real numbers

Q) The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change together next? Ans: The three traffic light will change together again when the time gap is perfect multiple of each light’s interval. Therefore, […]

Q) Using prime factorisation, find HCF and LCM of 96 and 120. Ans: By prime factorisation, we get: 96 = 25 x 3 and 120 = 23 x 3 x 5 ∴ LCM = 25 x 3 x 5 = 480 And HCF = 23 x3 = 24 Therefore, LCM of given numbers is 480

Q) Prove that √3 is an irrational number. Ans: Let us assume that √3 is a rational number Let √3 =  ; where q ≠ 0 and let p, q are co-primes.   3q2 = p2………………. (i) It means p2 is divisible by 3 p is divisible by 3 Hence, we can write that p = 3a,

Q) Prove that √5 is an irrational number. Ans: Let us assume that √5 is a rational number Let √5 =  ; where q ≠ 0 and let p, q are co-primes.   5q2 = p2………………. (i) It means p2 is divisible by 5 p is divisible by 5 Hence, we can write that p = 5a,

Q) The two numbers are in the ratio of 2 : 3 and their LCM is 180. What is the HCF of the these numbers? Ans: Let us consider, the two number be 2x & 3x in the ratio of 2 : 3. Therefore the LCM of these two numbers = 6x Since, its given

Q) Prove that 6 – √7 is an irrational number, given that √7 is an irrational number. Ans: Let us assume that 6 – √7 is a rational number Let 6 – √7 =  ; q ≠ 0 and p, q are integers √7 = Since, p and q are integers; Therefore 6q – p is

Q) Prove that 4n  can never end with digit 0, where n is a natural number. Ans:  Let’s assume that 4n  ends with 0. Since now it ends with zero, it is multiple of 10 and hence, it must be divisible by 2 and 5 both. This clearly means, that the factors of 4n  should include

Q) Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 a.m., when will they ring together again? Ans: VIDEO SOLUTION  STEP BY STEP SOLUTION The three bells will ring together again when the time gap is perfect multiple of each bell’s interval Therefore, we will take

Q) Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers. Ans: By prime factorisation, we get: 18180 = 22 x 32 x 5 x 101 and 7575 = 3 x 52 x 101 LCM = 22 x 32 x 52 x 101 LCM =

Q) Prove that 2 + √3 is an irrational number, given that √3 is an irrational number. Ans: Let us assume that 2 + √3 is a rational number Let 2 + √3 =  ; q ≠ 0 and p, q are integers √3 = Since, p and q are integers; Therefore p – 2q is