the irrational number. (a) 2  4 (b) ( 5)2 3. The factor tree shows the prime factorization (c) 9  4 (d) 2  3 of 1314. 9. Given a  3  2 . and b  3  2 , which of the following is correct? (a) a + b is irrational. (b) a - b is rational. (c) ab is raJtiOonIaNl. @iitwale in Telegram (d) a is rational. b Find the respective values of 'a' and 'b'. 10. Euclid's division lemma: For any two (a) 3, 37 (b) 3, 73 positive integers 'a' and 'b', there exist (c) 73, 3 (d) 9, 73 unique integers 'q' and 'r' such that a  bq  r 4. The following are the first and last steps in . What is the condition that 'r' must satisfy? finding the H.C.F. of 408 and 1032 using (a) 0  r  b (b) 0  r  b Euclid's algorithm. Step 1: 1032  4082  216 (c) 0  r  b (d) 0  r  b Step 2: ___________ Step 3: ___________ 11. Which of the following is a non-terminating Step 4: 192  248  0 repeating decimal? (a) 24 Choose the steps 2 and 3. 1600 (i) 408=2161+1921 (b) 171 (ii) 408=216+180+12 800 (iii) 216 =192 1 + 24 (c) 123 22  53 (iv) 192 = 24 8 + 0 (b) (i) and (iii) (d) 145 (a) (i) and (ii) 23  52  72 (c) (ii) and (iii) (d) (iii) and (iv) 5. For what value of 'x' does 6\" end with 5? (a) 0 (b) 1 (c) 5 (d) Never ends with 5.

12. Choose the terminating decimal. 19. Which of the following is an incorrect statement? (a) 141 (b) 17 (a) If a  b is an irrational number, then 1000 30 ab is also an irrational number. (c) 271 (d) 53 (b) The reciprocal of an irrational number is 90 343 always an irrational number. (c) There are infinitely many rational 13. Find the number which when divided by 43 numbers between any two irrational leaves a remainder 32 and gives a quotient numbers. 25. (d) 713 13 is a prime number. (a) 1045 (b) 1107 (c) 1150 (d) 1105 14. By what number must 1789 be divided to 20. Which of the following is true for two co- prime numbers? get a quotient 29 and remainder 49? (a) Their H.C.F. is 1. (a) 60 (b) 61 (b) TheirL.CM.is1. (c) 59 (d) 52 (c) Their H.C.F. is equal to their product. (d) Their L.C.M. is twice their H.C.F. 15. What is the L.C.M. of 140 and 605 if their H.CF. is H? (a) 8000 (b) 5500 21. The difference of the L.C.M. and H.C.F. of 210 and 55 is expressed as 210 x 6 + 55y. (c) 8400 (d) 7700 What is the value of y3 ? 16. 910 blue pens and 1001 red pens are (a) 361 (b) 19 (c) 55 JOIN @iit(wd)a6l8e59in Telegram distributed to students of class X so that each student gets the same number of pens of each kind. What is the maximum strength 22. Choose the methods that can be used to of the class? find the H.C.F. of any two numbers. (a) 91 (b) 80 (i) Euclid's division lemma (c) 94 (d) 86 (ii) Prime factorization (iii) Division of the numbers 17. Books in a library are stacked in such a way (iv) Product of numbers that they are stored subject wise and the (a) (i) and (iv) only stacks are of the same size. If there are 144 (b) (i), (ii) and (iii) only Geography books, 384 History books and (c) (i), (iii) and (iv) only 240 Economics books, in the library, in how (d) (ii), (iii) and (iv) only many stacks can the books be arranged? 23. A positive number 'n' when divided by 8 (a) 18 (b) 14 (c) 16 (d) 12 leaves a remainder 5. What is the remainder when 2n  4 is divided by 8? (a) 8 (b) 1 18. What is the L.C.M. of 6 and 2 ? (c) 6 (d) 0 14 7 (a) 3 (b) 6 24. The remainder when a number is divided by 7 7 143 is 31.What is the remainder when the (c) 4 (d) 5 same number is divided by 11? 7 7 (a) 5 (b) 7 (c) 6 (d) 9 3

25. Three ropes are 7 m, 12 m 95 cm and 3 m 31. 96 books of English, 240 books of hindi and 85 cm long. What is the greatest possible 336 books of mathematics have to be length which can be used to measure these packed in bundles with each bundle ropes? containing equal number of books of each (a) 35 cm (b) 55 cm of the subjects. What is the difference of the (c) 1 m (d) 65 cm largest number of books which can be packed in each bundle and the least number 26. Three bulbs are connected in such a manner of bundles which can be made? that they glow for every 24 seconds, 36 (a) 1 seconds and 54 seconds respectively. All of (b) 3 them glow at once at 8 a.m. When will they (c) 34 again glow simultaneously? (d) 48 (a) 8:30:36 a.m. (b) 8:03:36 a.m. (c) 8:36:03 a.m. (d) 8:36:30 a.m. 32. Which of the following is true about 27. Find The largest number which divides the 17 41 4361 43 ? numbers 120, 224 and 256. (a) It is a prime number. (a) 8 (b) 6 (b) It is a composite number. (c) 4 (d) 5 (c) It is an odd number. (d) Both (a) and (c) 28. A book seller purchased 117 books out of which 45 books are of mathematics and the 33. A circular field has a circumference of 360 remaining 72 books are of physics. Each book has the same size. Mathematics and km. Two cyclists Sumeet and John start together akJnmOd/hIcNrycrle@espaieticttwsivpaeeleylde,s aionfrTo1u2enldkemgt/hhrear m physics books are to be packed in separate and 15 bundles and each bundle must contain the same number of books. Find the least circular field. After how many hours will they meet again at the starting point? number of bundles which can be made of (a) 100 hours (b) 171 hours these 117 books. (a) 8 (b) 11 (c) 120 hours (d) 140 hours (c) 13 (d) 9 34. Find the H.C.F. of 6930 and 8085. 29. Sandeep donated 75 glucose biscuits and (a) 1155 (b) 2205 (c) 1515 (d) 2025 45 monaco biscuits to the students of a class. These are to be packed in identical packets. The two type of biscuits are to be 35. If 0.2317 is expressed in the form of p packed separately and each containing the q equal number of biscuits. Find the least where 'p' and 'q' are co-prime and also 'q' is number of glucose and monaco biscuit in the form 2n 5m what are the values of 'm' packets respectively. and 'n' respectively? (a) 4 and 3 (b) 4 and 5 (a) 5, 15 (b) 5, 3 (c) 4 and 4 (d) 3 and 4 (c) 2, 3 (d) 3, 2 30. An army contingent of 616 members is to 36. If 0.737373..... is expressed in the form of p march behind an army band of 32 members q in a parade. The two groups are to march in , where 'p' and 'q' are co-primes, what are the prime factors of 'q'? the same number of columns. What is the (a) 4 and 7 (b) 3 and 11 maximum number of columns in which they can march? (c) 7 and 11 (d) 4 and 3 (a) 3 (b) 8 (c) 12 (d) 4 4

37. Which of the following is correct about 44. The L.C.M. of 318 and 477 is expressed as 41 159 p  318 .What is the value of 'p'? 37500 (a) 2 (b) 4 (a) It is a non-terminating repeating decimal. (c) 3 (d) 0 (b) It is a terminating repeating decimal. (c) It is a terminating and not repeating 45. A rectangular metal piece of dimensions 360 decimal. cm by 280 cm is cut into some identical (d) It is a non-terminating and not repeating small squares. If the side of each square has decimal. the largest possible length, find the number of square pieces formed. (a) 126 (b) 20 38. Find the L.C.M. of 3465 and 5460. (c) 40 (d) 63 (a) 181080 (b) 180180 (c) 108108 (d) 108801 46. In a school, the duration of a period in junior section is 40 minutes and in senior 39. If the LCM Of (480,672) = 3360, find section is 60 minutes. If the first bell for each H.C.F. of (480,672). (a) 75 (b) 69 section rings at 9 a.m., when will the two bells ring together again? (c) 67 (d) 96 (a) 10:45 a.m. (b) 10:15 a.m. 40. Find the respective values of H.C.F. and (c) 12:00 p.m. (d) 11:00 a.m. L.C.M. of 5474, 9775 and 11730. (a) 391 and 410550 (b) 319 and 401550 47. M The prime factorization of two numbers (c) 410550 and 319 (d) 405150 and 193 are 32  73 11 and 3 72 113 17 . Which of 41. The circumferences of the front wheel and the followJinOgINis @a cioitmwmaolne fianctToreolef gthream the rear wheels of a tricycle are 120 cm and numbers? (a) 1683 (b) 5831 90 cm respectively. Before beginning to ride (c) 1089 (d) 539 the tricycle, Ruth marks the points where the tyres touch the ground as A and B respectively on the front and the rear wheels. How many revolutions do the front and rear wheel make when both A and B touch the ground again simultaneously? (a) 6, 8 (b) 3, 4 (c) 9, 12 (d) 1, 4 42. Which of the following is a correct statement (a)  is a natural number. (b)  is an irrational number. (c)  is not defined. (d) The value of  is 22 . 7 43. The product of L.C.M. and H.C.F. of two numbers is 88288. If one of the numbers is 248, find the other number. (a) 356 (b) 635 (c) 365 (d) 653 5

Answer - Keys 1. D 2. C 3. B 4. B 5. D 6. A 7. C 8. D 9. C 10. C 11. D 12. A 13. B 14. A 15. D 16. A 17. C 18. B 19. D 20. A 21. D 22. B 23. C 24. D 25. A 26. B 27. A 28. C JO2I9N. @Biitwale i3n0T. eleBgram 31. C 32. B 33. C 34. A 35. C 36. B 37. A 38. B 39. D 40. A 41. B 42. B 43. A 44. B 45. D 46. D 47. D 6

Solutions 1. (d) The product of two numbers is equal to 7. (c) The dimensions of a rectangle are the product of their L.C.M. and H.C.F. (257)cm and (25273) cm . Let Ajit's score be ' x ' . Its area  (25 7)(252  73) cm2 Then 1441 x  504012  26 52  74cm2  x  504012  420 144 8. (d) 2  4  2  2  0 (Rational number) 2. (c) The square of a positive integer 'm' is of the form 3m or 3m + 1 (for some 'm'). ( 5)2  5 (Rational number) Hence, the remainder obtained when a perfect square is divided by 3 is either 0 or 9  4 321 1. (Rational number) 3. (b) 219 3  657 Also, the sum or difference of two irrational numbers is irrational. Thus, 2  3 is and 3 73  219 irrational. Hence, a = 3 and b = 73. 9. (c) Not available 4. (b) Using Euclid's algorithm, H.C.F. of 408 10. (c) Not available and 1032 is: Step 1: 1032  408  2  216 JOIN @iitwale in Telegram Step 2: 408  216  1  192 Step 3: 216  192  1  24 11. (d) Not available Step 4: 192  24  8  0 Hence, the required steps are (i) and (iii) 12. (a) Not available only. 13. (b) Not available 5. (d) If 6x ends with 5, then 6\" would contain the prime 5. 14. (a) By Euclid's lemma, But 6x  (23)x  2x 3x . 1789  29x  49 , where ' x ' is the divisor.  The only prime numbers in the 1789  49  29x factorization of 6x are 2 and 3.  x  1740  60 By uniqueness of fundamental theorem, 29 there are no primes other than 2 & 3 in 6x . So, 6\" will never end with 5. 15. (d) The product of two numbers = The product of their L.C.M. and H.C.F. 6. (a) If 'p' divides x3 , then 'p' divides ' x ' .  140605 11 L.C.M. Here, 1728 = 12s.  L.C.M .  140 605  7700 So, 4 divides 1728 means 4 divides 123. 11 Thus, 4 divides 12 is the required statement. 16. (a) The maximum strength of class X = H.C.F. (910, 1001) 1001  9110  91 910  9110  0 7

Hence the maximum strength of class X is 120 104116 91. 104 166  8and16  8 2  0 Thus, H.C.F. 17. (c) H.C.F. of 144, 384 and 240 is 48. (224, 120) = 8. Number of stacks Now, find the H.C.F. of 8 and the third number 256.   144  384  240   3  8  5  16 256  832  0  48 48 48  i.e., H.C.F. (256, 8) = 8 Hence, the largest required number is 8. 18. (b) L.C.M. of a and c  L.C.M .(a,c) b d H.C.F.(b,d ) 28. (c) Not available  L.C.M. of 6 and 2 29. (b) Not available 14 7 30. (b) Not available  L.C.M .(6, 2)  6 H.C.F.(14,7) 7 31. (c) Not available 19. (d) 71313 104  2313 32. (b) Not available The product of two prime numbers is composite. 33. (c) Not avJaiOlabINle @iitwale in Telegram  713 13 is not a prime number. 34. (a) Not available 20. (a) The H.C.F. of two co-prime numbers is 1. 35. (c) Not available 36. (b) Not available 21. (d) Not available 37. (a) Not available 22. (b) Not available 38. (b) 3465  32 5711 23. (c) Not available 5460  22357l3 24. (d) Not available L. C. M. (3465, 5460)  22 32 571113 180180 25. (a) Not available 39. (d) The two numbers are 480 and 672. 26. (b) All the three bulbs glow at once at 8 a.m. Their L.C.M. = 3360. The time when they glow  H.C.F.  Product of the numbers simultaneously again = L.C.M. (24, 36, 54) L.C.M . seconds =216 seconds = 3 minutes 36 seconds 40. (a) 5474  271723  The time when the three bulbs glow 9775  52 17 23 together again is at 8 : 03 : 36 a.m. 11730  23517 23  H. C. F. (5474, 9775, 11730) 27. (a) Let us find H.C.F. of 120 and 224. 17 23  391and 224 1201104 8

L. C. M.(5474, 9775, 11730) JOIN @iitwale in Telegram  2352 7l7 23  410550 41. (b) Not available 42. (b)  is an irrational number. 22 is the nearest value of 71. Apart from 22 77 ,  also has other nearest values 43. (a) Not available 44. (b) Not available 45. (d) Not available 46. (d) The L.C.M. of 40 and 60 will give the number of minutes after which the two bells will ring together again. Now, 40  23 5 60  22 35 L. C. M. (40, 60)  22 35 2 120 Hence, the two bells ring together again after 120 minutes i.e., after 2 hours i.e., at 11: 00 a.m. 47. (d) Not available 9

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Pair of Linear Equations in Two Variables 1. Which of the following is a linear equation? 8. What is a system of simultaneous equations called if its graph has intersecting lines? (a) x  2y  7 (b) x3 1  0 (a) Inconsistent system (b) Consistent system (c) x  6  12 (d) x  3  14 (c) Dependent system x 2x (d) Independent system 2. Which of the following is a solution of 2p  3q  5 ? 9. IHI What is the nature of the graphs of a (a) p  1,q 1 (b) p  1, q  1 dependent system? (a) Parallel lines (c) p 1,q 1 (d) p  1,q  1 (b) Perpendicular lines (c) Intersecting lines 3. Which of the following is the other name for (d) Coincident lines a pair of linear equations in two variables? (a) Consistent equations 10. What is the nature of the graphs of a system (b) Simultaneous equations of linear equations with exactly one (c) Inconsistent equations solution? (d) Dependent equations (a) Parallel lines (b) Perpendicular lines 4. What is the condition that a system of (c) Coincident lines simultaneous equations a1x  b1y  c1  0 and (d) Intersecting lines a2x  b2 y  c2  0 must satisfy to have exactly one solution? JOIN @iitwale in Telegram (a) a1  b1 (b) a1  b1 11. What is the number of solutions of the pair a2 b2 a2 b2 of linear equations 4p  6q 18  0 and (c) a1  c1 (d) b1  c1 2p  3q  9  0 ? a2 c2 b2 c2 (a) 0 5. How many solutions do the equations (b) 1 (c) 2 satisfying a1  b1  c1 have? (d) Infinitely many a2 b2 c2 12. Which of the following is a consistent system (a) One (b) Two of simultaneous equations? (c) Three (d) Infinitely many (a) m  3n  6 (b) a  3b  6 6. What is a system of simultaneous equations 2m  6n 12 2a  3b 12 called if it has no solution? (c) x  4y  6 (d) l  2m  6 (a) Consistent system (b) independent system 2x 8y 12 3l  6m 12 (c) Inconsistent system (d) Dependent system 13. Find the unique solution of the system of simultaneous equations 2x  y  2 and 7. How many solutions does the system of 4x  y  4 . equations p  2q  4 and 2p  4q 12  0 (a) x  0, y 1 (b) x  0, y  0 have? (b) 1 (c) x 1, y  0 (a) 0 (d) x 1, y 1 (c) 2 (d) 3

14. The sum of a two-digit number and the (a) k  2 (b) k  2 number obtained by reversing its digits is 3 3 154. If the digits differ by 4, find the number. (c) k  2 (d) k  2 (a) 95 (b) 73 9 9 (c) 84 (d) 62 21. If the length of a rectangle is increased by 2 m and breadth is reduced by 2 m, its area 15. Choose the dependent system from the decreases by 28 sq. m. If the length is following. reduced by 1 m and the breadth is (a) m  n  7 (b) 3x  2y  5 increased by 2 m, the area increases by 33 sq m. Find the actual measurements of the 3m  3n  21 2x  3y  7 rectangle. (a) l 13m,b 11 m (c) 3x  3y 18 (d) 2x  y  6 (b) l  23 m, b  11 m x  y 10 4x  2y  4 (c) l  23m,b  20m 16. Five years ago, a father's age was seven times his son's age. Five years from now, the (d) l 12 m, b  10 m father's age will be thrice the son's age. What are the respective present ages of 22. The side of a square is 4m more than the father and son? side of another square. The sum of their (a) 40 years, 10 years areas is 208 sq. m. What is the side of the (b) 10 years, 40 years larger square? (c) 25 years, 5 years (a) 12m (b) 8m (d) 30 years, 8 years (c) 9m JOIN @iit(wd)a5lme in Telegram 17. Rajesh buys 7 books and 6 pens for `3800 23. Two numbers are in the ratio 2 :7. If 6 is and Amar buys 3 books and 5 pens of the same kind for `1750. What are the added to each of the numbers, the ratio becomes 1 :3. Find the numbers. respective costs of a book and a pen? (a) 14, 49 (b) 16, 56 (a) `350, `50 (b) `500, `75 (c) `250, `100 (d) `500, `50 (c) 18, 63 (d) 24, 84 18. The system of simultaneous equations 24. If the pair of equations 3x  5y  k and 3m  n 1and(2k 1)m  (k 1)n  2k 1 , is 9x 12y  6 has infinitely many solutions, inconsistent. What is the value of 'k'? which of the following is true? (a) k = 2 (b) k = 6 (a) 3 (b) 1 (c) k  6 (d) k = 3 (c) 2 (d) 0 19. What type of a system of equations is the 25. If the pair of linear equations 3x  5y  3 and pair of linear equations 2x - 3y = 8 and 4x - 6y = 9? 6x  ky  6 do not have any solution, which (a) Consistent system (b) Inconsistent system of the following is true? (b) k=10 (c) Dependent system (a) k = 5 (d) Independent system (c) k  10 (d) k  5 26. When is the pair of linear equations 20. If the pair of linear equations 2x  ky  3  0 7x  3y  4;3x k y  4 consistent? 7 and 6x  2 y  7  0 has a unique solution, 3 (a) k = 9 (b) k  9 which of the following is true? (c) k  9 (d) k  7 3

27. When does the pair of linear equations 35. If 12a  3b 1 and 7b  2a  9 , find the 7x  ky  k;14x  2y  k 1 have infinitely average (arithmetic mean) of 'a' and 'b'. (a) 2.5 (b) 1 many solutions? (c) 0.1 (d) 0.5 (a) k = 1 (b) k  1 (c) k = 2 (d) k = 4 36. If 4x  6y  32 and 4x  2y  4 , find the value 28. The pair of linear equations x  y  3; of 8y. (a) 24 (b) 28 2x  5y 12 has a unique solution x  x1, (c) 36 (d) 42 y  y1 Find the value of x1 . (a) 1 (b) 2 ( p  2) q 1 (c) -1 (d) -2 2  37. For the equations   pq  5 29. Which of the following solutions does the and ( p  2)  q  1   pq  5 , find the solution pair of linear equations x  2y  5 ; 2  3x 12y 10 have? set (p, q). (a) A unique solution (a)  10,  1  (b)  10, 1 (b) No solution  2   2  (c) More than two solutions (d) Infinitely many solutions (c) 10,  1  (d) 10, 1  2  2  30. If the sum of the ages (in years) of a father and his son is 65 and twice the difference of their ages (in years) is 50, what is the age of 38. Identify the solution of 2 x  3 y  1 and the father? (a) 45 years (b) 40 years 8x 9y 1J6O. IN @iit3wal4e in Telegram (c) 50 years (d) 55 years (a) x  6, y  4 31. Three chairs and two tables cost `1850.Five (b) Infinitely many solutions chairs and three tables cost `2850. Find the (c) x  4, y  6 (d) No solution total cost of one chair and one table. 39. Find the values of 'x' and y for x  y  0.9 ; (a) `800 (b) `850 (c) `900 (d) `950 11  2 x y 32. If a  b  5and 3a  2b  20 , find 3a  b . (a) 3.2, 5.6 (b) 3.2, 2.3 (a) 25 (b) 20 (c) 5.6, 2.3 (c) 15 (d) 10 (d) 4.5, 6.4 33. Which of the respective values of 'x' and 'y' 40. What is the solution of the equations, satisfy the following equations I and II? (I) 3x  y 19 (II) 3x  y 1  2x  y  2  3x  2y 1? x y 9 35 6 (a) 7, 2 (b) 7, -2 (a) x  1, y  1 (c) -7, 2 (d) -7, -2 (b) x 1, y 1 (c) x 1, y  2 34. If 3x  5y  5and x  5 , what is the value (d) x  2, y 1 xy 7 of x  y ? (a) 9 (b) 6 (c) 4 (d) 3 4

41. The course of an enemy submarine as 47. cm Find 'x' and 'y' for the equations plotted on a set of rectangular axes gives the equation 2x  3y  5 . On the same axes. the x y 4 34 course of a destroyer is indicated by the equation x  y 10 . Find the point (x, y) at and 5x  y  8 . 34 which the submarine can be destroyed. (b) x  3, y  4 (a) (-7, 3) (b) (7, -3) (a) x  8, y  6 (c) (-3, 7) (d) (3, -7) (c) x  6, y  8 (d) x  4, y  6 42. Find the solution of the equations 48. How many solutions does the system of 8x  9y  6xy 10x  6y 19xy . equations, 3x  4y  5 and 12x 16y  20 (a) x  3 , y  3 (b) x  2 , y  3 have? 22 32 (a) More than two solutions (b) Exactly two solutions (c) x  3 , y  2 (d) x  3, y  2 (c) Exactly one solution 23 (d) No solution 43. Find the values of 'x' and y, for the 49. Find the number of solutions of the equations a2  b2  0; a2b  b2a  a  b equations x  1  2 and 2xy  3y  2 . xy x y y where x, y  0 . (a) 0 (b) 1 (c) 2 (d) Infinitely many (a) x  a2, y  b2 (b) x  b2, y  a2 (c) x  b , y  a (d) x  1 , y  1 50. Which ofJOtheINfo@llowiiitnwg asloelutiinonTs edloegthream ab ba system of equations 2x  y  5 and x  2y  4 44. If x  1  5 and 2x  3  13 , what is the have? yy (a) Consistent and a unique solution (b) Consistent and infinitely many solutions value of (2x  3y) ? (c) Inconsistent (d) No solution (a) 1 (b) 2 (c) 3 (d) 5 51. If the equations 4x  7 y 10 and 45. Which of the following is the solution of the 10x  ky  25 represent coincident lines, what system of equations 4  5y  7 and is the value of 'k'? x (a) 5 (b) 17 3 4y 5? 2 x (c) 27 (d) 35 (a) x   1 , y  1 (b) x  1 , y  1 2 2 3 3 (c) x   1 , y  1 (d) x  1 , y  1 52. For what value of 'k', will the equations 3 3 4x  6y 11 and 2x  ky  7 be inconsistent? 46. The solution of 2x  3y  2 and 3x  2y  2 (a) 2 (b) 3 (c) 4 (d) 8 can be represented by a point. In which of 53. For what value of 'k' will the system of the following parts of the coordinate plane does the point lie? equations 3x  5y  2 and kx 10y  0 have a (a) First quadrant (b) Second quadrant non zero solution? (c) Third quadrant (d) Fourth quadrant (a) 0 (b) 2 (c) 6 (d) 8 5

54. If the cost of 3 audio cassettes and 2 VCDs (a) ` 900 (b) ` 350 is ` 350 and that of 2 audio cassettes and 3 (c) ` 650 (d) ` 700 VCDs is `425, what is the cost of a VCD? 61. The angles A, B, C and D in order in a (a) ` 140 (b) ` 125 cyclic quadrilateral are (2x  y)o,(2(x  y))o , (c) ` 115 (d) ` 110 (3x  2y)o , and (4x  2y)o . Find their 55. The difference between two numbers is 5 and the difference between their squares is measures in the same order. 65. Find the larger number. (a) 70, 110, 80, 100 (a) 9 (b) 10 (b) 70, 80, 110, 100 (c) 11 (d) 12 (c) 70, 80, 100, 110 (d) 80, 100, 110, 70 56. `49 was divided among 150 children. Each 62. The smallest angle of a triangle is one-fifth girl got 50 paise and a boy 25 paise. How many boys were there? (a) 100 (b) 102 the sum of the other two and the largest angle exceeds the sum of the other two by (c) 104 (d) 105 20 . Find the largest angle of the triangle. 57. The area of a rectangle increases by 76 (a) 100 (b) 90 square units, if the length and breadth are (c) 120 (d) 110 each increased by 2 units. However, if the 63. The sum of Raju's age and half of Sameer's length is increased by 3 units and breadth is decreased by 3 units, the area gets reduced age is 4. One-third Raju's age added to twice Sameer's age is 5. Find the sum of by 21 square units. Find the sum of the their aygeeasr.sJOIN length and breadth of the rectangle. (a) 7 @iit(wb)a3leyeainrs Telegram (a) 40 units (b) 42 units (c) 5 years (d) 2 years (c) 4 units (d) 36 units 58. What number must be added to each of the numbers, 5, 9, 17, 27 to make them proportionate? (a) 2 (b) 1 (c) 3 (d) 5 59. Two numbers differ by 3 and their product is 54. Find the numbers. (a) 9 and 6 (b) -9 and -6 (c) Both (a) and (b) (d) 9 and -4 60. A part of the monthly expenses of a family is constant and the remaining varies with the price of wheat. When the price of wheat is ` 250 per quintal, the monthly expenses of the family is ` 1000 and when it is ` 240 per quintal, the monthly expenses is ` 980. Find the monthly expenses of the family on wheat when the cost of wheat is ` 350 a quintal. 6

Answer - Keys 1. A 2. C 3. B 4. B 5. D 6. C 7. A 8. B 13. C 14. A 9. D 10. D 11. D 12. B 19. B 20. D 25. B 26. C 15. A 16. A 17. D 18. C 31. B 32. A 37. D 38. D 21. B 22. A 23. D 24. A 43. A 44. D 49. A 50. A 27. A 28. A 29. A 30. A 55. A 56. C 33. B 34. D 61. B 62. A JOIN @iitwale in Telegram 35. D 36. B 39. B 40. B 41. B 42. C 45. B 46. A 47. C 48. A 51. D 52. B 53. C 54. C 57. D 58. C 59. C 60. D 63. C 7

Solutions 1. (a) A linear equation is of degree 1. The To find if the system of simultaneous standard form of a linear equation is equations is consistent check if it. ax  by  c  0 . satisfies the condition a1  b1  c1 2. (c) Substitute the given values of 'p' and 'q' a2 b2 c2 in 2p + 3q = 5 and check whether the equation is satisfied. 12. (b) Given, system of equations are a  3b  6 Substituting p = 1 and q = 1 in 2p + 3q = and 2a  3b 12 . 5, makes it true.  a1  1 ; b1  1; 3. (b) The pair of linear equations in two a2 2 b2 variables is also known as simultaneous equations.  a1  b1 a2 b2 Hence the system is consistent. 13. (c) 2x  y  2 ....... (1) 4x  y  4 ....... (2) 4. (b) The system of simultaneous equations a1x  b1y  c1  0 and From (1), x  2  y a2x  b2 y  c2  0 , have exactly one (unique) 2 solution if a1  b1 . a2 b2  4 2 yJO IyN 4@iitwayle 0in Telegram 2 5. (d) Not available  x  20 1 6. (c) Not available 2 x 1 and y = 0 is the unique solution of the given system of simultaneous equations. 7. (a) Not available 14. (a) Not available 8. (b) Not available 15. (a) Not available 9. (d) Not available 16. (a) Not available 10. (d) Not available 17. (d) Not available 11. (d) Given: 4p  6q 18  0 and 18. (c) Not available 2p  3q  9  0 19. (b) Not available  a1  2; b1  2 and c1  2 a2 b2 c2 20. (d) Not available  a1  b1 21. (b) Not available a2 b2 22. (a) Not available Thus, the given system of equations has infinitely many solutions. 8

23. (d) Not available 37. (d) Not available 24. (a) Not available 38. (d) Not available 25. (b) Not available 39. (b) Not available 26. (c) Not available 40. (b) Given 3x  y 1  2x  y  2 35 27. (a) Not available  3x  2y 1 28. (a) Not available 6 29. (a) Given linear equations are Equate any two equations: x  2y  5 and 3x 12y 10 Here, a1 1,b1  2,c1  5  3x  y 1  2x  y  2 …….(1) and a2  3,b2 12,c2  10 35  a1  1 , b1  1 , c1  1  9x 8y 1 0 a2 3 b2 6 c2 2 And 2x  y  2  3x  2y 1 .....(2)  a1  b1  c1 56 a2 b2 c2  3x  4y  7  0 Hence, the given system of linear equations has a unique solution. Solving eq (1) and (2), we get y = 1 and x 1 . The solutiJonOoIfNeq@uatioitnws aisle in Telegram x 1, y 1. 30. (a) Let the age of father be 'x' years. 41. (b) Not available Let the age of son be 'y' years. 42. (c) Not available Given, x  y  65 ...... (1) and 2(x  y)  50 43. (a) Not available x  y  25 ......(2) Adding eq. (1) and eq. (2), we get 44. (d) Not available  x  45 45. (b) Not available Hence, the age of father = 45 years. 31. (b) Not available 46. (a) Not available 32. (a) Not available 47. (c) The given equations can be written as 33. (b) Not available 4x  3y  48 ..... (i) 20x  3y  96 ..... (ii) 34. (d) Not available Adding (i) and (ii), we get 24x 144  x6 35. (d) Not available Substituting x = 6 in (i). we get y 8. 36. (b) Not available 9

48. (a) Given equations are 3x  4y  5 52. (b) Not available and 12x 16y  20 Here a1  3,b1  4,c1  5 53. (c) Not available and a2 12,b2  16,c2  20 54. (c) Not available  a1  1 ; b1  1 ; c1  1 a2 4 b2 4 c2 4 55. (a) Not available  a1  b1  c1 56. (c) Let the no. of girls be 'x' a2 b2 c2 and the no. of boys be 'y'. Hence, the given equations have infinitely many solutions i.e., more than two Given, 0.50x  0.25y  49 solutions. and x  y 150 .... (1) 49. (a) Given equations become  x  x  49 …. (2) 2y x 1 2 .....(i) y From (1) & (2) x  46, y 104 and 2xy  3y  2 Hence, number of boys (y) = 104.  2x  2  3 ……(2) 57. (d) Let the length of the rectangle be 'x' y Let 1  z . then the equations are units, and the breadth be 'y' units. Then in y the first caJseO, IN @iitwale in Telegram x  z  2 and 2x  2z  3 . (x  2)( y  2)  xy  76 i.e., Here, a1 1,b1 1,c1  2 and 2x  2y  4  76 ....(1) a2  2,b2  2,c2  3 i.e., x  y  4 ..... (1)  a1  1 ; b1  1 ; c1  2 In the second case, a2 2 b2 2 c2 3 (x  3)( y  3)  xy  21  a1  b1  1 ; c1  2 i.e., x  y  4 a2 b2 2 c2 3 .....(2) Hence, the equations do not have any solution. Adding eq. (1) and eq. (2), we get 2x  40  x  20 units x  y  4  y  20  4 16 units 50. (a) Given equations are Hence, the length of rectangle is 20 units 2x  y  5 and the breadth is 16 units. and x  2y  4 Here a1  2,b1 1,c1  5 Their sum =20+16 and a2 1,b2  2,c2  4 = 36 units  a1  b1  c1 a2 b2 c2 58. (c) Four numbers are in proportion if First  Fourth = Second  Third Hence, the given equations have a unique Let 'x' be added to each of the given solution and are consistent. numbers to make the numbers proportionate. Then, 51. (d) Not available 10

(5  x)(27  x)  (9  x)(17  x)  x3 59. (c) Not available 60. (d) Not available 61. (b) Not available 62. (a) Not available 63. (c) Not available JOIN @iitwale in Telegram 11

Quadratic Equations 1. Identify the quadratic equation from the 7. What are the roots of following. 17a2  20a 10 10a2  2a  7 ? (a) p  1  1, p  0 (b) p2  1  1, p  0 (a) 1 ,3 (b) 3, 1 p p 7 7 (c) x2  1  1, x  0 (d) x2  2 x 1  0 (c) 1, 3 (d) 3,  1 x 7 7 2. Find the roots of the quadratic equation 8. Identify the factors of 4x2  4x  5 . 5 (a) 1,1 (b) 1, 1 2 2 (a) 2 , 2 (b) 5 , 5 55 22 (c) 1 ,1 (d) 1, 1 2 2 (c) 5 , 5 (d) 2 , 2 22 55 3. Which of the following statements is correct? (a) x 1 is a root of 2x2  3x 1  0 . 9. The age of a man is the square of his son's (b) x  2 is not a root of 6x2  7x  5  0 . age. A year ago, the man's age was eight (c) x  1is a root of 3x2  x 1  0 . times the age of his son. What is the present age of the man? (d) x   2 is not a root of 5x2 8x  4  0 . (a) 47 years (b) 49 years 5 (c) 36 years (d) 48 years 4. Find the value of 'p' for which m  1 is a JOIN @iitwale in Telegram 3 10. Find two consecutive even numbers whose root of the equation product is double that of the greater pm2  ( 3  2)m 1  0 number. (a) 3 (b) 2 (a) 1, 3 (b) 4, 6 (c) 2, 4 (d) 6, 8 (c) 6 (d) 7 5. For what respective values of 'm' and 'n' are 11. The length and breadth of a rectangle are (3k + 1) cm and (2k - 1) cm respectively. x  2 and x  5 the roots of Find the perimeter of the rectangle if its area 53 is144cm2 . mx2  nx 10  0 ? (a) 15,-19 (a) 50 cm (b) 10 cm (b) -19, 15 (c) 32 cm (d) 25 cm (c) 19,-15 (d) -15, 19 12. The sum of squares of two consecutive positive even numbers is 340. Find them. (a) 12, 14 (b) 12, 10 6. The sides of two square plots are (2x 1)m (c) 10, 8 (d) 14, 16 and (5x  4)m . The area of the second square plot is 9 times the area of the first 13. Find two consecutive positive odd numbers, square plot. Find the side of the larger plot. the sum of whose squares is 514. (a) 15m (a) 11, 13 (b) 13m (b) 15, 17 (c) 31 m (c) 11, 9 (d) 39m (d) 13, 15

14. The area of a rectangular cardboard is 21. The quadratic equation ax2  bx  c  0 has no real root. Which of the following is true? 80 cm2 . If its perimeter is 36 cm, find its length. (a) b2  4ac  0 (b) b2  4ac  0 (a) 40 cm (b) 10 cm (c) 20 cm (d) 8 cm (c) b2  4ac  0 (d) b2  4ac  0 15. Find two consecutive integers whose 22. What is the nature of the roots of the product is 600. (b) 50, 12 quadratic equation 25x2  49  0 ? (a) 30, 20 (a) Real and distinct (b) Real and equal (c) Irrational (d) No real roots (c) 15, 40 (d) 24, 25 16. Find the present age of a boy whose age 12 23. When are the roots of a quadratic equation years from now will be the square of his real and equal? (a) When the discriminant is positive. present age. (b) 7 years (b) When the discriminant is zero. (a) 5 years (c) When the discriminant is negative. (c) 4 years (d) 6 years (d) When the discriminant is non-negative. 17. Identify the correct statement. 24. How are the roots of 3x2  7x  8  0 ? (a) The roots of the quadratic equation (a) Real and unequal (b) Real and equal 2y2  9y  0 are 0 and 9 . (c) Not real 2 (d) Cannot be determined. (b) The value of 'k' for which 25. What is thJeOvaIlNue @of 'kii'tfworawlehicihnthTeeroleotgs roaf m 4m2  k 15  0 has a root m = 3 is 7. the quadratic equation 3x2  2kx  27  0 are (c) The quadratic equation (4x 11)2  0 has real and equal? (b) -9 only two distinct roots. (a) 9 only (d) 7x2 12x 18  0 is not a quadratic (c) 9 or-9 (d) Neither 9 nor-9. equation. 18. Find the roots of 3x2  2 6x  2  0 . 26. Find the sum of the roots of (a) 2 , 2 (b) 2 , 2 x2  x  210  0 (b) 29 33 33 (a) -2 (d) -1 (c) 20 (c) 2 , 3 (d) 2 , 3 32 33 19. Divide 63 into two parts such that their 27. In the quadratic equation 9x2  x  2  0 , product is 962. (b) 28, 35 find the value of a for which x  1 is its (a) 24, 3C 3 (c) 26, 37 (d) 27, 36 solution. (a) -2 (b) 3 (c) -4 (d) 6 20. Which of the following is a quadratic equation? 28. The ratio of the length and breadth of a (a) x  5  x2 rectangular photo frame is 3 : 2. Find its x length if its area is 864 cm2 . (b) x2  2 1 (a) 34 cm (b) 26 cm x2 (c) 24 cm (d) 36 cm (c) 2x2  3 x  4  0 (d) x2 1  2x2  4 3

29. A two digit number is 4 times the sum of its 36. Find the value of 'k' for which x2  4x  k  0 digits and also 16 more than the product of digits. Find the number. has coincident roots. (a) 4 (b) -4 (a) 48 (b) 36 (c) 0 (d) -2 (c) 44 (d) 32 30. A quadratic equation  x2  5x    0 has 37. If the roots of x2  4mx  4m2  m 1  0 are real, which of the following is true? (a) m  1 (b) m  1 two roots x  1 and x  2 . Find the (c) m  1 (d) m  0 3 respective values of and  . 38. What is the ratio of the sum and the product (a) 3, 2 (b) 2, -5 of roots of 7x2 12x 18  0 (c)-3, 5 (d) 3, -2 (a) 7:12 (b) 2:3 31. Find the common root of the equations (c) 3:2 (d) 7:18 x2  7x 10  0 and x2 10x 16  0 . 39. Which of the following is the quadratic (a) - 2 (b) 3 equation one of whose roots is 3  2 3 ? (c) 5 (d) 2 32. If the product of the roots of (a) x2  6x  3  0 (b) x2  6x  3  0 x2  3x  k 10 is - 2, what is the value of' k'? (c) x2  6x  3  0 (d) x2  6x  3  0 (a) -2 (b) 8 40. If a and Rare the roots of the equation (c) 12 (d)-8 x2  8x  p  0 such that  2   2  40 , find 33. If 2a2  a  2 1 and a >0, find 'a'. the value JofO'pI'.N @iitwale in Telegram (a) 8 (b) 10 (c) 12 (d) 14 (a) 3 (b) 1 2 (d) -1 41. Which of the following quadratic (c) 3 polynomials can be factorized into a product of real linear factors? 34. Find 'a' if a  3  10 . a (a) 2x2  5x  9 (b) 2x2  4x  5 (a) 5, 2 (c) 3x2  4x  6 (d) 5x2  3x  2 (b)  7,7 42. If  and  are the roots of the equation (c) 7,7 x2  3x  2  0 , which of the following is the (d) 5,2 equation whose roots are ( 1) and ( 1) ? 35. Find the value of 'p' so that x2  5 px 16  0 (a) x2  5x  6  0 has no real root. (b) x2  5x  6  0 (a) Greater than 8 (c) x2  5x  6  0 5 (d) x2  5x  6  0 (b) Less than 8 5 43. Which of the following equations has 2 as a (c) Lies between 8 and 8 root? 55 (a) 2x2  7x  6  0 (d) Less than 15 (b) x2  4x  5  0 8 (c) 3x2  6x  2  0 (d) x2  3x 12  0 4

44. If the equation ax  5x  c  0 has 10 as the sum of the roots and also as the product of the roots, which of the following is true? (a) a  c  5 (b) a  2,c  3 (c) a  5, c 1 (d) a  3, c  2 45. Find the product of the roots of the quadratic equation 9m2  24 m 16  0 . (a) 4 (b) 9 3 16 (c) 16 (d) 3 9 4 46. What is the nature of the roots of 3x2  x  6  0 ? (a) Real and equal (b) Real and distinct (c) Not real (d) Cannot be determined. 47. The perimeter and area of a rectangular JOIN @iitwale in Telegram park are 80 m and 400 m2 . What is its length? (a) 20m (b) 15m (c) 30m (d) 40m 48. If a and P are the roots of the equation x2  kx 12  0 such that    1 , what is the value of 'k'? (b) ± 5 (a) 0 (c) ±1 (d) ± 7 49. What is the value of 'k' for which 2x2  kx  k has equal roots? (a) 4 only (b) 0 only (c) 8 only (d) 0, 8 50. Which of the following statements is true? (a) x2  x 1  0 has no real roots. (b) x2  4x  3  0 and x2  x  2  0 have two common roots. (c) x2  3x  4  0 have real and equal roots. (d) The roots of ax2  bx  c  0,a  0 are reciprocal to each other if a  c . 5

Answer - Keys 1. A 2. D 3. B 4. C 5. A 6. D 7. A 8. C 9. B 10. C 11. A 12. A 13. B 14. B 15. D 16. C 17. A 18. B 19. C 20. D 21. A 22. A 23. B 24. C 25. C 26. D 27. B 28. D JO2I9N. @Aiitwale i3n0T. eleDgram 31. D 32. B 33. B 34. A 35. C 36. A 37. B 38. B 39. B 40. C 41. B 42. D 43. A 44. A 45. C 46. C 47. A 48. D 49. D 50. A 6

Solutions 1. (a) A quadratic equation has a degree 2. 10. (c) Not available In (b) and (c), the degree of the polynomial 11. (a) Not available is 3. 12. (a) Not available 13. (b) Not available In (d), x2  2 x 1 is not a polynomial as 14. (b) Not available x  x1/2 the power of the variable is not an integer. In (a), p  1  1 p2  p 1  0 is a p quadratic equation. 2. (d) 2m2  3m 1  0 15. (d) Let the two consecutive integers be 'x'  (m 1)(2m 1)  0 and x 1 .  m  1or  1 According to the problem, 2 x(x 1)  600  x  25 or 24 3. (b) Not available  x 1  24 1  25 The required numbers are 24 and 25. 4. (c) Not available JOIN @iitwale in Telegram 5. (a) Not available 16. (c) Let the present age of the boy be 'x' 6. (d) Not available years. 12 years from now, his age will be (x 12) 7. (a) Not available years. According to the problem, (x 12)  x2 8. (c) Not available  (x  4)(x  3)  0  x  4or  3 9. (b) Let the present age of the son be 'x' Since age cannot be negative, the required years. Then the father's age is x2 years. present age of the boy is 4 years. One year ago, the son's age was (x 1) years and the father's age was (x2 1) years. 17. (a) Not available According to the problem, (x2 1)  8(x 1) 18. (b) Not available  (x  7)(x 1)  0  x 1or 7 19. (c) Not available If x 1, x2 1  the father's age is 1 year is ridiculous. 20. (d) Not available If x  7 ,then x2  49 Hence, the present age of the father is 49 21. (a) Not available years. 22. (a) b2  4ac  4(25)(49)  4900  0 7

The roots of the given quadratic equation  p lies between 8 and 8 . are real and distinct. 55 23. (b) When the value of the discriminant is 36. (a) Since the roots are coincident, zero, the roots of quadratic equation are real b2  4ac  0 and equal.  (4)2  (1)(k)  0  k  4 24. (c) Not available 37. (b) Given, x2  4mx  4m2  m 1  0 have real roots. 25. (c) Not available  b2  4ac  0  (4m2)  4(1)(4m2  m 1)  0 26. (d) Not available  4m  4  0  m  1 27. (b) Not available 28. (d) Not available 38. (b) Given, 7x2 12x 18  0 Sum of roots  b  12 29. (a) Let the digits in the tens place and the a7 ones place be x and y respectively. Then, Product of roots  c  18 according to the problem, a7 10x  y  4(x  y)  y  2x  The required ratio  12  2  2 : 3 and 10x  y  xy 16  x  4 or 2 JOIN @iit1w8ale3 in Telegram If x  2 ,then y  2x  4 The number is 24. 39. (b) Let  3  2 3 and   3  2 3 If x  4 then y  8 . The number is 48.   32 332 3 6 Hence, the required number is 48.   (3)2  (2 3)2  9 12  3 is 30. (d) Not available  The required quadratic equation x2  6x 3  0 . 31. (d) Not available 40. (c) The given equation is x2  8x  p  0 32. (b) Not available     8,  p  2   2  40 (Given) 33. (b) Not available  (   )2  2  40  p 12 34. (a) Not available 41. (b) Except for the equation in option (b) all 35. (c) Given x2  5 px 16  0 has no real root options have discriminant < 0 and hence  b2  4ac  0 cannot have real linear factors.  (5 p2)  4(1)(16)  0  p8 42. (d) Given  and  are the roots of 5 x2  3x  2  0 .     3and  2 8

 ( 1)  ( 1)      2 JOIN @iitwale in Telegram  3 2  5 (   )( 1)      1  231 6 Hence, the required equation is x2  5x  6  0 43. (a) Not available 44. (a) Not available 45. (c) Not available 46. (c) Not available 47. (a) Not available 48. (d) Given equation is x2  kx 12  0 .     b  k and 12 a    1 (Given)  1    (1  ) 12   2   12  0    3,4 If   3,  4 , and   4,  3     7 or 7  k 49. (d) Not available 50. (a) Not available 9

Areas Related to Circles 1. When the circumference and area of a circle 8. The difference between circumference and are numerically equal, find the numerical radius of a circle is 37 m. Find the value of its diameter. circumference of the circle. (a) 7 m (b) 44 m (a)  (b) 8 (c) 154 m (d) 186 m 2 (d) 4 9. If the ratio of areas of two circles is 16:25, (c) 2 2. The radius of a circle is 14 m. Find the what is the respective ratio of their circumferences? circumference of the circle. (a) 25:16 (b) 5:4 (a) 616m (b) 88m (c) 154m (d) 176m (c) 4:5 (d) 3:5 3. If the radius of a circle is 7 cm , find the 10. If the ratio of circumference of two circles is  4:9, what is the ratio of their respective areas? area of the circle. (a) 9:4 (b) 16:81 (a) 154cm2 (b) 49 cm2 (c) 4:9 (d) 2:3  11. If the area of a circle is A, radius is 'r' and (c) 22 cm2 circumference is C. which of the following (d) 49 cm2 relations is true? 4. If the circumference of a circle is 30 , what is (a) rC JOIN @iit(wb)aCleinr Telegram   2A A2 the diameter of the circle? (c) AC  r2 (d) A  C (a) 60 (b) 15 4 r  12. Find the area of the sector of a circle, whose (c) 30 (d) 30 radius is 6 m when the angle at the centre is 2 42 . 5. If the circumference of a circle is 44 m, what (a) 13.2m2 (b) 14.2m2 is the area of the circle? (c) 13.4m2 (d) 14.4m2 (a) 6084.5 m2 (b) 276.5 m2 13. What is The area of a sector of a circle of radius 16 cm cut off by an arc which is 18.5 (c) 154m2 (d) 44 m2 cm long? 6. A circular grass lawn of 35 m radius, has a (a) 168cm2 (b) 148cm2 path 7 m wide running around it on the outside. Find the area of the path. (c) 154cm2 (d) 176cm2 (a) 1496m2 (b) 1450m2 (c) 1576m2 (d) 1694 m2 14. Find the area of a segment of a circle of radius 21 cm if the angle made by the arc of 7. How many plants will be there in a circular the segment has a measure of 60 . bed whose outer edge measures 30 cm (a) 45.27 cm2 (b) 40.27 cm2 allowing 4 cm2 for each plant? (c) 40.8 cm2 (d) 44.27 cm2 (a) 18 (b) 750 (d) 24 (d) 120

15. A sector of120 cut out from a circle has an (a)   4x2 sq.units 4 area of 9 3 . What is the radius of the circle? 7 (b) (  4) x2 sq.units (a) 3cm (b) 2.5 cm 4 (c) 3.5 cm (d) 3.6 cm (c)   8  2  4  x sq.units 16. The length of the minute hand of a wall (d) 2(  4)x2 sq.units clock is 21 cm long. What is the area swept by it in 10 minutes? 22. If a square of side 's' units is carved out from (a) 231 cm2 (b) 221 cm2 a circle, find the ratio of the areas of two (c) 210cm3 (d) 200cm2 figures. 17. The diameters of the wheels of a car are (a) 2 : (b)  : 2 each 63 cm. Find the distance travelled by (c)  : 4 (d) 4 : the car when the wheels make 1000 23. The figure shows an isosceles triangle and a revolutions. semicircle with centre O. (a) 1890 m (b) 1980 m (c) 1900 m (d) 1800 m 18. The radius of a circle is 20 cm. Three more Given that the radius of the semicircle is 2.8 concentric circles are drawn inside it in such cm, find the perimeter of the given figure. a manner that it is divided into four parts of equal area. Find the radius of one of the (a) 15.6 cmJOIN @iit(wb)a1l8e.8incmTelegram three concentric circles. (c) 16.8 cm (d) 20.4 cm (a) 8 3 cm (b) 2 3 cm (c) 10 3 cm (d) 14 3 cm 24. The given figure shows two identical circles and a rectangle. 19. A wire bent in the form of a circle of radius 42 cm is cut and again bent in the form of a square. What is the ratio of the regions enclosed by the circle and the square? (a) 11 : 12 (b) 21 : 33 (c) 22 : 33 (d) 14 : 11 What is the unshaded area of the figure in 20. What is the area of the largest triangle that cm2 ? (Take   3.14 .) can be inscribed in a semicircle whose (a) 40 (c) 25 (b) 43 (d) 33 radius is 'r' cm? (a) 2r cm2 (b) r2 cm2 25. By The shaded part of the given figure shows a running track in a stadium. (c) 2r2 cm2 (d) r cm2 2 21. The top of a dining table is rectangular, 2x Find the area of the track. units long and x units wide, with two (Take   22 .) semicircles along the breadth. Find the area 7 of the table. 3

(a) 2058m2 Find the area of the remaining piece of (b) 1546m2 (c) 3514m2 paper. (Use   22 .) (d) 1923m2 7 (a) 296.1 cm2 26. The figure shows a right-angled triangle and (c) 221.5cm2 (b) 265.4 cm2 a semicircle. PQ is the diameter of the (d) 201.7cm2 semicircle. 31. Mrs. Vidya bought a piece of cloth as shown in the figure. The portion of the cloth that is not coloured consists of 6 identical semicircles. Find the perimeter of the whole figure. Find the area of the coloured portion. (a) 13  8 (b) 30  5 (a) 144 cm2 (b) 126 cm2 (c) 18  6 (d) 18 10 (c) 195cm2 (d) 243 cm2 27. The outer diameter and the inner diameter 32. The given figure shows a plot of land, which of a circular path are 728m and 700 m is made up of 2 semicircles, a rectangle and respectively. Find the area of the 22 circular an isosceles triangle. The diameter of the bigger semicircle is 7 m longer than that of path. (Use   22 .) (b) 25012m2 7 the smalleJr [email protected](tUwsaelei2n2 T.) elegram (a) 45260m2 7 (c) 31416m2 (d) 19541 m2 28. If the circumference of a circle is increased by 50%, by what percent will its area be increased? (a) 75% (b) 100% (c) 125% (d) 150% If the shaded region is covered with grass, find the area of the land covered with grass. 29. Each wheel of a car makes 5 revolutions per (a) 502 m2 (b) 497 m2 second. If the diameter of a wheel is 84 cm, (c) 433 m2 (d) 564 m2 find the speed of the car in km/h. (Give your 33. The figure given is made up of a rectangle, answer correct to the nearest km.) 2 identical semicircles and a quadrant. (a) 48 km / h (b) 32 km / h (c) 41 km / h (d) 25 km / h 30. The figure given shows two identical Find the unshaded area of the figure. semicircles cut out from a piece of coloured paper. (Use   22 .) (b) 1154cm2 7 (a) 1350cm2 (c) 1400cm2 (d) 1260cm2 4

34. In the given figure, a circle with centre B (Use   3.14 ) (b) 197.59 cm2 overlaps another circle with centre A and a (a) 383.68 cm2 (d) 242.52 cm2 square. The ratio of areas of P and Q is 5 :4 (c) 173.45 cm2 and the area of Q is 1 the area of circle B. 8 38. The figure given shows two identical The radii of circle A and circle B are 10 cm semicircles inside a square. (Use   22 .) and 8 cm respectively. 7 Find the area of the unshaded part of the What is the shaded area of the region? figure. (Take   3.14 .) (a) 449.75 cm2 (b) 520.60 cm2 (a) 15cm2 (b) 21 cm2 (c) 563.72 cm2 (d) 450.92 cm2 (c) 16cm2 (d) 23cm2 35. In the given figure, O is the centre of the 39. A regular hexagon is inscribed in a circle of circle whose diameter is 14 cm. radius 14 cm. Find the area of the circle falling outside the hexagon. (a) 106.79 cm2 (b) 241.8 cm2 (c) 79.27 cm2 (d) 173.9 cm2 Find the perimeter of the figure. 40. Kill In theJgOiveInNfig@urei,itAwBaCleis ainrigThet-alengglerdam (Use   22 .) (b) 124cm triangle in which ABC  90 , AB = 6 cm 7 (d) 160cm and BC = 8 cm. 0 is the centre of the in circle. (a) 134cm (c) 112cm 36. A bucket is pulled from a well by means of a Find the area of shaded region. rope which is wound round a wheel of diameter 77 cm. Given that the bucket (Use   22 .) (b) 11.42cm2 ascends in 1 minute 28 seconds with a 7 uniform speed of 1.1 m/s, find the number of complete revolutions that the wheel (a) 12.56cm2 makes in raising the bucket. (Take   22 .) (c) 13.65cm2 (d) 10.57cm2 7 (a) 10 (b) 55 (c) 25 (d) 40 41. Arjun drew a figure as shown in figure, where a circle is divided into 18 equal parts. 37. David cut out the given figure during one of He then shaded some of the parts. (Take his art classes. The figure is made up of   3.14 .) rectangles and quadrants. If David wanted to colour the figure red. Find the total area that he needs to colour. 5

Find the total area that Arjun shaded. 45. The figure given is made up of a circle and 3 identical semicircles. 0 is the centre of the (a) 25.12 cm2 (b) 29.25 cm2 circle and XY is the diameter of the circle. (c) 36.4 cm2 (d) 45.2 cm2 42. In the adjoining figure, ABC is an equilateral triangle of side 14 cm. M is the centre of the circumcircle. Given that XY is 28 cm, find the perimeter of the shaded part of the figure. Find the area of the shaded region. (Use   22 .) (b) 50 cm 7 (d) 15 cm (a) 67 cm (c) 80 cm (a) 115.27 cm2 (b) 96.63 cm2 46. The figure given shows a rectangle with a semicircle and 2 identical quadrants inside (c) 120.46 cm2 (d) 146.72 cm2 it. 43. 4 identical semicircles are drawn inside a big square as shown. Each side of the big square is 14cm long. JOIN @iitwale in Telegram What is the shaded area of the figure? Find the area of the shaded region. (Use   22 .) (b) 259 cm2 7 (d) 216cm2 (Use   22 .) 7 (a) 363 cm2 (c) 305cm2 (a) 125cm2 (b) 112 cm2 47. The minute hand of a clock is 7 cm long. Find the area traced out by the minute hand (c) 173cm2 (d) 159 cm2 of the clock between 4:15 p.m. and 4:35 44. The figure given is made up of a square with p.m. on a day. a circle inscribed in it. (a) 59 cm2 (b) 65 cm2 (c) 52 cm2 (d) 45 cm2 What is the area of the shaded region? 48. A square shaped bus shelter is supported on (Take   3.14 .) four circular poles-The circumference of (a) 6.28 cm2 each pole is 'x' m and the length of each (b) 1.42 cm2 side of the shelter is 'y' m. Find the area of (c) 4.91 cm2 the unsupported part of the shelter. (d) 7.36 cm2 (a)  x 2  y2  m2 (b)  y2  x2  m2  p   p      (c)  x2  y2  m2 (d)  y 2  x2  m2  p   p      6

49. The circumference of a circle exceeds its (c) 42 (d) 40 diameter by 30 cm. Find the radius of the 53. If the length of the arc of a circle having a circle. central angle 36° is 22 cm, what is the area (a) 7 cm (b) 14 cm of the circle? (c) 21 cm (d) 28 cm (a) 3580 cm2 (b) 3850 cm2 50. The given figure shows 2 identical circles (c) 3058 cm2 inside a rectangle in cm2 . (d) 3805 cm2 54. If the perimeter of an arc of a circle is 43 cm and the radius of the circle is 21 cm, find the Find the shaded area of the figure. angle subtended at the centre. (Take   3.14 .) (a) 1300 (a) 3.72 (b) 2.73 (b) 1154 (c) 1092 (c) 2.37 (d) 3.27 (d) 1256 55. The given figure is made up of a rectangle 51. A copper wire, when bent in the form of a and 2 identical circles. Given that the length of the rectangle is 28 cm. square, encloses an area of 484 cm2 . If the same wire is bent in the form of a circle, find the area in cm2 . (b) 661 What is thJeOarIeNa o@f thieitswhaadleed irnegiTone?legram (a) 606 (d) 610 (c) 616 (Use   22 .) 52. Four equal circles are described about the 7 (b) 105 cm2 four corners of a square so that each of (d) 123 cm2 (a) 119 cm2 them touches two of the others. If each side (c) 152cm2 of the square measures 14 cm, find the area of the remaining portion of the square apart from four circles in cm2 . (a) 20 (b) 24 7

Answer - Keys 1. D 2. B 3. D 4. C 5. C 6. D 7. A 8. B 9. C 10. B 11. A 12. A 13. B 14. B 15. A 16. A 17. B 18. C 19. D 20. B 21. C 22. A 23. B 24. B 25. A 26. C 27. C 28. C JO2I9N. @Aiitwale i3n0T. eleCgram 31. B 32. B 33. D 34. D 35. A 36. D 37. A 38. B 39. A 40. B 41. A 42. C 43. B 44. A 45. C 46. B 47. C 48. D 49. A 50. D 51. C 52. C 53. B 54. B 55. A 8

Solutions 1. (d) Not available 10. (b) Not available 2. (b) Not available 11. (a) Not available 3. (d) Not available 12. (a) Not available 4. (c) Not available 13. (b) Not available 5. (c) Not available 14. (b) 6. (d) Area of sector OAB  x r2 360o  60o  22  21 21  231cm2 360o 7 r  35m Area of OAB  3 r2  190.73 cm2 R  35  7  42 m 4 Area of circular path   (R  r)(R  r) 1694m2 Area ofJseOctIoNr =@23i1itw- 1a9l0e.7i3n Telegram  40.27 cm2 7. (a) Circumference = 30 cm 15. (a) Area of sector Area  C2  30 30  71.6 cm2  x  r2  9 3 (Given) 4 4 22 360o 7 7  120o  22  r2  66 Number of plants  A  71.6 360o 7 7 44 66 360o 7  r2  7  120o  22  9 17.9  18  r  9  3cm 8. (b) 2 r  r  37 r  37  37  7m 16. (a) The angle swept by the minute hand in 2 1 22 60 minutes  360 2  1 The angle swept by the minute hand in 10 7 Circumference  2 r  44 m 9. (c) We have, minutes   360 10 o  60o A1 : A2 16 : 25  60  C1  A1  16  4 Sector angle  60 C2 A2 25 5 The ratio of circumferences is 4 : 5. The area swept by the minute hand in 10 minutes = The area of a sector of a circle of radius 21 cm and sector angle 60 . 60  r2 360o 9

 60  22  21 21cm2  231cm2 Area of shaded region 360o 7  23.145cm5 cm  157 cm2 Breadth of rectangle = Diameter of circle 17. (b) One revolution made by the wheel of  5 cm 2 10 cm the car is equal to the circumference of the Area of rectangle = Diameter of circle wheel.  20 cm10 cm  200 cm2 Given diameter of the wheel, Area of the unshaded region d = 63 cm  200 cm2 157 cm2  43 cm2 Hence, the unshaded area of the figure is  The radius, r  d  63 cm 43 cm2 . 22 25. (a) Not available Circumference, C  2 r 26. (c) Not available  2 22  63  229 198cm 27. (c) Not available 72 28. (c) Not available 1.98 m  The distance travelled by the car in one 29. (a) Not available revolution = 1.98 m The distance travelled by the car in 1000 30. (c) Not avJaiOlabINle @iitwale in Telegram revolutions 31. (b) Not available 1.981000 m 1980 m 32. (b) Not available 18. (c) Not available 33. (d) 19. (d) Not available Area of quadrant  1  22  42 42 20. (b) Not available 47 21. (c) Not available 1386 cm2 Area of rectangle  42(42  21)  2646 cm2 22. (a) Not available Unshaded area  2846 cm2 1386 cm2 1260 cm2 23. (b) 34. (d) Not available Circumference of circle  2 r  2 22  2.8  17.6cm 35. (a) Radius of circle = 14 cm - 2 = 7 cm One side of the figure opposite to 35 cm = 7 35 cm - 7 cm = 28 cm Circumference of semicircle = 8.8 cm Perimeter of the given figure = (5 + 5 + 8.8) = 18.8 cm 24. (b) Radius of each circle  20 cm  4  5 cm 10