problem archive
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Prove that an equilateral triangle can be constructed on any given line segment. -
Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can pass through a single point. (ii)... -
Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? ... -
Consider two ‘postulates’ given below: (i) Given any two distinct points A and B, there exists a third point C which is in between A and B. (ii) There... -
If a point C lies between two points A and B such that AC = BC, then prove that AC = \frac{1}{2} AB. Explain by drawing the figure. -
Prove that every line segment has one and only one mid-point. -
Factorize 32760 as a product of primes. -
Check if 3803 and 3607 are primes. -
Consider the numbers 4^n, where n is a natural number. Check whether there is any value of n for which 4^n ends with the digit zero. -
Find the LCM and HCF of 6 and 20 by the prime factorisation method. -
Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM. -
Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 -
Find the LCM and HCF of the following pairs of integers and verify that LCM x HCF = product of the two numbers. (i) 26 and 91 (ii) 510 and 92 (iii) 33... -
Find the LCM and HCF of the following integers by applying the prime factorisation method. (i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25 -
Given that HCF (306, 657) = 9, find LCM (306, 657). -
Check whether 6^n can end with the digit 0 for any natural number n. -
Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers. -
There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Supp... -
Prove that \( \sqrt{3} \) is irrational. -
Show that \(5 - \sqrt{3}\) is irrational.