CAT Quantitative Aptitude MCQ - 2 - CAT MCQ

Since two women are always together, 6 women can be arranged in 5! x 2! = 240 ways.
Now, 5 men can stand at 6 different gaps created between the women.
They can be arranged in ⁶P5 = 6! = 720 ways

So, total number of ways = 240 x 720 = 172800

Hence, option 2.

48= 24 *3

Highest power of 3 in 32! = 32/3+10/3+3/3 

=10+3+1

= 14

Highest power of 2 in 32! = 32/2+16/2+8/2+4/2+2/2

= 16+8+4+2+1

=31

Highest power of  24 in 32!= 7

Number of zeroes in a number can be found by calculating the highest power of 10.

10 = 2 x 5

Number of zeroes will be highest power of 5 in 40!.
Highest power of 5 in 40!
Answer: 9

Let us assume D is the distance. 
Upstream Speed = 4.8 km/h 
Downstream speed = 7.2 km/h 
According to the question, D/4.8 + D/7.2 = 10 
So, D = 28.8 km and 
Hence the total distance = 57.6 km 
Alternatively, the ratio of downstream speed: upstream speed = 3:2 
Ratio of the downstream time: upstream time = 2:3 The time taken in the downstream movement = 4 h and the time taken in the upstream movement = 6 h 
So, the distance covered = 4 x 7.2 = 6 x 4.8 = 28.8 km 
Hence the total distance = 57.6 km 
In the case of escalators, moving staircase works like an external agent as the river works for boats and streams. The speed of an escalator and the person will be added when the staircase is going up and the person walking up with it have the same direction of the movement. Now if the direction of the movement of an escalator and the person is opposite, then the resultant speed (or, the relative speed) will be equal to the speed of the person — to the speed of an escalator.

Let us define a function from a set P of 3 elements (a, b, c) and a set Q with 4 elements (w, x, y, z). Now, for a function to be defined from set P to set Q, every element of P must connect with exactly one element of Q while every element of Q can connect to multiple elements of P.
Thus, the three elements of set P each has four options from Q. Thus, the total number of relations in this case = 4³ = 64.
Now, the number of relations given is 4096. x^y = 4096

(x, y) = (4096, 1); (2, 12); (4, 6); (8, 4); (16, 3); (64, 2)

Hence, there are 6 such pairs possible.
Hence, option 4.

Let the income of X and Y be 6x and 5x respectively.
Let he expenses of X and Y be 9y and ly respectively.

6x- 9y = 3000 ... (1)

5x- 7y = 3000 ... (2)

By solving these two equations we get,

42x - 45x = -6000

x = 2000

Therefore, income of x = 2000 x 6 = 12000 And income of Y = 2000 x 5 = 10000

Sum of their incomes = 12000 + 10000 = 22000

Answer: 22000

If a + b + c = 0, a* + 63 + c3 = 3 abc

The sum of the cubes of three numbers = -120 x 3 = -360

Hence, option 3.

Let the number of employees who left be x.
The total funds allocated towards the salary after increment = (46 + x) x 20000 x 1.15

The new salary per employee = 20000 x 1.15 + 2000 = 25000

(46 + x) x 20000 x 1.15 = 25000 x 46 Solving for x, we get

x = 4

Hence, option 1.

Let the 1st term (t₁) of an A.R be a and common difference be d.

t₃ = a + 2d = 23 ... (i)

Similarly,

t₈ = a + 7d = 58 ... (ii)

Solving these equation we get,

a = 9 and d = 7

t₈= a + 79d = 9 + 79(7) = 562

Hence, option 2.

Consider the clock at 14 AM in the night.

The hour hand and minute hand are opposite to each other once before the time is 1 AM. They are again opposite to each other once before the time is 2 AM. And so on in every hour till 7 AM.

Between 6 AM and 8 AM, the hands are opposite to each other only once, at 7 AM. Again the hands are opposite to each other once every hour from 9 AM to 14 noon.

So, in a span of 14 hours, the number of times that the two hands are opposite to each other is 13.

So, in 28 hours, the number of times that the two hands are opposite to each other is 13*2 = 26

The equation can be written as, (1 + 2 + 3 + ...+ 10)log₂x = 440
(11 x 10/2)log₂x = 440
log₂X = 8
 x = 2⁸ = 256
Thus, x = 256 and √x = 16
∴ x + √x = 272
Hence, option 3.

= 45 km/hr.

Let the speed of the train be x km/hr. Then, relative speed = (x - 5) km/hr.

 x - 5 = 45          x = 50 km/hr.

Let one of the digits be x.
∴ The other digit will be x + 4
Now,

6x + 12 = 2x² + 8x
x² + x - 6 = 0
x = 2, -3
Since, -3 cannot be one of the digits the digits are 2 and 6.
Thus, 62 is the number with highest value which can be formed using the two digits.
Hence, option 2.

Let (a, b) is the required co-ordinates of the foot of the perpendicular.

4a + 2b - 9 = 0 ... (i)

Slope of the above line = -2

Slope of the line perpendicular to the above line =1/2 (b - 4)/(a - 3) = 1/2

a-2b = —5 ... (ii)

Solving the equations (i) and (ii) we get,

a = 4/5 and b = 29/10

Hence, option 2.

 y² + 2y - 3 = (y - l)(y + 3) < 0

-3 < y < 1

y can take the integral values -2, -1, 0

Roots o f the given equation can be - 2 , - 1 , 0

As roots are non-zero, therefore, the roots of the equation x² + bx + c = 0 are -2,-1.
x² + bx + c = (x + 2)(x + 1) = 0

x² + bx + c = x² + 3x + 2 = 0

b = 3 and c = 2

b + c = 5

Hence, option 2.

Original per kg cost of grocery = 3000/15 = Rs. 200

New rate = 1.2 x 200 = Rs. 240

Grocery purchased after increase in price = 3000/240 =12.5kg

Reduction in consumption = 15 - 12.5 = 2.5 kg

Answer: 2.5

Area of rhombus = (1/2) x 12 x 14 = 84 cm² Let the diagonal of the square be d

(1/2) x d² = 84

Diagonal of the square = d = 2√42 cm Hence, option 1.

As we do not know in which direction the other train is running, we cannot determine answer.
Answer: 0

Selling price after successive discount = 700 x 0.8 x 0.75 x 0.7 = Rs. 294

Hence, option 1.

AB touches smaller circle at C. So, OC ⊥ AB.
Now, in AOCB, OB = 20 cm and BC =10 cm. By pythagoras theorem we get, OC = 10√3 cm
Hence, option 1.

Let the number of officers and non-officers be n and m respectively.

Total salary of officers = 6800n And, Total salary of non-officers = 2000m

Total salary of all the employees = 6800n + 2000m = 3200(m + n)

3600n = 1200m

3n = m

m = 3 x 5 = 15

Number of non-officers =15

Answer: 15

We need to select exactly three points from the given 17 points to form a triangle.
However, no triangle can be formed if all the three points selected are collinear.
Number o f triangles formed = ¹⁷C₃ - ⁵C₃ = 670

Hence, option 4.

Let 5x be the lateral side of the triangle and 4x be the base of the triangle.
5x + 5x + 4x= 14
x =1
On solving the two equation we get y = 4 and x = 5.
We get the following triangle.

By Pythagoras theorem,
Height of the triangle = √5² - 2² = √21 cm
Hence, area of the triangle = 1/2 x 4 x √21 = 2√21 cm²
Hence, option 4.

On substituting the values from the options given above, we see that x = 3 is the only value that satisfies the given equation.
Hence, option 2.

2f[xy) = [f(x)]^y + [f(y)]^x for all real x, y Replacing ‘y’ by 1, we get,
∴ 2 x f(x) = [f(x)]+ [f(1)]^x 
∴ f(x) = 2^x 
∴ f(2) = 2² = 4,f(3) = 2³ = 8 and so on.
∴ f(1) +f(2) + ... + f(10) = 2¹ + 2² + 2³ + ... + 2¹⁰
The above series is in G.R with ratio = 2 and initial term = 2

Answer: 2046

343^x+2 - 49^x+3 + 7^x+4 = 7^{3x + 6} - 7^2x+6 + 7^x+4
∴ 7^x+4(7^{2x +2} - 7^x+2 + 1) = 0
Now, 7^x+4 ≠ 10
∴ 7^{2x + 2} - 7^x+2 + 1 = 0
Let 7^x+1 = y
∴ y²  - 7y + 1 = 0


Taking log7 on both sides,


Hence, option 3.

Consider the given diagram.
∠ OBA ≌ ∠O'BA'... (Vertically opposite angles)
∠ OAB ≌ Z.O'A'B ... (Radius is perpendicular to the tangent)
∴ ΔOAB ~ AO'A'B ... (AA test of similarity)

Given that OO ' = 39 cm and AA' = 15 cm

O'B = 26 cm and A'B = 10 cm

O'A' = (O'B² - A'B²)^1/2 = 24 cm

Area of the larger circle = 576π

Hence, option 2.

= 1
Hence, option 3.

The angle of elevation is 45°.
So, height of the tower will be same as the distance of the tower from Sushil.
The height of the tower is 25 m

Answer: 25

Let the total work be LCM of number of days taken by X and Y individually.

Total work = LCM (5, 7) = 35 unit

In 2 days, X & Y will complete (12/35)th of the total work.
In this manner they will complete (24/35)th work in 4 days.
After this X will work for 1 day and he completes 1/5th work.
After 5th day, work left = (11/35 - 1/5) = 4/35 On 6th day, Y will do the remaining 4/35th work in 4/5th of days time.
So total time taken is 5 + 4/5 days.
Hence, option 4.