1. If 2x = 8y+1 and 9y = 3x-9. What is the value of xy? | Easy |
A. 4412 B. 4413 C. 215 D. None of these |
View Answer
Answer: Option B
Explanation:
2x = 23(y+1)
x = 3y+3 —– ❶
9y = 3x-9
32y = 3x-9
2y = x-9 —– ❷
x = 21, y = 6
Therefore, xy = 216 = (212)3 = 4413
2. If ap=bq=cr where x,y,z ≠ 0 and c = ab, then which of the following is true? | Medium |
A. r = pq/(p+q) B. r = pq/(p-q) C. r = (p+q)/pq D. None of these |
View Answer
Answer: Option A
Explanation:
Let ap=bq=cr=k
a = k1/p, b = k1/q, c = k1/r
Now, c = ab (given)
Therefore, k1/r = k1/p. k1/q
-> k1/r = k1/p+1/q
-> 1/r = 1/p + 1/q
-> r = pq/(p+q)
3. If 2x=4y=8z, what is the least value of (x+y+z), if x,y and z are natural numbers? |
Difficult |
A. 10 B. 11 C. 12 D. Can’t be determined |
View Answer
Answer: Option B
Explanation:
Given, 2x = 4y = 8z
-> if x =1, then y =1/2 and z = 1/3
Now, the LCM of 2 and 3 (denominators of the above values of y and z) is 6 -> the first set of values of x, y and z if they are all natural numbers, will be 6, 3 and 2 respectively.
Thus, 26 = 43 = 82
-> x+y+z = 11
4. If x raised to the power x5 equals 5, then what is the value of x? | Medium |
A. 51/5 B. 21/2 C. 31/3 D. None of these |
View Answer
Answer: Option A
Explanation:
Let x5 = y
Then x = y1/5
Therefore, (y1/5)y = 5
-> yy/5 = 5
-> yy = 55
-> y = 5
-> x5 = 5
-> x = 51/5
Alternatively
Check the value of x from the options, which satisfies the equation
5. If xpqr = xp.xq.xr where p, q, r and x are all positive integers, what is the value of p3+q3+r3 ? | Medium |
A. 48 B. 14 C. 36 D. None of these |
View Answer
Answer: Option C
Explanation:
Given, xpqr = xp.xq.xr = xp+q+r
-> pqr = p+q+r
By trial and error, p, q, r are 1, 2, 3 (respective values are not important as each variable undergoes the same treatment).
-> 13+23+33 = 36
6. 293-292-291 is the same as: | Easy |
A. 292 B. 290 C. 291 D. None of these |
View Answer
Answer: Option C
Explanation:
293-292-291 = 291[22-2-1]
= 291
7. Which of the following is true? | Easy |
A. Nine raised to the power 87 equals 98 raised to the power 7. B. Nine raised to the power 87 is less than 98 raised to the power 7 C. Nine raised to the power 87 is more than 98 raised to the power 7 D. None of these |
View Answer
Answer: Option C
Explanation:
Nine raised to the power 87 = 98*8*8…..7 times
98 raised to the power 7 = (98)7 = 956
A. Therefore, nine raised to the power 87 is more than 98 raised to the power 7.
8. If x, y, and z are natural numbers such that x<y≤z and 3x+4y+5z is the largest five-digit number that satisfies the given conditions, find x4+y3+z2. | Difficult |
A. 608 B. 1688 C. 8168 D. Can’t be determined |
View Answer
Answer: Option B
Explanation:
A five-digit value of 3x+4y+5z is only possible if z is 6 or 7. This expression will not be a five-digit number for z values less than 6 and more than 7. However, since 3x+4y+5z is the largest five-digit number, we take the value of z as 7.
Again, for 3x+4y+5z to be maximum under the given conditions, x= 6 and y=7. For x=6, y=7, z=7, 3x+4y+5z is still a five-digit number. Thus, plug in these values to find x4+y3+z2.
Desired value = 64 + 73 + 72 = 1296 +343 +49 = 1688
9. Which of the following might be true if p = q2 and q = pq? | Difficult |
A. p2 +q2 = 5/16 B. p/q3 = 2 C. p = q = 1 D. All of the above |
View Answer
Answer: Option D
Explanation:
p = q2 —– ❶
q = pq —– ❷
Substituting ❶ in ❷
q = (q2)q = q2q -> q = 1 or ½
when q = 1, p = 1 -> Option C is possible
when q = ½, p = ¼
We check other options using the above values.
Option A is possible when p = ¼, and q = ½ as (1/4)2 + (1/2)2 = 5/16
Option B is possible when p = ¼ and q = ½ as (1/4)/(1/2)3 = 8/4 =2
Therefore, all of them might be true.
10. Let x to the power 2n4 equals y, and x to the power 4n equals z. If yn = z4, where x and n are positive integers, find the value of (2n+1)√n20.
A. 64 |
Difficult |
View Answer
Answer: Option C
Explanation:
Given, yn = z4
-> x raised to 2n4+1 = x raised to 4n+1
-> 2n4+1 = 4n+1
-> 2n5 = 22n+2
-> 2n5 = 2.22n+1
-> n5 = 22n+1
By trial and error, we find that n = 2
-> (2n+1)√n20 = 5√220 = (220)1/5 = 24 =16