Test: CAT Quantitative Aptitude- 6 - CAT MCQ

To find the number of trailing zeros in 60!, we can use the formula that counts how many times 5 is a factor in the numbers from 1 to 60. This is because the number of trailing zeros in a factorial is determined by the number of pairs of factors of 2 and 5, and there are always more factors of 2 than 5.

1. Identify the formula: The number of trailing zeros in n! can be calculated using the formula:

- Since 125>60, we stop here.

3. Add the results:
Total trailing zeros=12+2+0=14

12^th term of an A.P. = Sum of first 12 terms - Sum of first 11 terms = 555 - 420= 135

4^thterm of G.P. = 12^th term of an A.P. =135

Let the first term and common ratio of G.P. be a and r.

4^th term pf G.P. = ar³ = 135 .

∴ r = 135/5 = 27

∴ r = 3 Answer: 3

Option 3

As per the given condition,
hog(x) + fog(x) = 30
x² + 13|x| - 30 = 0
(For x > 0, |x| = x)
When x = +ve,
x² + 13x - 30 = 0
(x + 15)(x - 2) = 0
x = 2, x = -15 (rejected as x > 0)
When x = -ve,
x² - 13x - 30 = 0
(x + 2)(x - 15) = 0
x = -2, x = 15 (rejected because x < 0)
Sum = -2 + 2 = 0

The digits at thousands place can be anything from 1 to 9 and the digit at hundreds place can be anything from 0 to 9.

The sum of the two digits will be either a single or double digit.

∴ The number of required numbers = 9 x 10 = 90

Hence, option 4.

Let T be the position of tree.

m∠TPR = 2 x m∠TPS

∴ PS is the angle bisector.

 ∴  x = 6 meters

Hence, option 2.

The speed of the train = 270 km/hr = 75 m/s The time after which the difference in position of A is 1800 m = 1800/(75 - 3) = 25 seconds Now, in these 25 seconds,

• A walks 25 x 3 = 75 m towards rearmost part of the train.
• B walks 25 x 4 = 100 m towards foremost point of the train.

At his stage, there is distance of 2100 m between them.

∴ The length of the train = 75 + 2100+100 = 2275 m

Hence, option 4.

Answer: 4

When two numbers divided by a common number N leave the same remainder, then their difference must be divisible by N.
Thus must be divisible by N.
194 = 97 x 2
Since N is of two digits, the number is 97.

Total purchase price = 45.45 x 100 = Rs. 4545 Let the tax paid be Rs. x

∴ (6817.5 - 4545 - x)/(4545 + x) = 0.2

∴ 2272.5 - x = 909 + 0.2x

∴ x = 1136.25
The tax percentage levied over cost price = (1136.25/4545) x 100 = 25% 
Answer: 25

When 3¹,3², 3³ and 3⁴ divided by 12 leaves remainder 3, 9, 3 and 9 respectively.

As the pattern repeats itself after 2 steps, 3 odd number leaves remainder 3 and 3even number leaves remainder 9.

∴ R em ainder of 3¹³⁶ when divided by 12 = 9

Hence, option 3.

From the given conditions;

We can see that P_m increases exponentially as m increases.

Option 3 is the only possible answer.

Hence, option 3.

Let g(x) = a and g(x - 1 ) = b

Let x = x + 1

Similarly, g(x + 3) = l/a, g(x + 4) = bla, g(x + 5) = b, g(x + 6) = a .. .and so on.
∴ g(x + 6) = g(x)

Answer: 6 optin 3

log₂(x² - y²) - log₂(x + y) = 1

x - y = 2 ....(1)
log₃x + log₃y = 1
log₃(x . y) = 1
xy = 3¹ = 3
Put the value of x from equation (1) in (2):
(2 + y)y = 3
y² + 2y - 3 = 0
y = 1, -3
y = -3 rejected
So, the value of y is 1.

The correct option is Option B.

Let the capacity of Container 1 be ‘a’ litres and that of Container 2 be ‘6’ litres.

∴ Petrol in container 1 = 0.3a 

Petrol in container 2 = 0.456

Similarly, petrol in the new container = 0.35(a + b)

∴ 035 a + 0.45 b = 0.35(a + b)

∴ a = 2b

Since a = 8, b = 4.

Hence, the capacity of container 2 is 8 litres.

Hence, the answer is 4

As shown in the figure the square PQRS cuts the axes at A(2,0), B(0,2), C (-2,0) and D(0, - 2).
When rotated by 60° the axes will cut the square at A', B', C' and D' respectively.

PD’ = AP - AD'

Hence, option 3.

Let free cab service is for a fixed distance for z km.
Let per kilometer charge = Rs. y
Let the distance covered by Biswas and Amit is 'a' km and 'b' km, respectively.
So,
(a + b - 2z )y = 110 ...(1)
(2a - z)y = 200 …(2)
(2b - z )y = 100 …(3)
(3) + (2) - (1)
We get
(a + b) y = 190
Also, if we add (1) and (3)
(a + b - z ) y = 150 ..(4)
Simplifying (1) and (4), we get
19z = 4a + 4b.. (5)
Also,
2a - 4b = - z …(6)
From (1) and (4)zy = 40
Also,
a = 3z
Biswas initially paid (3a - z)y, if he travelled thrice the distance travelled initially by him which is 8zy = Rs. 320

• Start of 2009:

  • Aging: The original population (P) ages by 1 year. So, their new average age becomes 31 years.
  • Adding Migrants: 0.2P migrants arrive, each with an average age of 25 years.
  • Total Population: P (original) + 0.2P (migrants) = 1.2P
  • Total Age:
    • Original population: P * 31 = 31P
    • Migrants: 0.2P * 25 = 5P
    • Total: 31P + 5P = 36P
  • New Average Age: Total Age divided by Total Population = 36P / 1.2P = 30 years

• Start of 2010:

  • Aging: The population is now 1.2P with an average age of 30 years. Aging by 1 year increases the average age to 31 years.
  • Adding Migrants: Another 0.2P migrants arrive, each with an average age of 25 years.
  • Total Population: 1.2P + 0.2P = 1.4P
  • Total Age:
    • Existing population: 1.2P * 31 = 37.2P
    • Migrants: 0.2P * 25 = 5P
    • Total: 37.2P + 5P = 42.2P
  • New Average Age: 42.2P / 1.4P ≈ 30.1429 years

• Start of 2011:

  • Aging: The population is now 1.4P with an average age of approximately 30.1429 years. Aging by 1 year increases the average age to about 31.1429 years.
  • Adding Migrants: Another 0.2P migrants arrive, each with an average age of 25 years.
  • Total Population: 1.4P + 0.2P = 1.6P
  • Total Age:
    • Existing population: 1.4P * 31.1429 ≈ 43.6P
    • Migrants: 0.2P * 25 = 5P
    • Total: 43.6P + 5P = 48.6P
  • New Average Age: 48.6P / 1.6P = 30.375 years

The correct option is Option D.

The number of chocolates received by Siya is multiple of 3. 

Among the given options, only 15 is multiple of 3.

So, Siya gets 15 chocolates.

Let x = 200,000,000
Then, the factors become (x – 3)(x + 3) = x² – 9.
So, the product is 40,000,000,000,000,000 – 9 = 39,999,999,999,999,991, whose digit sum is 3 + 15(9) + 1 = 139.

Since each user receive files in multiple of three, let us assume that they receive 3a, 3b, 3c, 3d and 3e number of files, respectively, where a, b, c, d and e are natural numbers.
Therefore, 3a + 3b + 3c + 3d + 3e = 36
a + b + c + d + e = 12
So, we have to find in how many ways the sum of five numbers can be 12.
Therefore, number of ways in which the sum of five numbers is equal to 12 = ^{(12 - 1)}C_{(5 - 1)} = 330

Let the unit of work done by 1 man in 1 hour and 1 day be 1 MDH unit (Man, Day, Hour).
According to question,
Work done by N workers working 8 hours per day in 20 days = N × 8 × 20 MDH units.
Since this work is equal to 70% of the total work,
i.e. 70% of the total work = 160N MDH units
Therefore, total work = 
So, remaining 30% of the total work = 
This work is completed by (N - 15) workers working 12 hours in 10 days = (N - 15) (10)(12) = 120(N - 15)

So, 

 

 

So, N = 35

Given: Speed of bus B = 75% of speed of A = 3/4 of speed of A
i.e. Ratio of speeds of A and B = 4 : 3
Let their speeds be 4V and 3V.

Also, the distance between S₁ and S₃ is 60% of that between S₁ and S₂.
i.e. S₁S₃ = 60/100 of S₁S₂ = 3/5 of S₁S₂
So, S₁S₃ = 3d, S₂S₃ = 2d and S₁S₂ = 5d

Now, time taken by bus A to reach S₃ = 3d/4v hours ---(1)
Time taken by bus B to reach S₃ = 2d/3v hours ---(2)

We need to find the time taken by A to travel from S₁ to S₂, i.e. 5d/4v.

Now, bus B started one hour later than bus A.

i.e. The time taken by bus A to reach S₃ is one hour more than the time taken by bus B to reach S_3.
i.e. 

Let d/v = k


=> k = 12

So,  hours