Question 1 of 5
The smallest irrational number by which \( \sqrt{20} \) should be multiplied so as to get a rational number, is:
A
\( \sqrt{20} \)
B
\( \sqrt{2} \)
C
5
D
\( \sqrt{5} \)
Why: \( \sqrt{20} = 2\sqrt{5} \), which is irrational. To make it rational, multiply by \( \sqrt{5} \): \( 2\sqrt{5} \times \sqrt{5} = 2 \times 5 = 10 \), which is rational. Among the options, \( \sqrt{5} \) is irrational and the smallest that works. \( \sqrt{20} \) gives 20 (rational but larger), \( \sqrt{2} \) gives \( 2\sqrt{10} \) (irrational), 5 gives \( 10\sqrt{5} \) (irrational). Thus, option D is correct.
Question 2 of 5
Find the least number which when divided by 12, 16, and 24 leaves the remainder 7 in each case.
Why: This requires finding the least number N such that N ≡ 7 (mod 12), N ≡ 7 (mod 16), N ≡ 7 (mod 24). This means N - 7 is divisible by 12, 16, and 24, so N - 7 is a multiple of LCM(12, 16, 24). Prime factors: 12=2²×3, 16=2⁴, 24=2³×3. LCM=2⁴×3=48. Thus, N - 7 = 48k, N = 48k + 7. Smallest positive N is for k=1: 48+7=55? Wait, check: 55÷12=4*12=48, rem 7; 55÷16=3*16=48, rem 7; 55÷24=2*24=48, rem 7. But least is actually 48*1 +7=55? Standard answer from sources is LCM-1? No: actually sources confirm 48k+7, smallest > divisors is 55, but many PYQs have 79? Wait, recheck: actually for these divisors LCM=48, yes 55 works: 55-7=48 divisible by all. But some sources say 79, perhaps misrecall. Verified: 55÷12=4 r7, 55÷16=3 r7, 55÷24=2 r7. Yes, 55 is correct least positive.
Question 3 of 5
Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.
Why: The proof uses division algorithm to consider all cases modulo 3, showing x² mod 3 ≠ 2.
Question 4 of 5
Find the value of \( (4^0 + 4^{-1}) \times 2^2 \).
Why: Using laws of exponents: zero exponent rule and negative exponent rule. First simplify inside parentheses, then multiply. This tests basic exponent properties as per NCERT Class 8 syllabus.[1]
Question 5 of 5
Express \( 4^{-3} \) as a power with base 2.
Why: This question tests the power of power rule and base conversion. Standard Class 8 level problem from NCERT important questions.[1]
