[Download PDF] CMAT 2021 Slot-1 Paper with Solutions

Instructions

For the following questions answer them individually

Question 1

Two pipes A and B can fill a tank in 5 hours and 20 hours respectively. Both pipes together can fill the same tank in:

Video Solution

Solution

Let the volume be 20 units
So A fills at 4 units/hr
B fills at 1 unit/hr
Now together they fill 4+1 = 5 units/ hour
Now time taken to fill completely : 20/5 = 4 hrs

Question 2

In June-2020, the ratio of boys to girls in a college was 3 : 2. In September-2020, there were 80 fewer boys and 20 fewer girls in the college and the ratio of boys to girls was 7 : 5. What was the total number of students in the college in June-2020?

Video Solution

Solution

In June 2020
Let the number of boys be 3x
So number of girls will be 2x
Now number of boys in September 2020: 3x-80 and girls in September 2020 : 2x-20
Now as per given 3x-80:2x-20 =7:5
we get 15x-400 =14x-140
we get x=260
So number of students in June 2020 = 3x+2x =5x = 1300

Question 3

M is a 4-digit number. If the left most digit is removed, then the resulting three digit number is $$\frac{1}{9}^{th}$$ of M. How many such M's are possible?

Video Solution

Solution

According to question,

Number AXYZ/9 = XYZ

It can be written as

A000/9 + XYZ/9 = XYZ

A000/9 + XYZ/9 = 9XYZ/9

A000/9 = 8XYZ/9

A000 = 8XYZ

We have to find the maximum value of A so that when A000 is divided by 8 gives three digit numbers.

7000/8= 875 and 8000/8=1000.

So, answer is 7.

Question 4

L and M together can complete a piece of work in 72 days, M and N together can complete it in 120 days, and L and N together in 90 days. In what time can L alone complete the work?

Video Solution

Solution

Let L does a units of work/day
M does b units of work/ day
N does c units of work/ day
Now as per given conditions :
Let work be W
So we get
W = 72(a+b) =120(b+c) =90(a+c)
we get
6a+6b = 10b +10c
we get 6a-4b-10c=0 (1)
also
4a+4b =5a+5c
we get a-4b+5c =0 (2)
From (1) and (2)
we get a=3 , b=2 and c=1
Now therefore W = 360
Now time taken by L alone to complete the work = 360/3 = 120 days

Question 5

Given below are two statements
Statement I : The set of numbers(5,6, 7, p, 6, 7, 8, q) has an arithmetic mean of 6 and mode (most frequently occurring number) of 7. Then $$p\times q=16$$.
Statement II: Let p and q be two positive integers such that $$p+q+p\times q=94$$. Then $$p+q=20$$.
In light of the above statements, choose the correct answer from the options given below

Video Solution

Solution

Statement 1:

It is given that the mode of the set is 7. Thus, either p or q(or both) will be 7.

Now, p*q will be a multiple of 7. But it is given that p*q = 16. Thus, this statement is false.

Statement 2:

It is given that p + q + p*q = 94

If p + q = 20, then p*q = 74.

Thus, either (p,q) will be (2,37), (37,2), (1,74) or (74,1).

Since none of the values will give a sum of 20, this statement is also false.

Thus, the correct option is B.

Question 6

A milkman adds 10 litres of water to 90 litres of milk. After selling $$\frac{1}{5}^{th}$$ of the total quantity, he adds water equal to the quantity he sold. The proportion of water to milk he sells now would be:

Video Solution

Solution

Initially Milkman has 90 liters of milk
Now amount of water added = 10 liters
So total quantity becomes : 100 liters
He sold 1/5 of quantity
so he sold 2 liters of water and 18 liters of milk (A total of 20 liters is sold )
Now he is left with 10-2 = 8 liters of water and 90-18 = 72 liters of milk.
Now he adds 20 liters of water
so the amount of milk and water with him will be : 72 and 28 respectively.

Question 7

In a $$\triangle ABC,D,E$$ and $$F$$ are the mid-points of the sides AB, BC and CA respectively. Then the ratio of the area of a $$\triangle DEF$$ and the area of a $$\triangle ABC$$ is:

Video Solution

Solution

We are given :

Now In triangle ABC
using the mid point theorem
we get DE || BC and DE = $$\frac{1}{2}\left(BC\right)$$
Now ADE is similar to ABC
So Area of ADE : Area of ABC = (DE/BC)^2 = 1:4
Similarly Area of BDF :Area of ABC = 1:4
and Area of CEF : Area of ABC = 1:4
Now we can say if area of ABC = 4A
then area of ADE = area of BFD = area of CEF = a
Therefore area of DEF = 4a-3a =a
so area of DEF : area of ABC = 1:4

Question 8

A 100-meter long train crosses a 200-meter long and 20-meter wide bridge in 20 seconds. What is the speed of the train in Km/hour?

Video Solution

Solution

We will only consider length of bridge
Now total distance = 100+200 = 300m
Now time taken to cross = 20 seconds
Therefore speed = 15m/sec = 15(18/5) = 54km/hr

Question 9

Given below are two statements:
Statement I: $$\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}=2$$
Statement II: If $$a+b+c=0$$, then $$(a^{3}+b^{3}+c^{3})\div abc=3$$
In light of the above statements, choose the correct answer from the options given below

Video Solution

Solution

From statement 1 we get
Adding taking LCM
$$\frac{\left(\sqrt{\ 7}+\sqrt{\ 5}\right)^2+\left(\sqrt{\ 7}-\sqrt{\ 5}\right)^2}{7-5}$$
= 12
So statement 1 is false
Now statement 2 is an identity
so 2 is true .

Question 10

Given below are two statements:
Statement I : A coin is tossed three times. The probability of getting exactly two heads is 3/8
Statement II: In tossing of 10 coins, the probability of getting exactly 5 heads is 63/256

In the light of the above statements, choose the correct answer from the options given below.

Video Solution

Solution

Statement I:

Probability of getting head = P(H) = $$\frac{1}{2}$$

Probability of getting tail = P(T) = $$\frac{1}{2}$$

3 coins are tossed and probability of getting 2 heads = $$3_{C_2}\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)$$ = $$\frac{3}{8}$$

Statement I is true.

Statement II:

10 coins are tossed and probability of getting exactly 5 heads = $$10_{C_5}\left(\frac{1}{2}\right)^5\left(\frac{1}{2}\right)^5$$ = $$\frac{63}{256}$$

Statement II is true

Answer is option A.

CAT Content

CMAT 2021 Slot-1

Two pipes A and B can fill a tank in 5 hours and 20 hours respectively. Both pipes together can fill the same tank in:

In June-2020, the ratio of boys to girls in a college was 3 : 2. In September-2020, there were 80 fewer boys and 20 fewer girls in the college and the ratio of boys to girls was 7 : 5. What was the total number of students in the college in June-2020?

M is a 4-digit number. If the left most digit is removed, then the resulting three digit number is $$\frac{1}{9}^{th}$$ of M. How many such M's are possible?

L and M together can complete a piece of work in 72 days, M and N together can complete it in 120 days, and L and N together in 90 days. In what time can L alone complete the work?

A milkman adds 10 litres of water to 90 litres of milk. After selling $$\frac{1}{5}^{th}$$ of the total quantity, he adds water equal to the quantity he sold. The proportion of water to milk he sells now would be:

In a $$\triangle ABC,D,E$$ and $$F$$ are the mid-points of the sides AB, BC and CA respectively. Then the ratio of the area of a $$\triangle DEF$$ and the area of a $$\triangle ABC$$ is:

A 100-meter long train crosses a 200-meter long and 20-meter wide bridge in 20 seconds. What is the speed of the train in Km/hour?

Given below are two statements:
Statement I: $$\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}=2$$
Statement II: If $$a+b+c=0$$, then $$(a^{3}+b^{3}+c^{3})\div abc=3$$
In light of the above statements, choose the correct answer from the options given below

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CAT Content

CMAT 2021 Slot-1

Two pipes A and B can fill a tank in 5 hours and 20 hours respectively. Both pipes together can fill the same tank in:

In June-2020, the ratio of boys to girls in a college was 3 : 2. In September-2020, there were 80 fewer boys and 20 fewer girls in the college and the ratio of boys to girls was 7 : 5. What was the total number of students in the college in June-2020?

M is a 4-digit number. If the left most digit is removed, then the resulting three digit number is $$\frac{1}{9}^{th}$$ of M. How many such M's are possible?

L and M together can complete a piece of work in 72 days, M and N together can complete it in 120 days, and L and N together in 90 days. In what time can L alone complete the work?

A milkman adds 10 litres of water to 90 litres of milk. After selling $$\frac{1}{5}^{th}$$ of the total quantity, he adds water equal to the quantity he sold. The proportion of water to milk he sells now would be:

In a $$\triangle ABC,D,E$$ and $$F$$ are the mid-points of the sides AB, BC and CA respectively. Then the ratio of the area of a $$\triangle DEF$$ and the area of a $$\triangle ABC$$ is:

A 100-meter long train crosses a 200-meter long and 20-meter wide bridge in 20 seconds. What is the speed of the train in Km/hour?

Given below are two statements:Statement I: $$\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}=2$$Statement II: If $$a+b+c=0$$, then $$(a^{3}+b^{3}+c^{3})\div abc=3$$In light of the above statements, choose the correct answer from the options given below

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CAT Quant Questions | CAT Quantitative Ability

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CAT Verbal Ability Questions | VARC Questions For CAT

Given below are two statements:
Statement I: $$\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}+\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}=2$$
Statement II: If $$a+b+c=0$$, then $$(a^{3}+b^{3}+c^{3})\div abc=3$$
In light of the above statements, choose the correct answer from the options given below