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CAT Practice Test - 33 - CAT MCQ


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30 Questions MCQ Test CAT Mock Test Series and 500+ Practice Tests 2024 - CAT Practice Test - 33

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CAT Practice Test - 33 - Question 1

The sum of the present ages of Varun and Kapil is 42 yrs. The ratio of their ages after 5 yrs will be 15:11. What is the present age of Kapil?

Detailed Solution for CAT Practice Test - 33 - Question 1

Let Varun's age = V
Kapil's age = K
V + K = 42
⇒ V = 42 - K


⇒ 11(V + 5) = 15(K + 5)
⇒ 11(42 - K + 5) = 15K + 75
⇒ 462 - 11K + 55 = 15K + 75
⇒ - 11K - 15K = 75 - 517
⇒ - 26K = - 442
⇒ K = 17

CAT Practice Test - 33 - Question 2

If the product of n positive real numbers is unity, then their sum is necessarily

Detailed Solution for CAT Practice Test - 33 - Question 2

The numbers must be reciprocals of each other.

Hence the sum is greater than the product of numbers.

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CAT Practice Test - 33 - Question 3

Let x,y and z be distinct integers x and y are odd positive, and z is even and positive. Which one of the following statements cannot be true?

Detailed Solution for CAT Practice Test - 33 - Question 3

 (x - z)2 y is even cannot be true.
x is odd and z is even. Therefore, x - z is odd.
And y is odd.
Therefore, (x - z)2 will be odd and (x - z)2 y will be odd.

CAT Practice Test - 33 - Question 4

In the following questions two equations numbered I and II are given. You have to solve both the equations and
I. x2 - x - 12 = 0
II. y2 + 5y + 6 = 0

Detailed Solution for CAT Practice Test - 33 - Question 4

I. x2 - x - 12 = 0
⇒ x2 - 4x + 3x - 12 = 0
⇒ x(x - 4) + 3(x - 4) = 0
⇒ (x - 4) (x + 3) = 0
∴ x = 4 or -3
II. y2 + 5y + 6 = 0
⇒ y2 + 3y + 2y + 6 = 0
⇒ y(y + 3) + 2(y + 3) = 0
⇒ (y + 3) (y + 2) = 0
∴ y = -3 or -2
x ≥ y

CAT Practice Test - 33 - Question 5

I. x2 - 8x + 15 = 0
II. y2 - 3y + 2 = 0

Detailed Solution for CAT Practice Test - 33 - Question 5

I. x2 - 8x + 15 = 0
⇒ x2 - 5x - 3x + 15 = 0
⇒ x(x - 5) - 3(x - 5) = 0
⇒ (x - 3) (x - 5) = 0
∴ x = 3 or 5
II. y2 - 3y + 2 = 0
⇒ y2 - 2y - y + 2 = 0
⇒ y(y - 2) - 1(y - 2) = 0
⇒ (y - 1) (y - 2) = 0
∴ y = 1 or 2
x > y

CAT Practice Test - 33 - Question 6

Which of the following statements is/are true ?
1. A quadrilateral is inscribed in a circle of area 154 cm². If the diagonals of the quadrilateral are 14 cm and 10 cm, then its area is 70 cm² .
2. A parallelogram is inscribed in a circle of area 38.5 cm² . If a diagonal of the parallelogram is 7 cm and the sum of its adjacent sides is 7√2 cm . then the area of the parallelogram is 24.5 cm². 
3. An isosceles trapezium is inscribed in a circle of area 346.5 cm². If the median of the trapezium is 21 cm and the parallel sides are 10 cm apart , the area of the trapezium is 210 cm².

CAT Practice Test - 33 - Question 7

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is

Detailed Solution for CAT Practice Test - 33 - Question 7

A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is

CAT Practice Test - 33 - Question 8

The average speed of Howarh express inclusive of various halts at stations is 60kmph. However, if we do not take the time of halts into consideration the speed of Howrah express works out to be 72kmph. What is the average time per hour of journey?

Detailed Solution for CAT Practice Test - 33 - Question 8

Average time per hour = (72-60) / 72 
= 12/72 
= 1/6

CAT Practice Test - 33 - Question 9

The function f (x) = | x - 2 | + | 2.5 - x| + |3.6 -x|, where x is a real number, attains a minimum at:

CAT Practice Test - 33 - Question 10

Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?

Detailed Solution for CAT Practice Test - 33 - Question 10

Let the sides of the regular hexagon ABCDEF be s.
The area of ABCDEF = area of 6 equilateral triangles of side, s = 6[s*s*sin 60]/2 = 2.598076211 s2 sq units.
Applying cosine for the triangle ABC,
AC2 = AB2 + BC2 – 2*AB*BC*cos120o
= s2 + s2 – 2s2 * (-0.5) = s2 + s2 + s2 
= 3s2
or AC = s√3
The area of ACE, which is an equilateral triangle of side, s√3
= (s√3*s√3*sin 60)/2 = 1.299038106s2 sq units which is half the area of the hexagon.
So the ratio of the area of triangle ACE to hexagon ABCDEF is 1 : 2.
 

CAT Practice Test - 33 - Question 11

Based on the figure below, what is the value of x, if y = 10?

Detailed Solution for CAT Practice Test - 33 - Question 11

Suppose x = 11
x+4 = 15, x-3=8
Hypotenuse = (225+64)1/2
= 17
⇒ 17 - 11 = 6
(6)2 + (x-3)2 = y2
x-3=8
Hence y = 10

CAT Practice Test - 33 - Question 12

Given that −1 ≤ v ≤ 1,−2 ≤ u ≤ −0.5 and −2 ≤ z ≤ −0.5 and  , then which of the following is necessarily true?

Detailed Solution for CAT Practice Test - 33 - Question 12

Substitute the extreme values in the equation :
v = 1, u = -0.5, z = -2
Then w = vz/u 
= 1*(-2)/-0.5
= 4
Option (B) satisfies.

CAT Practice Test - 33 - Question 13

Ravi invested a certain amount in a 15 years fixed deposit scheme. An interest of Rs. 1,250 was accrued for the 5th year whereas Rs. 2,450 was accrued for the 11th year. If the interest is compounded annually , what is the maturity value of Rs. 1, 00,000 invested today, at the end of the 15 years of the fixed deposit scheme?

Detailed Solution for CAT Practice Test - 33 - Question 13

 P[1+(r/100)5] - P = A
P[1+(r/100)11] - P = B
B/A = (r/100)6
2450/1250 = (r/100)6
49/25 = (r/100)6
7/5 = (r/100)3
now P=100000
Amount after 15 years be M
M = 100000[1+(r/100)15]
M = 100000[ 1+ {(r/100)3}5]
M = 100000[1+ (7/5)5]
M = 537824

CAT Practice Test - 33 - Question 14

The curved surface of the cone inscribed in a given sphere is maximum if h=

Detailed Solution for CAT Practice Test - 33 - Question 14

The curved surface of the cone inscribed in a given sphere is maximum 
if h= maximum, if h is height of cone
because the curved surface area of cone is equal to πrh
when h is maximum , surface area is maximum even when it is inscribed in a given sphere.

CAT Practice Test - 33 - Question 15

A solution of sugar syrup has 15% sugar. Another solution has 5% sugar.How many litres of the second solution must be added to 20 litres of the first solution to make a solution of 10% sugar?

Detailed Solution for CAT Practice Test - 33 - Question 15

Hence both the types should be added in the ratio of 1 : 1 to obtain the required strength.
Hence 20 litres of first type should be added to the 20 litres of the second type to get the desired solution

CAT Practice Test - 33 - Question 16

Let n! = 1 × 2 × 3 ×...× n for integer n ≥ 1. If p = 1! + (2 × 2!) + (3 × 3!) + ....+ (10 × 10!), then p + 2 when divided by 11! leaves a remainder of

Detailed Solution for CAT Practice Test - 33 - Question 16

if p = 1! then p + 2 = 3 when divided by 2! gives a remainder 1
if p = 1! + (2 × 2!) = 5 then p + 2 = 7 when divided by 3! gives a remainder 1
The result is the same for p = 1! + (2 × 2!) + (3 × 3!) + .....
So the answer for the given sum should be1

CAT Practice Test - 33 - Question 17

A, B and C are defined as follows:
A = (2.000004) ÷ [(2.000004)² + (4.000008)] 
B = (3.000003) ÷ [(3.000003)² + (9.000009)] 
C = (4.000002) ÷ [(4.000002)² + (8.000004)] 
Which of the following is true about the values of the above three expressions?

Detailed Solution for CAT Practice Test - 33 - Question 17

Given expressions can be reduced as follows
A = 1/4.000004
B = 1/6.000003
C = 1/6.000002
Among all of them B is smallest.

CAT Practice Test - 33 - Question 18

For the product n(n+1)(2n+1),n∈N which one of the following is not necessarily true?

Detailed Solution for CAT Practice Test - 33 - Question 18

sum of 1st n squares but without the "6" in the denominator.
And about 237  
The above expression will obviously be divisible by 237
When n= 237 as it will be 237*238*475 In fact even for n = 236 and n = 118 it will be divisible by 237.
 

CAT Practice Test - 33 - Question 19

Which of the following is the greatest?

Detailed Solution for CAT Practice Test - 33 - Question 19

40% of 30 = 0.4*30 = 12
b) 3/5 of 25 = 1.5*25 = 15
c) 6.5% of 200 = 0.065*200
⇒ 6.5*2 = 13
e) (1/2)-4 = (2)4 = 16

CAT Practice Test - 33 - Question 20

The period of the function tan(3x+5)is

Detailed Solution for CAT Practice Test - 33 - Question 20

Let, f(x)=tan(3x+5)
By property of Periodic function.
period of f(x)= period of f(x+λ),λϵR
∴ period of tan(3x)= period of tan(3x+λ),λϵR
∴ period of tan(3x+5)= π/3
[Period of tanx=π, ∴ period of tanλx = π/∣λ∣ and period of tan(x)=period of tan(x+a)]

CAT Practice Test - 33 - Question 21

The letters of the word 'RANDOM' are written in all possible orders and these words are written out as in a dictionary .Find the rank of the word 'RANDOM'.

Detailed Solution for CAT Practice Test - 33 - Question 21

Arranging RANDOM in dictionary order gives us
ADMNOR.
Consider the words starting with A. If the starting letter is fixed as A, then the remaining 5 letters can be rearranged in 5! ways.
Similarly with the words starting with letters D,M,N,O.
Hence 5×5!
=5(120)
=600 ...(i)
Now consider the words starting with R,
R−−−−−
In Dictionary order the first word starting with R will be
RADMNO
Keeping the positions of RAD fixed we can rearrange the letters MNO in 3! ways.
Similarly for RAMDNO.
Hence
600+2(3!)
= 612.
The successive word will be
RANDMO and RANDOM.
Hence the rank of the word RANDOM is 612+2 = 614
 

CAT Practice Test - 33 - Question 22

5 coins are tossed. If two of them show heads , then the probability that all 5 coins show head is

Detailed Solution for CAT Practice Test - 33 - Question 22

Total outcome:25 = 32
Outcome of at least two of them show heads:
Total-(1 head)-(0 head)= 32 - 5 - 1 = 26
Outcome of 5 heads = 1

Probability = 1/26

CAT Practice Test - 33 - Question 23

Mohan bought a cycle for Rs. 475 and then sold it at a loss of 8% of the cost price. For how much did he sell the cycle?

CAT Practice Test - 33 - Question 24

What will be the quadratic equation in x when the roots have their airthmetic mean as A and geometric mean G ?

Detailed Solution for CAT Practice Test - 33 - Question 24

let the roots be a and b 
now
a + b = 2A
ab = G2
quadratic equation is given by
x2 - (sum of roots)x + product of roots = 0
x2 - 2Ax + G2 = 0

CAT Practice Test - 33 - Question 25

'A' can run a km in 4 minutes 54 secs and 'B'can run this distance in 5 minutes. How many mts ahead of 'A' should 'B' stand at the start of the km Race so that both of them reach the finish point together?

Detailed Solution for CAT Practice Test - 33 - Question 25

Distance covered by B in 6 sec = (1000/300 * 6)
= 20 m
Therefore, A beats B by 20 m.
For a dead-heat race, A must give B a start of 20 m.

CAT Practice Test - 33 - Question 26

The number of employees in Obelix Menhir Co. is a prime number and is less than 300. The ratio of the number of employees who are graduates and above, to that of employees who are not, can possibly be

Detailed Solution for CAT Practice Test - 33 - Question 26

The addition of numerator and denominator should gives a prime number.
3 is a factor of 189 and 183 => A and D eliminated
17 is a factor of 187 and 221 => B and C eliminated
181 is a prime.

CAT Practice Test - 33 - Question 27

Find the sum of all the integers which are multiples of 7 and lie between 200 and 400.

Detailed Solution for CAT Practice Test - 33 - Question 27

The first term after 200 divisible by 7 is 203. The last term before 400 divisible by 7 is 399. The number of terms = ?
=> l = a + ( n - 1 ) d
=> 399 = 203 + ( n - 1 ) 7
=> 399 - 203 = ( n - 1 ) 7
=. 196 = ( n - 1 ) 7
=> 196 / 7 = ( n - 1 )
=> 28 = ( n - 1 )
=> n = 28 + 1 = 29
Hence the number of terms is 29.
Applying Sum formula we get,
Sn = n/2[a+l]
S29 = 29/2[203 + 399]
= 29*301
S29 = 8729

CAT Practice Test - 33 - Question 28

A condition for a function y = f (x) to have an inverse is that it should be

CAT Practice Test - 33 - Question 29

Ram and Shyam attempted to solve a quadratic equation. Ram made a mistake in writing down the constant term. He ended up with the roots (4, 3). Shyam made a mistake in writing down the coefficient of x. He got the root as (3, 2). What will be the exact roots of the original quadratic equation?

Detailed Solution for CAT Practice Test - 33 - Question 29

ax2 + bx + c
Sum of roots : -b/a => 7
Product = c/a => 6
x2 - (sum of roots)x + Product
x2 - 7x + 6 = 0
x2 - 6x - x + 6 = 0
x(x-6) -1(x-6) = 0
(x-1)(x-6) = 0
x = 1,6
 

CAT Practice Test - 33 - Question 30

If 1 cm on a map corresponds to an actual distance of 40 kms. And the distance on the map between Bombay and Calcutta is 37.5 cms., the actual distance between them is:

Detailed Solution for CAT Practice Test - 33 - Question 30

1cm = 40 Kms
37.5cm = 37.5×40kms
actual distance between Bombay and Calcutta is 1500kms
 

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