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CAT Practice Test - 31 - CAT MCQ


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30 Questions MCQ Test - CAT Practice Test - 31

CAT Practice Test - 31 for CAT 2024 is part of CAT preparation. The CAT Practice Test - 31 questions and answers have been prepared according to the CAT exam syllabus.The CAT Practice Test - 31 MCQs are made for CAT 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for CAT Practice Test - 31 below.
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CAT Practice Test - 31 - Question 1

The sum of the present ages of the father and the son is 60 yrs. Six yrs ago, the father was 5 times as old as his son. The age of the son after 6 yrs will be

Detailed Solution for CAT Practice Test - 31 - Question 1

Let the age of son be x and the age of father be y
According to question,
x + y = 60 .....(1)
and
y - 6 = 5(x - 6)
⇒ y - 6 = 5x - 30
⇒ 5x - y = 24 .....(2)
Solving equation (1) and (2)
6x = 84
⇒ x = 14 years
Age of son after 6 yesrs
= (14 + 6) years = 20 years

CAT Practice Test - 31 - Question 2

The digits of a three-digit number A are written in the reverse order to form another three digit number B. If B > A and B - A is perfectly divisible by 7, then which of the following is necessarily true ?

Detailed Solution for CAT Practice Test - 31 - Question 2

Let A = 100x + 10y + z
⇒ B = 100z + 10y + x
B - A = 99 (z - x)
For B - A to be divided by 7, z - x has to be divisible by 7. Only possibility is z = 9, x = 2
∴ Biggest number A can be 299

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CAT Practice Test - 31 - Question 3

A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs. 250 and Rs. 300 per day respectively. In addition, a male opeartor gets Rs. 15 per call he answers and a female operator gets Rs. 10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ assuming he has to employ more than 7 of the 12 female operators avilable for the job ?

Detailed Solution for CAT Practice Test - 31 - Question 3

let x be no. of male and y be no. of female operators.
We have 40x + 50y = 1000
x = 25 - (5*y/4)
=> y = 7,12
So y can be 8 or 12
If y = 8 then x = 15
If y = 12 then x = 10
Cost is minimum in 2nd case when no. of males are 10.

CAT Practice Test - 31 - Question 4

In the following questions two equations numbered I and II are given. You have to solve both the equations and give answer.

I. x2 + 28 = 11 x
II. 2y2 + 18 = 12y

Detailed Solution for CAT Practice Test - 31 - Question 4

I. x2 - 11x + 28 = 0
⇒ x2 - 7x - 4x + 28 = 0
⇒ x(x - 7) - 4(x - 7) = 0
⇒ x = 7, 4
Also,
II. 2y2 + 12y + 18 = 0
⇒ 2y2 + 6y + 6y + 18 = 0
⇒ 2y(y + 3) + 6(y + 3) = 0
⇒ y = -3, -3
Hence, x > y

CAT Practice Test - 31 - Question 5

I. x2 + 5x = 7x + 8
II. y2 + 7y = 12y + 6

Detailed Solution for CAT Practice Test - 31 - Question 5

I. x2 - 2x - 8 = 0
⇒ x2 - 4x + 2x - 8 = 0
⇒ x(x - 4) + 2(x - 4) = 0
⇒ x = 4, -2
Also,
II. y2 - 5y - 6 = 0
⇒ y2 - 6y + y - 6 = 0
⇒ y(y - 6) + 1(y - 6) = 0
⇒ x = 6, - 1
Hence, x < y

CAT Practice Test - 31 - Question 6

Three circles A, B and C have a common center O. A is the inner circle, B middle circle and C is outer circle. The radius of the outer circle C, OP cuts the inner circle at X and middle circle at Y such that OX = XY = YP. The ratio of the area of the region between the inner and middle circles to the area of the region between the middle and outer circle is

CAT Practice Test - 31 - Question 7

A plot of land is in the shape of a trapezium whose dimensions are given in the figure below :

Hence the perimeter of the field is

Detailed Solution for CAT Practice Test - 31 - Question 7


The perimeter of the field =(13m+9m+20m+30m) = 72m

CAT Practice Test - 31 - Question 8

Consider a sequence of seven consecutive integers. The average of the first five integers is n. The average of all the seven integers is :

Detailed Solution for CAT Practice Test - 31 - Question 8

Let the seven consecutive integers be x−3,x−2,x−1,x,x+1,x+2,x+3
Given that average of first five integers is n
Therefore, n = [(x−3)+(x−2)+(x−1)+x+(x+1)]/5
⟹5x−5 = 5n
⟹x−1 = n
⟹x = n+1 ------(1)
Average of seven integers is  [(x−3)+(x−2)+(x−1)+x+(x+1)+(x+2)+(x+3)]/7
⇒ 7x/7 = x = n+1 (from (1))

CAT Practice Test - 31 - Question 9

If f(x) = log x,then

CAT Practice Test - 31 - Question 10

In the figure given below, ABOP is a rectangle and O is the centre of the circle. It is also given that AB = BC and the measure of the angle ∠ABC is 60. Find the measure of the angle ∠OPN.

CAT Practice Test - 31 - Question 11

In ΔABC, ∠B is a right angle, AC = 6 cm, and D is the mid-point of AC. The length of BD is :

Detailed Solution for CAT Practice Test - 31 - Question 11

In a right angled triangle, the length of the median to the hypotenuse is half the length of the hypotenuse.

CAT Practice Test - 31 - Question 12

Consider the following two curves in the x-y plane
y = x3 + x2 + 5
y = x2 + x + 5
Which of the following statements is true for −2 ≤ x ≤ 2?

Detailed Solution for CAT Practice Test - 31 - Question 12

y=x³ + x² + 5
y = x² + x + 5
We equate each other.
x³ + x² + 5 = x² + x + 5
Subtract x² both the side
x³ + 5 =  x + 5
Subtract 5 both the side
x³  =  x
x³ - x = 0
x(x² - 1) = 0
x(x + 1)(x - 1) = 0
We get
x = 0    &   x - 1 = 0   & x + 1 = 0
x = 0, x = 1 & x = -1
We get the three values of x in interval [-2,2]
So, both the curves cuts each other exactly three times.
 

CAT Practice Test - 31 - Question 13

A father left a will of Rs.35 lakhs between his two daughters aged 8.5 and 16 such that they may get equal amounts when each of them reach the age of 21 years. The original amount of Rs.35 lakhs has been instructed to be invested at 10% p.a. simple interest. How much did the elder daughter get at the time of the will?

Detailed Solution for CAT Practice Test - 31 - Question 13

Let Rs x be the amount that the elder daughter got at the time of the will. Therefore, the younger daughter got (3500000 - x)
The elder daughter’s money earns interest for (21 - 16) = 5 years @ 10% p.a simple interest
The younger daughter’s money earns interest for (21 - 8.5) = 12.5 years @ 10% p.a simple interest
As the sum of money that each of the daughters get when they are 21 is the same,


CAT Practice Test - 31 - Question 14

If   then minimum value of P/Q2

Detailed Solution for CAT Practice Test - 31 - Question 14


Clearly, the minimum does not exist.

CAT Practice Test - 31 - Question 15

There are 3 containers : A, B and C which contain water, milk and acid respectively in equal quantities. 10% of the content of A is taken out and poured into B. Then the same amount from B is transferred to C, from which again the same amount is transferred to A. What is the proportion of milk in container A at the end of the process?

Detailed Solution for CAT Practice Test - 31 - Question 15

Suppose A contains 100 litre water,
B contains 100 litre milk,
C contains 100 litre acid.
10% from A (i.e., 10 litre) is poured to B. Then,
Quantity of milk in B 
= 100 litre
Total quantity of the mixture in B 
= 110 litre.
Concentration of milk in B = 100/110
= 10/11
10 litre from B is poured to C. Then,
Quantity of milk in C 
= (10×10/11)= 100/11 litre.
Total quantity of the mixture in C = 110 litre.
Concentration of milk in C 
= (100/11)/110
= 10/121
10 litre from C is poured to A. Then,
Quantity of milk in A 
= 10×(10/121) = 100/121 litre.
Total quantity of the mixture in A = 100 litre.
Concentration of milk in A 
= (100/121)/100
= 1/121

CAT Practice Test - 31 - Question 16

Let a, b, c be distinct digits. Consider a two-digit number 'ab' and a three-digit number 'ccb', both defined under the usual decimal number system, if ab2 = ccb > 300, then the value of b is

Detailed Solution for CAT Practice Test - 31 - Question 16

I am assuming that the question is (ab)*(ab) = ccb.
Ok so ab as well as (ab)*(ab) also has its unit's digit as b as (ab)*(ab) = ccb. This can only be possible when b is one of these: 1, 5, 6, 0.
25 is the only 2 digit number whose square 625 is 3 digit and is greater than 300. 15,35,45… don't satisfy the conditions.
Similarly 26*26=676 is the only one which satisfies the above conditions.
Similarly 20*20= 400 and 30*30= 900 are the only squares satisfying the above conditions.
Similarly 21*21= 441 is the only square satisfying the conditions.
Out of 625,676,400,900 and 441, only 441 is of the form ccb as the first digit and the second digit are same.
Hence ccb= 441 and b=1.
 

CAT Practice Test - 31 - Question 17

A positive whole number M less than 100 is represented in base 2 notation, base 3 notation, and base 5 notation. It is found that in all three cases the last digit is 1, while in exactly two out of the three cases the leading digit is 1. Then M equals

Detailed Solution for CAT Practice Test - 31 - Question 17

Since in all three cases the last digit is 1, the number should give remainder 1 when divided individually by 2,3,5
So the no. may be 31 or 91 
Now 31 in base 2,3 and 5 give first digit as 1 in all the 3 cases while 91 gives exactly two out of the three cases the leading digit as 1

CAT Practice Test - 31 - Question 18

If a1 = 1 and an+1−3an+2 = 4n for every positive integer n, then a100 equals

Detailed Solution for CAT Practice Test - 31 - Question 18

a1 = 1.
an+1 =  3an - 2 + 4n
a2 = 3 – 2 + 4 = 5
a3 = 15 – 2 + 8 = 21
a4 = 63 – 2 + 12 = 73.
Now, our options involve 3n followed by some addition or subtraction.
So a2 = 5, can be written as 31 + 2 and 32 - 4.
a3 = 21 can be written as 32 + 12 and 33 – 6.
a4 = 63 can be written as 33 + 36 and 34 – 8.
We see that the subtraction trend is repeating, as 3n – 2*n. So, a100 is 3100 – 200.

CAT Practice Test - 31 - Question 19

A student took five papers in an examination, where the full marks were the same for each paper. His marks in these papers were in the proportion of 6:7:8:9:10. In all papers together, the candidate obtained 60% of the total marks then, the number of papers in which he got more than 50% marks is :

Detailed Solution for CAT Practice Test - 31 - Question 19

Let the marks obtained in five subjects be 6x, 7x, 8x, 9x and 10x. 
Total marks obtained= 40x
Max. Marks of the five subjects = 40x/0.6 [40x is 60% of total marks] 
Max. Marks in each subject = 40x/0.6*5 = 13.33x
Hence, % of each subject = 6x*100/13.33 = 45.01%
Or, 7x*100/13.33 = 52.51
In same way other percentage are 60.01%, 67.52%, 75.01%.
Hence, number of subjects in which he gets more than 50% marks = 4.

CAT Practice Test - 31 - Question 20

The period of sin3x + cos 3x is

CAT Practice Test - 31 - Question 21

Number of triangles formed in a decagon by joining its vertices is

Detailed Solution for CAT Practice Test - 31 - Question 21

 In a decagon, by joining one vertex to the remaining vertices you can have 8 triangles. If you are considering all the vertices independently you will have a total of 8*10 = 80 triangles.

CAT Practice Test - 31 - Question 22

If S(n) is the set of all factors of n , then what is the probability that a randomly chosen element of S(1050) is a multiple of 5 ?

CAT Practice Test - 31 - Question 23

The cost of 13 kgs of sugar is Rs 195. The cost of 17 kg of rice is Rs 544 and the cost of 21 kgs of wheat is Rs 336. What is the total cost of 21 kgs of sugar, 26 kgs of rice and 19 kgs of wheat?

Detailed Solution for CAT Practice Test - 31 - Question 23

Let the number be x,
13 kg sugar costs 195. So 1 kg costs 15.
17 kg rice costs 544, so 1 kg costs 32.
21 kg wheat costs 336, so 1 kg costs 16.
Hence 21 kg sugar + 26 kg rice + 19 kg wheat = (21 × 15) + (26 × 32) + (19 × 16) = 315 + 832 + 304 = Rs. 1451/-.

CAT Practice Test - 31 - Question 24

if α, β are the roots of the equation ax² + bx + c = 0, then the value of α³ + β³ is

Detailed Solution for CAT Practice Test - 31 - Question 24

we know (a+b)³=a³+3a²b+3ab²+b³
which can be written as (a+b)³=a³+b³+3ab(a+b).
=》a³+b³ = (a+b)³- 3ab(a+b).
similarly
α³+β³= (α+β)³ - 3αβ(α+β).
here The first term is,  ax^2  its coefficient is  a .
The middle term is,  bx  its coefficient is  b.
The last term, "the constant", is  c .
we know in a quadratic equation α+β = -b/a.
and αβ = c/a.
so, substitute these values in (α+β)³ - 3αβ(α+β).
=》(-b/a)³ - 3(c/a)(-b/a).
=》-b³/a³+3bc/a^2.
by taking the LCM OF BOTH,WE GET.
(-b³+3abc)/a³.
therefore α³+β³ = (-b³+3abc)/a³.
 

CAT Practice Test - 31 - Question 25

A circular running path is 726 mts in circumference. Two men start from the same point and walk in opposite directions at 3.75 km/hr and 4.5 km/hr respectively. When will they meet for the first time?

Detailed Solution for CAT Practice Test - 31 - Question 25

Let both of them meet after T min.
4500 m are covered by Suresh in 60m
In T min = 4500T/60
In T min Suresh's wife will cover = 3750T/60
Given, 4500T/60 + 3750T/60 = 726
=> 8250T = 726
T = 5.28 min

CAT Practice Test - 31 - Question 26

A group of workers can do a piece of work in 24 days. However as 7 of them were absent it took 30 days to complete the work. How many people actually worked on the job to complete it?

Detailed Solution for CAT Practice Test - 31 - Question 26

Let the original number of workers in the group be 'x'
Therefore, actual number of workers = x - 7.
We know that the number of manhours required to do the job is the same in both the cases.
Therefore, x (24) = (x - 7).30 
24x = 30x - 210
6x = 210
x = 35.
Therfore, the actual number of workers who worked to complete the job = x - 7 = 35 - 7 = 28.

CAT Practice Test - 31 - Question 27

Let S denote the infinite sum 2 + 5x + 9x2 + 14x3 + 20x4 + ...., where |x| < 1 and the coefficient of xn - 1 is 

Detailed Solution for CAT Practice Test - 31 - Question 27

Let us assume 

By Taking x as common factor.
So, S = [2(1-x) + x)]  

CAT Practice Test - 31 - Question 28

Let function f : R → R be defined by f (x) = 2x + sin x for x ∈ R. Then F is

Detailed Solution for CAT Practice Test - 31 - Question 28

f(x) = 2x + sin x
Differentiating wrt x on both sides, we get
f'(x) = 2 + cos x
−1 ≤ cos x ≤ 1
−1 + 2 ≤ 2 + cos x ≤ 1 + 2
1 ≤ 2 + cos x ≤ 3
So, f'(x) > 0 , for all real values of x 
So, f(x) is one − one
f(x) → ∞ as x→∞ and f(x) → −∞ as x→−∞
So, f(x) is onto.
 

CAT Practice Test - 31 - Question 29

f(x)= 2x3 + px2 + qx - 4 and f(2)= 0 , find the value of p + q , where p and q are non zero. If f(x)= 0 has three real roots, all of them being integers and further two of three roots are equal.

CAT Practice Test - 31 - Question 30

Rakesh traveled from city A to city B covering as much distance in the second part as he did in the first part of this journey. His speed during the second part was twice as that of the speed during the first part of the journey. What is his average speed of journey during the entire travel?

Detailed Solution for CAT Practice Test - 31 - Question 30

The first part is 1/3rd of the total distance and the second part is 2/3rd of the total distance. He travels at s km/hr speed during the first half and 2s km/hr speed during the second half.
If 3 km is the total distance, then 1 km was travelled at s km/hr and 2 kms was travelled at 2s km/hr speed.
Hence, average speed = Total Distance/Total Time
= [3(1/s + 2/s)]
= 3/(4/2s) = 3s/2
This, however = s + 2s/2
= 3s/2
 which is the arithmetic mean of the speeds of the two parts.

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