MATHEMATICS IN EVERYDAY LIFE 8


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1 MATHEMATICS IN EVERYDAY LIFE Chapter : Square and Square Roots ANSWER KEYS EXERCISE.. We know that the natural numbers ending with the digits,, or are not perfect squares. (i) ends with digit. ends with digit. (iii) ends with digit. (iv) ends with digit. Therefore, all these numbers are not perfect squares.. If a number has or in the unit place, then its square ends with, if the number has or in the unit place, then its square ends with, and the number or in the unit place, its square ends with. Therefore, (i) ends with, its square ends with. ends with, its square ends with. (iii) ends with, its square ends with.. The units digits of the square of a number having digits at units places as or is. Therefore, () and () will have as their units digits. and (iii) have as their units digit.. (i) being an even number () will also be an even number. (iii) (iv) being an odd number, () will also be on odd number. being an even number, () will also be an even number. being an even number, () will also be an even number.. Since, between n and (n + ), there are n nonperfect square numbers. Therefore, (i) () and () there are = natural number. () and () there are = natural numbers.. (i) () = ( ) hundred + = () = ( ) hundred + = (iii)() = ( ) hundred + = EXERCISE.. We know that m, m, m + is a m = Pythagorean triplet m = m = () = = m + = () + = + =,, is a Pythagorean triplet.. Let m = m = = Now, m = () = = and m + = () + = + = Hence. Let m = m =,,, is a Pythagorean triplet. m = () = = m + = () + = + =,, is Pythagorean triplet.. If a, b, c are three number where c > a, b such that a + b = c, then (a, b, c) is called Pythagorean triplet. Thus, () = () + () = + = () = () + (), and is a Pythagorean triplet. EXERCISE.. (i) Since, ones digit of is, the possible ones digit of the square root may be or. Mathematics In Everyday Life
2 (iii) (iv) Since, ones digit of is, the possible ones digit of the square root may be or. Since, ones digit of is, the possible ones digit of the square root may be or. Since, ones digit of is, the possible ones digit of the square root may be or.. We know that if the units digit of a number is,, or, then it does not have a square root in the set of natural numbers, hence it will not be a perfect square. Hence (i), and (iii) is not a perfect squares. = ( ) ( ). The number is left unpaired. So, is not a perfect square. Now, If we multiplied with it should become a perfect square. ie. = ( ) ( ) ( ) = ( ) ( ) ( ) Square root of ( ) ( ) ( ) = = square root of the new number is.. (i) We have = = = = = = = = = = = = We have performed subtraction times. (iii). (i) We have = = = = = = = = = = = = = = = = = We have performed subtraction times. =. We have = = = = = = = = = = = = = We have performed subtraction times. Hence =. = = = = = Answer Keys
3 = = = = = (iii) = = (iv) = = = = (v) = = = = = = (vi) = = (vii) = = = = (viii) = = = = (ix) = = = = (x) = = = = = = Mathematics In Everyday Life
4 . = ( ) ( ) ( ) The number is left unpaired. So, it is not a perfect square. If we multiplied by it should become a perfect square. So, = ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) =. (i) = = = ( ) ( ) The number is left unpaired. If multiplied by it should be a perfect square number. So, = ( ) ( ) ( ) = = = = if multiplied by, then the given number makes a perfect square and its square root is. = ( ) ( ) ( ) The number is left unpaired. If multiplied by, it should be a perfect square number. So, ( ) = ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) = = = if multiplied by, then the given number make a perfect square and its square root is. (iii) = ( ) ( ) ( ) ( ) The number is left unpaired. If we multiplied by, it should be a perfect square number. So, = ( ) ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) = = = if multiplied by, then the given number make a perfect square number and its square root is.. We know that, if the units digit of a number is,, or, then it does not a perfect square number. Therefore, (i), and (iii) are not a perfect square number. While is a perfect square number. EXERCISE.. The least number divisible by each of the numbers, and is their L.C.M. L.C.M of, and is,, =,, Now, = ( ) The numbers and are not in pair for the number to be a perfect square each factor of the must have a pair. So, To make pairs of and, the number has to be multiplied by ie.. = is the required square number.. = ( ) ( ) The number is left unpaired. So the given number must be divided by to get a square number. = Now, square root of new number is. Answer Keys
5 . The least number which is divisible by each of the numbers, and is their L.C.M. L.C.M. of, and = = Now, = ( ) ( ),,,, The number is left unpaired. For the number to be a perfect square, each factor of the number must have a pair, to make pair of the number, has to be multiplied by. the required number is =.. The prime factors of = ( ) ( ) The number is left unpaired. So, the given number must be divided by to get a square number. = is a square number. is the required smallest number.. Let the one number be x, then the other number will be x. Then, x x = x = x = = x = = x = One number is and the other number is. required numbers are and.. Let the number be x. Other number will be x. Then, x x = x = x = x = = x = x = x = One number is and the other number is. required numbers are and.. Length of a rectangular field = m Breadth of a rectangular field = m Area of rectangular field = length breadth = m m = ( )m = m Since, area of square = area of rectangular field (Side) = m side = m = m = ( )m = m side of the square is m.. Let number of students in a school be x, Then, x x = x = x = x = x = x = Mathematics In Everyday Life there were students in the school.. = ( ) ( ) ( ) The number is left unpaired. So, is multiplied to the number to make a perfect square number.. Let the number be x. Then, the other number is. x x Therefor, = x x = x = = x =
6 x = x = = x = the required numbers are and.. Let the number be x. Then, the other number is x. Therefore, x x x = x = = x = x = x = x = x = the required numbers are and.. (i) = ( ) ( ) The number is left unpaired. So, the given is divided by to get a square number. So, = = = = = ( ) ( ) ( ) The number is left unpaired. So, the given number is divided by to get a square number. So, is a square number... (iii) = = = = ( ) The number is left unpaired. So, the given number is divided by to get a square number. = EXERCISE. Answer Keys
7 Mathematics In Everyday Life
8 . EXERCISE.. (i), square root will have digits., square root will have digits. (iii), square root will have digits. (iv), square root will have digits. (v), square root will have digits. (vi), square root will have digits. (vii), square root will have digits.. The greatest five digit number is. Let us find square root of. From the square root of, we can notice that () is less then by. If we subtracted the remainder from the number, we get a perfect square number. = is the required number. the greatest five digit number is. Least number of fourdigits is.. (i) also, () = Now, () = = We notice that () <, Thus, if we added to, it becomes a perfect square. the smallest four digit square number is. (iii) () < by. So, in order to get a perfect square number we subtract from. () is less than by. So, in order to get a perfect square number, we subtract from. () is less than by. So, in order to get a perfect square number, we subtracted from. Answer Keys
9 (iv) () < by. So, in order to get a perfect square number, we subtracted from.. (i) The given number is.. (i) () > > () Thus, () = = the given number to be added is. EXERCISE. () =, () = () > > () Thus, () = = the number to be added is. The given number is. Also () = () = () > > () Thus, () = = the number to be added is. (iii) The given number is. Since, () = and () =. (iii) Mathematics In Everyday Life
10 (iv). (i) The given fraction is. On simplifying, (iv) (v) (vi) (By Simplification) (iii). Area of a square field = (side) (side) = sq.m Answer Keys
11 side = side of a square field = m m. side of the square field = Area = = side of the square field = m m. (i) (iii). (i). (i) Thus, Thus, (On simplification) Mathematics In Everyday Life
12 . (i) Thus, Thus, EXERCISE..... (iv) (v) (vi) (iii)... (i) Answer Keys
13 (iii). (i) (iii)... (iv)... We have, =. and = or... Mathematics In Everyday Life
14 . (i) Let the fraction be x. Then x x =. x =. x =. x =. = = the required fraction is.. Let the fraction be x. Then, x x =. x =. x =. x =. = = required fraction is Area of a square field =. sq.m Area of a square field = (Side) (side) =. side =. side =. m.. side of the square field =. m. Area of a square field =. sq.m Area of square field = (Side) (side) =. side =. side =. m.. side of the square field =. m. EXERCISE. Look at the table, the entry in the column of is..... Look at the table, the entry in the column of is... Answer Keys
15 . Look at the table, the entry in the column of is..... Look at the table, the entry in the column of is..... = =. =. =. (from table =.). = =. =. =.. = =. =. =.. = =. =. =.. = =. =. =. (from table. ) (from table. ) (from table. ) (from table. ).. = = =. =... =... = = = =. (from table. ) =.. =.. =... = = = = (..) (from table. &. ) =.. =. =.. Mathematics In Everyday Life
16 .. = For., we find approximate difference between and. =. and =. (from table) =.. =. For the difference of (= ), the difference between and =. For the difference. =.. =.. =. +. =.. =... = = = = (..) =. =.. =... = = = =. =....,. MULTIPLE CHOICE QUESTIONS., square root will have digits. option (a) is correct.. We have, between n and (n + ), there are n natural numbers. So, between () and () there are = natural numbers. option (c) is correct.. () = = option (a) is correct.... =.. =. option (b) is correct =.... option (c) correct. () = & () = > () by. So, () = = Thus, the number must be subtracted from to make it a perfect square. option (a) is correct... =. option (b) is correct. (from table. ) Answer Keys
17 Since, () = and () = So, () < < () () = = Thus, the number to be added is. option (d) is correct.. (,, ) () = + = + = () < + (,, ) () = + = + = () = +, (It is a Pythagorean triplet) (,, ) () = () + () = + = () > () + () And (,, ) = + = + = > + option (b) is correct.. We have the numbers ends with,,,, are perfect squares. So, is a perfect square. option (b) is correct.. = ( ) ( ) There are the number is left unpaired So, the given number should be divided by to make it perfect square. option (c) is correct.. In the following number end with digit (even). So its square ends with even digit. option (b) is correct.. If a number has or in the unit place, then its square ends with. () and () end with unit digit. Hence option (b) and (d) is correct.. = ( ) ( ) The number is left unpaired. The given number must be multiplied by obtain a perfect square.. Area of a square field = sq.m Area of square field = (side) Mathematics In Everyday Life (side) = sq.m side = m = side = m option (d) is correct. A. True or False: MENTAL MATHS CORNER. The square of a prime number is prime. (False). There is no square number between and. (True). All square numbers are positive. (True). The product of two square numbers is a square number. (True). The difference between two square numbers is a square number. (False). The sum of two square numbers is a square number. (False). A number ending with even number of zeros is always a perfect square. (False). The square of a natural number is either a multiple of or exceeds. The multiple of by. (True) B. Fill in the blanks:. If x y ; then y = x.. Upto, there are only numbers which are perfect squares.. A rational number whose square is is. The number and when divided by leave the remainder and respectively.. The sum of first odd natural numbers is times = () = =
18 . A number ending with an odd number of zeros is never a perfect square.. Negative numbers have no square root in the system of rational numbers.. A number ending with,, or is never a perfect square. Review Exercise. (i) = = = = = = = = = = (iii) = = = = = We have (n ) = n n = n = n = and n = () = =. () = = () + () = + = () = () + () (,, ) is a Pythagorean triplet.. We have between (n ) and (n + ), there are n natural numbers. (i) between and, there are = natural numbers. between and, there are = natural numbers. (iii) between and, there are = natural numbers.. (i) (iii).... Answer Keys
19 . x = x = = x = = the required number is.. (i) () = & () = () < < () () = = Thus, the number must be added to given number to make a perfect square. Now, + = () () () () () () = =. Let the number of chairs in a row be x. Then x x = x = x = = there are chairs in a row.. The least number which is exactly divisible by each of the numbers,, and is their L.C.M.,,, L.C.M. of,, and,,, = =,,, HOTS QUESTIONS. The largest digit number is. The square root of. () =, To make it least square number be multiplied in,,, number by i.e.,,, So, the required least number = =. Let the number be x. Then x x = Mathematics In Everyday Life < by, in order to digit largest square number. Subtracting from. Thus the required number is =. Let the number be x, Then x x = x = x = = x = = the required number is.. Let m = m = m = () = = and m + = () + = + = Thus, the required Pythagorean triplet is (,, ).
20 Illustration for first ten lockers and first ten students is given below : Students Locker Number O O C O C O C O O C O C O C O C O C O C O C O O C O C O Opening the locker C Closing the locker If a number is a perfect square, it will have an odd number of factors e.g., has three (odd) factors, and. So, locker number will be visited by students ( st, rd, th ). First student will open the locker, third will close it and ninth will open it again. (see illustration) Whereas if a number is a nonperfect square, it will have an even number of factors e.g., has four (even) factors,, and. So, locker number will be visited by students ( st, nd, th, th ). First student will open the locker, second will close it, fourth student will open it and eighth will close it. (see illustration) We see that, if a particular locker is visited an odd number of times (in case of perfect squares) it will be open at the end of the procedure, otherwise it will be closed (in case of nonperfect squares). So, the open lockers are numbered,,,,,,,,,, all of which are perfect squares. Total number of lockers opened is. Answer Keys
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