power indices and surds Model Questions & Answers, Practice Test for ssc mts tier 1 2024

Answer: (d)

$1/{a - b} - 1/{a + b} - {2b}/{a^2 + b^2} - {4b^3}/{a^4 + b^4} - {8b^7}/{a^8 - b^8}$

= ${(a + b) - (a - b)}/{(a - b)(a + b)} - {2b}/{a^2 + b^2} - {4b^3}/{a^4 + b^4} - {8b^7}/{a^8 - b^8}$

= ${2b}/{a^2 - b^2} - {2b}/{a^2 + b^2} - {4b^3}/{a^4 + b^4} - {8b^7}/{a^8 - b^8}$

= ${2b(a^2 + b^2) -2b (a^2 - b^2)}/{(a^2 - b^2)(a^2 + b^2)} - {4b^3}/{a^4 + b^4} - {8b^7}/{a^8 - b^8}$

= ${2a^2b + 2b^3 - 2a^2b + 2b^3}/{(a^4 - b^4)} - {4b^3}/{a^4 + b^4} - {8b^7}/{a^8 - b^8}$

= ${4b^3(a^4 + b^4) - 4b^3(a^4 - b^4)}/{(a^4 - b^4)(a^4 + b^4)} - {8b^7}/{a^8 - b^8}$

= ${4a^4b^3 + 4b^7 - 4a^4b^3 + 4b^7}/{a^8 - b^8} - {8b^7}/{a^8 - b^8}$

= ${8b^7}/{a^8 - b^8} - {8b^7}/{a^8 - b^8}$ = 0

Answer: (a)

(A) $√{{5041}/{6889}} = {71}/{83}$ = rational number

(R) Now, 2 is a rational number but $√2$ is not a rational number. i.e., irrational number.

Answer: (a)

If $√{x + √{x + √{x + ......}}}$ then its value is ${√{4x + 1} + 1}/2$

Therefore, according to the question,

${√{4x + 1} + 1}/2 = {√{4.1 + 1} + 1}/2 = {√5 + 1}/2 ={2.236 + 1}/2$

= ${3.236}/2$ = 1.618 which lies between 1 and 2.

Answer: (a)

${(243 + 647)^2 + (243 - 647)^2}/{243 × 243 + 647 × 647}$

= ${(243)^2 + (647)^2 +2(243).(647) + (243)^2 + (647)^2 -2.(243)(647)}/{(243)^2 + (647)^2}$

= ${2(243)^2 + 2(647)^2}/{(243)^2 + (647)^2}$

= ${2[(243)^2 + (647)^2]}/{(243)^2 + (647)^2}$

= 2

∴ Option (c) is correct.

Answer: (b)

$√2, ^3√3, ^6√6 and ^12√{12}$

LCM of 2, 3, 6 and 12 is 12

It can be written as

$^12$$√{2^6}, ^12√{3^4}, ^12√{6^2} and ^12√{12}$.

So $^3$$√3$ is largest number.