show that any positive odd integer in the form of 4q+1 and 4q+3
Answers
Answer:
class 10
Step-by-step explanation:
Assignment of Real Number
MATHS ASSIGNMENT NO-1, TERM-1 (2020-21)
CLASS-X
Real Numbers
Section – A (Short Answer Type Questions)
1. Write down the decimal expansion of 33/(22 x 5), without actual division.
( Ans : 0.3 )
2. The decimal expansion of the rational number 43/ 24 x 53 will terminate after how many places of decimal ?
(Ans : 4)
3. Write the HCF of smallest composite number and smallest prime number.
(Ans : 2)
4. Given that H.C.F. of 35 and 45 = 5, L.C.M. of 35 and 45 = 63 x a . Find the value of a. ( Ans : 5 )
5. Use Euclid’s division algorithm to find HCF of 867 and 255.
( Ans : 51 )
6. The H.C.F and L.C.M. of two numbers are 33 and 264 respectively. When the first
number is completely divided by 2 the quotient is 33. Find the other number.
( Ans : 132)
7. If H.C.F (6,a) =2 and L.C.M(6,a) = 60. Find a.
(Ans : 20)
8. State whether the following rational numbers have a terminating decimal expansion OR non terminating repeating decimal :
i)19 over 128 ii) 36 over 405 iii) 15 over 320 iv) 18 over 96 v)22 over 75
9. Using fundamental theorem of arithmetic, find the HCF and LCM of:
1) 510 and 92 ii)72 and 90 iii) 225 and 240 iv) 35, 48, 56
[Ans : i) 2; 23460 ii )18; 360 iii) 15 ;3600 iv) 1 ; 1680 ]
10. Use Euclids Division lemma to find the HCF of :
i) 84 and 105 ii) 4052 and 12576 iii) 616 and 1300
[Ans: i)21 ii) 4 iii) 4]
11. Without dividing , write the decimal expansion of :
i) 3 over 8 ii) 7 over 80 iii) 223 over 40 iv) 26 over 1600 v) 241 over 200 vi) 154 over 175
[Ans : 0.375 ii) .0875 iii) 5.575 iv) 0.01625 v) 1.205 vi) 0.88 ]
Section –B (Long Answer Type Questions)
1. Find the H.C.F of 65 and 117 and express it in the form 65m + 117n
( Ans : HCF 13, m=2 and n= -1)
2. If the H.C.F. of 210 and 55 is expressible in the form 210 x 5 + 55y. Find y
3. Write whether every positive integer can be of the form 4q+2, where q is an integer. Justify your answer.
4. Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep again for the first time.
(Ans : 12.20 pm)
5. Prove that ( 3 + √2) is an irrational number.
6. Prove that the square of any positive integer is of the form 5q, 5q+1, 5q+4 for some
integer q.
7. Show that the square of any odd positive integer can be of the form 6q+1 or 6q+3, for some integer q.
8. What will be the least possible number of the planks, if three pieces of timber 42m, 49m and 63m long have to be divided into planks of the same length? ( 7m, 42 planks)
(Ans : 22)
9. Show that only one of the numbers n, n+2 and n+4 is divisible by 3.
10. Show that p2 will leave a remainder 1 when divided by 8, if p is an odd positive integer.
11. Find the H.C.F and L.C.M. of 288, 360 and 384 by prime factorization method.
( Ans :24, 5760)
12. Three sets of physics, chemistry and mathematics books have to be stacked in such a way that all the books are stored topic wise and the number of books in each stack is the same. The number of physics books is 192, the number of chemistry books is 240 and the number of mathematics books is 168. Determine the number of stacks of physics, chemistry and mathematics books.
( Ans : 8, 10,7 )
Section – C (HOTS)
1. Prove that 15 + 17√3 be an irrational number
2. The H.C.F of 2472, 1284 and a third number N is 12. If their L.C.M. is 23 X 32 X 5 X 103 X 107. Then find the number N.
(Ans : 180)
3. Find the smallest number which leaves remainder 8 and 12 when divided by 28 and 32 respectively.
(Ans : 204)
4. If remainder of (5m+1)(5m+3)(5m+4)/5 is a natural number then find it.
(Ans: 2)
5. Prove that x2 – x is divisible by 2 for all positive integer x.
Step-by-step explanation:
let us start with taking s where a is positive odd integers we apply the division algorithm with a and b = 4
since 0 <_ r < the possible remainder are 0 ,1,2 and 3
this can be 4q or 4q +1 or 4q +2 or 4q +3
where q is the quotient how ever , since odd a cannot be 4q or 4q + 2
any odd integers is form of 4q + 1 or 4 q + 3