| 1. If (4+√12)/(√75-√48+√108 -√147+2√3) = x√3 + 1 then find x. | Easy |
| A. 2/3 B. 3 C. 2 D. None of these |
View Answer
Answer: Option A
Explanation:
(4+2√3)/(5√3-4√3+6√3-7√3+2√3) = (4+2√3)/(2√3) = 2/√3 +1 = (2√3)/3 +1
Hence, x = 3
| 2. If (√(2k+x)+ √(2k-x))/(√(2k+x)- √(2k-x)) = 2k/x then find the value of x in terms of k. | Medium |
| A. ±k B. ±2k C. k2 D. None of these |
View Answer
Answer: Option B
Explanation:
Rationalize the given expression by multiplying both numerator and denominator by (√(2k+x)+ √(2k-x)) to get:
((√(2k+x)+ √(2k-x))(√(2k+x)+ √(2k-x)))/((2k+x)-(2k-x)) = 2k/x
-> ((2k+x+ 2k-x)+2√((2k+x)(2k-x)))/2x = a/x
-> 2k+√(4k2– x2) = 2k
-> √(4k2– x2) = 0
-> 4k2– x2 = 0
-> x = ±2k
| 3. If the following expression can be written as k(√2-1),find k. 1/(√10-√9) – 1/(√9-√8) + 1/(√8-√7) – 1/(√7-√6) + 1/(√6-√5) |
Easy |
| A. √5 B. 1 C. 1/√5 D. None of these |
View Answer
Answer: Option A
Explanation:
Rationalize the given expression to get:
(√10+√9)/(10-9) – (√9+√8)/(9-8) + (√8+√7)/(8-7) – (√7+√6)/(7-6) + (√6-√5)/(6-5)
=√10+ √9-√9-√8+√8+√7-√7-√6+√6-√5
= √10-√5
= √5 (√2-1)
| 4. If √(10+2√24) = √x +√y, and x>y, then what are the respective values of x and y? | Medium |
| A. √6, √4 B. 6, 4 C. 6, √4 D. None of these |
View Answer
Answer: Option B
Explanation:
√(10+2√24) = √[(6+4)+2√(6*4)] = √(√6+√4)2 = √6+√4
-> x = 6, y=4
Note = √[(m+n)+2√mn] = √m+√n
| 5. If x = 2+√2 what is the value of x3-6x2+12x-2(4+√2)? | Difficult |
| A. 1 B. 0 C. 2 D. None of these |
View Answer
Answer: Option B
Explanation:
Given, x = 2+√2
-> x2 = 4+2+4√2 = 6+4√2
-> x3 = x2.x = (6+4√2)(2+√2) = 12+6√2+8√2+8 = 20+14√2
Therefore, x3-6x2+12x-2(4+√2) = 20+14√2-36-24√2+24+12√2-8-2√2 = 0
| 6. If x = √(3&2)+ √(3&4) what is the value of x3-6x? | Difficult |
| A. 3 B. 6 C. 2 D. None of these |
View Answer
Answer: Option B
Explanation:
Cubing both sides, we get
x3 = (∛2)3 + (∛4)3 + 3(∛2)(∛4)(∛2+(∛4) = 2+4+3.∛8.x
-> x3 = 6+6x
-> x3-6x = 6
| 7. Find the square root of (1 + 1/(√2+1) + 1/(√3+√2) + 1/(√4+√3) ….. 1/(√484+ √483)) | Difficult |
| A. 3√7 B. 2√5 C.2√7 D. None of these |
View Answer
Answer: Option C
Explanation:
Rationalize the given expression to get:
1 + (√2-1)/(2-1) + (√3-√2)/(3-2) + (√4-√3)/(4-3) + ….. (√784-√783)/(784-783)
= 1+√2-1+√3-√2 —– +√784-√783 = √784
= 28 (all terms cancel except √784, which is 28)
Now √28 = 2√7
| 8. If 1/(2-√2+√3-√6) = -(√2+1)(√x+√y), where x and y are positive integers and x<y, find respective values of x and y. | Medium |
| A. 2, 5 B. 3, 2 C. 2, 3 D. None of these |
View Answer
Answer: Option C
Explanation:
1/(2-√2+√3-√6) = 1/(√2(√2-1)+√3(1-√2))
= 1/((√2-1)(√2-√3))
= ((√2+1)(√2+√3))/((2-1)(2-3))
= -(√2+1)(√2+√3)
| 9. If √(9+ 2√20) = x+√y, and √(16+ 2√63) = a+√b, and then find ax +by. It is given that a, b, x and y are positive integers. | Difficult |
| A. 586 B. 16807 C. 16816 D. None of these |
View Answer
Answer: Option C
Explanation:
√(9+2√20) = √[(4+5)+2√(4*5)] = √(√4+√5)2 = √4+√5 = 2+√5
-> x = 2; y = 5
√(16+2√63) = √[(7+9)+2√(7*9)] = √(√7+√9)2 = √7+√9 = 3+√7
-> a = 3; b = 7
-> ax +by = 32 +75 = 16816
| 10. If the sixth root of (8a * 42(b+2) * 18(c+3)) is a natural number , then which of the following may be true?
A. a=6, b=4, c=3 |
Medium |
View Answer
Answer: Option A
Explanation:
Let x = (8a * 42(b+2) * 18(c+3))
Then 6th root of x will be x1/6, which will be a natural number if x can be expressed as the sixth power of a natural number.
x =(8a * 42(b+2) * 18(c+3))
= (8a * 6(b+2) * 7(b+2) * 3(c+3) * 6(c+3))
= (8a * 6(b+c+5) * 7(b+2) * 3(c+3))
Check options.
For a=6, b=4, c=3, x = (86 * 612 * 76 * 36)
-> x1/6 = (8*62*7*3), which is a natural number.