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7th NMTC Allen

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7th NMTC Allen

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Ayush Srivastav
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Excellent Mathematics-1 Target : NMTC Round-1 Class - VII ConTENTS EXxcELLENT MATHEMATICS NMTC Round-1 Target fol iEy st Basic Number Operation Pattern Detection Ratio and Percentage Clock, Time and Distance Indices Algebra Geometry Logical Arguments Number Theory Data Handling NMTC : Round-1 GneTEaR Basic Number Operation 1, Given a sequence of two digit numbers grouped in brackets as follows (10), (11, 20), (12, 21, 30), (13, 22, 31, 40) ... (89, 98), (99). The digital sum of the numbers in the bracket having maximum numbers is, ae 2) 10 (3) 9 or 10 18 ing the digit 2 \d 7, and addition or subraction operations only, the number 2010 is written. The maximum number of 7 that can be used, so that the total numbers used is a minimum is, (a) 284 (2) 286 (3) 288 (4) 290 3. You join a job. Your pay for the first day is Rs. 5/-. Bach day after that your pay will be twice as much as it was the day before. Your pay on the tenth will be = (1) Rs. 100 (2) Rs. 250 (3) Rs. 5120 (4) Rs. 2560 4, Using all the digits 1 to 9 only once, how many nine digit prime numbers can you write ? a. (2) None a9 (4) More than 100 5. You have five pieces of 6 cm rods and 4 pieces of 7 cm rods. Using some or all of them, which one of the following lengths you cannot measure ? (a) 30 (2) 29 (3) 31 (4) 33 How many two digit numbers divide 109 with a remainder of 4? (2 a4 @)3 (4) None 7. The highest power of 2 that divides the sum of the numbers 4 + 44 + 444 +... + 444. is M2 @3 a4 a5 8. Which of the following can never be a common factor of 287 + x and 378 + x where x can be any natural number? (1) 26 @) 13 (3) 91 a7 How many 4 digit numbers with middle digits 97 are divisible by 45? ao @2 @a4 @i 10. The product of three consecutive odd numbers is 357627. What is their sum ? a) 213 (2) 243, (3) 153 (4) 209 11. The number of positive integers whose square is a factor of 2000 is a3 26 (3) 10 @r 12, Sum of all integers less than 100 which leave a remainder 1 when divided by 3 and leave a remainder 2 when divided by 4 is () 416 (2) 1717 (3) 1250 (4) 1314 TTT TT TT TTT EEE Class VII 13. 14, 16. 17. 18, 19. 20. 21. 22. 24, 25. 2 ALLEN ‘The number of prime numbers less than I lakh, whose digital sum is 2 (digital sum of a number is the sum of its digits) is as @4 @)3 (4) None of these ‘The quotient of 100! and 50% is (ay 50 (2) 501” (3) 200% (4) 400% ‘The least number of numbers to be deleted from the set (1, 2, of the remaining numbers is a perfect square is 13, 14, 15} so that the product at 22 @)3 aa The digit 1 is attached to the right of a 3 digit number making it a 4 digit number which is 7777 more than the given number. The sum of the digits of the number is a) 23 (2) 18 @17 (4) 16 A number is formed by writing the first 10 primes in the increasing order. Half of the digits are now crossed out, so that the number formed by the remaining digits without changing the order, is as large as possible. The second digit from the left of the new number is a2 3 @)5 7 Nine numbers are written in ascending order, The middle number is also the average of the nine numbers. The average of the 5 larger number is 68 and the average of the 5 smaller numbers is 44. ‘The sum of all the number is (a) 540 (2) 450 (3) 504 (4) 501 The largest positive integer n for which n°" < 6% is, air 213 @)17 aia ‘The sum of two numbers is 1215 and their GCI 81. How many pairs of such numbers are possible? a2 Q)4 (3)6 as ‘The number of digits in 8!554 (when written in base 10 form) is (a2 2) 45 3) 55 (4) 2007 ‘The number of two digits numbers whose digit sum is divisible by 6 is qu Qs 7 (4) 22 ‘A number when divided by 899 gives a remainder 63, What is the remainder when the number is divided by 297 ays (2) 28 G) 16 @ 12 A six digit number is formed by repeating a three digit number twice (like 245245). Such numbers are always divisible by ) 1001 (2) 25 (3) 101 @ i ‘The number AT389B where A, B are digits is divisible by 72, then (A, B) is () 6.3) 2) 6.6) aan 47.5) NMTC : Round-1 26. The remainder when the number (2 x 3 x 4 x 2014 x 2015 x 2016) ~ 2015 is divided by 2016 is ayo (2) 2008 @)1 (4) 2007 27. The number of non negative integers which are less than 1000 and end with only one zero is a) 90 2) 9 @) 91 (4) 100 28. Ina queue, Amar is 10" from the front while Akbar is 25! from behind and Antony is just in the middle of the two. If there be 50 persons in the queue, then the position occupied by Antony from the front is (a) 16 (2) 18 (3) 19 (4) 30" 29. The digits of the year 2000 add up to 2. In how many years has this happened since the year I till the year 2004 ? a3 Q2)6 a9 (4) 10 30. The sum of seven consecutive natural numbers is 84. The difference between the largest and the smallest among them is a7 26 14 @u 31. The sum of the digits of the number 10" — 1 is 3798; The value of n is (431 2) 673 (3) 422 4) 501 32. The units digit of the sum of the first 99 whole numbers (that is, 0+ 1 +2 +..... + 98) is ao Qi (@)-5 wo 33. Here is a magic square, created using the four numbers 10, 20, 30, 40 once in each row, column or diagonal. The value of A + B is a) 10 C}to} DIE (2) 20 A}30]) F| G ee B |x [20] k 4) 40 34. The number of whole numbers between V8 and V80 is :~ LIMIN]10 as 2)6 @)7 as 1 3 ) ( ( ) ( 2001 ) 35. The value of the product { 9={1}}x{>_(3}}x{2-(3} x... x (2- wae of tnt (25(2)}2-(5))2-(5)) =» (2-(3oa)) _ : 4) 2008, a) 2005 @ Os ®) sos o> 36. I multiplied a natural number by 18 and another natural number by 21. Then I added the products. Which one of the following could be the sum of the two products ? (a) 2002 (2) 2003 (3) 2004 (4) 2005 37. Ram divides a number by 1209 and gets a remainder 62. If Shyam divides the sa then what is the remainder?” M3 Q7 a0 a4 38. A transport company’s vans each carry a maximum load of 12 tonnes. 24 sealed boxes each weighing 5 tonnes have to be transported to a factory. The number of van loads needed to do this is ao Q) 10 au ar 39. ‘The GCD and LCM of two numbers a, b are respectively 27 and 2079. If a is divided by 9, the quotient is 21. Then b is (a) 243 (2) 297 (3) 189 @u 40. 146 is a natural number whose “digit-product” is 1 x 4 x 6 = 24. How many such three-digit numbers are possible with “digit-product” 24? aa Q) 18 @) 2 (4) 16 1e number by 31, 3 Class VII 41. 44. 4s. 46. 47. 48. 49. 50. 52. 53. 54. 58. ALLEN A four digit number of the from aba (a’s and b’s are the digits of the four digit number) is divisible by 33. The number of such four digit numbers is (a) 36 Qo 3)3 @l ‘The number of three- (2 @4 @)7 (4) 10 ‘Three boys P, Q, R agree to divide a bag of marbles as follows = it number each of which leaves a reminder 6 when divides 2002 is P takes one more than half of the marbles; Q takes a third of the remaining marbles; R takes the marbles left out now in the bag. ‘The original number of marbles found at the beginning in the bag must be (1) a multiple of 6 (2) one more than a multiple of 6 (3) two more than a multiple of 6 (4) three more than a multiple of 6 eile =} We multiply the consecutive even positive integers until the product 2 x 4 x 6X 8X vou. Xm, Where nis the first number, for which the product is divisible by 2015. Find the value of n.? By using the digits 1,2,3 and 4 we can form numbers of four digits number such as 1234, 2134, 4321, 3142 and so on, There are 24 four-digit numbers with distinct digits that can formed, by using each of the digits 1, 2, 3 and 4, Find out the sixth largest number. Amrita has written down four whole numbers. If she chooses thre€ of her numbers at a time and adds up each triple, she obtains total of 115, 153, 169 and 181. What is the largest of Amrita’s number What is the remainder when 10% is divided by 9? My house number is a three-digit number. The sum of this number and its three individual digits is 429. What is the product of the three digits which make up the house number ? If we add 5 with the ten’s digit and subtract 3 from the unit digit of a two digit number then the resulting number is twice the original number. Find out the original two digit number ? What is the quotient if the least common multiple of the first 40 positive integers divided by the least common multiple of the first 30 positive integers. A square of area 125 is divided into five parts of equal area-four squares and one L-shaped figure as shown in the picture, What is the length of the shortest side of the L-shaped figure ? In a football tournament 10 teams participated, Each team played with every other team exactly twice. Find the total number of games played Find the unit digit of 277. How can you use four 4's to create an expression that has a value equal to 1? The sum of the first 100 positive whole number is 5050. Using this find out the sum of the first 100 positive odd whole numbers, We have to find the sum 1 +3 +5... + 199 ; 5 j i i i i 87. 58. 59. 60. 61. 62. 63. 64. 65. 66. NMTC : Round-1 How many digits will be there when (999999999999)? is expanded ? How many zeros are there in the product of the first 100 positive integers. How many positive numbers less than 10000 are both squares of integers and divisible by 10 ? The number 49800 is said to end in two '0's. In how many ‘0's does 2! x 51% x 10° end in ? For some number n, the sum of the first n positive integer is 240 less then the sum of the Ist (n + 5) positive integer. Then n itself is the sum of how many positive integer ? ‘The sum of the odd positive numbers from 1 to n is 9409. What is the value of n ? It takes 852 digits to number the pages of a book consecutively. How many pages are there in the book Find out the sum of all natural numbers tess than 45, which are not divisible by 3. Let ‘m' be the smallest four digit number such that the three digit number obtained by removing the left most digit is one ninth of the original number. What is the value of 'm! 2 L Lot 1 Find the sum of x3 3x4 4x5 10x11 Find the last digit of 57% when it is expanded. Es 9 np 2 2] 3s 1 28 | 29 | 30 | 31 | 32 2 3] 2 See} 45, 4123 OL a1 7 50, 2292 |. 5V5 - 10 52.90 Many solutions Foreg. 44 = 1 5. 10000 56, 24 digits 44 59, 3998 zeros | First nine numbers 61. n= 193 64.m=1125 os. 10 66.7 |, m= 112 i : Class VII ol Nea ins 1, A student starts at the year 2001 and counts backwards, 8 at a time, giving the sequence of years 2001, 1993, 1985, A year which she will count is Pattern Detection sum of the number (a) 1841 (2) 1901 (3) 1923 (4) 1903 2. In the sequence 1, 22, 333, 4444, ~~», the n'™ term has number of n’s. then the of digits in the 100 term is (a) 100 2) 300 (3) 10000 &) 100 3. The adjacent table defines an operation *. For example, form the table we find el en a*c=dandb*d=b.Ifb*x=a, then x * x is ape pea (a Q)b elafalofec Be @a dlalble|a 4. Figure out from amongst the four alternatives as to how the pattern would appear when the transparent sheet is folded at the dotted line 4 What comes next in the series given below? KKK? 2-2 8 «a 6. Choose the suitable figure, so that a series is formed by the figures A, B, C, D 9 |@a @ 8) (4) none of these taken in order NMTC : Round-1 7, Natural numbers are written in a sequence as follows 1234567891011 1213 1415 16.. What is the 2003" digit in this sequence ? 8. The diagram shows 6 small squares made with matchsticks. How many matchsticks must be removed to leave precisely 3 small squares which touch only at comers ? 9. The numbers 1 to 9 are to be placed so that there is one number in each square and the row and column totals are as shown ? What number goes in the central square ? Is B 24 7 4 20 10. Pradipta counted upto 1000 by 6's beginning with 6, Dropadi counted upto 1000 by 4's beginning with 4 and Maricha counted upto 1000 by 5's beginning with 5. How many of the numbers were counted by all of them 11, The integers from 1 to 20 are listed below in such a way that the sum of each adjacent pair is a prime number. Missing numbers are marked as _ 20, _ 16, 15, 4, 12, _ 10, 7, 6, _ 2, 17, _ 14,9, 8, 5, 18, _. Which number goes in place of _ 12. In this unusal game of noughts and crosses the first play to form a line or three Os or three Xs loses. It is X's turn, Where should she place her cross to make sure that she does not lose Alo|s c[x[p E[x[o 13. Find out the 200th term in the sequence 1, 1, 1, 2, 1,3, 1, 4, 1,5 14. A contest among n > 2 players is held over a period of 4 days. On each day each player receives a score of 1 or 2 0F3...... oF m, points with no two players gett he same score on the same day, At the end of the contest it is discovered that every player recived the same total of 26 points. How many players participated ? ANSWERS Sele 6 Que[i [27s] 4]5 ‘ans 1[if2[4]i SECTION B 7.0 8.5 9.4 10.16 1.a=3,b=19,c=1,d=1,e=11 12, The correct place isto start with A 13. 100 14. 12 Class VII CHAPTER) Ratio and Percentage Sule ‘The percentage of natural numbers form 10 to 99 both incl natural number is ¢ which are the product of consecutive id 7 Woy QI @) 10 ao ‘Two regular polygons have the number of sides in the ratio 3 : 2 and/the interior angle in the ratio 10 : 9 in that order. The number of sides of the polygon are respectively (1) 6 and 4 (2) 9 and 6 (3) 12 and 8 (4) 15 and 10 n=, on complete simplification has the denominator ami 22 a3 a4 10a The biggest value of (a © N) is never greater than 10+a (10 Qo Qs (4) None of these The value of z satisfying the equation a2 Qe @)-1 @i 20% of 50% is what percent of 25% of 40% ? (1) 80% 2) 60% (3) 65% (4) 100% 16 16, A boy on being asked what += of a fraction was made the mistake of dividing the fraction by +>, 33 and got an answer which exceeded the correct answer by 37>. The correct answer is o% oa @2 (4) None of th es 2) ay a 7 ) None of these 10. uu. 12. 14. 15. 16. 17. NMTC : Round-1 Ina school, there are 5 times as many boys as girls, and 6 times as many girls as teachers. If b, g, t represent the boys, girls and teachers respectively the total number of boys, girls and teachers in the school is (1) 37b (3) 30g (378 When the price of an article is increased by 15%, the number of articles sold decreases by 20%, What is the percentage change in the sales revenue ? (Sales revenue = price of each article x number of articles sold). (1) 5% increase (2) 3% decrease (3) 8% increase (4) 89% decrease By selling a cap for Rs. 34.40, a man gains 7.5%, What will be the CP of the cap? (1) Rs. 32.80 (2) Rs. 32, (3) Rs. 32.40 (4) Rs. 28.80 6 men can do piece of work in 12 days. How many men are needed to do the work in 18 days? (1) 3 men (2) 6 men (3) 4 men (4) 2 men 4 men and 3 women finish a job in 6 days. And 5 men and 7 women can do the same job in 4 days. How long will 1 man and 1 women take to do the work? 2 1 1 1 a 22{2)aws (2) 25|>Jdays 3) 5[ 5] days) 2 Fass EI si AA student multiplied a number by = instead) of [. What is the percent (D) 34% (2) 44% (3) 54% (4) 64% If x is 80% of y, then what percent of 2x is y? (1) 40% @ 023% 8) 663% (4) 80% Ajay sold two. motorbikes for Rs, 40000 each. He sold one at 20% profit and the other at 20% loss. Find the profit or loss percentage in the whole transaction, (1) 2% profit (2) 3% loss (3) 4% loss (4) No profit, no loss _ day ~3bx. Was b=22:3.and x: y= 3:4, then the value of Se ppy 8 ys » 5 oe ye qa) 3 (2) 6 QB) 5 (4) 7 If 2x + 3y) : Gx + Sy) = 18 = 29 then x : y is equal to (2:3 3:4 (33:5 @i:s Class VII 18. 19. 20. 2. 22. 25. 26. 27. 28. 29. 10 a What number must be taken from each term of the fraction $= that it may 2:3 7 ao Q) 10 gay @ 2 20 men working 9 hours per day can complete a work in 12 days. To complete the same work in 15 days working 12 hours a day, the number of men required is ay 1s Qy14 ar @u If 6 men or 8 women can finish a work in 12 days, 9 men and 12 women will finish the work in nearly (1) 4 days (2) 3 days (3) 6 days (4) 5 days Ifa + 2a + 3a +--+ + 1000a = 2b + 4b + Ob +--+ + 2000b = 3c + 6e + 9¢ ++ ++ + 3000C then a:bicisas () 1:23 (2) 3:2:1 (3) 2: 4) Scientist A has invented a device for cars to save petrol by 20%; Scientist B’s invention saves 30%: invention of scientist C save 40%. The three inventions are independent in effect. if all the three devices are used, how much petrol can one save ? (1) 90% (2) 140% (3) 66.4% (4) 33.6% Given X is 50% larger than Z and y is 25% larger than Z, Then X is what percent larger than Y ? 2 (1) 20% (2) 25% (3) 15% (4) 50% A chocolate drink is 8% pure chocolate, by volume. If 5 litres of pure milk are added to 25 litres of this drink, the percent of chocolate in the new drink is nearly «10 Q7 (3) 15 @ 13 If the price of an article is increased by 15% and the total number of articles sold is decreased by 10%, the profit on income is (1) 5% (2) 1.5% (3) 3.5% (4) 2.5% In an examination 60% passed in English, 52% passed in Maths while 32% failed in both. If 220 candidates passed in both, then the total number of candidates was; (a) 200 2) 300 (3) 400 (4) 500 a 5 If x is F oF 490, then > of x is (a) 350 2) 250 (3) 70 ) 420 Four points P, Q, R, S are on a line segment, as shown, -——_. If PQ = QR= 1:2, QR: RS =8: 5, then PQ: QS is (3:13 Q)1:7 @ 1:13 4:13 S,, Ss, S; are three sums of money such that S, is the simple interest on S, and S, is the simple interest on S, for the same rate and same period. Then @S?=S,8, Q)82=S,S, (8) 82 =S,S, (4) $,8,S. NMTC : Round-1 ALLEN 30, If 5 men can build a wall in 12 days. How many men can build it in 10 days? (1) 6 men (2) 7 men (3) 8 men (4) 4 men x3 oxy 31. S=3, then the value of 2 is wt ae as a OY or D or 32. Candles A and B are lit together. Candle A lasts eleven hours, candle B lasts seven hours. After three hours the two candles have equal lengths remaining. Find the ratio of their original lengths. 33. The monthly income of A is 20% more than the monthly income of B. Then how much percent B's monthly income is less than that of A? 34. A car salesman sells two types of cars : Maruti and Indica. A Maruti is sold at 40% profit, whereas an Indica is sold at 60% profit. The salesman has calculated than if he sells the same number of each car his overall profit will be 48%. In reality, he sells $0% more Indica than Maruti. What is the 9% of profit ? (If a car is bought for 200,000 and is sold for 300,000, he has 50% profit as the difference is 50% of the price he paid.) 35. There are 2 red, 3 blue and 4 green marbles in a bag. I take one marble at a time out of the bag without looking to it, What is the least number of marbles I must take out to be sure that I have 3 marbles of the same colour. What is the least number of marbles I must take to get 3 green marbles ? 36. Given A is 50% larger than C and B is 25% larger than C then A is what percent larger than B ? 37. A store prices an item im rupees and paise so that when 4% sales tax is added no rounding is necessary because the result is exactly n dollars, where n is a positive integer. Find the smallest value of n ? rs S 9 [10] 11] 12] 18 igs) be) ist sata 29| 30| 31 3fifa SEcTION B 5/6 S 1[4 2 25 | 26| 27 | 28 3/4 4 32 Ms Bxx 162% 34. 50% 38.8 marbles 36, 20% 37. n= 13 Class VII Clock, Time and Distance ol Nea ins 1. The time on an electronic digital watch is 11.11. How many minutes before this would the watch have shown a time with all digits identical ? (72 (2) 144 (3) 216 (4) 316 2. A watch is set right at 3 pm. It loses 20 minutes in 24 hours. The true time when the watch shows 2 pm on the fourth day is (2) 3pm (2) 4.19pm. (3) 3.19pm. (4) 4pm 3. Ina kilometer race, A beats B by 1 minute and beats C by 375 meters. If B beats C by 30 seconds, the time taken by C to run 1 km is (1) 150 sec (2) 210 sec (3) 240 sec (4) 200 sec 4. Sound travels at 330 m/s. How many kilometers away is a thunder cloud when its sound follows the flash after 10 s (33 (3) 0.33 @) 3.33 5. Which one of the following distance time level graphs is not possible ? a 6. A clock is started at noon. By 10 minutes past 5, the hour hand has turned through (a) 145° (2) 150° (3) 155° (4) 160° 7. A watch which gains 5 seconds in 3 minutes was set right at 7 a.m. In the aftemoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is: 9 9 (1) 59-5 min past 3(2) 4 pm G) 587; min past 3 (4) 27min past 3 1 8. How much does a watch lose per day, if its hands coincide every 64 minutes? (3 2) 36: (3) 90 min (4) 96 min 9. At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but, not together? ws 7, 25> @ 52 714) 5 ) 5 min, past 7 (2) 5 min 3) Sz min. past 7(4) 5 min 10. At what time between 5.30 and 6 will the hands of a clock be at right angles 5 (1) 4357 min, past 5 (2) 437] min, past 5 (3) 40 min, past 5 (4) 45 min, past 5 11. The angle between the minute hand and the hour hand of a clock when the time is 4.20, is: aor (2) 10° as (4) 20° DL i j : i e 8 14. 15. 16. 18. 19. 20. 21. 22. NMTC : Round-1 At what angle the hands of a clock are inclined at 15 minutes past 5? aay sete (2) 64° @) oe (4) 7 At 3.40, the hour hand and the minute hand of a clock form an angle of: (1) 120° (2) 125° (3) 130° (4) 135° How many times are the hands of a clock at right angle in a day? (a 22 (2) 24 3) 44 (4) 48 The angle between the minute hand and the hour hand of a clock when the time is 8.30, is: (a) 80° 2) 75° (3) 60° (4) 105° How many times in a day, are the hands of a clock in straight line but opposite in direction? (a) 20 2) 22 3) 24 (448 At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions? (1) 45 min, past 4 (2) 40 min, past 4 4 6 (3) 507] min. past 4 (4) 547 min. past 4 At what time between 9 and 10 o'clock will the hands of a watch be together? (1) 45 min, past 9 (2) 50 min, past 9 l 2 @) 49> min, past 9 (4) 48min, past 9 At what time, in minutes, between 3 o'clock and 4 o'clock, both the needles will coincide each other? 1 4 4 4 w 55 Q) 1 @) BF @ 16> How many times do the hands of a clock coincide in,a day? (a) 20 (2) 21 (3) 22 24 How many times in a day, the hands of a clock are straight? ay 22 (2) 24 ay 44 (4) 48 A watch which gains uniformly is 2 minutes low at noon on Monday and is 4 min. 48 see fast at 2 pation the following Monday. When was it correct? (1) 2 p.m, of Tuesday (2) 2 p.m. on Wednesday (3) 3 pam. on Thursday (4) 1 p.m. on Friday ANSWERS Class VII 10. 12. 14 ol Nea ins Indices What is half of 44° 2 ay 20 40 2” (4) 29 1 The value of 3 of 1577 is ays? ~ Q) 15° (3) 5 x 15° (4) 5x3 Which one of the following number is perfect square ? (1) 495466 (2) 48546" (3) 445368 (4) 445% If abc} = 5° and ab? = 56 then abe equals as Qs as as 1644 4 44 4 a 4 44 = 48 then x is (4s (2) 44 (3) 176 an Baded ed C4646 te «SOS = a then, mis wi Q)2 @)3 a4 a 162" x 3%= 576 then 2 is * > (2 @3 ae ) ) OZ Given n is a natural number, Ie tata nts? = 644 then gtntentt nts t at =? times ay 512 (2) 256 (3) 1024 (4) 2048 The total number of powers of prime factors in the expression 67 x 35! x 3327 is (24 (2) 44 G3) 121 (4) 102 Text = 44 4 54 then x is a? @t ae @tt =) 25 ?) 20 9 ‘The expression yt 2) 9 at 4) 5 ws e 5 as The value of [2-3(2—3)'}! is ___. @t 3 4) 5 as @G) -! (4) 5 NMTC : Round-1 M e. Then (PEL -(P=LY 13. pis an odd prime. Then ( a y (& 7) is (1) a fraction less than p (2) a fraction greater than p (3) a natural number not equal to p (4) a natural number equal to p 1 1 1 14=||1+-— 1 ofS ][ alae (na+m Q)n+m+I GB) m (4) None of these 3 a4 @o (3) Both (1) & (2) (4) None of these ANSWERS Que. 3 anf 3 [3fifs[i[4[2]s]4 15 Class VII 16 ol Nea ins ale waN a, b, ¢, d are natural numbers such that a= be, b = ed, ¢ (¢ +d) +a) is equal to la and d = ab. Then (a +b) (b +c) (@+b+e+dP 2) (a+ bP ++ dy (3) a+ dP + b+) (4) (a+ oF + (b +d? 0 then a? + — isa @ ata (1) Positive integer (2) Positive fraction which is not an integer (3) Negative integer (4) Negative fraction which is not an integer If 36a* = a, then is equal to (2) 6a" (A) 60° Let n be a 3 digit number such that n = sum of the squares of the digits of n. The number of such nis Mo @1 32 (4) more than two It is given that ¥4., which one of the following is incorrect y 5 xty 9 xty?_4ll oO vrs ) A wom ay 20) Ifa, b, c, d are positive integers such that a = bed, b = dab and d = abe, then the value of fatbto+dyt (abs be Fededay aa @2 (3) 16 If 60% = 3 and 60° = 5 then the value of 12°" is equal to () Jeo 2) 3 @B)5 (4) 2 10. 12. 13. 14. 15. NMTC : Round-1 Consider the equation x? + y? = 2007, of solutions of the equation is fen x isa real number and y is a natural number, the number mo (2) 2006 (3) 88 (4) 44 (1) loses by the transaction (2) will not lose by the transaction (3) loss or gain depends on whether a > b or a b respectively A three digit number ab7 = a3 + b’ + 75, Then a is a6 a4 @7 (48 aid = where a, natural numbers. at pris Where a b are natural number Aas 26 (B)a=13,b= 13 x14 (a=l4,b= 13x14 Of these statement the correct statements are (1) (A) and (B) (2) (A) and (©) (3) (BY and (C)_ (A) (A) (B) and (C) p is a prime number greater than 3. When p? is divided by 12 the remainder is (1) always an odd number greater than 2 (2) always 1 @)Lor I (4) alwaysaneven number a is a real number such that a° + 4a ~ 8 = 0. Then the value of a’ + 64a? is (1) 128 (2) 164 (3) 256 (4) 180 3. on 35x..n—"_, on complete simplification has the denominator 4 nel a : aa at 2 @)3 aa In the addition problem shown, different letters represent different digits. If the carry over from adding the units digit is 2, then (A + 1) cannot be DID +1 +say MATH a2 Qa @)7 ws | e177 Class VII 16. 17. 18. 19. 20. 24. 22. 23. 24. 28. 26. 27. 18 All Tha-1=b+2=c-3=d+4 then the largest among a, b,c, dis Wa @b Be aa Ita = a4 1 and ta = a= 1, then 1* #1 + 2* = #2 43493 + wa + 1000 ~ *1000 is equal to (1) 1000 (2) ~ 1000 (3) 2000 4) -2000 atbee wo 1 @2 4) 3b If x > 2y and z< © then 6 6 () x> 62 @)x<6 @y a+l @) > 8 8 Instead of multiplying a given number by 75+ a student divided it by [5'. His answer was 297 more than the correct answer. The given number was (ys @ 19 (3) 152 (4) 64 L 3 If x+—=-I then the value of x x mo @1 @-1 (2 NMTC : Round-1 b?) is a prime number. The number 28. a,b where a> b are natural numbers each less than 10 such that (a? of such pairs (a,b) is as 26 @)7 8 29. f(x) = ax? + bx? + ex — 5 where a, b, c are constants. If f(-7) = 7, then f(7) equals to ()-17 Qa 3) 14 @) 21 30. Ify 2 and y—4 are the factors of py? + Sy + + then (par Q)p=r Q)psy > 0 and S*2= V3, the value of y ny as @4 @t a6 36. a, b, ¢ are real numbers and none of them zero and is. c= (tot Bad me) mea (1) 2012 when a 2012 (2) 2012 when ab = 2012 (3) 4 for all real values of a and b (4) 2012 for all real values of a and b 37. The angles of a triangle are in the ratio 2:3 : 7. The length of the smallest side is 2012 em. The radius of the circum circle of the triangle (in em) is (a) 2013 (2) 2011 (3) 4024 (4) 2012 bt bre cra at () 2012 Qt 3) 0 (4) (2012) 39. If one root of Va—x+Vb+x=Va+ Jb is 2012, then a possible value of a, b is. (1) (2000, 2012) (2) (4024, 2012) (3) (1000, 1012) (4) (1012, 1000) 40. Ifa = 2012, b = 2011, ¢ = 2010 then the value of a? + b? + c? — ab — be ~ ca is ao (2) 2012 @)3 (4) 4024 41, Ifa +b +c=0 where a, b, ¢ are non zero real numbers, then the value of (a? — be)? = (b? = ca) (c? - ab) is at (2) abe @aetb+e Ho 42. Aruna, Bhanu and Rita have some amount of money. The ratio of the money of Aruna to that of Bhanu is 7: 15 and the ratio of the money of Bhanu and Rita is 7 : 16. If Atuna has Rs. 490, the amount of money Rita has is (in Rupees) (1) 1500 (2) 1600 (3) 2400 (4) 3600 38. If a = 2012, b =-1005, e = =1007, then the value of +3abe is es | Class VII a value of ~+2. yuzux ALL! 43. x and y are real numbers such that 7* ~ 16y = 0 and 4* ~ 49y = 0, then the value of (y ~ x) is 5 19 us 1569 e ay 12 poo) M5 az @ op ) Fea 44. The number of real solutions of the equation x+Vx? +Vx' +1 =1 is at Q)2 3 ao ele =} 4S a= 14345474 + 2009 b=2+4+6+8 +... + 2010 then the value of (a - b) is — 46. The number of terms in the expansion (a +b + € 47. A predator beast weighs 2008 kg at the beginning of a year. During the first month of the year its 1 weight is increased by 335% and in the second month decreased by 25% and in the third month increased by 50%, in the fourth month decreased by 332% the fifth month increased by 62% and in the sixth month decreased by 2139p. This increase and decrese of the weight continued in the next 6 months in the same order with the same percentage. Find the weight of the predator at the end of the year. 48. When a barrel is 40% empty it contains 80 litres more than when it is 20% full. Find the full capacity of the barrel (in litres). 49. If a? — b? = 2011 where a, b are integers, then find the most negative value of (a + b). 50. x and y are real numbers such that xy.= x + y = ~ (x #0, y #0). Then find the numerical value of 2 (ry). SI. _p is the difference between a real number and its reciprocal. q is the difference between the square of the same real number and the square of the reciprocal. Then the value of p' + q? + 4p? is. 52. Ifx y the value of xy +2) + % +2 when a = 2012 is. yel p yx 53. If apb,.c, d, satisfy the equations a+ Th + 3c + 5d.= 0, 8a + 4b + 6c + 2d = — 16, 2a + 6b + de + 8d = 16, Sat 3b+7e+d=-16 4 then find the value of @ 4d) (b +c). 7 54. x, y, z are realnumbers such that (x + y)? = 16, (y + 2)? = 36, (2 +x)? = 81, x +y +2>3. Then number 5 of possible values of (x + y + 2) is. j 55. The combined age of a man and his wite is six times the combined ages of their children, Two years ago # their united ages were ten times the combined ages of their children. Six years henee their % combined age will be three times the combined age ofthe children. The number of children they have is. § 56. The sum of the roots of the equation x’ 3 57. There are four non-zero numbers x, = u,2=u-x, then calculate the 4 3 & 8 20 NMTC : Round-1 58. A sum of money is divided between two persons in the ratio 3 : 5. If the share of one person is Rs. 2000 more than that of the other, then find the sum of money (in rupees). eee [ a ee at | is wh is Ge where a 60, There is a famine in a pla e. But there is sufficient food for 400 people for 31 days. After 28 days 280 of them left the place. Assuming that each person consumes the same amount of food per day, the number of days for which the rest of the food would last for the remaining people is. 61. Given a #0, b #0. Find the number of real numbers pairs (ay b) which satisfy the equation at + b= (a+ by! 62. The sum of five unequal whole number is 90. What can be the second largest number of the five atmost? 63. In a running competition of 200 meters A defeats B by 40 meters and B defeats C by 10 meters. Find out in how many meters A will defeat C in a running competition of 375 meters ? 64, x= 142434 .. + 2008and y= 1x342K443K5+ ofx-y=? + 2004 x 2006, then the value 65. Five years ago, the average age of A, B, C and D was 45 years. age of all the five is 49 years. Find the present age of E ? E joining them now, the average 66. x men working x hours a day for each of x days produce x articles, How many articles will be produced by y men working y hours a day for each of y days ? 67. In.a 300 meter race, A defeats B by 40 meters and in a 500 meter race, B defeats C by 50 meter. Then in a 100 meter race Aidefeat C by how many meter ? 68. In her purse, Jayasree has 20 coins, with a total value of Rs.5/-. There are three denominations of coin-10p, 20p and S0p- in her house and she has more 50 p coins than 10p coins. How many of each type of coin-does she have ? 69. (i) Seven Women and five men attended a party. At this party each man shake hands with every other person. Each women shakes hands only with men, How many handshakes took place at the party ? Gi) At a party, every two people shake hands once. How many people attended the party if there were 66 handshares. 70. Three friends have a tradition of buying each other a gift once a year. Each person is presented a gift by the other two persons who split the price equally. So A and B each pay half the cost of the gift for C, B and C each pay half the cost of gift for A and C and A pay half the cost of the gift for B. The final rule is that each person spends all the money he brings. If A brings Rs. 50, B brings Rs.58 and C bring Rs.62 what is the purchase amount for A's gift ? 71. A small school has 100 students and rooms A, B and C. After the first period, half of the students in room A move to room B, one-fifth of the students students in room C move to room A. After the move, the total number of students in each room is room B move to room C, and one-third of the the same as its was before. How many students are in room A ? es 2] Class VII ALLEN 72. The machines P, Q and R, working together, can do a job in x hours. When working alone P needs an additional 6 hours to do the job ; Q, one additional hour ; and R, x additional hours. Find the value of x? The number 13 is prime. If you reverse it you also obtain a prime number 31. What is the larger of the pair of primes that satisfies this condition and has a sum 110. PE ai) 7s 7] 3 22 [23 3 | 3 37 [38 aval 45,1005 46.10 1687 hg 48.200 lites 49. -2011 51.24) 52, 2012 -16 3 55.3 57. U2 58, 8000 2013 0 63.90 meters 64, 2005 65.45 years 7 1. 22 meters 68. 2 > 10p coins, 14 20p coins, 4 50p coins, 69. (i) 45 handshakes, (ii) 12 people 2 70.Rs. 70 71.20 T.5 73.73 is larger number NMTC : Round-1 ol Nea ins Geometry In the adjoining rangoli design each of the four sided figures is a thombus and the d any two dots is 1 unit, The total area of the design is (1) 36,3 sq. units (2) 93 sq. units (3) 2443 sq. units (4) 18.3 sq. units In the adjoining figure, ABCD is a square of side 4 units. Semicircles are drawn outside the squares with diameter 2 units as shown, The area of the shaded portion in square units is as 4 (2) 16 (3) 16 - 20 @ 8-5 A cube of edge 4 cm is painted extemally with red colar, If is then cut into one cm cubes. How many of these do not have red point on any face ? M4 8 (3) 56 (4) 16 In the adjacent figure BA and BC are produced to meet CD and AD produced is E and F. Then ZAED + ZCED is (1) 80° (2) 50° (3) 40° (4) 160° In an isosceles triangle ABC, AC = BC, ZBAC is bisected by AD where D lies on BC. It is found that AD = AB. Then ZACB equals ay 72 (2) sa (3) 36° (4) none of these es 23 Class VII 6. 10. 24 ALLEN ABC is a right angled triangle with ZBAC = 90°. AH is drawn perpendicular to BC where H lies on BC. If AB = 60 and AC = 80, then BH = (a) 36 (2) 32 (3) 24 (4) 30 Three identical rectangles are overlapping as in the diagram, The length and breadth of each rectangle are respectively 2007 cm and 10 cm, The area of each of the shaded square portions is 16 em? ‘The perimeter of the outer boundary of the figure in cm is 1) 10070 a 2007 em (2) 12070 10cm| (3) 14070 (4) 11070 In the adjoining figure A = 60°, C = 50°, ZBDG = 30°, ZGEF = 20", Then A (1) EG = 2FG d, hon (2) EG > FG SG @G) EG = FG p—arA B (4) BG < FG The lengths of the altitudes of a triangle are in the ratio 1: 2: 3. Then (1) one angle of the triangle must be 60° (2) the triangle is a right angled triangle (3) the triangle is an obtuse angled triangle (4) such a triangle does not exist Three circles Cy, C,, C with radi () a= Q farm Where r, <1, @ @5 ws wo Suppose a mirror stands on the Y-axis drawn on a graph sheet. Then the reflection of the point (-2,4) the mirror will be 4Y (1) (2.4) a) —t+ ea) 2) 2,4) tx. (3) G42) | 4) 4,2 In the adjoining figure 1, and J, are parallel lines. r is a transversal which cuts /,, [y at A.B respectively. ‘The angles atA, B are trisected. The measures of the angles ACB and ADB are respectively (1) 60°, 120° (2) 120°, 60° Q@)x-y.xty 4) 2x-y), Ax +y) 27 Class VII 32. 33. 34. 36. 28 ALLEN ABC is an isosceles triangle with AB = AC = 2008 cm. ADC is drawn as an equilateral triangle on AC outside AABC. AD is parallel to BC. The bisector of D meets AB in E, say. Then BE is equal to (1) 1004 em (2) 2008 em ao (4) 502 em cm and DE In the adjoining figure ABC, DEF are equilateral triangles. AB value of AE + BD + CF is em. Then the possible (1) 6.9 em 2) 7.1 om (3) 5.2. em (4) 8.3 om ‘The measure of one of the angles of a right triangle is five times that of a second angle. Then the possibility of the second largest angle is a7 A rectangular sheet of paper is folded so that the comers A, B go to A’, B’ as in the figure. Then ZZXY is (2) 75° (3) 72° or 75° (4) None of these “oF Wd (J) an acute angle (2) an obtuse angle (3) a right angle (4) avariable angle depending on the point X On a line segment AB = 2008 cm a square and a regular hexagon are drawn as shown in the diagram. The distance between their centres P,Q, in em is () 10043 (2) 1004(/3 +1) (3) 20083 (4) 1004 (3-1) NMTC : Round-1 37. Two sqaures 17 cm x 17 cm overlap to form a rectangle 17 cm x 30 cm. The area of the over lapping Heeger (1) 289, (2) 68 (3) 510 (4) 85 38. In the figure, PQRS is a rectangle of area 2011 square units. K, L, M, N are the mid points of the respective sides. O is the midpoint of MN. The area of the triangle OKL is equal to (in square units) s a R K M P L Ea gy 2200 oa en 39, PQRS is a parallelogram. MP and NP divide ZSPQ into three equal parts (ZMPQ > ZNPQ) and MQ and NQ divide ZRQP into 3 equal parts (ZMQP > ZNQP). If k (ZPNQ) = (ZPMQ) then k = L a (21 a) 2 a) 4 z @ @s @5 40. A point P inside a rectangle ABCD is joined to the angular points. Then (1) Sum of the areas of two of the triangles so formed is equal to the sum of the other two (2) The Sum of the areas of the triangles so formed is a whole number whatever may be the dimensions of the rectangle. (3) The sum of the areas of a pair of opposite triangles is greater than half the area of the rectangle, (4) None of these 41, In the adjoining figure ABCD and BGFE are rhombus. AB = 10 cm, GF = 3 em. GE meets DC at H. ZA = 60°. The perimeter of ABEHD (47 (2) 40 (3) 39 29 Class VII 42. The perimeter of an isosceles right angled triangle is 2012. Its area is (1) 2012 -y2) (2) (1006)? (3 - 22) (3) (2012)? (4) (1006)? 43. Two regular polygons of same number of sides have sides 40 cm and 9 cm in legnth. The lengths of the side of another regular polygon of the same number of sides and whose area is equal to the sum of the areas of the given polygons is (in cm) (dl) 49 (2) 31 (3) 41 (4) 360 44, AX and BX are two adjacent sides of a regular polygon. If ZABX = 4 Zax, then the number of sides of the polygon is 6 @7 a9 as 45. A rectangular block of dimensions 16 x 10 x 8 units is painted. It is cut into cubes of dimensions 1x 1x L. The number of cubes which are not painted at all is (a) 945 (2) 672 (3) 812 (4) 796 46. The radius of a circle is increased by 4 units and the ratio of the areas of the original and the increased e is 4 : 9. The radius of the original circle is a6 4 3) 12 (48 47. Two sides of a triangle are 10 cm and 5 cm in length and the length of the median to the third side is ot em. The area of the triangle is 6Vxcm?. The value of x is 7 A B D c a 12 2)13 @) 14 @) 15 48. ABCD is a rectangle, Through C a variable line is drawn so as to cut AB at X and DA produced at Y. Then BX x DY is YX D c (1) Twice the area of the rectangle ABCD (2) Equal to the area of the rectangle ABCD (3) A variable quantity which lies between the area of rectangle ABCD and twice the area of the rectangle ABCD (4) Always a constant less than the area of rectangle ABCD 0 NMTC : Round-1 49. In the adjoining figure, O is the centre of the circle. ACOB is a square with A on the circle, Through B a line parallel to OA is drawn to cut the circle at D nearer to A. Then ZBOD = &) (1) 20° (2) 18° @) 15° @ ae 50, In the figure below, AB is a diameter of a circle. AB is produced to P such that BP = radius of the circle. PC is a tangent to the circle. The tangent at B and AC produced cut a E. Then ACDE is (1) isosceles with EC = ED (2) isosceles with EC = CD (3) equilateral (4) a scalene triangle 51, ABC is a right angled triangle with B = 90°. BDEF is a square. BE is perpendicular to AC, The measure of ZDEC is ; : 2 52, From a point within an equilateral triangle perpendiculars are drawn to the three sides and are 5, 7 and Sems is length. The perimeter of the triangle is om. 53. ABCD is a rectangle rotated clockwise about A by 90° as shown, The rotation takes D to D’. AB = 6 cm, BC’ = 10 cm, The breadth of the rectangle ABCD is A B BwB, CoC, Dy 31 Class VII 54. 56. 37. 32 ALL! AB is a line segment 2000 cm long. The following design of semicircle is drawn on AB, with AP 5 cm and repeating the designs. The area enclosed by the semicircular designs from A to B is P. Al B ABCD is a square. E and F are points respectively on BC and CD such that ZEAF = 45°, AE and AF Areaof AAEF cut the diagonal BD at P, Q respectively. Then find value of AOL AAEF . ae ‘Areaof AAPQ In the adjoining figure BAC is a 30° - 60° ~ 90° triangle with AB = 20. D is the midpoint of AC. The perpendicular at D to AC meets the line parallel to AB through © at E. The line through E perpen- dicular to DE meets BA produced at F. If DF = 5 , find x. ZQPR PR and PQ are tangents to a circle and QS is a diameter. Find and PQ are tangents to a cirele and QS is a diameter. Find S375 ABCD is a trapezium such that ABI|CD. AB = 10 cm; BC = 8 em ; CD = DA = 5 cm. Find the height of the trapezium and also find AC and BD. ABCD is a convex pentagon with ZA = 2B = 2D = 90° ; ZC = ZE, Sides AB, BC, CD, DE and EA are extended to K, L, M, N and O respectively. The exterior angles ZOAB, ZKBC, ZLCD, ZMDE and ZNEA are bisected and the bisectors are produced in either direction to form the pentagon PQRST. Find the angles of this pentagon. NMTC : Round-1 60. All sides of the convex pentagon ABCDE are equal in length ZA = 2B = 90°. Then find the value of ZB. 61. A bar code is formed using 25 black and certain white bars. White and black bars altemate. The first and the last are black bars. Some of the black bars are thin and others are wide. The number of white bars is 15 more than the thin black bars. Then find the number of thick black bars, 62. By drawing 10 lines of which 4 are horizontal and 6 are vertical crossing each other as in the figure, one can get 15 cells. With the same 10 lines of which 3 are vertical and 7 horizontal we get 12 cells. Find the maximum number of cells that could be got by drawing 20 lines (some horizontal and some vertical) is 63. The angle of a polygon are in the ratio 2: 4 : 5 : 6: 6: 7. Calculate the difference between the greatest and least angle of the polygon. 64. The perimeter of a right angled triangle is, 132. The sum of the square of all its sides is 6050. Find the sum of the legs of the triangle. 65. Nine squares are arranged to form a rectangle ABCD. The smallest square P has an area 4 sq.units. Find the areas of Q and R. A B ) (R) © AO | ay @ D 66. ABCD is a trapezium with AB and CD parallel, If AB = 16 em, BC = 17 em, CD = 8 em, DA = 15 em then the area of the trapezium (in cm?) will be ? 67. A square is inscribed in another square each of whose four vertices lines on each side of the square. ‘The area of the smaller square is [= times the area of the bigger one. Then calculate the ratio with which each vertex of the smaller square divides the side of the bigger square. 68. PSR is an isosceles triangle in which PS = PR. SP is produced to O such that PO = SP. Then the value of ZSRO is equal to ? 69. CAB is an angle whose measure is 70°. ACFG and ABDE are squares drawn outside the angle. The diagonal FA meets BE at H. Then calculate the measure of the ZBAH. 33 Class VII 70. nm. 14. 15. 16. 34 ALLEN In rectangle ABCD, AB = 2BC = 4 cm E and F are midpoints of AB and CD respectively. ESD and ETC are ares of circles centred at A and B respectively. If the perpendicular bisector line ! of EF cuts the arcs at S and T as in the diagram, then ST is equal to (in cm) A D F | Triangle ABC is equilateral of side length 8 em. Each are shown in the diagram is an are of a circle with the opposite vertex of the triangle as its centre. Calculate the total area enclosed within the entire figure shown (in em?), A & ABCD is a square. A line AX meets the diagonal BD at X and AX = 2016 cm. The length of CX (in cm) is equal to ? O is the centre of a circle of radius 15 em. M is a point at a distance of 5 em from O. AMB is any chord of the circle through M, then calculate the value of AM x MB. D A B ‘An isosceles trapezoid is circumscribed about a circle of radius 2 cm and the area of the trapezoid is 20 cm?. Find the length of equal sides of the trapezoid. ‘A triangle has sides with lengths 13 cm, 14 cm, and 15 em. A circle whose centre lies on the longest side touches the other two sides. Find the radius of the circle (in em). ABC and ADE are two secants of a circle of radius 3 em. A is at a distance of 5 em from the centre of the circle. The secants include an angle of 30°. The area of the AACE is 10 em?. Then calculate the area of the ADB (in em?). NMTC : Round-1 .N, QNM, RNL are congruent 77. In the figure, ABCD is a rectangle with sides 24 units, right angled triangles, then calculate the area of each triangle. 78. AB and AC are two straight line segments enclosing an angle 70°. Squares ABDE and ACFG are drawn outside the angle BAC. The diagonal FA is produced to meet the diagonal EB in H, calculate the value of ZEAH. 79. Inthe pentagon ABCDE, 2D = 2 ZB and the other angles/are each equal to half the sum of the angles ZB and 2D. What will be the largest interior angle of the pentagon. 80. A triangle whose sides are integers has a perimeter 8. The area of the 81. ABC and ADC are isosceles triangles with AB = AC = AD, ZBAC = 40°, ZCAD = 70°. Calculate the value of ZBCD + ZBDC gle is. 82. In the figure below two equal eles S,, S, of radii 2 units each touch each other. AB is the common diameter. The tangent at B meets the tangent from A to the circle S, at C as shown. If BC = Ky2 then the value of K is. 83. In the figure below, AABC is equilateral AD, BE and CF are respectively perpendicular to AB, BC Areaof ADEF and AC. Calculate = —— ‘Areaof ABC 35 Class VII AP 84. ABCD is a parallelogram P is a point on AD such that —=——~. AD 2015 AQ AC and BP. Calculate “2. ‘AC Qis the point of intersection of ES [7] 8] 9 [x0] aifiz 1[2{3|4[4lifz 26 | 27| 28 | 29| 30| 31 | 32 3[4[3}4[il4[sa 46 | 47| 48 | 49| 50 4/3[2|3[3 51. 45° $2, 42/3 cm 53.2 em $4, 1250 5q. cm, 56.7 $7.2 24 ° $8, Height= em, AC=6em, B= 3VITem 5%. 112 60, ZE = 150 61. 16 62. 81 63, 120° 64, 77 units 65, Area of Q=8, Area of R= 1296 66, 180¢0m? 67. 413 68. 90° 69, 25° 70. (4-203) 71. 32(n-3) 72.2016 73.200 74.Sem 75. 56/9 16. 815 71. 48 .q, units 78. 25° 79, 144° \ 81. 145° 82. K=2 83.3 84. 1/2016 NMTC : Round-1 Snes Logical Arguments Rahim wants to arrange a party of a certain number of people such that no two of participants (whose dates of birth he knows) of the party will have birthday in the same month, The number of people to be invited for the party cannot exceed. ar Q) 13 (3) 364 4) 23 ‘Today is Saturday. What day of the week is 100 days from now? (1) Friday (2) Tuesday (3) Monday (4) Sunday If 15% January 1997 was a Wednesday, then 15 February 2006 will be a (1) Wednesday (2) Tuesday (3) Sunday (4) Friday One hundred and twenty students take an examination which is marked out of a total 100 (with no fractional marks). No three students are awarded the same mark. The smallest possible number of pairs of students who are awarded the same rank is = 2) 10 (3) 20 @19 two-litre bottle of juice is 5 times the cost of 1 cup of juice. If a single two tre bottle of juice costs Rs.6 more than 3 cups of juice, how much does a single two-litre bottle of juice cost? () Rs. 10 (2) Rs. 12 (3) Rs. 15) (4) Rs. 18, Which one of the following is true? (® If A is a brother of B, then Bis a brother of A. (ii) IFA likes B and B likes C, then A likes © (ii) IfA 4B, and/B + C, then A # C (1) Both @ and Gi) (2) Gi) only (3) All the three (4) None of the three For an integer n, a student states the following: 1. If m is odd, (n + 1)? is even IL If mis even, (n-1)? is odd IHL If nis even, (a= is imrational Which of the above would be true? (ie mt @i&u @) All @uél 37 Class VII 8. 10. un. 12. 1B. 14. 15. Ina magic square, each row and each column and both main diagonals have the same total. The number that should replace x in this partically completed magic square is (1) more information needed a oe 3 15 (3) 10 — @r = The image of INMO when reflected in a mirror is :- (a) Imwo (2) OMNI (3) INWO (4) OWNI Nine dots are arranged such that they are equally spaced horizontally and vertically as in the figure, ‘The number of triangles which are not right angled triangles that can be formed with the above dots as vertices is :- ay 18 @2 fi: @) 40 eee (4) 32 Rengu and Dingu are friends. One of them lies on Mondays, Tuesdays and Wednesdays and tells the truth on the other days of the week. The other fellow lies on Thursdays, Fridays and Saturdays and tells the truth on the other days of the week. On a particular noon, the two had the following conversation Rengu : I lie on Saturdays Dingu : I will lie tomorrow Rengu : I lie on Sundays On which day of the week did this conversation take place ? (1) Monday (2) Tuesday (3) Thursday (4) Wednesday The last two digits of 3°°!, when represented in decimal notation, will be ast Q2)01 @) 41 @2 In a foot ball league, a particular team played 50 games in a season. The team never lost three games consecutively and never won four games consecutively, in that season. If N is the number of games the team won in that season, then N satisfies. ()25s.N < 30 (2) 25.5 N< 36 @) 16s N38 (4) 16 < Ns 30 Here is a sequence of composite numbers having only one prime factor, written in ascending order 4, 8, 9, 16, 25, 27, 32 press. The 15% number of this sequence is (342 (2) 348 (3) 242 (4) 343 A says: "Lam a 6-digit number and all my middle digits are made of zeros." B says to A: "Iam your successor. My digit in the tens place is the same as your starting digit”. The value of the whole number Ais____. (1) 100008 (2) 100009 (3) 100006 (4) 10009 PUSS 9 NMTC : Round-1 Snes Number Theory A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35. ‘The number is (1) 105 (2) 210 (3) 420 (4) 525 The largest positive integer which cannot be written in the form 5m + 7n, where m and n are positive integers is (ay 25 (2) 35 (3) greater than 100 (4) greater than 350 a2 Qa 3)6 8 Three people each think of a number, which is the product of two different primes. The product of the three numbers which are thought of is (120 (2) 12100 (3) 240 (4) 3000 The number of three digit numbers that are divisible by 2 but not divisible by 4 is (a) 200 (2) 225 (3) 250 (4) 450 Two sequences S, and S, areas under The nth term of S, is = ‘The value of the difference between the 2016" terms of S, and S, is 4023 8063 8063 8047 eae @ ape 8 mae =~ mine 2015x2016 4032%4031 40244023 4031 «4032 The least number which when divided by 25, 40 and 60 leaves a remainder 7 in each case is 2 7 and the nth tem of S, is = a (1) 607 (2) 1007 (3) 807 (4) 507 +1 1) p isan ot prime. Then (224) (2) ig (0) a faction less than p (2) a fraction greater than p (3) a natural mumber not equal to p (4) natural umber equal to p Ix = 2 and y = 222 the value of x(y +2) + 242 when a = 2012 is yal yx nt at a) 4 2012 a 2 Q) 3 (3) Z (4) 2 37 Class VII 1 then the value of x? + 1 10, If x+— x Mo @1 G)-1 (42 11. The number of natural numbers a(< 100) such that (a—a*) is a square of a natural number is (7 (28 a9 10 12. Find the smallest number which when divided by 10 leaves a remainder of 9, when divided by 9 leaves a remainder of 8, when divided by 8 leaves a remainder of 7,.... ete. down to where, when divided by 2 it leaves a remainder 1 (4) 2500 (2) 2519 (3) 3500 (4) 3519 ‘The years of 20" century and 21" century are of 4 digits. The number of years which are divisible by the product of the four digits of the year is a7 Qs @)9 (4) none of these 14. The sum of two natural numbers is 100. Their product cannot be greater than: (a) 2500 (2) 1500 (3) 100 (4) 3151 15. The number of digits when 2008 is written in the decimal form is (1) 2008 (2) 1004 3) 74 (4) 19 16. The units digit of the number 2°" is a6 @2 @4 @s 17. Ifa, b are natural numbers such that a +b = 2008, then (<1)* + (1)? is at Q)-1 (32 (4) 2 or -2 18. If n = 10! -1, the number of digits in n° is (a) 30 (2) 28 (3) 32 a7 19. Anu, Babu and Chitra have 51 balls altogether. Babu gives 7 balls to Chitra and Chitra gives 5 balls to Anu and Anu gives 4 balls to Babu. If all the three have finally equal number of balls, then the number of balls Anu had ab the start is «i7 2) 20 (3) 16 @) 19 20. The number of ordered triples (a, b, c) 1 < a, b, ¢ <9 such that ac = b? — 1 is ao @Q7 a4 @) 18 21, Let 'm’ be the number of perfect squares in 1,2,3, ....... 1000000. Let 'n' be the number of perfect cubes in I, 2, 3, «1000000. Then the value of = is () 0.01 2) 0.1 (3) 10 (4) 100 22, The difference of the squares of two consecutive natural numbers is 2008. (1) Only one such pair exists (2) Infinitely many such pairs exist (3) No such pair exists (4) Exactly two such pairs exist 23. The value of a commodity is increased by x% first and again increased by x%. The total increase is aa 2 (1) 2x% (2) 4x% B) (a) (4) °% ee ee 25. 26. 27, 28. 29. 30. 31. 34. 35. NMTC : Round-1 1" 1 and y is |2> of 2=. en ays (22) eet oe (1) 2x =y Qy b, 2

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