Surds and Indices - Quantitative Aptitude Notes | Study Quantitative Aptitude (Quant) - CAT

Q 1. What will come in place o f question mark in 

ANSWER: 12 

Explanation:.

Q 2. What is value of (0.000001)^⅓ ?

ANSWER: 1/100

Explanation: 

If a cube has 6 decimals, then its cube
root will have 6/3 = 2 decimals

Q 3. What will be value of (684)²= 467856?  

ANSWER: 6.84 

Explanation: 

I f a square has 4 decimals, then its
square root will have 4/2 = 2 decimals

Q 4. If value of is approximately 5.2915, then value of is approximately  

ANSWER: 1.3228 

Explanation:  

Q 5. How many zeroes are there in 2000¹⁰?  

ANSWER: 30 

Explanation:  

2000¹⁰ = 2000 x ... 10 times
2000 = 2 x 1000 i.e it has 3 zeroes.

So that means 3 zeroes x 10 times= 30 zeroes

Q 6. What is value of (36)^⅙?  

b.

d.

ANSWER: 

Explanation:  

Q 7. What is value of M in (p/q)^2M+2 = (q/p)^9-M  

ANSWER: -11 

Explanation:  

Since the base is same i.e. p/q, we can equate the powers of both
∴ 2M+2 = -(9-M)

∴ 2M+2 = -9+M

∴ M = -11

Q 8. What is value of 2^2(-2)  

ANSWER: 

Explanation: 

Q 9. What is value of 2^(-2)2  

ANSWER: 16 

Explanation: 

 Negative number raised to even power gives positive answer

Q 10. There are four expressions given below separated by ‘=’ sign. Which one of the 4 expressions is not the same as the other three?
6² x 6 = (6¹)³ = 2³+3³ = 8³ - 14² - 10² 

ANSWER: 2³+3³ 

Explanation: 

This is very easy to solve
The first expression is 6² x 6 = 6³ = 216.

The second expression is (6¹)³ = 6^{1 x 3} = 6³ = 216
The third expression is 2³+3³ = 8+27 = 35 = ≠ other two
So answer is Option C

Question 11:   = ?

a) 5
b) 6
c) 7
d) None of these
Answer (D) 
Explanation:  


= 14/3

Question 12: Find the value of x in 2^{|5x - 6|} = 8^{3x - 2}
a) 2/3
b) 6/7
c) 9/5
d) cannot be determined 
Answer (B) 
Explanation:  

Taking log on both sides we get, 

⇒ |5x – 6| = 3(3x – 2) = 9x – 6
When x > 6/5, |5x – 6| = 5x – 6
Thus, 5x – 6 = 9x – 6
⇒ 5x = 9x which is not possible
When x < 6/5, |5x – 6| = 6 – 5x
Thus, 6 – 5x = 9x – 6
14x = 12
⇒ x = 6/7 

Question 13: If log₁₅₂ 729 = x, then what is the value of log₈₁ 5625 ? 

a)
b)
c)
d)
Answer (A) 
Explanation:  log₁₂₅ 729
= log₅ 9
= x
⇒
⇒

=  

Question 14: Find the number of digits in the number 2401²⁷ when written in base 7. 
a) 117
b) 181
c) 98
d) 109 
Answer (D) 
Explanation: 
The formula for finding the number of digits in a number ‘m’ in base ‘n’ is: 
[log_n(m)] + 1 , where [A] is greatest integer less than or equal to A. 
2401 = 7⁴
⇒
Thus, the answer is D. 

Question 15: What is the value of  
a) -2
b) -1
c) 0
d) None of these
Answer (A) 
Explanation:

 

Question 16: If  3^2x+3-244*3^x = -9, then which of the following statements is true? 
a) ‘x’ is a positive number
b) ‘x’ is a negative number
c) ‘x’ can be either a positive number or a negative number
d) None of the above
Answer (C) 
Explanation:
The equation can be written as follow: 

Let 3^x = t
⇒ 27t² - 244t + 9 = 0
⇒ 27t² - 243t-t + 9 = 0
⇒ 27t(t - 9)-1(t - 9) = 0
⇒ t = 9 or t = 1/27
⇒ 3^x = 3² or 3^x = 3^-3
So, x = 2 or x = -3
So, ‘x’ can be either a positive number or a negative number. Option c) is the correct answer 

Question 17: If x,y are natural numbers, log8+log5 = log(x+y)+log(x-y) and log8+log11 = log(x-y)+log(1+xy), then find the value of (x³ - y³)
a) 325
b) 316
c) 643
d) 602 
Answer (B)
Explanation:
log(x+y)(x-y) = log40
(x+y)(x-y) = 40
x+y = 10 and x-y = 4 or x+y = 20 and x-y = 2
x = 7 and y = 3 or x = 11 and y = 9.
log88 = log(x-y)(1+xy)
x = 7 and y = 3 satisfy the equation. 
(x³ - y³) = 7³ - 3³ = 316 

Question 18: If log₁₆ 5 = x and log₅ 3 = y, find the value of log₃ 6 in terms of x and y?
a) 
b)
c)
d) 1 + 4xy
Answer (A) 
Explanation:

log₁₆ 5 = x

log₅ 3 = y

We have to find the value of log₃ 6

= 1+ 
= 1+ 
= 1+1/4xy = 

Question 19: Find the value of log₁₀10 + log₁₀10² + ..... +log₁₀10ⁿ

a) 
b)
c)
d)
Answer (B) 
Explanation:

⇒
⇒
⇒
⇒

Question 20: If . Find x
a) 512
b) 289
c) 170
d) None of these
Answer (D)  
Explanation:

⇒
⇒
⇒
⇒
⇒ x = 28900
Since there is no such option, the correct option to choose is D – None of these.

Logarithms, Surds and Indices questions will be asked in CAT quantitative aptitude section,